## The Tiger Explained

Discussion of shapes with curves and holes in various dimensions.

### Re: The Tiger Explained

ICN5D wrote:
I'm currently compiling document about slices and rotations of 6D toratopes Aw, sweet! Any illustrations? I think you mentioned Mathematica before, so probably. Would it be any better for renders, do you think? I fear that treading into 7 and 8D waters will be beyond the capacity of CalcPlot3D. Though, I'm not sure, since I haven't gone there yet. But, if it handled Triger really well, then maybe some higher derivatives of it will be just as easy. As in, those that are a trace of spheres.

I've been thinking of those 6D toratopes that are spheres along a tigroid rim, like spheric tiger ((II)(II)I), spheric torus-tiger (((II)I)(II)I), and the like. It is interesting that the trace of spheres is actually from the lowest circle-array trace, that then had a bisecting rotation of its minor diameter. It's also cool to see how it plays around with the hidden minor diameter of tigroids. It seems to be already defined by the outermost brackets, in the positions of the two markers in (II). Adding another marker, as in ((II)(II)I) has the identity of (III), a sphere. (((II)I)(II)I) ought to have some wild looking obliques, wouldn't you say?

Well, no pictures, it looks like my 5D analysis a few posts back. I'm not at 21-torus 21-tiger (the shape you've mentioned), but for example:

12. 221-tiger 1-torus (((II)(II)I)I)
3 5D slabs: minor pair of 221-tigers (((II)(II)I)), tiger torus (((II)(II))I), two 221-ditoruses stacked in medium dimension (((II)(I)I)I) and (((I)(II)I)I)
5 4D slices: minor pair of tigers (((II)(II))), vertical stack of two minor pairs of spheritoruses (((II)(I)I)) and (((I)(II)I)), two ditoruses stacked in medium dimension (((II)(I))I) and (((I)(II))I), empty slice (((II)()I)I) and ((()(II)I)I), 2x2 array of torispheres (((I)(I)I)I)
Minor pair of tigers slice can evolve into minor pair of 221-tigers slab or tiger torus slab.
Vertical stack of two minor pairs of spheritoruses slice can evolve into minor pair of 221-tigers slab and two 221-ditoruses stacked in medium dimension slab.
Two ditoruses stacked in medium dimension slice can evolve into tiger torus slab or two 221-ditoruses stacked in medium dimension slab.
Empty slice can only evolve into two 221-ditoruses stacked in medium dimension slab.
2x2 array of torispheres slice can only evolve into two 221-ditoruses stacked in medium dimension slab, but in two different ways.
6 3D cuts: vertical stack of two minor pairs of toruses (((II)(I))) and (((I)(II))), empty cut A (((II)()I)) and ((()(II)I)), 2x2 array of pairs of spheres (((I)(I)I)), empty cut B (((II)())I) and ((()(II))I), 2x2 array of toruses (((I)(I))I), empty cut C (((I)()I)I) and ((()(I)I)I)
Vertical stack of two minor pairs of toruses cut can evolve into minor pair of tigers slice, vertical stack of two minor pairs of spheritoruses slice or two ditoruses stacked in medium dimension slice.
Empty cut A can evolve into vertical stack of two minor pairs of spheritoruses slice (x2) or empty slice.
2x2 array of pairs of spheres cut can evolve into vertical stack of two minor pairs of spheritoruses slice (in two different ways) or 2x2 array of torispheres slice.
Empty cut B can evolve into two ditoruses stacked in medium dimension slice (x2) or empty slice.
2x2 array of toruses cut can evolve into two ditoruses stacked in medium dimension slice (in two different ways) or 2x2 array of torispheres slice.
Empty cut C can evolve into empty slice or 2x2 array of torispheres slice (x2).
4 rotations are animations of tiger torus.
7 new rotations: vertical stack of two minor pairs of toruses - empty cut A (((II)(x)x)), vertical stack of two minor pairs of toruses - 2x2 array of pairs of spheres (((Ix)(I)x)), empty cut A - 2x2 array of pairs of spheres (((Ix)(x)I)), 2x2 array of pairs of spheres - 2x2 array of toruses (((I)(I)x)x), 2x2 array of pairs of spheres - empty cut C (((I)(x)I)x), 2x2 array of toruses - empty cut C (((I)(x)x)I), empty cut C - alternate empty cut C (((x)(x)I)I)
3 empty rotations: empty cut A - empty cut B (((II)()x)y), empty cut A - empty cut C (((Ix)()I)x), empty cut B - empty cut C (((Ix)()x)I)
2 double rotations are animations of tiger torus.
8 new double rotations: vertical stack of two minor pairs of toruses - alternate empty cut A (((xy)(Ix)y)), vertical stack of two minor pairs of toruses - empty cut C (((Ix)(y)x)y) or (((Ix)(y)y)x), vertical stack of two minor pairs of toruses - alternate empty cut C (((xy)(I)x)y), empty cut A - alternate empty cut A (((xy)(xy)I)), empty cut A - 2x2 array of toruses (((Ix)(x)y)y) or (((Ix)(y)y)x), empty cut A - alternate empty cut C (((xy)(x)I)y), 2x2 array of pairs of spheres - empty cut B (((Ix)(x)y)y) or (((Ix)(y)x)y), empty cut B - alternate empty cut C (((xy)(x)y)I)
1 triple rotation: empty cut A - alternate empty cut B (((xy)(xy)z)z) or (((xy)(xz)z)y)
Marek14
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### Re: The Tiger Explained

Not sure if rotations that simultaneously turn into and out of the same set of dimensions are different...

Example: 221-ditorus (((II)II)I) has rotation (((Ix)I)x) between major and minor pair of toruses. Would a double rotation between the same, (((Ix)xy)y), work differently?
Marek14
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### Re: The Tiger Explained

Hmm, not sure. I'll have to investigate. Are you referring to bisecting rotations? It seems like it, or a combo....
in search of combinatorial objects of finite extent
ICN5D
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### Re: The Tiger Explained

ICN5D wrote:Hmm, not sure. I'll have to investigate. Are you referring to bisecting rotations? It seems like it, or a combo....

I'm almost done with my 6D analysis Marek14
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### Re: The Tiger Explained

And here it is:

Code: Select all
`1. Hexasphere (IIIIII)1 5D slab: pentasphere (IIIII)1 4D slice: glome (IIII)Glome slice can only evolve into pentasphere slab.1 3D cut: sphere (III)Sphere cut can only evolve into glome slice.No rotations.2. 51-torus ((IIIII)I)2 5D slabs: pair of pentaspheres ((IIIII)), 41-torus ((IIII)I)2 4D slices: pair of glomes ((IIII)), torisphere ((III)I)Pair of glomes slice can evolve into pair of pentaspheres slab or 41-torus slab.Torisphere slice can only evolve into 41-torus slab.2 3D cuts: pair of spheres ((III)), torus ((II)I)Pair of spheres cut can evolve into pair of glomes slice or torisphere slice.Torus cut can only evolve into torisphere slice.1 rotation is animation of torisphere.3. 411-ditorus (((IIII)I)I)3 5D slabs: minor pair of 41-toruses (((IIII)I)), major pair of 41-toruses (((IIII))I), 311-ditorus (((III)I)I)4 4D slices: quartet of glomes (((IIII))), minor pair of torispheres (((III)I)), major pair of torispheres (((III))I), ditorus (((II)I)I)Quartet of glomes slice can evolve into minor pair of 41-toruses slab or major pair of 41-toruses slab.Minor pair of torispheres slice can evolve into minor pair of 41-toruses slab or 311-ditorus slab.Major pair of torispheres slice can evolve into major pair of 41-toruses slab 311-ditorus slab.Ditorus slice can only evolve into 311-ditorus slab.4 3D cuts: quartet of spheres (((III))), minor pair of toruses (((II)I)), major pair of toruses (((II))I), two toruses (((I)I)I)Quartet of spheres cut can evolve into quartet of glomes slice, minor pair of torispheres slice or major pair of torispheres slice.Minor pair of toruses cut can evolve into minor pair of torispheres slice (x2) or ditorus slice.Major pair of toruses cut can evolve into major pair of torispheres slice (x2) or ditorus slice.Two toruses cut can only evolve into ditorus slice.3 rotations are animations of ditorus.2 rotations are animations of 311-ditorus.1 double rotation is animation of 311-ditorus.4. 3111-tritorus ((((III)I)I)I)4 5D slabs: minor pair of 311-ditoruses ((((III)I)I)), medium pair of 311-ditoruses ((((III)I))I), major pair of 311-ditoruses ((((III))I)I), tritorus ((((II)I)I)I)7 4D slices: minor quartet of torispheres ((((III)I))), major/minor quartet of torispheres ((((III))I)), minor pair of ditoruses ((((II)I)I)), major quartet of torispheres ((((III)))I), medium pair of ditoruses ((((II)I))I), major pair of ditoruses ((((II))I)I), two ditoruses ((((I)I)I)I)Minor quartet of torispheres slice can evolve into minor pair of 311-ditoruses slab or medium pair of 311-ditoruses slab.Major/minor quartet of torispheres slice can evolve into minor pair of 311-ditoruses slab or major pair of 311-ditoruses slab.Minor pair of ditoruses slice can evolve into minor pair of 311-ditoruses slab or tritorus slab.Major quartet of torispheres slice can evolve into medium pair of 311-ditoruses slab or major pair of 311-ditoruses slab.Medium pair of ditoruses slice can evolve into medium pair of 311-ditoruses slab or tritorus slab.Major pair of ditoruses slice can evolve into major pair of 311-ditoruses slab or tritorus slab.Two ditoruses slice can only evolve into tritorus slab.8 3D cuts: octet of spheres ((((III)))), minor quartet of toruses ((((II)I))), major/minor quartet of toruses ((((II))I)), two minor pairs of toruses ((((I)I)I)), major quartet of toruses ((((II)))I), two major pairs of toruses ((((I)I))I), four toruses ((((I))I)I), empty cut (((()I)I)I)Octet of spheres cut can evolve into minor quartet of torispheres slice, major/minor quartet of torispheres slice or major quartet of torispheres slice.Minor quartet of toruses cut can evolve into minor quartet of torispheres slice, minor pair of ditoruses slice or medium pair of ditoruses slice.Major/minor quartet of toruses cut can evolve into major/minor quartet of torispheres slice, minor pair of ditoruses slice or major pair of ditoruses slice.Two minor pairs of toruses cut can evolve into minor pair of ditoruses slice (x2) or two ditoruses slice.Major quartet of toruses cut can evolve into major quartet of torispheres slice, medium pair of ditoruses slice or major pair of ditoruses slice.Two major pairs of toruses cut can evolve into medium pair of ditoruses slice (x2) or two ditoruses slice.Four toruses cut can evolve into major pair of ditoruses slice (x2) or two ditoruses slice.Empty cut can only evolve into two ditoruses slice.15 rotations are animations of tritorus.6 double rotations are animations of tritorus.3 new rotations: octet of spheres - minor quartet of toruses ((((IIx)x))), octet of spheres - major/minor quartet of toruses ((((IIx))x)), octet of spheres - major quartet of toruses ((((IIx)))x)3 new double rotations: octet of spheres - two minor pairs of toruses ((((Ixy)x)y)), octet of spheres - two major pairs of toruses ((((Ixy)x))y), octet of spheres - four toruses ((((Ixy))x)y)1 triple rotation: octet of spheres - empty cut ((((xyz)x)y)z)5. Tetratorus (((((II)I)I)I)I)5 5D slabs: minor pair of tritoruses (((((II)I)I)I)), tertiary pair of tritoruses (((((II)I)I))I), secondary pair of tritoruses (((((II)I))I)I), major pair of tritoruses (((((II))I)I)I), two tritoruses (((((I)I)I)I)I)11 4D slices: minor quartet of ditoruses (((((II)I)I))), medium/minor quartet of ditoruses (((((II)I))I)), major/minor quartet of ditoruses (((((II))I)I)), two minor pairs of ditoruses (((((I)I)I)I)), medium quartet of ditoruses (((((II)I)))I), major/medium quartet of ditoruses (((((II))I))I), two medium pairs of ditoruses (((((I)I)I))I), major quartet of ditoruses (((((II)))I)I), two major pairs of ditoruses (((((I)I))I)I), four ditoruses (((((I))I)I)I), empty slice ((((()I)I)I)I)Minor quartet of ditoruses slice can evolve into minor pair of tritoruses slab or tertiary pair of tritoruses slab.Medium/minor quartet of ditoruses slice can evolve into minor pair of tritoruses slab or secondary pair of tritoruses slab.Major/minor quartet of ditoruses slice can evolve into minor pair of tritoruses slab or major pair of tritoruses slab.Two minor pairs of ditoruses slice can evolve into minor pair of tritoruses slab or two tritoruses slab.Medium quartet of ditoruses slice can evolve into tertiary pair of tritoruses slab or secondary pair of tritoruses slab.Major/medium quartet of ditoruses slice can evolve into tertiary pair of tritoruses slab or major pair of tritoruses slab.Two medium pairs of ditoruses slice can evolve into tertiary pair of tritoruses slab or two tritoruses slab.Major quartet of ditoruses slice can evolve into secondary pair of tritoruses slab or major pair of tritoruses slab.Two major pairs of ditoruses slice can evolve into secondary pair of tritoruses slab or two tritoruses slab.Four ditoruses slice can evolve into major pair of tritoruses slab or two tritoruses slab.Empty slice can only evolve into two tritoruses slab.14 3D cuts: minor octet of toruses (((((II)I)))), major/minor/minor octet of toruses (((((II))I))), two minor quartets of toruses (((((I)I)I))), major/major/minor octet of toruses (((((II)))I)), two major/minor quartets of toruses (((((I)I))I)), four minor pairs of toruses (((((I))I)I)), empty cut A ((((()I)I)I)), major octet of toruses (((((II))))I), two major quartets of toruses (((((I)I)))I), four major pairs of toruses (((((I))I))I), empty cut B ((((()I)I))I), eight toruses (((((I)))I)I), empty cut C ((((()I))I)I), empty cut D ((((())I)I)I)Minor octet of toruses cut can evolve into minor quartet of ditoruses slice, medium/minor quartet of ditoruses slice or medium quartet of ditoruses slice.Major/minor/minor octet of toruses cut can evolve into minor quartet of ditoruses slice, major/minor quartet of ditoruses slice or major/medium quartet of ditoruses slice.Two minor quartets of toruses cut can evolve into minor quartet of ditoruses slice, two minor pairs of ditoruses slice or two major pairs of ditoruses slice.Major/major/minor octet of toruses cut can evolve into medium/minor quartet of ditoruses slice, major/minor quartet of ditoruses slice or major quartet of ditoruses slice.Two major/minor quartets of toruses cut can evolve into medium/minor quartet of ditoruses slice, two minor pairs of ditoruses slice or two medium pairs of ditoruses slice.Four minor pairs of toruses cut can evolve into major/minor quartet of ditoruses slice, two minor pairs of ditoruses slice or four ditoruses slice.Empty cut A can evolve into two minor pairs of ditoruses slice (x2) or empty slice.Major octet of toruses cut can evolve into medium quartet of ditoruses slice, major/medium quartet of ditoruses slice or major quartet of ditoruses slice.Two major quartets of toruses cut can evolve into medium quartet of ditoruses slice, two medium pairs of ditoruses slice or two major pairs of ditoruses slice.Four major pairs of toruses cut can evolve into major/medium quartet of ditoruses slice, two medium pairs of ditoruses slice or four ditoruses slice.Empty cut B can evolve into two medium pairs of ditoruses slice (x2) or empty slice.Eight toruses cut can evolve into major quartet of ditoruses slice, two major pairs of ditoruses slice or four ditoruses slice.Empty cut C can evolve into two major pairs of ditoruses slice (x2) or empty slice.Empty cut D can evolve into four ditoruses slice (x2) or empty slice.42 rotations: minor octet of toruses - major/minor/minor octet of toruses (((((II)x)x))), minor octet of toruses - two minor quartets of toruses (((((Ix)I)x))), minor octet of toruses - major/major/minor octet of toruses (((((II)x))x)), minor octet of toruses - two major/minor quartets of toruses (((((Ix)I))x)), minor octet of toruses - major octet of toruses (((((II)x)))x), minor octet of toruses - two major quartets of toruses (((((Ix)I)))x), major/minor/minor octet of toruses - two minor quartets of toruses (((((Ix)x)I))), major/minor/minor octet of toruses - major/major/minor octet of toruses (((((II))x)x)), major/minor/minor octet of toruses - four minor pairs of toruses (((((Ix))I)x)), major/minor/minor octet of toruses - major octet of toruses (((((II))x))x), major/minor/minor octet of toruses - four major pairs of toruses (((((Ix))I))x), two minor quartets of toruses - two major/minor quartets of toruses (((((I)I)x)x)), two minor quartets of toruses - four minor pairs of toruses (((((I)x)I)x)), two minor quartets of toruses - empty cut A ((((x)I)I)x)), two minor quartets of toruses - two major quartets of toruses ((((I)I)x))x), two minor quartets of toruses - four major pairs of toruses (((((I)x)I))x), two minor quartets of toruses - empty cut B (((((x)I)I))x), major/major/minor octet of toruses - two major/minor quartets of toruses (((((Ix)x))I)), major/major/minor octet of toruses - four minor pairs of toruses (((((Ix))x)I)), major/major/minor octet of toruses - major octet of toruses (((((II)))x)x), major/major/minor octet of toruses - eight toruses (((((Ix)))I)x), two major/minor quartets of toruses - four minor pairs of toruses (((((I)x)x)I)), two major/minor quartets of toruses - empty cut A (((((x)I)x)I)), two major/minor quartets of toruses - two major quartets of toruses (((((I)I))x)x), two major/minor quartets of toruses - eight toruses (((((I)x))I)x), two major/minor quartets of toruses - empty cut C (((((x)I))I)x), four minor pairs of toruses - empty cut A (((((x)x)I)I)), four minor pairs of toruses - four major pairs of toruses (((((I))I)x)x), four minor pairs of toruses - eight toruses (((((I))x)I)x), four minor pairs of toruses - empty cut D (((((x))I)I)x), major octet of toruses - two major quartets of toruses (((((Ix)x)))I), major octet of toruses - four major pairs of toruses (((((Ix))x))I), major octet of toruses - eight toruses (((((Ix)))x)I), two major quartets of toruses - four major pairs of toruses (((((I)x)x))I), two major quartets of toruses - empty cut B (((((x)I)x))I), two major quartets of toruses - eight toruses (((((I)x))x)I), two major quartets of toruses - empty cut C (((((x)I))x)I), four major pairs of toruses - empty cut B (((((x)x)I))I), four major pairs of toruses - eight toruses (((((I))x)x)I), four major pairs of toruses - empty cut D (((((x))I)x)I), eight toruses - empty cut C (((((x)x))I)I), eight toruses - empty cut D (((((x))x)I)I)6 empty rotations: empty cut A - empty cut B ((((()I)I)x)x), empty cut A - empty cut C ((((()I)x)I)x), empty cut A - empty cut D ((((()x)I)I)x), empty cut B - empty cut C ((((()I)x)x)I), empty cut B - empty cut D ((((()x)I)x)I), empty cut C - empty cut D ((((()x)x)I)I)39 double rotations: minor octet of toruses - four minor pairs of toruses (((((Ix)y)x)y)) or (((((Ix)y)y)x)), minor octet of toruses - empty cut A (((((xy)I)x)y)), minor octet of toruses - four major pairs of toruses (((((Ix)y)x))y) or (((((Ix)y)y))x), minor octet of toruses - empty cut B (((((xy)I)x))y), minor octet of toruses - eight toruses (((((Ix)y))x)y) or (((((Ix)y))y)x), minor octet of toruses - empty cut C (((((xy)I))x)y), major/minor/minor octet of toruses - two major/minor quartets of toruses (((((Ix)x)y)y)) or (((((Ix)y)y)x)), major/minor/minor octet of toruses - empty cut A (((((xy)x)I)y)), major/minor/minor octet of toruses - two major quartets of toruses (((((Ix)x)y))y) or (((((Ix)y)y))x), major/minor/minor octet of toruses - empty cut B (((((xy)x)I))y), major/minor/minor octet of toruses - eight toruses (((((Ix))y)x)y) or (((((Ix))y)y)x), major/minor/minor octet of toruses - empty cut D (((((xy))I)x)y), two minor quartets of toruses - major/major/minor octet of toruses (((((Ix)x)y)y)) or (((((Ix)y)y)x)), two minor quartets of toruses - major octet of toruses (((((Ix)x)y))y) or (((((Ix)y)x))y), two minor quartets of toruses - eight toruses (((((I)x)y)x)y) or (((((I)x)y)y)x), two minor quartets of toruses - empty cut C (((((x)I)y)x)y) or (((((x)I)y)y)x), two minor quartets of toruses - empty cut D (((((x)y)I)x)y) or (((((x)y)I)y)x), major/major/minor octet of toruses - empty cut A (((((xy)x)y)I)), major/major/minor octet of toruses - two major quartets of toruses (((((Ix)x))y)y) or (((((Ix)y))y)x), major/major/minor octet of toruses - four major pairs of toruses (((((Ix))x)y)y) or (((((Ix))y)x)y), major/major/minor octet of toruses - empty cut C (((((xy)x))I)y), major/major/minor octet of toruses - empty cut D (((((xy))x)I)y), two major/minor quartets of toruses - major octet of toruses (((((Ix)x))y)y) or (((((Ix)y))x)y), two major/minor quartets of toruses - four major pairs of toruses (((((I)x)x)y)y) or (((((I))x)y)y)x), two major/minor quartets of toruses - empty cut B (((((x)I)x)y)y) or (((((x)I)y)y)x), two major/minor quartets of toruses - empty cut D (((((x)y)x)I)y) or (((((x)y)y)I)x), four minor pairs of toruses - major octet of toruses (((((Ix))x)y)y) or (((((Ix))y)x)y), four minor pairs of toruses - two major quartets of toruses (((((I)x)x)y)y) or (((((I)x)y)x)y), four minor pairs of toruses - empty cut B (((((x)x)I)y)y) or (((((x)y)I)y)x), four minor pairs of toruses - empty cut C (((((x)x)y)I)y) or (((((x)y)y)I)x), empty cut A - two major quartets of toruses (((((x)I)x)y)y) or (((((x)I)y)x)y), empty cut A - four major pairs of toruses (((((x)x)I)y)y) or (((((x)y)I)x)y), empty cut A - eight toruses (((((x)x)y)I)y) or (((((x)y)x)I)y), major octet of toruses - empty cut B (((((xy)x)y))I), major octet of toruses - empty cut C (((((xy)x))y)I), major octet of toruses - empty cut D (((((xy))x)y)I), two major quartets of toruses - empty cut D (((((x)y)x)y)I) or (((((x)y)y)x)I), four major pairs of toruses - empty cut C (((((x)x)y)y)I) or (((((x)y)y)x)I), empty cut B - eight toruses (((((x)x)y)y)I) or (((((x)y)x)y)I)4 triple rotations: minor octet of toruses - empty cut D (((((xy)z)x)y)z) or (((((xy)z)x)z)y) or (((((xy)z)z)x)y), major/minor/minor octet of toruses - empty cut C (((((xy)x)z)y)z) or (((((xy)x)z)z)y) or (((((xy)z)z)x)y), major/major/minor octet of toruses - empty cut B (((((xy)x)y)z)z) or (((((xy)x)z)z)y) or (((((xy)z)x)z)y), empty cut A - major octet of toruses (((((xy)x)y)z)z) or (((((xy)x)z)y)z) or (((((xy)z)x)y)z)6. Tiger ditorus ((((II)(II))I)I)3 5D slabs: minor pair of tiger toruses ((((II)(II))I)), medium pair of tiger toruses ((((II)(II)))I), two tritoruses stacked in secondary dimension ((((II)(I))I)I) and ((((I)(II))I)I)5 4D slices: minor quartet of tigers ((((II)(II)))), two minor pairs of ditoruses stacked in medium dimension ((((II)(I))I)) and ((((I)(II))I)), two medium pairs of ditoruses stacked in medium dimension ((((II)(I)))I) and ((((I)(II)))I), empty slice ((((II)())I)I) and (((()(II))I)I), 2x2 array of ditoruses ((((I)(I))I)I)Minor quartet of tigers slice can evolve into minor pair of tiger toruses slab or medium pair of tiger toruses slab.Two minor pairs of ditoruses stacked in medium dimension slice can evolve into minor pair of tiger toruses slab or two tritoruses stacked in secondary dimension slab.Two medium pairs of ditoruses stacked in medium dimension slice can evolve into medium pair of tiger toruses slab or two tritoruses stacked in secondary dimension slab.Empty slice can only evolve into two tritoruses stacked in secondary dimension slab.2x2 array of ditoruses slice can only evolve into two tritoruses stacked in secondary dimension slab, but in two different ways.6 3D cuts: vertical stack of two minor torus quartets ((((II)(I)))) and ((((I)(II)))), empty cut A ((((II)())I)) and (((()(II))I)), 2x2 array of minor torus pairs ((((I)(I))I)), empty cut B ((((II)()))I) and (((()(II)))I), 2x2 array of major torus pairs ((((I)(I)))I), empty cut C ((((I)())I)I) and (((()(I))I)I)Vertical stack of two minor torus quartets cut can evolve into minor quartet of tigers slice, two minor pairs of ditoruses stacked in medium dimension slice or two medium pairs of ditoruses stacked in medium dimension slice.Empty cut A can evolve into two minor pairs of ditoruses stacked in medium dimension slice (x2) or empty slice.2x2 array of minor torus pairs cut can evolve into two minor pairs of ditoruses stacked in medium dimension slice (in two different ways) or 2x2 array of ditoruses slice.Empty cut B can evolve into two medium pairs of ditoruses stacked in medium dimension slice (x2) or empty slice.2x2 array of major torus pairs cut can evolve into two medium pairs of ditoruses stacked in medium dimension slice (in two different ways) or 2x2 array of ditoruses slice.Empty cut C can evolve into empty slice or 2x2 array of ditoruses slice (x2).10 rotations: vertical stack of two minor torus quartets - alternate vertical stack of two minor torus quartets ((((Ix)(Ix)))), vertical stack of two minor torus quartets - empty cut A ((((II)(x))x)), vertical stack of two minor torus quartets - 2x2 array of minor torus pairs ((((Ix)(I))x)), vertical stack of two minor torus quartets - empty cut B ((((II)x)))x), vertical stack of two minor torus quartets - 2x2 array of major torus pairs ((((Ix)(I)))x), empty cut A - 2x2 array of minor torus pairs ((((Ix)(x))I)), 2x2 array of minor torus pairs - 2x2 array of major torus pairs ((((I)(I)))x)x), 2x2 array of minor torus pairs - empty cut C ((((I)(x))I)x), empty cut B - 2x2 array of major torus pairs ((((Ix)(x)))I), 2x2 array of major torus pairs - empty cut C ((((I)(x))x)I)3 empty rotations: empty cut A - empty cut B ((((II)())x)x), empty cut A - empty cut C ((((Ix)())I)x), empty cut B - empty cut C ((((Ix)())x)I)10 double rotations: vertical stack of two minor torus quartets - alternate empty cut A ((((xy)(Ix))y)), vertical stack of two minor torus quartets - alternate empty cut B ((((xy)(Ix)))y), vertical stack of two minor torus quartets - empty cut C ((((Ix)(y))x)y) or ((((Ix)(y))y)x), vertical stack of two minor torus quartets - alternate empty cut C ((((xy)(I))x)y), empty cut A - alternate empty cut A ((((xy)(xy))I)), empty cut A - 2x2 array of major torus pairs ((((Ix)(x))y)y) or ((((Ix)(y))y)x), empty cut A - alternate empty cut C ((((xy)(x))I)y), 2x2 array of minor torus pairs - empty cut B ((((Ix)(x))y)y) or ((((Ix)(y))x)y), empty cut B - alternate empty cut B ((((xy)(xy)))I), empty cut B - alternate empty cut C ((((xy)(x))y)I)1 triple rotation: empty cut A - alternate empty cut B ((((xy)(xy))z)z) or ((((xy)(xz))z)y)7. 2211-tritorus ((((II)II)I)I)4 5D slabs: minor pair of 221-ditoruses ((((II)II)I)), medium pair of 221-ditoruses ((((II)II))I), tritorus ((((II)I)I)I), two 311-ditoruses ((((I)II)I)I)8 4D slices: minor quartet of spheritoruses ((((II)II))), minor pair of ditoruses ((((II)I)I)), two minor pairs of torispheres ((((I)II)I)), medium pair of ditoruses ((((II)I))I), two major pairs of torispheres ((((I)II))I), major pair of ditoruses ((((II))I)I), two ditoruses ((((I)I)I)I), empty slice (((()II)I)I)Minor quartet of spheritoruses slice can evolve into minor pair of 221-ditoruses slab or medium pair of 221-ditoruses slab.Minor pair of ditoruses slice can evolve into minor pair of 221-ditoruses slab or tritorus slab.Two minor pairs of torispheres slice can evolve into minor pair of 221-ditoruses slab or two 311-ditoruses slab.Medium pair of ditoruses slice can evolve into medium pair of 221-ditoruses slab or tritorus slab.Two major pairs of torispheres slice can evolve into medium pair of 221-ditoruses slab or two 311-ditoruses slab.Major pair of ditoruses slice evolves only into tritorus slab.Two ditoruses slice can evolve into tritorus slab or two 311-ditoruses slab.Empty slice evolves only into two 311-ditoruses slab.10 3D cuts: minor quartet of toruses ((((II)I))), two quartets of spheres ((((I)II))), major/minor quartet of toruses ((((II))I)), two minor pairs of toruses ((((I)I)I)), empty cut A (((()II)I)), major quartet of toruses ((((II)))I), two major pairs of toruses ((((I)I))I), empty cut B (((()II))I), four toruses ((((I))I)I), empty cut C (((()I)I)I)Minor quartet of toruses cut can evolve into minor quartet of spheritoruses slice, minor pair of ditoruses slice or medium pair of ditoruses slice.Two quartets of spheres cut can evolve into minor quartet of spheritoruses slice, two minor pairs of torispheres slice or two major pairs of torispheres slice.Major/minor quartet of toruses cut can evolve into minor pair of ditoruses slice (x2) or major pair of ditoruses slice.Two minor pairs of toruses cut can evolve into minor pair of ditoruses slice, two minor pairs of torispheres slice or two ditoruses slice.Empty cut A can evolve into two minor pairs of torispheres slice (x2) or empty slice.Major quartet of toruses cut can evolve into medium pair of ditoruses slice (x2) or major pair of ditoruses slice.Two major pairs of toruses cut can evolve into medium pair of ditoruses slice, two major pairs of torispheres slice or two ditoruses slice.Empty cut B can evolve into two major pairs of torispheres slice (x2) or empty slice.Four toruses cut can evolve into major pair of ditoruses slice or two ditoruses slice (x2).Empty cut C can evolve into two ditoruses slice (x2) or empty slice.15 rotations are animations of tritorus.7 new rotations: minor quartet of toruses - two quartets of spheres ((((Ix)Ix))), two quartets of spheres - two minor pairs of toruses ((((I)Ix)x)), two quartets of spheres - empty cut A ((((x)II)x)), two quartets of spheres - two major pairs of toruses ((((I)Ix))x), two quartets of spheres - empty cut B ((((x)II))x), two minor pairs of toruses - empty cut A ((((x)Ix)I)), two major pairs of toruses - empty cut B ((((x)Ix))I)3 empty rotations: empty cut A - empty cut B (((()II)x)x), empty cut A - empty cut C (((()Ix)I)x), empty cut B - empty cut C (((()Ix)x)I)6 double rotations are animations of tritorus.12 new double rotations: minor quartet of toruses - empty cut A ((((xy)Ix)y)), minor quartet of toruses - empty cut B ((((xy)Ix))y), two quartets of spheres - major/minor quartet of toruses ((((Ix)xy)y)), two quartets of spheres - major quartet of toruses ((((Ix)xy)))y), two quartets of spheres - four toruses ((((I)xy)x)y), two quartets of spheres - empty cut C ((((x)Iy)x)y) or ((((x)Iy)y)x), major/minor quartet of toruses - empty cut A ((((xy)xy)I)), two minor pairs of toruses - empty cut B ((((x)Ix)y)y) or ((((x)Iy)y)x), empty cut A - two major pairs of toruses ((((x)Ix)y)y) or ((((x)Iy)x)y), empty cut A - four toruses ((((x)xy)I)y), major quartet of toruses - empty cut B ((((xy)xy))I), empty cut B - four toruses ((((x)xy)y)I)2 triple rotations: major/minor quartet of toruses - empty cut B ((((xy)xy)z)z) or ((((xy)xz)z)y), empty cut A - major quartet of toruses ((((xy)xy)z)z) or ((((xy)xz)y)z)8. 320-tiger 1-torus (((III)(II))I)3 5D slabs: minor pair of 320-tigers (((III)(II))), two 311-ditoruses stacked in medium dimension (((III)(I))I), tiger torus (((II)(II))I)5 4D slices: vertical stack of two minor torisphere pairs (((III)(I))), minor pair of tigers (((II)(II))), empty slice (((III)())I), two ditoruses stacked in medium dimension A (((II)(I))I), two ditoruses stacked in medium dimension B (((I)(II))I)Vertical stack of two minor torisphere pairs slice can evolve into minor pair of 320-tigers slab or two 311-ditoruses stacked in medium dimension slab.Minor pair of tigers slice can evolve into minor pair of 320-tigers slab or tiger torus slab.Empty slice can only evolve into two 311-ditoruses stacked in medium dimension slab.Two ditoruses stacked in medium dimension slice A can evolve into two 311-ditoruses stacked in medium dimension slab or tiger torus slab.Two ditoruses stacked in medium dimension slice B can only evolve into tiger torus slab.6 3D cuts: empty cut A (((III)())), vertical stack of two minor torus pairs A (((II)(I))), vertical stack of two minor torus pairs B (((I)(II))), empty cut B (((II)())I), 2x2 array of toruses (((I)(I))I), empty cut C ((()(II))I)Empty cut A can evolve into vertical stack of two minor torisphere pairs slice (x2) or empty slice.Vertical stack of two minor torus pairs cut A can evolve into vertical stack of two minor torisphere pairs slice, minor pair of tigers slice or two ditoruses stacked in medium dimension slice A.Vertical stack of two minor torus pairs cut B can evolve into minor pair of tigers slice (x2) or two ditoruses stacked in medium dimension slice B.Empty cut B can evolve into empty slice or two ditoruses stacked in medium dimension slice A (x2).2x2 array of toruses can evolve into two ditoruses stacked in medium dimension slice A (x2) or two ditoruses stacked in medium dimension slice B.Empty cut C can only evolve into two ditoruses stacked in medium dimension slice B.7 rotations are animations of tiger torus.1 new rotation: empty cut A - vertical stack of two minor torus pairs A (((IIx)(x)))1 empty rotation: empty cut A - empty cut B (((IIx)())x)3 double rotations are animations of tiger torus.2 new double rotations: empty cut A - vertical stack of two minor torus pairs B (((Ixy)(xy))), empty cut A - 2x2 array of toruses (((Ixy)(x))y)1 triple rotation: empty cut A - empty cut C (((xyz)(xy))z)9. Torus tiger torus ((((II)I)(II))I)4 5D slabs: minor pair of torus tigers ((((II)I)(II))), two tritoruses stacked in tertiary dimension ((((II)I)(I))I), major pair of tiger toruses ((((II))(II))I), two tiger toruses((((I)I)(II))I)8 4D slices: two minor pairs of ditoruses stacked in minor dimension ((((II)I)(I))), major/minor quartet of tigers ((((II))(II))), two minor pairs of tigers ((((I)I)(II))), empty slice A ((((II)I)())I), two major pairs of ditoruses stacked in medium dimension ((((II))(I))I), two medium stacks of two ditoruses ((((I)I)(I))I), four ditoruses stacked in medium dimension ((((I))(II))I), empty slice B (((()I)(II))I)Two minor pairs of ditoruses stacked in minor dimension slice can evolve into minor pair of torus tigers slab or two tritoruses stacked in tertiary dimension slab.Major/minor quartet of tigers slice can evolve into minor pair of torus tigers slab or major pair of tiger toruses slab.Two minor pairs of tigers slice can evolve into minor pair of torus tigers slab or two tiger toruses slab.Empty slice A can only evolve into two tritoruses stacked in tertiary dimension slab.Two major pairs of ditoruses stacked in medium dimension slice can evolve into two tritoruses stacked in tertiary dimension slab or major pair of tiger toruses slab.Two medium stacks of two ditoruses slice can evolve into two tritoruses stacked in tertiary dimension slab or two tiger toruses slab.Four ditoruses stacked in medium dimension slice can evolve into major pair of tiger toruses slab or two tiger toruses slab.Empty slice B can only evolve into two tiger toruses slab.10 3D cuts: empty cut A ((((II)I)())), vertical stack of two major/minor quartets of toruses ((((II))(I))), two vertical stacks of two minor pairs of toruses ((((I)I)(I))), vertical stack of four minor pairs of toruses ((((I))(II))), empty cut B (((()I)(II))), empty cut C ((((II))())I), empty cut D ((((I)I)())I), 4x2 array of toruses ((((I))(I))I), empty cut E (((()I)(I))I), empty cut F (((())(II))I)Empty cut A can evolve into two minor pairs of ditoruses stacked in minor dimension slice (x2) or empty slice A.Vertical stack of two major/minor quartets of toruses cut can evolve into two minor pairs of ditoruses stacked in minor dimension slice, major/minor quartet of tigers slice or two major pairs of ditoruses stacked in medium dimension slice.Two vertical stacks of two minor pairs of toruses cut can evolve into two minor pairs of ditoruses stacked in minor dimension slice, two minor pairs of tigers slice or two medium stacks of two ditoruses slice.Vertical stack of four minor pairs of toruses cut can evolve into major/minor quartet of tigers slice, two minor pairs of tigers slice or four ditoruses stacked in medium dimension slice.Empty cut B can evolve into two minor pairs of tigers slice (x2) or empty slice B.Empty cut C can evolve into empty slice A or two major pairs of ditoruses stacked in medium dimension slice (x2).Empty cut D can evolve into empty slice A or two medium stacks of two ditoruses slice (x2).4x2 array of toruses cut can evolve into two major pairs of ditoruses stacked in medium dimension slice, two medium stacks of two ditoruses slice or four ditoruses stacked in medium dimension slice.Empty cut E can evolve into two medium stacks of two ditoruses slice (x2) or empty slice B.Empty cut F can evolve into four ditoruses stacked in medium dimension slice (x2) or empty slice B.19 rotations: empty cut A - vertical stack of two major/minor quartets of toruses ((((II)x)(x))), empty cut A - two vertical stacks of two minor pairs of toruses ((((Ix)I)(x))), vertical stack of two major/minor quartets of toruses - two vertical stacks of two minor pairs of toruses ((((Ix)x)(I))), vertical stack of two major/minor quartets of toruses - vertical stack of four minor pairs of toruses ((((Ix)))(Ix))), vertical stack of two major/minor quartets of toruses - empty cut C ((((II))(x))x), vertical stack of two major/minor quartets of toruses - 4x2 array of toruses ((((Ix))(I))x), two vertical stacks of two minor pairs of toruses - vertical stack of four minor pairs of toruses ((((I)x)(Ix))), two vertical stacks of two minor pairs of toruses - empty cut B ((((x)I)(Ix))), two vertical stacks of two minor pairs of toruses - empty cut D ((((I)I)(x))x), two vertical stacks of two minor pairs of toruses - 4x2 array of toruses ((((I)x)(I))x), two vertical stacks of two minor pairs of toruses - empty cut E ((((x)I)(I))x), vertical stack of four minor pairs of toruses - empty cut B ((((x)x)(II))), vertical stack of four minor pairs of toruses - 4x2 array of toruses ((((I))(Ix))x), vertical stack of four minor pairs of toruses - empty cut F ((((x))(II))x), empty cut C - 4x2 array of toruses ((((Ix))(x))I), empty cut D - 4x2 array of toruses ((((I)x)(x))I), empty cut D - empty cut E ((((x)I)(x))I), 4x2 array of toruses - empty cut E ((((x)x)(I))I), 4x2 array of toruses - empty cut F ((((x))(Ix))I)6 empty rotations: empty cut A - empty cut C ((((II)x)())x), empty cut A - empty cut D ((((Ix)I)())x), empty cut B - empty cut E (((()I)(Ix))x), empty cut B - empty cut F (((()x)(II))x), empty cut C - empty cut D ((((Ix)x)())I), empty cut E - empty cut F ((((Ix)(Ix))I)18 double rotations: empty cut A - vertical stack of four minor pairs of toruses ((((Ix)y)(xy))), empty cut A - empty cut B ((((xy)I)(xy))), empty cut A - 4x2 array of toruses ((((Ix)y)(x))y) or ((((Ix)y)(y))x), empty cut A - empty cut E ((((xy)I)(x))x), vertical stack of two major/minor quartets of toruses - empty cut B ((((xy)x)(Iy))), vertical stack of two major/minor quartets of toruses - empty cut D ((((Ix)x)(y))y) or ((((Ix)y)(y))x), vertical stack of two major/minor quartets of toruses - empty cut E ((((xy)x)(I))y), vertical stack of two major/minor quartets of toruses - empty cut F ((((xy))(Ix))y), two vertical stacks of two minor pairs of toruses - empty cut C ((((Ix)x)(y))y) or ((((Ix)y)(x))y), two vertical stacks of two minor pairs of toruses - empty cut F ((((x)y)(Ix))y) or ((((x)y)(Iy))x), vertical stack of four minor pairs of toruses - empty cut C ((((Ix))(xy))y), vertical stack of four minor pairs of toruses - empty cut D ((((I)x)(xy))y), vertical stack of four minor pairs of toruses - empty cut E ((((x)x)(Iy))y) or ((((x)y)(Iy))x), empty cut B - empty cut D ((((x)I)(xy))y), empty cut B - 4x2 array of toruses ((((x)x)(Iy))y) or ((((x)y)(Ix))y), empty cut C - empty cut E ((((xy)x)(y))I), empty cut C - empty cut F ((((xy))(xy))I), empty cut D - empty cut F ((((x)y)(xy))I)2 triple rotation: empty cut A - empty cut F ((((xy)z)(xy))z) or ((((xy)z)(xz))y), empty cut B - empty cut C ((((xy)x)(yz))z) or ((((xy)z)(xy))z)10. 321-ditorus (((III)II)I)3 5D slabs: minor pair of 32-toruses (((III)II)), 311-ditorus (((III)I)I), 221-ditorus (((II)II)I)5 4D slices: minor pair of torispheres (((III)I)), minor pair of spheritoruses (((II)II)), major pair of torispheres (((III))I), ditorus (((II)I)I), two torispheres (((I)II)I)Minor pair of torispheres slice can evolve into minor pair of 32-toruses slab or 311-ditorus slab.Minor pair of spheritoruses slice can evolve into minor pair of 32-toruses slab or 221-ditorus slab.Major pair of torispheres slice can only evolve into 311-ditorus slab.Ditorus slice can evolve into 311-ditorus slab or 221-ditorus slab.Two torispheres slice can only evolve into 221-ditorus slab.6 3D cuts: quartet of spheres (((III))), minor pair of toruses (((II)I)), two pairs of spheres (((I)II)), major pair of toruses (((II))I), two toruses (((I)I)I), empty cut ((()II)I)Quartet of spheres cut can evolve into minor pair of torispheres slice (x2) or major pair of torispheres slice.Minor pair of toruses cut can evolve into minor pair of torispheres slice, minor pair of spheritoruses slice or ditorus slice.Two pairs of spheres cut can evolve into minor pair of spheritoruses slice (x2) or two torispheres slice.Major pair of toruses cut can evolve into major pair of torispheres slice or ditorus slice (x2).Two toruses cut can evolve into ditorus slice (x2) or two torispheres slice.Empty cut can only evolve into two torispheres slice.2 rotations are animations of 311-ditorus.4 rotations are animations of 221-ditorus.3 rotations are animations of ditorus.1 double rotation is animation of 311-ditorus.3 double rotations are animations of 221-ditorus.1 new double rotation: quartet of spheres - two pairs of spheres (((Ixy)xy))1 triple rotation: quartet of spheres - empty cut (((xyz)xy)z)11. 2121-tritorus ((((II)I)II)I)4 5D slabs: minor pair of 212-ditoruses ((((II)I)II)), tritorus ((((II)I)I)I), major pair of 221-ditoruses ((((II))II)I), two 221-ditoruses ((((I)I)II)I)8 4D slices: minor pair of ditoruses ((((II)I)I)), major/minor quartet of spheritoruses ((((II))II)), two minor pairs of spheritoruses ((((I)I)II)), medium pair of ditoruses ((((II)I))I), major pair of ditoruses ((((II))I)I), two ditoruses ((((I)I)I)I), four torispheres ((((I))II)I), empty slice (((()I)II)I)Minor pair of ditoruses slice can evolve into minor pair of 212-ditoruses slab or tritorus slab.Major/minor quartet of spheritoruses slice can evolve into minor pair of 212-ditoruses slab or major pair of 221-ditoruses slab.Two minor pairs of spheritoruses slice can evolve into minor pair of major pair of 212-ditoruses slab or two 221-ditoruses slab.Medium pair of ditoruses slice can only evolve into tritorus slab.Major pair of ditoruses slice can evolve into tritorus slab or major pair of 221-ditoruses slab.Two ditoruses slice can evolve into tritorus slab or two 221-ditoruses slab.Four torispheres slice can evolve into major pair of 221-ditoruses slab or two 221-ditoruses slab.Empty slice can only evolve into two 221-ditoruses slab.10 3D cuts: minor quartet of toruses ((((II)I))), major/minor quartet of toruses ((((II))I)), two minor pairs of toruses ((((I)I)I)), four pairs of spheres ((((I))II)), empty cut A (((()I)II)), major quartet of toruses ((((II)))I), two major pairs of toruses ((((I)I))I), four toruses ((((I))I)I), empty cut B (((()I)I)I), empty cut C (((())II)I)Minor quartet of toruses cut can evolve into minor pair of ditoruses slice (x2) or medium pair of ditoruses slice.Major/minor quartet of toruses cut can evolve into minor pair of ditoruses slice, major/minor quartet of spheritoruses slice or major pair of ditoruses slice.Two minor pairs of toruses cut can evolve into minor pair of ditoruses slice, two minor pairs of spheritoruses slice or two ditoruses slice.Four pairs of spheres cut can evolve into major/minor quartet of spheritoruses slice, two minor pairs of spheritoruses slice or four torispheres slice.Empty cut A can evolve into two minor pairs of spheritoruses slice (x2) or empty slice.Major quartet of toruses cut can evolve into medium pair of ditoruses slice or major pair of ditoruses slice (x2).Two major pairs of toruses cut can evolve into medium pair of ditoruses slice or two ditoruses slice (x2).Four toruses cut can evolve into major pair of ditoruses slice, two ditoruses slice or four torispheres slice.Empty cut B can evolve into two ditoruses slice (x2) or empty slice.Empty cut C can evolve into four torispheres slice (x2) or empty slice.15 rotations are animations of tritorus.7 new rotations: major/minor quartet of toruses - four pairs of spheres ((((Ix))Ix)), two minor pairs of toruses - four pairs of spheres ((((I)x)Ix)), two minor pairs of toruses - empty cut A ((((x)I)Ix)), four pairs of spheres - empty cut A ((((x)x)II)), four pairs of spheres - four toruses ((((I))Ix)x), four pairs of spheres - empty cut C ((((x))II)x), four toruses - empty cut C ((((x))Ix)I)3 empty rotations: empty cut A - empty cut B (((()I)Ix)x), empty cut A - empty cut C (((()x)II)x), empty cut B - empty cut C (((()x)Ix)I)6 double rotations are animations of tritorus.12 new double rotations: minor quartet of toruses - four pairs of spheres ((((Ix)x)xy)), minor quartet of toruses - empty cut A ((((xy)I)xy), major/minor quartet of toruses - empty cut A ((((xy)x)Iy)), major/minor quartet of toruses - empty cut C ((((xy))Ix)y), two minor pairs of toruses - empty cut C ((((x)y)Ix)y) or ((((x)y)Iy)x), four pairs of spheres - major quartet of toruses ((((Ix))xy)y), four pairs of spheres - two major pairs of toruses ((((I)x)xy)y), four pairs of spheres - empty cut B ((((x)x)Iy)y) or ((((x)y)Iy)x), empty cut A - two major pairs of toruses ((((x)I)xy)y), empty cut A - four toruses ((((x)x)Iy)y) or ((((x)y)Ix)y), major quartet of toruses - empty cut C ((((xy))xy)I), two major pairs of toruses - empty cut c ((((x)y)xy)I)2 triple rotations: minor quartet of toruses - empty cut C ((((xy)z)xy)z) or ((((xy)z)xz)y), empty cut A - major quartet of toruses ((((xy)x)yz)z) or ((((xy)z)xy)z)12. 221-tiger 1-torus (((II)(II)I)I)3 5D slabs: minor pair of 221-tigers (((II)(II)I)), tiger torus (((II)(II))I), two 221-ditoruses stacked in medium dimension (((II)(I)I)I) and (((I)(II)I)I)5 4D slices: minor pair of tigers (((II)(II))), vertical stack of two minor pairs of spheritoruses (((II)(I)I)) and (((I)(II)I)), two ditoruses stacked in medium dimension (((II)(I))I) and (((I)(II))I), empty slice (((II)()I)I) and ((()(II)I)I), 2x2 array of torispheres (((I)(I)I)I)Minor pair of tigers slice can evolve into minor pair of 221-tigers slab or tiger torus slab.Vertical stack of two minor pairs of spheritoruses slice can evolve into minor pair of 221-tigers slab and two 221-ditoruses stacked in medium dimension slab.Two ditoruses stacked in medium dimension slice can evolve into tiger torus slab or two 221-ditoruses stacked in medium dimension slab.Empty slice can only evolve into two 221-ditoruses stacked in medium dimension slab.2x2 array of torispheres slice can only evolve into two 221-ditoruses stacked in medium dimension slab, but in two different ways.6 3D cuts: vertical stack of two minor pairs of toruses (((II)(I))) and (((I)(II))), empty cut A (((II)()I)) and ((()(II)I)), 2x2 array of pairs of spheres (((I)(I)I)), empty cut B (((II)())I) and ((()(II))I), 2x2 array of toruses (((I)(I))I), empty cut C (((I)()I)I) and ((()(I)I)I)Vertical stack of two minor pairs of toruses cut can evolve into minor pair of tigers slice, vertical stack of two minor pairs of spheritoruses slice or two ditoruses stacked in medium dimension slice.Empty cut A can evolve into vertical stack of two minor pairs of spheritoruses slice (x2) or empty slice.2x2 array of pairs of spheres cut can evolve into vertical stack of two minor pairs of spheritoruses slice (in two different ways) or 2x2 array of torispheres slice.Empty cut B can evolve into two ditoruses stacked in medium dimension slice (x2) or empty slice.2x2 array of toruses cut can evolve into two ditoruses stacked in medium dimension slice (in two different ways) or 2x2 array of torispheres slice.Empty cut C can evolve into empty slice or 2x2 array of torispheres slice (x2).4 rotations are animations of tiger torus.7 new rotations: vertical stack of two minor pairs of toruses - empty cut A (((II)(x)x)), vertical stack of two minor pairs of toruses - 2x2 array of pairs of spheres (((Ix)(I)x)), empty cut A - 2x2 array of pairs of spheres (((Ix)(x)I)), 2x2 array of pairs of spheres - 2x2 array of toruses (((I)(I)x)x), 2x2 array of pairs of spheres - empty cut C (((I)(x)I)x), 2x2 array of toruses - empty cut C (((I)(x)x)I), empty cut C - alternate empty cut C (((x)(x)I)I)3 empty rotations: empty cut A - empty cut B (((II)()x)y), empty cut A - empty cut C (((Ix)()I)x), empty cut B - empty cut C (((Ix)()x)I)2 double rotations are animations of tiger torus.8 new double rotations: vertical stack of two minor pairs of toruses - alternate empty cut A (((xy)(Ix)y)), vertical stack of two minor pairs of toruses - empty cut C (((Ix)(y)x)y) or (((Ix)(y)y)x), vertical stack of two minor pairs of toruses - alternate empty cut C (((xy)(I)x)y), empty cut A - alternate empty cut A (((xy)(xy)I)), empty cut A - 2x2 array of toruses (((Ix)(x)y)y) or (((Ix)(y)y)x), empty cut A - alternate empty cut C (((xy)(x)I)y), 2x2 array of pairs of spheres - empty cut B (((Ix)(x)y)y) or (((Ix)(y)x)y), empty cut B - alternate empty cut C (((xy)(x)y)I)1 triple rotation: empty cut A - alternate empty cut B (((xy)(xy)z)z) or (((xy)(xz)z)y)13. 231-ditorus (((II)III)I)3 5D slabs: minor pair of 23-toruses (((II)III)), 221-ditorus (((II)II)I), two 41-toruses (((I)III)I)5 4D slices: minor pair of spheritoruses (((II)II)), two pairs of glomes (((I)III)), ditorus (((II)I)I), two torispheres (((I)II)I), empty slice ((()III)I)Minor pair of spheritoruses slice can evolve into minor pair of 23-toruses slab or 221-ditorus slab.Two pairs of glomes slice can evolve into minor pair of 23-toruses slab or two 41-toruses slab.Ditorus slice can only evolve into 221-ditorus slab.Two torispheres slice can evolve into 221-ditorus slab or two 41-toruses slab.Empty slice can only evolve into two 41-toruses slab.6 3D cuts: minor pair of toruses (((II)I)), two pairs of spheres (((I)II)), empty cut A ((()III)), major pair of toruses (((II))I), two toruses (((I)I)I), empty cut B ((()II)I)Minor pair of toruses cut can evolve into minor pair of spheritoruses slice (x2) or ditorus slice.Two pairs of spheres cut can evolve into minor pair of spheritoruses slice, two pairs of glomes slice or two torispheres slice.Empty cut A can evolve into two pairs of glomes slice (x2) or empty slice.Major pair of toruses cut can only evolve into ditorus slice.Two toruses cut can evolve into ditorus slice or two torispheres slice (x2).Empty cut B can evolve into two torispheres slice (x2) or empty slice.4 rotations are animations of 221-ditorus.3 rotations are animations of ditorus.1 new rotation: two pairs of spheres - empty cut A (((x)IIx))1 empty rotation: empty cut A - empty cut B ((()IIx)x)3 double rotations are animations of 221-ditorus.2 new double rotations: minor pair of toruses - empty cut A (((xy)Ixy)), empty cut A - two toruses (((x)Ixy)y)1 triple rotation: empty cut A - major pair of toruses (((xy)xyz)z)14. 420-tiger ((IIII)(II))2 5D slabs: vertical stack of two 41-toruses ((IIII)(I)), 320-tiger ((III)(II))3 4D slices: empty slice ((IIII)()), vertical stack of two torispheres ((III)(I)), tiger ((II)(II))Empty slice can only evolve into vertical stack of two 41-toruses slab.Vertical stack of two torispheres slice can evolve into vertical stack of two 41-toruses slab or 320-tiger slab.Tiger slice can only evolve into 320-tiger slab.3 3D cuts: empty cut ((III)()), vertical stack of two toruses A ((II)(I)), vertical stack of two toruses B ((I)(II))Empty cut can evolve into empty slice or vertical stack of two torispheres slice (x2).Vertical stack of two toruses cut A can evolve into vertical stack of two torispheres slice (x2) or tiger slice.Vertical stack of two toruses cut B can only evolve into tiger slice.1 rotation is animation of 320-tiger.1 rotation is animation of tiger.1 double rotation is animation of 320-tiger.15. 31-torus 20-tiger (((III)I)(II))3 5D slabs: two 311-ditoruses stacked in minor dimension (((III)I)(I)), major[sphere] pair of 320-tigers (((III))(II)), torus tiger (((II)I)(II))5 4D slices: empty slice (((III)I)()), vertical stack of two major pairs of torispheres (((III))(I)), two ditoruses stacked in minor dimension (((II)I)(I)), major pair of tigers (((II))(II)), two tigers (((I)I)(II))Empty slice can only evolve into two 311-ditoruses stacked in minor dimension slabVertical stack of two major pairs of torispheres slice can evolve into two 311-ditoruses stacked in minor dimension slab or major[sphere] pair of 320-tigers slab.Two ditoruses stacked in minor dimension slice can evolve into two 311-ditoruses stacked in minor dimension slab or torus tiger slab.Major pair of tigers slice can evolve into major[sphere] pair of 320-tigers slab or torus tiger slab.Two tigers slice can only evolve into torus tiger slab.6 3D cuts: empty cut A (((III))()), empty cut B (((II)I)()), vertical stack of two major pairs of toruses (((II))(I)), two vertical stacks of two toruses (((I)I)(I)), vertical stack of four toruses (((I))(II)), empty cut C ((()I)(II))Empty cut A can evolve into empty slice or vertical stack of two major pairs of torispheres slice (x2).Empty cut B can evolve into empty slice or two ditoruses stacked in minor dimension slice (x2).Vertical stack of two major pairs of toruses cut can evolve into vertical stack of two major pairs of torispheres slice, two ditoruses stacked in minor dimension slice or major pair of tigers slice.Two vertical stacks of two toruses cut can evolve into two ditoruses stacked in minor dimension slice (x2) or two tigers slice.Vertical stack of four toruses cut can evolve into major pair of tigers slice (x2) or two tigers slice.Empty cut C can only evolve into two tigers slice.7 rotations are animations of torus tiger.1 new rotation: empty cut A - vertical stack of two major pairs of toruses (((IIx))(x))1 empty rotation: empty cut A - empty cut B (((IIx)x)())3 double rotations are animations of torus tiger.2 new double rotations: empty cut A - two vertical stacks of two toruses (((Ixy)x)(y)), empty cut A - vertical stack of four toruses (((Ixy)(xy))1 triple rotation: empty cut A - empty cut C (((xyz)x)(yz))16. Ditorus tiger ((((II)I)I)(II))4 5D slabs: two tritoruses stacked in minor dimension ((((II)I)I)(I)), medium pair of torus tigers ((((II)I))(II)), major[torus] pair of torus tigers ((((II))I)(II)), two[torus] torus tigers ((((I)I)I)(II))8 4D slices: empty slice A ((((II)I)I)()), two medium pairs of ditoruses stacked in minor dimension ((((II)I))(I)), two major pairs of ditoruses stacked in minor dimension ((((II))I)(I)), two minor stacks of two ditoruses ((((I)I)I)(I)), major quartet of tigers ((((II)))(II)), two major[A] pairs of tigers ((((I)I))(II)), four tigers ((((I))I)(II)), empty slice B (((()I)I)(II))Empty slice A can evolve only into two tritoruses stacked in minor dimension slab.Two medium pairs of ditoruses stacked in minor dimension slice can evolve into two tritoruses stacked in minor dimension slab or medium pair of torus tigers slab.Two major pairs of ditoruses stacked in minor dimension slice can evolve into two tritoruses stacked in minor dimension slab or major[torus] pair of torus tigers slab.Two minor stacks of two ditoruses slice can evolve into two tritoruses stacked in minor dimension slab or two[torus] torus tigers slab.Major quartet of tigers slice can evolve into medium pair of torus tigers slab or major[torus] pair of torus tigers slab.Two major[A] pairs of tigers slice can evolve into medium pair of torus tigers slab or two[torus] torus tigers slab.Four tigers slice can evolve into major[torus] pair of torus tigers slab or two[torus] torus tigers slab.Empty slice B can only evolve into two[torus] torus tigers slab.10 3D cuts: empty cut A ((((II)I))()), empty cut B ((((II))I)()), empty cut C ((((I)I)I)()), vertical stack of two major quartets of toruses ((((II)))(I)), two vertical stacks of two major pairs of toruses ((((I)I))(I)), four vertical stacks of two toruses ((((I))I)(I)), empty cut D (((()I)I)(I)), vertical stack of eight toruses ((((I)))(II)), empty cut E (((()I))(II)), empty cut F (((())I)(II))Empty cut A can evolve into empty slice A or two medium pairs of ditoruses stacked in minor dimension slice (x2).Empty cut B can evolve into empty slice A or two major pairs of ditoruses stacked in minor dimension slice (x2).Empty cut C can evolve into empty slice A or two minor stacks of two ditoruses slice (x2).Vertical stack of two major quartets of toruses cut can evolve into two medium pairs of ditoruses stacked in minor dimension slice, two major pairs of ditoruses stacked in minor dimension slice or major quartet of tigers slice.Two vertical stacks of two major pairs of toruses cut can evolve into two medium pairs of ditoruses stacked in minor dimension slice, two minor stacks of two ditoruses slice or two major[A] pairs of tigers slice.Four vertical stacks of two toruses can evolve into two major pairs of ditoruses stacked in minor dimension slice, two minor stacks of two ditoruses slice or four tigers slice.Empty cut D can evolve into two minor stacks of two ditoruses slice (x2) or empty slice B.Vertical stack of eight toruses can evolve into major quartet of tigers slice, two major[A] pairs of tigers slice or four tigers slice.Empty cut E can evolve into two major[A] pairs of tigers slice (x2) or empty slice B.Empty cut F can evolve into four tigers slice (x2) or empty slice B.19 rotations: empty cut A - vertical stack of two major quartets of toruses ((((II)x))(x)), empty cut A - two vertical stacks of two major pairs of toruses ((((Ix)I))(x)), empty cut B - vertical stack of two major quartets of toruses ((((II))x)(x)), empty cut B - four vertical stacks of two toruses ((((Ix))I)(x)), empty cut C - two vertical stacks of two major pairs of toruses ((((I)I)x)(x)), empty cut C - four vertical stacks of two toruses ((((I)x)I)(x)), empty cut C - empty cut D ((((x)I)I)(x)), vertical stack of two major quartets of toruses - two vertical stacks of two major pairs of toruses ((((Ix)x))(I)), vertical stack of two major quartets of toruses - four vertical stacks of two toruses ((((Ix))x)(I)), vertical stack of two major quartets of toruses - vertical stack of eight toruses ((((Ix)))(Ix)), two vertical stacks of two major pairs of toruses - four vertical stacks of two toruses ((((I)x)x)(I)), two vertical stacks of two major pairs of toruses - empty cut D ((((x)I)x)(I)), two vertical stacks of two major pairs of toruses - vertical stack of eight toruses ((((I)x))(Ix)), two vertical stacks of two major pairs of toruses - empty cut E ((((x)I))(Ix)), four vertical stacks of two toruses - empty cut D ((((x)x)I)(I)), four vertical stacks of two toruses - vertical stack of eight toruses ((((I))x)(Ix)), four vertical stacks of two toruses - empty cut F ((((x))I)(Ix)), vertical stack of eight toruses - empty cut E ((((x)x))(II)), vertical stack of eight toruses - empty cut F ((((x))x)(II))6 empty rotations: empty cut A - empty cut B ((((II)x)x)()), empty cut A - empty cut C ((((Ix)I)x)()), empty cut B - empty cut C ((((Ix)x)I)()), empty cut D - empty cut E (((()I)x)(Ix)), empty cut D - empty cut F (((()x)I)(Ix)), empty cut E - empty cut F (((()x)x)(II))18 double rotations: empty cut A - four vertical stacks of two toruses ((((Ix)y)x)(y)) or ((((Ix)y)y)(x)), empty cut A - empty cut D ((((xy)I)x)(y)), empty cut A - vertical stack of eight toruses ((((Ix)y))(xy)), empty cut A - empty cut E ((((xy)I))(xy)), empty cut B - two vertical stacks of two major pairs of toruses ((((Ix)x)y)(y)) or ((((Ix)y)y)(x)), empty cut B - empty cut D ((((xy)x)I)(y)), empty cut B - vertical stack of eight toruses ((((Ix))y)(xy)), empty cut B - empty cut F ((((xy))I)(xy)), empty cut C - vertical stack of two major quartets of toruses ((((Ix)x)y)(y)) or ((((Ix)y)x)(y)), empty cut C - vertical stack of eight toruses ((((I)x)y)(xy)), empty cut C - empty cut E ((((x)I)y)(xy)), empty cut C - empty cut F ((((x)y)I)(xy)), vertical stack of two major quartets of toruses - empty cut D ((((xy)x)y)(I)), vertical stack of two major quartets of toruses - empty cut E (((((xy)x))(Iy)), vertical stack of two major quartets of toruses - empty cut F ((((xy))x)(Iy)), two vertical stacks of two major pairs of toruses - empty cut F ((((x)y)x)(Iy)) or ((((x)y)y)(Ix)), four vertical stacks of two toruses - empty cut E ((((x)x)y)(Iy)) or ((((x)y)y)(Ix)), empty cut D - vertical stack of eight toruses ((((x)x)y)(Iy)) or ((((x)y)x)(Iy))2 triple rotations: empty cut A - empty cut F ((((xy)z)x)(yz)) or ((((xy)z)z)(xy)), empty cut B - empty cut E ((((xy)x)z)(yz)) or ((((xy)z)z)(xy))17. Double tiger (((II)(II))(II))2 5D slabs: two tiger toruses stacked in minor dimension (((II)(II))(I)), two torus tigers stacked in medium dimension (((II)(I))(II)) and (((I)(II))(II))4 4D slices: empty slice A (((II)(II))()), 2x2 medium/minor stack of ditoruses (((II)(I))(I)) and (((I)(II))(I)), empty slice B (((II)())(II)) and ((()(II))(II)), 2x2 array of tigers (((I)(I))(II))Empty slice A can only evolve into two tiger toruses stacked in minor dimension slab.2x2 medium/minor stack of ditoruses slice can evolve into two tiger toruses stacked in minor dimension slab or two torus tigers stacked in medium dimension slab.Empty slice B can only evolve into two torus tigers stacked in medium dimension slab.2x2 array of tigers slice can only evolve into two torus tigers stacked in medium dimension slab, but in two different ways.4 3D cuts: empty cut A (((II)(I))()) and (((I)(II))()), empty cut B (((II)())(I)) and ((()(II))(I)), 2x2 array of vertical stacks of two toruses (((I)(I))(I)), empty cut C (((I)())(II)) and ((()(I))(II))Empty cut A can evolve into empty slice A or 2x2 medium/minor stack of ditoruses slice (x2).Empty cut B can evolve into 2x2 medium/minor stack of ditoruses slice (x2) or empty slice B.2x2 array of vertical stacks of two toruses cut can evolve into 2x2 medium/minor stack of ditoruses slice (in two different ways) or 2x2 array of tigers slice.Empty cut C can evolve into empty slice B or 2x2 array of tigers slice (x2).6 rotations: empty cut A - empty cut B (((II)(x))(x)), empty cut A - 2x2 array of vertical stacks of two toruses (((Ix)(I))(x)), empty cut B - 2x2 array of vertical stacks of two toruses (((Ix)(x))(I)), empty cut B - empty cut C (((Ix)())(Ix)), 2x2 array of vertical stacks of two toruses - empty cut C (((I)(x))(Ix)), empty cut C - alternate empty cut C (((x)(x))(II))1 empty rotation: empty cut A - alternate empty cut A (((Ix)(Ix))())5 double rotations: empty cut A - alternate empty cut B (((xy)(Ix))(y)), empty cut A - empty cut C (((Ix)(x))(xy)), empty cut A - alternate empty cut C (((xy)(I))(xy)), empty cut B - alternate empty cut B ((((xy)(xy))(I)), empty cut B - alternate empty cut C (((xy)(x))(Ix))18. 22-torus 20-tiger (((II)II)(II))3 5D slabs: two 221-ditoruses stacked in minor dimension (((II)II)(I)), torus tiger (((II)I)(II)), two[sphere] 320-tigers (((I)II)(II))6 4D slices: empty slice A (((II)II)()), two ditoruses stacked in minor dimension (((II)I)(I)), two vertical stacks of two torispheres (((I)II)(I)), major pair of tigers (((II))(II)), two tigers (((I)I)(II)), empty slice B ((()II)(II))Empty slice A can only evolve into two 221-ditoruses stacked in minor dimension slab.Two ditoruses stacked in minor dimension slice can evolve into two 221-ditoruses stacked in minor dimension slab or torus tiger slab.Two vertical stacks of two torispheres slice can evolve into two 221-ditoruses stacked in minor dimension slab or two[sphere] 320-tigers slab.Major pair of tigers slice can only evolve into torus tiger slab.Two tigers slice can evolve into torus tiger slab or two[sphere] 320-tigers slab.Empty slice B can only evolve into two[sphere] 320-tigers slab.7 3D cuts: empty cut A (((II)I)()), empty cut B (((I)II)()), vertical stack of two major pairs of toruses (((II))(I)), two vertical stacks of two toruses (((I)I)(I)), empty cut C ((()II)(I)), vertical stack of four toruses (((I))(II)), empty cut D ((()I)(II))Empty cut A can evolve into empty slice A or two ditoruses stacked in minor dimension slice (x2).Empty cut B can evolve into empty slice A or two vertical stacks of two torispheres slice (x2).Vertical stack of two major pairs of toruses cut can evolve into two ditoruses stacked in minor dimension slice (x2) or major pair of tigers slice.Two vertical stacks of two toruses cut can evolve into two ditoruses stacked in minor dimension slice, two vertical stacks of two torispheres slice or two tigers slice.Empty cut C can evolve into two vertical stacks of two torispheres slice (x2) or empty slice B.Vertical stack of four toruses cut can evolve into major pair of tigers slice or two tigers slice (x2).Empty cut D can evolve into two tigers slice (x2) or empty slice B.7 rotations are animations of torus tiger.3 new rotations: empty cut B - two vertical stacks of two toruses (((I)Ix)(x)), empty cut B - empty cut C (((x)II)(x)), two vertical stacks of two toruses - empty cut C (((x)Ix)(I))2 empty rotations: empty cut A - empty cut B (((Ix)Ix)()), empty cut C - empty cut D ((()Ix)(Ix))3 double rotations are animations of torus tiger.6 new double rotations: empty cut A - empty cut C (((xy)Ix)(y)), empty cut B - vertical stack of two major pairs of toruses (((Ix)xy)(y)), empty cut B - vertical stack of four toruses (((I)xy)(xy)), empty cut B - empty cut D (((x)Iy)(xy)), vertical stack of two major pairs of toruses - empty cut C (((xy)xy)(I)), empty cut C - vertical stack of four toruses (((x)xy)(Iy))19. 42-torus ((IIII)II)2 5D slabs: 41-torus ((IIII)I), 32-torus ((III)II)3 4D slices: pair of glomes ((IIII)), torisphere ((III)I), spheritorus ((II)II)Pair of glomes slice can only evolve into 41-torus slab.Torisphere slice can evolve into 41-torus slab or 32-torus slab.Spheritorus slice can only evolve into 32-torus slab.3 3D cuts: pair of spheres ((III)), torus ((II)I), two spheres ((I)II)Pair of spheres cut can evolve into pair of glomes slice or torisphere slice (x2).Torus cut can evolve into torisphere slice (x2) or spheritorus slice.Two spheres cut can only evolve into spheritorus slice.1 rotation is animation of torisphere.1 rotation is animation of spheritorus.1 double rotation is animation of 32-torus.20. 312-ditorus (((III)I)II)3 5D slabs: 311-ditorus (((III)I)I), major pair of 32-toruses (((III))II), 212-ditorus (((II)I)II)5 4D slices: minor pair of torispheres (((III)I)), major pair of torispheres (((III))I), ditorus (((II)I)I), major pair of spheritoruses (((II))II), two spheritoruses (((I)I)II)Minor pair of torispheres slice can only evolve into 311-ditorus slab.Major pair of torispheres slice can evolve into 311-ditorus slab or major pair of 32-toruses slab.Ditorus slice can evolve into 311-ditorus slab or 212-ditorus slab.Major pair of spheritoruses slice can evolve into major pair of 32-toruses slab or 212-ditorus slab.Two spheritoruses slice can only evolve into 212-ditorus slab.6 3D cuts: quartet of spheres (((III))), minor pair of toruses (((II)I)), major pair of toruses (((II))I), two toruses (((I)I)I), four spheres (((I))II), empty cut ((()I)II)Quartet of spheres cut can evolve into minor pair of torispheres slice or major pair of torispheres slice (x2).Minor pair of toruses cut can evolve into minor pair of torispheres slice or ditorus slice (x2).Major pair of toruses cut can evolve into major pair of torispheres slice, ditorus slice or major pair of spheritoruses slice.Two toruses cut can evolve into ditorus slice (x2) or two spheritoruses slice.Four spheres cut can evolve into major pair of spheritoruses slice (x2) or two spheritoruses slice.Empty cut can only evolve into two spheritoruses slice.2 rotations are animations of 311-ditorus.4 rotations are animations of 212-ditorus.3 rotations are animations of ditorus.1 double rotation is animation of 311-ditorus.3 double rotations are animations of 212-ditorus.1 new double rotation: quartet of spheres - four spheres (((Ixy))xy)1 triple rotation: quartet of spheres - empty cut (((xyz)x)yz)21. 2112-tritorus ((((II)I)I)II)4 5D slabs: tritorus ((((II)I)I)I), medium pair of 212-ditoruses ((((II)I))II), major pair of 212-ditoruses ((((II))I)II), two 212-ditoruses ((((I)I)I)II)8 4D slices: minor pair of ditoruses ((((II)I)I)), medium pair of ditoruses ((((II)I))I), major pair of ditoruses ((((II))I)I), two ditoruses ((((I)I)I)I), major quartet of spheritoruses ((((II)))II), two major pairs of spheritoruses ((((I)I))II), four spheritoruses ((((I))I)II), empty slice (((()I)I)II)Minor pair of ditoruses slice can only evolve into tritorus slab.Medium pair of ditoruses slice can evolve into tritorus slab or medium pair of 212-ditoruses slab.Major pair of ditoruses slice can evolve into tritorus slab or major pair of 212-ditoruses slab.Two ditoruses slice can evolve into tritorus slab or two 212-ditoruses slab.Major quartet of spheritoruses slice can evolve into  medium pair of 212-ditoruses slab or major pair of 212-ditoruses slab.Two major pairs of spheritoruses slice can evolve into medium pair of 212-ditoruses slab or two 212-ditoruses slab.Four spheritoruses slice can evolve into major pair of 212-ditoruses slab or two 212-ditoruses slab.Empty slice can only evolve into two 212-ditoruses slab.10 3D cuts: minor quartet of toruses ((((II)I))), major/minor quartet of toruses ((((II))I)), two minor pairs of toruses ((((I)I)I)), major quartet of toruses ((((II)))I), two major pairs of toruses ((((I)I))I), four toruses ((((I))I)I), empty cut A (((()I)I)I), eight spheres ((((I)))II), empty cut B (((()I))II), empty cut C (((())I)II)Minor quartet of toruses cut can evolve into minor pair of ditoruses slice or medium pair of ditoruses slice (x2).Major/minor quartet of toruses cut can evolve into minor pair of ditoruses slice or major pair of ditoruses slice (x2).Two minor pairs of toruses cut can evolve into minor pair of ditoruses slice or two ditoruses slice (x2).Major quartet of toruses cut can evolve into medium pair of ditoruses slice, major pair of ditoruses slice or major quartet of spheritoruses slice.Two major pairs of toruses cut can evolve into medium pair of ditoruses slice, two ditoruses slice or two major pairs of spheritoruses slice.Four toruses cut can evolve into major pair of ditoruses slice, two ditoruses slice or four spheritoruses slice.Empty cut A can evolve into two ditoruses slice (x2) or empty slice.Eight spheres cut can evolve into major quartet of spheritoruses slice, two major pairs of spheritoruses slice or four spheritoruses slice.Empty cut B can evolve into two major pairs of spheritoruses slice (x2) or empty slice.Empty cut C can evolve into four spheritoruses slice (x2) or empty slice.15 rotations are animations of tritorus.7 new rotations: major quartet of toruses - eight spheres ((((Ix)))Ix), two major pairs of toruses - eight spheres ((((I)x))Ix), two major pairs of toruses - empty cut B ((((x)I))Ix), four toruses - eight spheres ((((I))x)Ix), four toruses - empty cut C ((((x))I)Ix), eight spheres - empty cut B ((((x)x))II), eight spheres - empty cut C ((((x))x)II)3 empty rotations: empty cut A - empty cut B (((()I)x)Ix), empty cut A - empty cut C (((()x)I)Ix), empty cut B - empty cut C (((()x)x)II)6 double rotations are animations of tritorus.12 new double rotations: minor quartet of toruses - eight spheres ((((Ix)x))xy), minor quartet of toruses - empty cut B ((((xy)I))xy), major/minor quartet of toruses - eight spheres ((((Ix))y)xy), major/minor quartet of toruses - empty cut C ((((xy))I)xy), two minor pairs of toruses - eight spheres ((((I)x)y)xy), two minor pairs of toruses - empty cut B ((((x)I)y)xy), two minor pairs of toruses - empty cut C ((((x)y)I)xy), major quartet of toruses - empty cut B ((((xy)x))Iy), major quartet of toruses - empty cut C ((((xy))x)Iy), two major pairs of toruses - empty cut C ((((x)y)x)Iy) or ((((x)y)y)Ix), four toruses - empty cut B ((((x)x)y)Iy) or ((((x)y)y)Ix), empty cut A - eight spheres ((((x)x)y)Iy) or ((((x)y)x)Iy)2 triple rotations: minor quartet of toruses - empty cut C ((((xy)z)x)yz) or ((((xy)z)z)xy), major/minor quartet of toruses - empty cut B ((((xy)x)z)yz) or ((((xy)z)z)xy)22. 220-tiger 2-torus (((II)(II))II)2 5D slabs: tiger torus (((II)(II))I), two 212-ditoruses stacked in medium dimension (((II)(I))II) and (((I)(II))II) 4 4D slices: minor pair of tigers (((II)(II))), two ditoruses stacked in medium dimension (((II)(I))I) and (((I)(II))I), empty slice (((II)())II) and ((()(II))II), 2x2 array of spheritoruses (((I)(I))II)Minor pair of tigers slice can only evolve into tiger torus slab.Two ditoruses stacked in medium dimension slice can evolve into tiger torus slab or two 212-ditoruses stacked in medium dimension slab.Empty slice can only evolve into two 212-ditoruses stacked in medium dimension slab.2x2 array of spheritoruses slice can only evolve into two 212-ditoruses stacked in medium dimension slab, but in two different ways.4 3D cuts: vertical stack of two minor pairs of toruses (((II)(I))) and (((I)(II))), empty cut A (((II)())I) and ((()(II))I), 2x2 array of toruses (((I)(I))I), empty cut B (((I)())II) and ((()(I))II)Vertical stack of two minor pairs of toruses cut can evolve into minor pair of tigers slice or two ditoruses stacked in medium dimension slice (x2).Empty cut A can evolve into two ditoruses stacked in medium dimension slice (x2) or empty slice.2x2 array of toruses cut can evolve into two ditoruses stacked in medium dimension slice (in two different ways) or 2x2 array of spheritoruses slice.Empty cut B can evolve into empty slice or 2x2 array of spheritoruses slice (x2).4 rotations are animations of tiger torus.3 new rotations: empty cut A - empty cut B (((Ix)())Ix), 2x2 array of toruses - empty cut B (((I)(x))Ix), empty cut B - alternate empty cut B (((x)(x))II)2 double rotations are animations of tiger torus.3 new double rotations: vertical stack of two minor pairs of toruses - empty cut B ((((Ix)(y))xy), vertical stack of two minor pairs of toruses - alternate empty cut B (((xy)(I))xy), empty cut A - alternate empty cut B (((xy)(x))Iy)23. 222-ditorus (((II)II)II)3 5D slabs: 221-ditorus (((II)II)I), 212-ditorus (((II)I)II), two 32-toruses (((I)II)II)6 4D slices: minor pair of spheritoruses (((II)II)), ditorus (((II)I)I), two torispheres (((I)II)I), major pair of spheritoruses (((II))II), two spheritoruses (((I)I)II), empty slice ((()II)II)Minor pair of spheritoruses slice can only evolve into 221-ditorus slab.Ditorus slice can evolve into 221-ditorus slab or 212-ditorus slab.Two torispheres slice can evolve into 221-ditorus slab or two 32-toruses slab.Major pair of spheritoruses slice can only evolve into 212-ditorus slab.Two spheritoruses slice can evolve into 212-ditorus slab or two 32-toruses slab.Empty slice can only evolve into two 32-toruses slab.7 3D cuts: minor pair of toruses (((II)I)), two pairs of spheres (((I)II)), major pair of toruses (((II))I), two toruses (((I)I)I), empty cut A ((()II)I), four spheres (((I))II), empty cut B ((()I)II)Minor pair of toruses cut can evolve into minor pair of spheritoruses slice or ditorus slice (x2).Two pairs of spheres cut can evolve into minor pair of spheritoruses slice or two torispheres slice (x2).Major pair of toruses cut can evolve into ditorus slice (x2) or major pair of spheritoruses slice.Two toruses cut can evolve into ditorus slice, two torispheres slice or two spheritoruses slice.Empty cut A can evolve into two torispheres slice (x2) or empty slice.Four spheres cut can evolve into major pair of spheritoruses slice or two spheritoruses slice (x2).Empty cut B can evolve into two spheritoruses slice (x2) or empty slice.4 rotations are animations of 221-ditorus.4 rotations are animations of 212-ditorus.3 rotations are animations of ditorus.1 empty rotation: empty cut A - empty cut B ((()Ix)Ix)3 double rotations are animations of 221-ditorus.3 double rotations are animations of 212-ditorus.3 new double rotations: two pairs of spheres - four spheres (((I)xy)xy)), two pairs of spheres - empty cut B (((x)Iy)xy), empty cut A - four spheres (((x)xy)Iy)24. 330-tiger ((III)(III))1 5D slab: 320-tiger ((III)(II)) and ((II)(III))2 4D slices: vertical stack of two torispheres ((III)(I)) and ((I)(III)), tiger ((II)(II))Vertical stack of two torispheres slice can only evolve into 320-tiger slab.Tiger slice can only evolve into 320-tiger slab, but in two different ways.2 3D cuts: empty cut ((III)()) and (()(III)), vertical stack of two toruses ((II)(I)) and ((I)(II))Empty cut can evolve only into vertical stack of two torispheres slice.Vertical stack of two toruses cut can evolve into vertical stack of two torispheres slice or tiger slice (x2).1 rotation is animation of 320-tiger.1 rotation is animation of tiger.1 double rotation is animation of 320-tiger.1 triple rotation: empty cut - alternate empty cut ((xyz)(xyz))25. 21-torus 30-tiger (((II)I)(III))3 5D slabs: torus tiger (((II)I)(II)), major[circle] pair of 320-tigers (((II))(III)), two[circle] 320-tigers (((I)I)(III))5 4D slices: two ditoruses stacked in minor dimension (((II)I)(I)), major pair of tigers (((II))(II)), two tigers (((I)I)(II)), vertical stack of four torispheres (((I))(III)), empty slice ((()I)(III))Two ditoruses stacked in minor dimension slice can only evolve into torus tiger slab.Major pair of tigers slice can evolve into torus tiger slab or major[circle] pair of 320-tigers slab.Two tigers slice can evolve into torus tiger slab or two[circle] 320-tigers slab.Vertical stack of four torispheres slice can evolve into major[circle] pair of 320-tigers slab or two[circle] 320-tigers slab.Empty slice can only evolve into two[circle] 320-tigers slab.6 3D cuts: empty cut A (((II)I)()), vertical stack of two major pairs of toruses (((II))(I)), two vertical stacks of two toruses (((I)I)(I)), vertical stack of four toruses (((I))(II)), empty cut B ((()I)(II)), empty cut C ((())(III))Empty cut A can evolve only into two ditoruses stacked in minor dimension slice.Vertical stack of two major pairs of toruses cut can evolve into two ditoruses stacked in minor dimension slice or major pair of tigers slice (x2).Two vertical stacks of two toruses cut can evolve into two ditoruses stacked in minor dimension slice or two tigers slice (x2).Vertical stack of four toruses cut can evolve into major pair of tigers slice, two tigers slice or vertical stack of four torispheres slice.Empty cut B can evolve into two tigers slice (x2) or empty slice.Empty cut C can evolve into vertical stack of four torispheres slice (x2) or empty slice.7 rotations are animations of torus tiger.1 new rotation: vertical stack of four toruses - empty cut C (((x))(IIx))1 empty rotation: empty cut B - empty cut C ((()x)(IIx))3 double rotations are animations of torus tiger.2 new double rotations: vertical stack of two major pairs of toruses - empty cut C (((xy))(Ixy)), two vertical stacks of two toruses - empty cut C (((x)y)(Ixy))1 triple rotation: empty cut A - empty cut C (((xy)z)(xyz))26. Duotorus tiger (((II)I)((II)I))2 5D slabs: major[circle] pair of torus tigers (((II)I)((II))) and (((II))((II)I)), two[circle] torus tigers (((II)I)((I)I)) and (((I)I)((II)I))5 4D slices: four ditoruses stacked in minor dimension (((II)I)((I))) and (((I))((II)I)), major[A]/major[B] quartet of tigers (((II))((II))), two major[B] pairs of tigers (((I)I)((II))) and (((II))((I)I)), empty slice (((II)I)(()I)) and ((()I)((II)I)), 2x2 major[A]/major[B] array of tigers (((I)I)((I)I))Four ditoruses stacked in minor dimension slice can evolve into major[circle] pair of torus tigers slab or two[circle] torus tigers slab.Major[A]/major[B] quartet of tigers can only evolve into major[circle] pair of torus tigers slab, but in two different ways.Two major[B] pairs of tigers can evolve into major[circle] pair of torus tigers slab or two[circle] torus tigers slab.Empty slice can only evolve into two[circle] torus tigers slab.2x2 major[A]/major[B] array of tigers can only evolve into two[circle] torus tigers slab, but in two different ways.5 3D cuts: empty cut A (((II)I)(())) and ((())((II)I)), vertical stack of four major pairs of toruses (((II))((I))) and (((I))((II))), two vertical stacks of four toruses (((I)I)((I))) and (((I))((I)I)), empty cut B ((()I)((II))) and (((II))(()I)), empty cut C (((I)I)(()I)) and ((()I)((I)I))Empty cut A can evolve into four ditoruses stacked in minor dimension slice (x2) or empty slice.Vertical stack of four major pairs of toruses cut can evolve into four ditoruses stacked in minor dimension slice, major[A]/major[B] quartet of tigers slice or two major[B] pairs of tigers slice.Two vertical stacks of four toruses cut can evolve into four ditoruses stacked in minor dimension slice, two major[B] pairs of tigers slice or 2x2 major[A]/major[B] array of tigers slice.Empty cut B can evolve into two major[B] pairs of tigers slice (x2) or empty slice.Empty cut C can evolve into empty slice or 2x2 major[A]/major[B] array of tigers slice (x2).11 rotations: empty cut A - vertical stack of four major pairs of toruses (((II)x)((x))), empty cut A - two vertical stacks of four toruses (((Ix)I)((x))), vertical stack of four major pairs of toruses - alternate vertical stack of four major pairs of toruses (((Ix))((Ix))), vertical stack of four major pairs of toruses - two vertical stacks of four toruses (((Ix)x)((I))), vertical stack of four major pairs of toruses - alternate two vertical stacks of four toruses (((Ix))((I)x)), vertical stack of four major pairs of toruses - alternate empty cut B (((II))((x)x)), two vertical stacks of four toruses - alternate two vertical stacks of four toruses (((I)x)((I)x)), two vertical stacks of four toruses - empty cut B (((x)I)((Ix))), two vertical stacks of four toruses - empty cut C (((I)I)((x)x)), two vertical stacks of four toruses - alternate empty cut C (((x)I)((I)x)), empty cut C - alternate empty cut C (((x)I)((x)I))3 empty rotations: empty cut A - alternate empty cut B (((II)x)(()x)), empty cut A - empty cut C (((Ix)I)(()x)), empty cut B - alternate empty cut C ((()I)((Ix)x))9 double rotations: empty cut A - alternate vertical stack of four major pairs of toruses (((Ix)y))((xy))), empty cut A - alternate two vertical stacks of four toruses (((Ix)y)((x)y)) and (((Ix)y)(y(x))), empty cut A - empty cut B (((xy)I)((xy))), empty cut A - alternate empty cut C (((xy)I)((x)y)), vertical stack of four major pairs of toruses - empty cut B (((xy)x)((Iy))), vertical stack of four major pairs of toruses - empty cut C (((Ix)x)((y)y)) or (((Ix)y)((y)x)), vertical stack of four major pairs of toruses - alternate empty cut C (((xy)x)((I)y)), two vertical stacks of four toruses - alternate empty cut B (((Ix)x)((y)y)) or (((Ix)y)((x)y)), empty cut B - empty cut C (((x)I)((xy)y))2 triple rotations: empty cut A - alternate empty cut A (((xy)z)((xy)z)) and (((xy)z)((xz)y)), empty cut B - alternate empty cut B (((xy)x)((yz)z)) or (((xy)z)((xy)z))27. 321-tiger ((III)(II)I)3 5D slabs: 320-tiger ((III)(II)), vertical stack of two 32-toruses ((III)(I)I), 221-tiger ((II)(II)I)5 4D slices: vertical stack of two torispheres ((III)(I)), tiger ((II)(II)), empty slice ((III)()I), vertical stack of two spheritoruses A ((II)(I)I), vertical stack of two spheritoruses B ((I)(II)I)Vertical stack of two torispheres slice can evolve into 320-tiger slab or vertical stack of two 32-toruses slab.Tiger slice can evolve into 320-tiger slab or 221-tiger slab.Empty slice can only evolve into vertical stack of two 32-toruses slab.Vertical stack of two spheritoruses A slice can evolve into vertical stack of two 32-toruses slab or 221-tiger slab.Vertical stack of two spheritoruses B slice can only evolve into 221-tiger slab.6 3D cuts: empty cut A ((III)()), vertical stack of two toruses A ((II)(I)), vertical stack of two toruses B ((I)(II)), empty cut B ((II)()I), 2x2 array of spheres ((I)(I)I), empty cut C (()(II)I)Empty cut A can evolve into vertical stack of two torispheres slice (x2) or empty slice.Vertical stack of two toruses cut A can evolve into vertical stack of two torispheres slice, tiger slice or vertical stack of two spheritoruses slice A.Vertical stack of two toruses cut B can evolve into tiger slice (x2) or vertical stack of two spheritoruses slice B.Empty cut B can evolve into empty slice or vertical stack of two spheritoruses slice A (x2).2x2 array of spheres cut can evolve into vertical stack of two spheritoruses slice A (x2) or vertical stack of two spheritoruses slice B.Empty cut C can evolve only into vertical stack of two spheritoruses slice B.1 rotation is animation of 320-tiger.6 rotations are animations of 221-tiger.1 rotation is animation of tiger.1 empty rotation: empty cut A - empty cut B ((IIx)()x)1 double rotations is animation of 320-tiger.3 double rotations are animations of 221-tiger.1 new double rotation: empty cut A - 2x2 array of spheres (((Ixy)(x)y)1 triple rotation: empty cut A - empty cut C ((xyz)(xy)z)28. 21-torus 21-tiger (((II)I)(II)I)4 5D slabs: torus tiger (((II)I)(II)), two 212-ditoruses stacked in minor dimension (((II)I)(I)I), major pair of 221-tigers (((II))(II)I), two 221-tigers (((I)I)(II)I)8 4D slices: two ditoruses stacked in minor dimension (((II)I)(I)), major pair of tigers (((II))(II)), two tigers (((I)I)(II)), empty slice A (((II)I)()I), vertical stack of two major pairs of spheritoruses (((II))(I)I), two vertical stacks of two spheritoruses (((I)I)(I)I), vertical stack of four spheritoruses (((I))(II)I), empty slice B ((()I)(II)I)Two ditoruses stacked in minor dimension slice can evolve into torus tiger slab or two 212-ditoruses stacked in minor dimension slab.Major pair of tigers slice can evolve into torus tiger slab or major pair of 221-tigers slab.Two tigers slice can evolve into torus tiger slab or two 221-tigers slab.Empty slice A can evolve only into two 212-ditoruses stacked in minor dimension slab.Vertical stack of two major pairs of spheritoruses slice can evolve into two 212-ditoruses stacked in minor dimension slab or major pair of 221-tigers slab.Two vertical stacks of two spheritoruses slice can evolve into two 212-ditoruses stacked in minor dimension slab or two 221-tigers slab.Vertical stack of four spheritoruses slice can evolve into major pair of 221-tigers slab or two 221-tigers slab.Empty slice B can only evolve into two 221-tigers slab.10 3D cuts: empty cut A (((II)I)()), vertical stack of two major pairs of toruses (((II))(I)), two vertical stacks of two toruses (((I)I)(I)), vertical stack of four toruses (((I))(II)), empty cut B ((()I)(II)), empty cut C (((II))()I), empty cut D (((I)I)()I), 4x2 array of spheres (((I))(I)I), empty cut E ((()I)(I)I), empty cut F ((())(II)I)Empty cut A can evolve into two ditoruses stacked in minor dimension slice (x2) or empty slice A.Vertical stack of two major pairs of toruses cut can evolve into two ditoruses stacked in minor dimension slice, major pair of tigers slice or vertical stack of two major pairs of spheritoruses slice.Two vertical stacks of two toruses cut can evolve into two ditoruses stacked in minor dimension slice, two tigers slice or two vertical stacks of two spheritoruses slice.Vertical stack of four toruses cut can evolve into major pair of tigers slice, two tigers slice or vertical stack of four spheritoruses sliceEmpty cut B can evolve into two tigers slice (x2) or empty slice B.Empty cut C can evolve into empty slice A or vertical stack of two major pairs of spheritoruses slice (x2)Empty cut D can evolve into empty slice A or two vertical stacks of two spheritoruses slice (x2).4x2 array of spheres cut can evolve into vertical stack of two major pairs of spheritoruses slice, two vertical stacks of two spheritoruses slice or vertical stack of four spheritoruses slice.Empty cut E can evolve into two vertical stacks of two spheritoruses slice (x2) or empty slice B.Empty cut F can evolve into vertical stack of four spheritoruses slice (x2) or empty slice B.7 rotations are animations of torus tiger.12 new rotations: vertical stack of two major pairs of toruses - empty cut C (((II))((x)x)), vertical stack of two major pairs of toruses - 4x2 array of spheres (((Ix))(I)x), two vertical stacks of two toruses - empty cut D (((I)I)(x)x), two vertical stacks of two toruses - 4x2 array of spheres (((I)x)(I)x), two vertical stacks of two toruses - empty cut E (((x)I)(I)x), vertical stack of four toruses - 4x2 array of spheres (((I))(Ix)x), vertical stack of four toruses - empty cut F (((x))(II)x), empty cut C - 4x2 array of spheres (((Ix))(x)I), empty cut D - 4x2 array of spheres (((I)x)(x)I), empty cut D - empty cut E (((x)I)(x)I), 4x2 array of spheres - empty cut E (((x)x)(I)I), 4x2 array of spheres - empty cut F (((x))(Ix)I)6 empty rotations: empty cut A - empty cut C (((II)x)()x), empty cut A - empty cut D (((Ix)I)()x), empty cut B - empty cut E ((()I)(Ix)x), empty cut B - empty cut F ((()x)(II)x), empty cut C - empty cut D (((Ix)x)()I), empty cut E - empty cut F ((()x)(Ix)I)3 double rotations are animations of torus tiger.15 new double rotations: empty cut A - 4x2 array of spheres (((Ix)y)(x)y) or (((Ix)y)(y)x), empty cut A - empty cut E (((xy)I)(x)y)), vertical stack of two major pairs of toruses - empty cut D (((Ix)x)(y)y) or (((Ix)y)(y)x), vertical stack of two major pairs of toruses - empty cut E (((xy)x)(I)y), vertical stack of two major pairs of toruses - empty cut F (((xy))(Ix)y), two vertical stacks of two toruses - empty cut C (((Ix)x)(y)y) or (((Ix)y)(x)y), two vertical stacks of two toruses - empty cut F (((x)y)(Ix)y) or (((x)y)(Iy)x), vertical stack of four toruses - empty cut C (((Ix))(xy)y), vertical stack of four toruses - empty cut D (((I)x)(xy)y), vertical stack of four toruses - empty cut E (((x)x)(Iy)y) or (((x)y)(Iy)x), empty cut B - empty cut D (((x)I)(xy)y), empty cut B - 4x2 array of spheres (((x)x)(Iy)y) or (((x)y)(Ix)y), empty cut C - empty cut E (((xy)x)(y)I), empty cut C - empty cut F (((xy))(xy)I), empty cut D - empty cut F (((x)y)(xy)I)2 triple rotations: empty cut A - empty cut F ((xy)z)(xy)z) or ((xy)z)(xz)y), empty cut B - empty cut C (((xy)x)(yz)z) or (((xy)z)(xy)z)29. 33-torus ((III)III)2 5D slabs: 32-torus ((III)II), 23-torus ((II)III)3 4D slices: torisphere ((III)I), spheritorus ((II)II), two glomes ((I)III)Torisphere slice can only evolve into 32-torus slab.Spheritorus slice can evolve into 32-torus slab or 23-torus slab.Two glomes slice can only evolve into 23-torus slab.4 3D cuts: pair of spheres ((III)), torus ((II)I), two spheres ((I)II), empty cut (()III)Pair of spheres cut can only evolve into torisphere slice.Torus cut can evolve into torisphere slice or spheritorus slice (x2).Two spheres cut can evolve into spheritorus slice (x2) or two glomes slice.Empty cut can only evolve into two glomes slice.1 rotation is animation of 23-torus.1 rotation is animation of torisphere.1 rotation is animation of spheritorus.1 double rotation is animation of 32-torus.1 double rotation is animation of 23-torus.1 triple rotation: pair of spheres - empty cut ((xyz)xyz)30. 213-ditorus (((II)I)III)3 5D slabs: 212-ditorus (((II)I)II), major pair of 23-toruses (((II))III), two 23-toruses (((I)I)III)5 4D slices: ditorus (((II)I)I), major pair of spheritoruses (((II))II), two spheritoruses (((I)I)II), four glomes (((I))III), empty slice ((()I)III)Ditorus slice can only evolve into 212-ditorus slab.Major pair of spheritoruses slice can evolve into 212-ditorus slab or major pair of 23-toruses slab.Two spheritoruses slice can evolve into 212-ditorus slab or two 23-toruses slab.Four glomes slice can evolve into major pair of 23-toruses slab or two 23-toruses slab.Empty slice can only evolve into two 23-toruses slab.6 3D cuts: minor pair of toruses (((II)I)), major pair of toruses (((II))I), two toruses (((I)I)I), four spheres (((I))II), empty cut A ((()I)II), empty cut B ((())III)Minor pair of toruses cut can only evolve into ditorus slice.Major pair of toruses cut can evolve into ditorus slice or major pair of spheritoruses slice (x2).Two toruses cut can evolve into ditorus slice or two spheritoruses slice (x2).Four spheres cut can evolve into major pair of spheritoruses slice, two spheritoruses slice or four glomes slice.Empty cut A can evolve into two spheritoruses slice (x2) or empty slice.Empty cut B can evolve into four glomes slice (x2) or empty slice.3 rotations are animations of ditorus.4 rotations are animations of 212-ditorus.1 new rotation: four spheres - empty cut B (((x))IIx)1 empty rotation: empty cut A - empty cut B ((()x)IIx)3 double rotations are animations of 212-ditorus.2 new double rotations: major pair of toruses - empty cut B (((xy))Ixy), two toruses - empty cut B (((x)y)Ixy)1 triple rotation: minor pair of toruses - empty cut B (((xy)z)xyz)31. Triger ((II)(II)(II))1 5D slab: two 221-tigers stacked in minor dimension ((II)(II)(I)), ((II)(I)(II)) and ((I)(II)(II))2 4D slices: empty slice ((II)(II)()), ((II)()(II)) and (()(II)(II)), 2x2 vertical stack of spheritoruses ((II)(I)(I)), ((I)(II)(I)) and ((I)(I)(II))Empty slice can only evolve into two 221-tigers stacked in minor dimension slab.2x2 vertical stack of spheritoruses slice can only evolve into two 221-tigers stacked in minor dimension slab, but in two different ways.2 3D cuts: empty cut ((II)(I)()), ((I)(II)()), ((II)()(I)), (()(II)(I)), ((I)()(II)) and (()(I)(II)), 2x2x2 array of spheres ((I)(I)(I))Empty cut can evolve into empty slice or 2x2 vertical stack of spheritoruses slice (x2).2x2x2 array of spheres cut can only evolve into 2x2 vertical stack of spheritoruses slice, but in three different ways.2 rotations: empty cut - alternate empty cut 2 ((II)(x)(x)), empty cut - 2x2x2 array of spheres ((Ix)(I)(x))1 empty rotation: empty cut - alternate empty cut 1 ((Ix)(Ix)())2 double rotations: empty cut - alternate empty cut 3 ((xy)(Ix)(y)), empty cut - alternate empty cut 4 ((xy)(I)(xy))32. 222-tiger ((II)(II)II)2 5D slabs: 221-tiger ((II)(II)I), vertical stack of two 23-toruses ((II)(I)II) and ((I)(II)II)4 4D slices: tiger ((II)(II)), vertical stack of two spheritoruses ((II)(I)I) and ((I)(II)I), empty slice ((II)()II) and (()(II)II), 2x2 array of glomes ((I)(I)II)Tiger slice can only evolve into 221-tiger slab.Vertical stack of two spheritoruses slice can evolve into 221-tiger slab or vertical stack of two 23-toruses slab.Empty slice can only evolve into vertical stack of two 23-toruses slab.2x2 array of glomes slice can only evolve into vertical stack of two 23-toruses slab, but in two different ways.4 3D cuts: vertical stack of two toruses ((II)(I)) and ((I)(II)), empty cut A ((II)()I) and (()(II)I), 2x2 array of spheres ((I)(I)I), empty cut B ((I)()II) and (()(I)II)Vertical stack of two toruses cut can evolve into tiger slice or vertical stack of two spheritoruses slice (x2).Empty cut A can evolve into vertical stack of two spheritoruses slice (x2) or empty slice.2x2 array of spheres cut can evolve into vertical stack of two spheritoruses slice (in two different ways) or 2x2 array of glomes.Empty cut B can evolve into empty slice or 2x2 array of glomes (x2).1 rotation is animation of tiger.3 rotations are animations of 221-tiger.2 new rotations: 2x2 array of spheres - empty cut B ((I)(x)Ix), empty cut B - alternate empty cut B ((x)(x)II)1 empty rotation: empty cut A - empty cut B ((Ix)()Ix)2 double rotations are animations of 221-tiger.3 new double rotations: vertical stack of two toruses - empty cut B ((Ix)(y)xy), vertical stack of two toruses - alternate empty cut B ((xy)(I)xy), empty cut A - alternate empty cut B ((xy)(x)Iy)33. 24-torus ((II)IIII)2 5D slabs: 23-torus ((II)III), two pentaspheres ((I)IIII)3 4D slices: spheritorus ((II)II), two glomes ((I)III), empty slice (()IIII)Spheritorus slice can only evolve into 23-torus slab.Two glomes slice can evolve into 23-torus slab or two pentaspheres slab.Empty slice can only evolve into two pentaspheres slab.3 3D cuts: torus ((II)I), two spheres ((I)II), empty cut (()III)Torus cut can only evolve into spheritorus slice.Two spheres cut can evolve into spheritorus slice or two glomes slice (x2).Empty cut can evolve into two glomes slice (x2) or empty slice.1 rotation is animation of 23-torus.1 rotation is animation of spheritorus.1 double rotation is animation of 23-torus.`
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### Re: The Tiger Explained

Wow, Marek, that's a very dense chunk of information! If you don't mind me asking, how long did it take? This thread should be called the Toratope Encyclopedia, or something. We've gone wayyyy past discussions of the tiger by now! And just think: it would take three/four times longer to make the full 7D list! No one expects that, but you are a potential candidate, by our votes Or, we could just pick them at random, and do a full cut analysis.

Not sure if rotations that simultaneously turn into and out of the same set of dimensions are different...

Example: 221-ditorus (((II)II)I) has rotation (((Ix)I)x) between major and minor pair of toruses. Would a double rotation between the same, (((Ix)xy)y), work differently?

Well, let's see, here: According to (((Ix)I)x), it's a rotation from a displaced pair to a cocircular pair. Then, we have (((Ix)xy)y), where X is between a displaced <--> concentric pair, and Y is between a concentric <--> cocircular pair. The goal is to go from displaced to cocircular. The rotation (((Ix)I)x) is doing it with one circle of rotation, where (((Ix)xy)y) does it with two orthogonal circles. It looks like it would be an identical rotation, so long as (((Ix)xy)y) rotates both X and Y at the same rate and time. It kind of feels like (((Ix)I)x) a diagonal line in the 2D cut array, when compared to (((Ix)xy)y), which is along two independent legs, and thus perpendicular axes in the cut array.

And, a few more questions:

1) How do you represent double tiger (((II)(II))(II)) in the new notation? It seems like it would be 220-tiger 20-tiger, or maybe ([(00)0]0)0-tigroid?

2) How about something like duotiger (((II)(II))((II)(II)))? I don't think it would be (0000)0-tigroid, which is more like ((II)(II)(II)(II)) tetriger. The trace is an array of 16 tigers, so maybe something like ([(00)0][(00)0])0-tigroid?
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### Re: The Tiger Explained

Join me for another session of Shape Talk.....

So, I found that deriving the arrays of any shape with a torus or sphere trace is getting kind of boring. I noticed that 4D has many more toratopes to choose from when building a shape . This unavoidably leads to shapes with all empty 3D cuts, when getting arrays with +2 on each leg. But, that's okay, it's a good exercise in empty cut evolutions. After peering into 10 and 12 dimensions, imagining the 4-axis array of 4D toratopes isn't much of a leap in difficulty. Once again, I owe it all to the toratope notation, and the cut algorithm. I came on here already familiar with 5D and some 6D rotopes and tapertopes. The toratopes were out of my range of sight, but the cut alg shows everything. That's what did it, it taught me about the driving force behind the intercept and cut arrays. It evolved my concept of 3D thinness, and what the huge 4x4x4 arrays of 64 spheres from a 9D (((II)I)((II)I)((II)I)) come from.

On that note, I had a look at a new 12D shape:

((((II)I)((II)I)(II))(((II)I)I)) - [[21,21,2]211]0-tigroid

Lowest dimensional trace in 4D
-----------------------------------------
((((I))((I))(I))(((I)))) - A 4x4x2x8 tesseractoid tower 4-axis array of 256 torispheres ((III)I)

Each leg of the 4-axis array corresponds to a 1-D puncture of a particular toratope,
4x4x2x8 referenced by X-Y-Z-W
---------------------------------------------------------
X: (TORUS-2) cut of 4 points along a line ((I))
Y: (TORUS-2) cut of 4 points along a line ((I))
Z: (CIRCLE-1) cut of 2 points along a line (I)
W: (DITORUS-3) cut of 8 points along a line (((I)))

Moving along the cut arrays of each leg
-----------------------------------------------------
X
• (TORUS-2): Rows # 1,2,3,4
- Axis 1 - ((I)i) : Merge both 1-2 and 3-4 into two in a row, then vanish
- Axis 2 - ((Ii)) : Sequentially merge 2-3 and vanish, then 1-4 and vanish

Y
• (TORUS-2): Rows # 1,2,3,4
- Axis 1 - ((I)i) : Merge both 1-2 and 3-4 into two in a row, then vanish
- Axis 2 - ((Ii)) : Sequentially merge 2-3 and vanish, then 1-4 and vanish

Z
• (CIRCLE-1): 4x4 Sheets # 1,2
- Axis 1 - (Ii) : Merge 1-2 and vanish

W
• (DITORUS-3): 4x4x2 Cuboids # 1,2,3,4,5,6,7,8
- Axis 1 - (((Ii))) : Sequentially merge 4-5 and vanish, then 3-6, then 2-7, and finally 1-8 and vanish
- Axis 2 - (((I)i)) : Merge both 2-3 and 6-7 and vanish, then 1-4 and 5-8 and vanish
- Axis 3 - (((I))i) : Merge all 1-2 , 3-4 , 5-6 , 7-8 into four in a column, then vanish

3D Empty Cuts and Evolution Sequence
--------------------------------------------------
• (((())((I))(I))(((I)))) and ((((I))(())(I))(((I))))
Empty zone, moving out will make a torisphere cut evolution of {point -> blob -> torus -> blob -> point} suddenly appear in a 4x2x8 cuboid wall array of 64 locations, sequenced 2 times

• ((((I))((I))())(((I))))
Empty zone, moving out will make a torisphere cut evolution of {point -> blob -> torus -> blob -> point} suddenly appear in a 4x4x8 cuboid tower array of 128 locations, sequenced only 1 time

• ((((I))((I))(I))((())))
Empty zone, moving out will make a torisphere cut evolution of {sphere -> 2 conc spheres -> sphere} suddenly appear in a 4x4x2 cuboid flat brick array of 32 locations, sequenced 4 times

Now, wasn't that some nice visualization stretching? My brain really needs this stuff. Nothing like a good, challenging high-D shape to explore that my computer can't handle in search of combinatorial objects of finite extent
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### Re: The Tiger Explained

It took a few days. Actually, I feel I could do 7D encyclopedia, IF I got rid of the rotation animations. Those are the main time-eaters since their number grows quadratically with number of 3D cuts.

As for double tiger, yes, its name would be 220-tiger 20-tiger -- and in fact, it will be for its higher-dimensional analogues. The duotiger (((II)(II))((II)(II))) might be better called duotiger tiger, or perhaps "hypertiger".

The important thing is you have many approaches for various 5-6D toratopes Marek14
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### Re: The Tiger Explained

A few days isn't bad. That's about how long it took me to do the 5D list of polynomial equations. Detailing the rotations is cool, but lack of renders don't do it justice. It's easy to picture a static array of intercepts. But, once I compile the library of renders, the rotation examples become straightforward. I'm starting to see some reflected morphings in higher-D shapes, such as an array of concentric to cocircular rotations, compared to just one pairing. For example, I haven't rendered it yet, but I suspect that Spheric Torus-Tiger (((II)I)(II)I) would be identical in its obliques as triger, when slightly merging the 4x2 array of spheres cut. But, with the addition of a concentric+stacked torus arrangement (((II))(I)) , it would have a more complex transformation cycle, when getting close to that cut. Actually, it sounds kind of cool! I think I might do that one next, you never know.......
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### Re: The Tiger Explained

ICN5D wrote:A few days isn't bad. That's about how long it took me to do the 5D list of polynomial equations. Detailing the rotations is cool, but lack of renders don't do it justice. It's easy to picture a static array of intercepts. But, once I compile the library of renders, the rotation examples become straightforward. I'm starting to see some reflected morphings in higher-D shapes, such as an array of concentric to cocircular rotations, compared to just one pairing. For example, I haven't rendered it yet, but I suspect that Spheric Torus-Tiger (((II)I)(II)I) would be identical in its obliques as triger, when slightly merging the 4x2 array of spheres cut. But, with the addition of a concentric+stacked torus arrangement (((II))(I)) , it would have a more complex transformation cycle, when getting close to that cut. Actually, it sounds kind of cool! I think I might do that one next, you never know.......

Well, the rotations were more-or-less intended as a guideline for you Marek14
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### Re: The Tiger Explained

Well, the rotations were more-or-less intended as a guideline for you Oh! Well, thank you for that! It'll be a good reference when making them. No more questionable rotations.

I noticed you use the word " slab " a lot ... what are you describing? A flat array?
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### Re: The Tiger Explained

Here's some more incredible Oblique Artifacts of Tiger Ditorus, along with the axials:

Tiger Ditorus : ((((II)(II))I)I)

A revisit of the Axial Midsection of 2 concentric along a 2x2 array of 8 toruses: ((((I)(I)))I) Axial Midsection of 2 cocircular along a 2x2 flat square of 8 toruses, trimmed open to reveal the interior: ((((I)(I))I)) Axial Midsection of 4 cocircular stacked 2 high of 8 toruses, trimmed open: ((((II)(I)))) The 4 cocircular stacked 2 high can rotate to another identical cut, by means of a Tiger rotation. This image is at the 45° inter-axial cut between them, making four cocircular Tiger Cages Same cut, trimmed down to the middle. This shows what the lowest trace of 4 concentric along a 2x2 square of 16 circles would look like, at the plane of the cut An assortment of the wild obliques of Tiger Ditorus. Keep in mind that these are straight-line cuts, right through the middle, from outside to outside. No curves were followed during these slices, they are naturally complex, because this shape is no joke....       This shape, Tiger Ditorus, has caused my computer to crash several times, when trying to render it in high resolution, because it's sooo badass. It's very complicated in its tangled-up looping madness of rings and holes. Basically, it's the skin of a torus inflated by a whole tiger, so yeah, it's gonna be kinda crazy Last edited by ICN5D on Sun Apr 27, 2014 4:57 pm, edited 1 time in total.
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### Re: The Tiger Explained

Well, basically I use "cut" for 3D, "slice" for 4D and "slab" for 5D to make it easier to differentiate them from each other.
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### Re: The Tiger Explained

Oh, okay. I noticed that the verbal language evolves much faster than anything else. There's always some better way to describe it! Would you then use " trace " for 2D midsections? How about 6D midcuts of +7D, we'll be there soon with renders. BTW, I explored Spheric Torus-Tiger, (((II)I)(II)I) last night. It's not as amazing as I thought it would be, but still pretty neat to see all the cuts. I didn't mess with obliques too much, just played with single rotations.
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### Re: The Tiger Explained

ICN5D wrote:Oh, okay. I noticed that the verbal language evolves much faster than anything else. There's always some better way to describe it! Would you then use " trace " for 2D midsections? How about 6D midcuts of +7D, we'll be there soon with renders. BTW, I explored Spheric Torus-Tiger, (((II)I)(II)I) last night. It's not as amazing as I thought it would be, but still pretty neat to see all the cuts. I didn't mess with obliques too much, just played with single rotations.

I'd use "trace" for any midsection, provided it's the lowest possible.
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### Re: The Tiger Explained

I was thinking about what shape would be a torus along the duoring of a tiger. It seems like it is tiger torus (((II)(II))I). Where this shape can actually have two descriptions as a tiger along edge of circle and torus along rim of tiger.
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### Re: The Tiger Explained

Well, I couldn't help myself. It was too tempting to build the equation and see what it looks like. It's a very simple equation. And, I'll be darned, the cut algorithm really does say what the cut will be Let's just call this example a proof of truth. Even in the most incomprehensible high-D toratope, the cut algorithm will shed light where there is darkness. And allow us to explore even higher dimensional toratopes than 9D, with confidence.

So, here you go, everybody. Here's a nine dimensional toratope rendered in its 3D midsection:

(((II)I)((II)I)((II)I)) - Triotorus Tiger , (111)0-tiger Axial Midsection of 64 spheres in a 4x4x4 cubic array : (((I))((I))((I))) Simultaneous merging of all 64 into one single, connected structure in search of combinatorial objects of finite extent
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### Re: The Tiger Explained

Very nice Crystallic tiger Marek14
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### Re: The Tiger Explained

Cool, huh? I've probably imagined that array a hundred times, but it's nice to see the function come to life for real! And, it's not put together piece by piece manually, that's the actual equation for the 9D toratope! Totally awesome.

On another note, I just got done with a full exploration of Cyltorintigroid Torus ((((II)I)(II))I) , (10)1-tigroid . I spent about 4/5 days using 7 different equations for the shape. That's four axial translation and three rotation equations. Took cool pics of the awesome axials, obliques, made a rotation montage, and a short rotation animation. This is the first shape I explored with all possible rotation equations. Now, I know I missed a lot, in many others I did before. And, let me tell you, this shape is much more complex than I anticipated. MUCH more complex. I found all that there was to discover in this shape, and I'll make a post a little later.
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### Re: The Tiger Explained

Here it is, one of the coolest and most complicated shapes I've explored so far:

((((II)I)(II))I) - Cyltorintigroid Torus , (10)1-tigroid Axial Translation Equations:

• ((((I))(I))I) - 4x2 array of 8 torii
((sqrt((sqrt(x^2 + a^2) - 3.75)^2 + b^2) - 1.9)^2 + (sqrt(y^2 + c^2) - 2.25)^2 - 1.75)^2 + z^2 - 1^2 = 0

• ((((II))(I))) - 2 cocirc by 2 conc stacked 2 high of 8 torii
((sqrt((sqrt(x^2 + y^2) - 3.75)^2 + a^2) - 1.9)^2 + (sqrt(z^2 + b^2) - 2.25)^2 - 1.75)^2 + c^2 - 1^2 = 0

• ((((I))(II))) - 2 cocirc stacked 4 high of 8 torii
((sqrt((sqrt(x^2 + a^2) - 3.75)^2 + b^2) - 1.9)^2 + (sqrt(y^2 + z^2) - 2.25)^2 - 1.75)^2 + c^2 - 1^2 = 0

• ((((I)I)(I))) - 2 cocirc in 2x1x2 vert square of 8 torii
((sqrt((sqrt(x^2 + a^2) - 3.75)^2 + y^2) - 1.9)^2 + (sqrt(z^2 + b^2) - 2.25)^2 - 1.75)^2 + c^2 - 1^2 = 0

Rotation Equations

• ((((Xy)z)(Yx))Z) - rotation A
((sqrt((sqrt((x*sin(c))^2 + (y*cos(a))^2) - 3.75)^2 + (z*cos(b))^2) - 1.9)^2 + (sqrt((y*sin(a))^2 + (x*cos(c))^2) - 2.25)^2 - 1.75)^2 + (z*sin(b))^2 - 1^2 = 0

• ((((Xz)y)(Yx))Z) - rotation B
((sqrt((sqrt((x*sin(c))^2 + (z*cos(a))^2) - 3.75)^2 + (y*cos(b))^2) - 1.9)^2 + (sqrt((y*sin(b))^2 + (x*cos(c))^2) - 2.25)^2 - 1.75)^2 + (z*sin(a))^2 - 1^2 = 0

• ((((Xy)x)(Yz))Z) - rotation C
((sqrt((sqrt((x*sin(b))^2 + (y*cos(a))^2) - 3.75)^2 + (x*cos(b))^2) - 1.9)^2 + (sqrt((y*sin(a))^2 + (z*cos(c))^2) - 2.25)^2 - 1.75)^2 + (z*sin(c))^2 - 1^2 = 0

In this shape, I found it interesting that all of the cuts for a regular 5D cyltorintigroid (((II)I)(II)) show up, but with an extra cocircular pairing. This is from the cutting plane slicing across the main circle of the torus, producing something analogous to the 2 concentric circles of a torus cut : ((II)) . Also noteworthy, is the cut of the 4x2 array of toruses. This comes from the 4x2 array of circles cut of cyltorintigroid, that then had a non-intersecting rotation around the frame of the minor diameter: the Cyltorinder Margin. This turned the 4x2 trace of circles into a 4x2 array of toruses. The entire minor diameter inflated margin had a non-intersecting rotation around it, to make Cyltorintigroid Torus.

Axial Midcut of 8 toruses in a 4x2 array : ((((I))(I))I) Axial Midcut of 8 toruses in 2 cocircular along a 2x1x2 vertical square : ((((I)I)(I))) Axial Midcut of 8 toruses in a 2 cocircular by 2 concentric stacked 2 high vertical column : ((((II))(I))) Axial Midcut of 8 toruses in a 2 cocircular stacked 4 high vertical column : ((((I))(II))) Here is a large collection of the obliques and some translations:          Three Concentric Toruses !!           A rotation montage from one empty axial midsection to another empty. This is a momentary evolution of our 3D plane passing along a ring structure between holes. And, notice the small circle in the middle of the two larger? It stays still while the larger circle morphs around it. This structure lies on a hexatangent plane, a case where all three cut planes are bitangent. It cuts along a line that is tangent to six points on the surface of Cyltorintigroid Torus. Notice the six narrow pinching-off points?      I haven't finished the rotation animation yet. It's a short and simple display of awesomeness that you shall see soon in search of combinatorial objects of finite extent
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### Re: The Tiger Explained

It looks very nice BTW, are you familiar with the "pair" terminology I've been using here recently? Basically, I now call two concentric toruses a "major pair" and the cocircular ones a "minor pair" (since they are differing in their major, resp. minor diameters). Similarly, "major quartet" are four concentric toruses and "minor quartet" four cocircular ones -- "major/minor" quartet are then four toruses from your "Axial Midcut of 8 toruses in a 2 cocircular by 2 concentric stacked 2 high vertical column" picture.

These describe elements of the array allowing to reduce the description of the array itself.
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### Re: The Tiger Explained

Yep, I noticed. I guess I haven't adopted it yet But, it does do a better job at describing the middle pairing of multiple ditoruses. It would be more verbose to call them " concentric in medium diameter ", where " medium pair " is more efficient. So, how about a stacking+pairing? I think I've seen you call it " stacked in medium diameter " or something.

1) How would you describe concentric and stacked in medium diameter, as in the ((((II)(I)))I) cut of tiger ditorus? A quartet of medium pairs stacked in medium dimension? A medium stack of medium pairs?

2) Or, how would you describe concentric in multiple parameters, as in ((((((II))I))((II)))), which is 16 cyltorintigroids paired in all possible diameters? A hexadecatet of torus parameter main/minor / circle parameter / minor pairs?
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### Re: The Tiger Explained

ICN5D wrote:Yep, I noticed. I guess I haven't adopted it yet But, it does do a better job at describing the middle pairing of multiple ditoruses. It would be more verbose to call them " concentric in medium diameter ", where " medium pair " is more efficient. So, how about a stacking+pairing? I think I've seen you call it " stacked in medium diameter " or something.

Well, I use the following terms:

"Two/four/eight" etc. shapes means that they lie in one line in their major (largest) dimension. For some tigroids, this can be ambiguous. That is, why I can qualify it with brackets. For example, "Two[torus] torus tigers" is ((((I)I)I)(II)), a cut of ditorus tiger, while "Two[circle] torus tigers" is (((II)I)((I)I)), a cut of duotorus tiger.
If more than one major dimension is used, we get array. Once again, qualifiers may be necessary. For example, a 2x2 array of tigers can be either (((I)(I))(II)), a cut of double tiger, or (((I)I)((I)I)), a cut of duotorus tiger. The second case would be "2x2 array [A/B] of tigers".

Now for stacks. A stack is a group of toratopes arranged without use of their major dimensions.
"Vertical" is a term used solely for toruses. "Vertical stack" is a stack in minor dimension of those toruses. It can evolve into "vertical array" if there are more than 1 minor dimensions, as in vertical 2x2 array of spheritoruses ((II)(I)(I)), a cut of triger.
For more complicated shapes, the dimension descriptors themselves are used. So, a stack of two ditoruses can be either a medium stack (((II)(I))I) or a minor stack (((II)I)(I)) -- I tend to call those "two ditoruses stacked in the medium dimension" and "two ditoruses stacked in the minor dimension", but "medium stack" and "minor stack" would work as well. Then, you could have a 2x2 medium/minor array of ditoruses (((II)(I))(I)) or 2x2 medium array of 221-ditoruses (((II)(I)(I))I) etc.
But with major/medium/minor descriptors we can only get as far as ditorus. That's why, for tritorus and higher, the "medium" descriptor is replaced by "secondary/tertiary/quaternary" etc. So a tritorus has two major dimensions, one secondary dimension, one tertiary dimension and one minor dimension, and each of these dimension types is associated with a diameter as well.

How would dimensions of, say, ditorus tiger ((((II)I)I)(II)), be named?
Well, they would have following structure:
((((MAJOR[ditorus])SECONDARY)TERTIARY)(MAJOR[circle])MINOR)
Major dimensions (and diameters) need a qualifier since there are two different kinds, but there's only one type of secondary and tertiary dimension. The ditorus tiger in its basic form doesn't have a minor dimension, but it has a minor diameter and higher analogues like 211-ditorus 21-tiger ((((II)I)I)(II)I) have a specific minor dimension.

So two ditorus tigers could be described as:
Two[ditorus] ditorus tigers (((((I)I)I)I)(II)), cut of tritorus tiger (((((II)I)I)I)(II))
Two[circle] ditorus tigers ((((II)I)I)((I)I)), cut of ditorus/torus tiger ((((II)I)I)((II)I))
Secondary stack of two ditorus tigers ((((II)(I))I)(II)), cut of tiger torus tiger ((((II)(II))I)(II))
Tertiary stack of two ditorus tigers ((((II)I)(I))(II)), cut of torus double tiger ((((II)I)(II))(II))
[Minor stack of two 211-ditorus 21-tigers ((((II)I)I)(II)(I)), cut of ditorus triger ((((II)I)I)(II)(II)) ]
Major[ditorus] pair of ditorus tigers (((((II))I)I)(II)), cut of tritorus tiger (((((II)I)I)I)(II))
Major[circle] pair of ditorus tigers ((((II)I)I)((II))), cut of ditorus/torus tiger ((((II)I)I)((II)I))
Secondary pair of ditorus tigers (((((II)I))I)(II)), cut of tritorus tiger (((((II)I)I)I)(II))
Tertiary pair of ditorus tigers (((((II)I)I))(II)), cut of tritorus tiger (((((II)I)I)I)(II))
Minor pair of ditorus tigers (((((II)I)I)(II))), cut of ditorus tiger torus (((((II)I)I)(II))I)

1) How would you describe concentric and stacked in medium diameter, as in the ((((II)(I)))I) cut of tiger ditorus? A quartet of medium pairs stacked in medium dimension? A medium stack of medium pairs?

This would be the medium stack of medium pairs of ditoruses.

2) Or, how would you describe concentric in multiple parameters, as in ((((((II))I))((II)))), which is 16 cyltorintigroids paired in all possible diameters? A hexadecatet of torus parameter main/minor / circle parameter / minor pairs?

This would be the major[torus]/major[circle]/medium/minor 16-plet of torus tigers. This would be a cut of ((((((II)I)I)I)((II)I))I), a 9D tritorus/torus tiger torus.
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### Re: The Tiger Explained

ICN5D wrote:Here it is, one of the coolest and most complicated shapes I've explored so far:

((((II)I)(II))I) - Cyltorintigroid Torus , (10)1-tigroid [...]

A rotation montage from one empty axial midsection to another empty. This is a momentary evolution of our 3D plane passing along a ring structure between holes. And, notice the small circle in the middle of the two larger? It stays still while the larger circle morphs around it. [...] Wow. This is totally mind-blowing! Looks like pairs of earrings. Amazing stuff!
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### Re: The Tiger Explained

So, I guess that the small ring is a torus-shaped hole which is cut by a plane into two circles no matter how the cut is oriented?
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### Re: The Tiger Explained

It's very cool, isn't it? Well, at first, I wasn't sure how or why these strange obliques appeared between empty cuts. Then, it dawned on me by imagining how a 1D cut of a tiger would look. All 1D cuts of a tiger are empty, as the thin cutting line passes through either orthogonal hole, in the minor stack ( vertical stack ) of toruses. But, when we rotate this 1D cut from one hole to another, the line cuts through the rings of both toruses at an oblique angle. Manifested in 1D, it's the sudden appearance of two fissing points while rotating through, then merge and disappear. Manifested in 6D->3D , we get these wild looking ring-structure slices, scanning along it from one empty to another. The rotation aspect adds another morphing parameter to the cut evolution. The small torus in the middle that sits still is stuck to the rim of the inflated margin. The larger torus that morphs around the smaller is what looks like the outer skin of a ditorus. At least that's what I feel. It definitely comes from a vertical column of a pair of shapes. Whatever makes it, it's cool as heck.

And, because of your interest quickfur, I'll add that rotation to the animation, since I'm still making it. These types are totally bizarre, and there's more. I saw them in ditorus tiger ((((II)I)I)(II)) , triger ((II)(II)(II)) , and now cyltorintigroid torus ((((II)I)(II))I). Though, I don't remember any in double tiger (((II)(II))(II)) , but I may have missed it. Maybe I should make rotation animations of all of them? That would be cool! All right, something to do.....

Now, as for the arrangement terminology.... Perhaps if it is just between you and I, we can get as technical as you want! But, I think that for other people who come to this thread, it may be better to use layman's descriptions of the complex arrays. It seems like if we convert the cut notation into a technical arrangement description, we would then again have to translate that description, into simpler language. Which I'm totally okay with, but it may introduce an initial comprehension barrier, when nothing makes any sense to the reader. Part of me is saying " heck yeah, it's way better! ", while the other is saying " no one will understand even the descriptions, now! ". But, obviously for complex arrangements like the previously mentioned ((((((II))I))((II)))), it requires technical verbage out of sheer necessity. Someone would already be adept to technical arrangement terminology by the time they analyze shapes like that. The terminology gets more complex when describing higher dimensional toratopes, because of the additional diameters to pair or stack in. Perhaps a Toratope Arrangement Glossary is in order, complete with all of the different terms for how to stack toratopes of arbitrary diameters.

I guess it boils down to this: there are 2 ways per diameter to arrange toratopes.

• Diameter Pairing : Smaller inside Larger - cocircular, concentric

• Diameter Stacking : Side by Side - displaced, vertical column, vertical square, cuboid rectangular array

On that topic, I see a common pattern with the arrangement " (I) " in place of any dimension marker. The way I have come to understand it, is it's a horizontal displacement in the respective diameter's hyperplane. If it's in the major diameter like (((I)I)I) , the two are parallel and displaced in the major hyperplane. Whereas in the minor diameter, ((II)(I)), they are parallel again, but displaced horizontally according to the minor diameter's hyperplane. It only appears vertical, because the minor diameter's hyperplane is perpendicular to the major's hyperplane, so a horizontal translation is 90 degrees vertical. And for (((II)(I))I), it's technically a horizontal displacement in the medium diameter's hyperplane. So, no matter which diameter the displacing (I) product is in, it's always along the same plane as the diameter.

Here's my non-empty cut analysis of it:

((((II)I)(II))I) : Cyltorintigroid Torus , (10)1-tigroid
-----------------------------------------------------------------
A ((((I)I)(II))I) : major1 stack of tiger torii
B ((((II))(II))I) : major1 pair of tiger torii
C ((((II)I)(I))I) : tertiary stack of tritoruses
D ((((II)I)(II))) : minor pair of cyltorintigroids
--------------------------------------------------------------------
A1 ((((I))(II))I) : medium stacked quartet of ditoruses
A2 ((((I)I)(I))I) : 2x2 maj/med array of 4 ditoruses
A3 ((((I)I)(II))) : major1 stacked minor pairs of 4 tigers
--------------------
B1 ((((I))(II))I) : medium stacked quartet of ditoruses
B2 ((((II))(I))I) : medium stacked major pairs of 4 ditoruses
B3 ((((II))(II))) : maj1/minor pairs of 4 tigers
--------------------
C1 ((((I)I)(I))I) : 2x2 maj/med array of 4 ditoruses
C2 ((((II))(I))I) : medium stacked major pairs of 4 ditoruses
C3 ((((II)I)(I))) : minor stacked minor pairs of 4 ditoruses
--------------------
D1 ((((I)I)(II))) : major1 stacked minor pairs of 4 tigers
D2 ((((II))(II))) : maj1/minor pairs of 4 tigers
D3 ((((II)I)(I))) : minor stacked minor pairs of 4 ditoruses
-------------------------------------------------------------------------

A1 = B1
A2 = C1
A3 = D1
B2 = C2
B3 = D2
C3 = D3

A1a ((((I))(I))I) : 4x2 array of 8 toruses
A1b ((((I))(II))) : minor stacked quartet of minor pairs of 8 toruses
--------------------
A2a ((((I))(I))I) : 4x2 array of 8 toruses
A2b ((((I)I)(I))) : maj/minor array of minor pairs of 8 toruses
--------------------
A3a ((((I))(II))) : minor stacked quartet of minor pairs of 8 toruses
A3b ((((I)I)(I))) : maj/minor array of minor pairs of 8 toruses
--------------------
B2a ((((I))(I))I) : 4x2 array of 8 toruses
B2b ((((II))(I))) : minor stack of maj/minor pairs of 8 toruses
--------------------
B3a ((((I))(II))) : minor stacked quartet of minor pairs of 8 toruses
B3b ((((II))(I))) : minor stack of maj/minor pairs of 8 toruses
--------------------
C3a ((((I)I)(I))) : maj/minor array of minor pairs of 8 toruses
C3b ((((II))(I))) : minor stack of maj/minor pairs of 8 toruses
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### Re: The Tiger Explained

Well, remember that I've posted all the cuts and rotation animations for 6D toratopes not too far back Actually, the rotation animations might be better understood if we won't tie them to particular toratope, but rather to a species:

Take torus species, (()). It has 4 possible 3D cuts: pair of spheres ((III)), torus ((II)I), two spheres ((I)II) and empty cut (()III) [individual toruses can lack some of these, but these 4 are present in general case]

Now, single rotation is basically a 4D slice where two I's are replaced by x's. And the two replacements must be in different "groups", otherwise the rotation won't change anything.
((IIx)x) - rotation between pair of spheres and torus
((Ix)Ix) - rotation between torus and two spheres
((x)IIx) - rotation between two spheres and empty cut

Double rotation is derived from 5D slab:
((Ixy)xy) - double rotation between pair of spheres and two spheres. Other assignment of initial values would lead to double rotation between torus and torus, not sure if this actually changes.
((xy)Ixy) - double rotation between torus and empty cut. Other assignment would lead to double rotation between two spheres and two spheres.

Triple rotation is derived from 6D:
((xyz)xyz) - triple rotation between pair of spheres and empty cut. Triple rotation between torus and two spheres is also possible.

In double/triple rotation, you can imagine the four initial conditions as vertices of square/cube and the rotation happens between the diagonals. So for ((Ixy)xy) the vertices are pair of spheres - torus - two spheres - torus, for ((xy)Ixy) they are torus - two spheres - empty cut - two spheres and for ((xyz)xyz) they are arranged with one pair of spheres, one empty cut and three of the other two.

The number of possible cut grows with the complexity of species, but we also start getting a number of empty rotations -- if a pair of parentheses is empty (), even in the rotation descriptor, than the rotation will be never anything more than an empty set.

So, a ditorus species ((())) has 10 cuts, with following single rotations (those correspond to taking one I and moving it elsewhere):
(((III))) - quartet of spheres -> [minor pair of toruses, major pair of toruses]
(((II)I)) - minor pair of toruses -> [quartet of spheres, major pair of toruses, two pairs of spheres, two toruses]
(((II))I) - major pair of toruses -> [quartet of spheres, minor pair of toruses, two toruses, four spheres]
(((I)II)) - two pairs of spheres -> [minor pair of toruses, two toruses, empty cut A, empty cut B]
(((I)I)I) - two toruses -> [minor pair of toruses, major pair of toruses, two pairs of spheres, four spheres, empty cut B, empty cut C]
(((I))II) - four spheres -> [major pair of toruses, two toruses, empty cut C, empty cut D]
((()III)) - empty cut A -> [two pairs of spheres, empty cut B]
((()II)I) - empty cut B -> [two pairs of spheres, two toruses, empty cut A, empty cut C]
((()I)II) - empty cut C -> [two toruses, four spheres, empty cut B, empty cut D]
((())III) - empty cut D -> [four spheres, empty cut C]

Note that any rotation between two empty cuts is also empty.
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### Re: The Tiger Explained

Yes, I remember! I did it for practice in new arrangement terminology.
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ICN5D
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### Re: The Tiger Explained

Marek14 wrote:Note that any rotation between two empty cuts is also empty.

Actually, that's not entirely accurate. In some shapes, structure does appear at an oblique angle between empties. That's what this montage illustrates: The far left and right are the two empty cuts of (((()I)(I))I) and ((((I)I)())I). The pieces you see are at 7.5° increments of rotation from 0° to 90°, moving left or right

This also happens in ditorus tiger, triger, and potentially some others. It comes from the cut ((((I)I)(I))I), which is a 2x2 vertical wall of ditoruses. By rotating around inside an array of 4D-vertically stacked shapes, we see two empties, with two diagonally opposed shapes at the corner of a square. A 3D axial cut slides cleanly between the four, through the gap separating them: (((()I)(I))I) and ((((I)I)())I) . When we rotate from one empty 3D hole to another, our line of sight passes by two shapes diagonally opposite, at an oblique cut angle.

Triger is also a good example to explore. In one rotation from empty to empty, we see two rings appear, inflate to minor stacked toruses, then deflate to thin rings and disappear. It's the translation evolution of two spheritoruses, but made by a rotation. In this cut: ((II)(I)(I)), we have a minor array of 4 spheritoruses that has two empty cut possibilities in 3D : ((II)(I)()) and ((II)()(I)) . So, by rotating from one empty cut to another, in the middle of the square array, we see our 3-plane scan along only two of the four spheritoruses, and make the spheritorus cut evolution of ((II)Ii)) : And then there's Ditorus Tiger , where we have the vertical square of 4 ditoruses : ((((I)I)I)(I)) . From this arrangement, we can have two empty 3D cuts of : (((()I)I)(I)) and ((((I)I)I)()). When we scan from one empty to the other, we see, at an oblique angle, our trimmed down line of sight pass along two of those ditoruses. The two are made into minor pairs and rotation trimmed very narrow:  Pics are cool, but I feel that animations are key in the display of awesomeness.
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ICN5D
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### Re: The Tiger Explained

I thought of another really cool shape today. I discovered it by its trace in 3D, as 32 toruses in all possible arrangements. They are paired and stacked in all possible diameters and dimensions. Might be cool to render some day!

(((((II)(II))I)(II))I) - tiger torus tiger torus , (((00)1)0)1-tiger

Lowest Dimensional Trace: (((((I)(I)))(I))) - A major/minor array of major/minor pairs of 32 toruses

So, I've been experimenting with your new notation system, and I found a good adaptation to it for describing traces of high-D shapes. The format (00)0-tigroid is sufficient for circles and (000)0 for spheres. It will also work well for lines or squares of toruses, since they can be further cut into circles. But for a toratope with a 3D array of toruses, there is no lower trace. Since the toruses have three stacking directions and can be paired in both diameters, I suggest using something like ((00)0,0)0-tigroid for torus traces in a 3D rectangular array.

Trace of Circles (II)
--------------------------
(ab)c-tigroid
--------------------------
2a+1 x 2b+1 array of 2c concentric pairs

Trace of Spheres (III)
----------------------------
(abc)d-tigroid
----------------------------
2a+1 x 2b+1 x 2c+1 array of 2d concentric pairs

Trace of Toruses ((II)I)
------------------------------
((ab)c,d)e - tigroid
------------------------------
2a+1 x 2b+1 x 2d+1 major/minor array of 2c major pairs / 2e minor pairs

Then, for traces of 4D toratopes:

Trace of Glomes (IIII)
-----------------------------
(abcd)e-tigroid
-----------------------------
2a+1 x 2b+1 x 2c+1 x 2d+1 array of 2e concentric pairs

Trace of Torispheres ((III)I)
------------------------------------
((abc)d,e)f-tigroid
------------------------------------
2a+1 x 2b+1 x 2c+1 x 2e+1 major/minor array of 2d major pairs / 2f minor pairs

Trace of Spheritoruses ((II)II)
--------------------------------------
((ab)c,de)f
--------------------------------------
2a+1 x 2b+1 x 2d+1 x 2e+1 major/minor array of 2c major pairs / 2f minor pairs

Trace of Ditoruses (((II)I)I)
-----------------------------------
(((ab)c,d)e,f)g
------------------------------------
2a+1 x 2b+1 x 2d+1 x 2f+1 major/medium/minor array of 2c major pairs / 2e medium pairs / 2g minor pairs

Trace of Tigers ((II)(II))
---------------------------------
((ab)c,(de)f)g-tigroid
---------------------------------
2a+1 x 2b+1 x 2d+1 x 2e+1 major/major array of 2c major pairs / 2f major pairs / 2g minor pairs

** Interesting that a tiger cannot be stacked in its minor dimension! Not until we get to Spheric Tiger ((II)(II)I).

Then, for 5D, we can use an expression like (((ab)c,d)e,(fg)h)i-tigroid for the lowest trace of (((II)I)(II)). Take an example, like a 17 dimensional

((((((((II)I)(II))I)I)(((II)I)I))(((((II)(II))I)I)) - (((10)2,2)0,(00)2)0-tigroid

(((10)2,2)0,(00)2)0-tigroid would then be a 4x2x8x2x2 major[torus]/major[circle]/medium array of 256 locations of major[torus] quartet / major[circle] quartet of 4,096 torus-tigers.

So for a trace of torus-tigers, we have five stacking dimensions plus four pairing diameters, requiring 9 parameters to describe all 9 possible arrangements of (((II)I)(II)) :

(((ab)c,d)e,(fg)h)i
---------------------------
2a+1 x 2b+1 x 2d+1 x 2f+1 x 2g+1 major[torus]/major[circle]/medium array of

2c major[torus] pairs / 2e medium pairs / 2h major[circle] pairs / 2i minor pairs of

2(5+a+b+c+d+e+f+g+h+i) cyltorintigroids.

-- Philip
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