The Tiger Explained

Discussion of shapes with curves and holes in various dimensions.

Re: The Tiger Explained

Postby quickfur » Wed May 21, 2014 2:30 pm

What would be really cool, is a projection or slice of a 6D (or higher) toratope that produces icosahedral symmetry. :P

Or better yet, a slice of an 8D toratope that produces 600-cell symmetry (though we'd have a hard time recognizing that in the 3D slices!).
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Re: The Tiger Explained

Postby ICN5D » Wed May 21, 2014 10:37 pm

quickfur wrote:What would be really cool, is a projection or slice of a 6D (or higher) toratope that produces icosahedral symmetry. :P

Or better yet, a slice of an 8D toratope that produces 600-cell symmetry (though we'd have a hard time recognizing that in the 3D slices!).



How does the icosahedral symmetry differ in the way of this 12-tangent cut? Apparently, those narrow pinching points are in the vertices of an icosahedron.


As for tangent points at the vertices of 4D shapes: my guess would be some toratope that has a 4D lowest non-empty cut, capable of a multitangent slice producing those proper locations. Which sounds like a pretty neat investigation! I'll have to get some book-learning going in order to do that.
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Re: The Tiger Explained

Postby ICN5D » Thu May 22, 2014 2:20 am

You know, Marek, I also have tigers in my background!

Image
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Re: The Tiger Explained

Postby Marek14 » Thu May 22, 2014 5:30 am

Eh, only three dimensions... :)
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Re: The Tiger Explained

Postby ICN5D » Thu May 22, 2014 5:40 am

Marek, you got me thinking about polytope inscription with multitangent planes. This one could have an octahedron traced out, by joining those six tangent points.

Image
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Re: The Tiger Explained

Postby Marek14 » Thu May 22, 2014 6:19 am

Yes, seems like it.
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Re: The Tiger Explained

Postby ICN5D » Sun May 25, 2014 1:27 am

It just dawned on me that this oblique cut of ((((II)I)(II))I) is tangent to 14 points. They seem to be arranged in three squares and a line, not sure what to make of that. Two of the squares are nested in the same axis, the 3rd is laid flat, and the line skewers the nested pair.

Image
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Re: The Tiger Explained

Postby ICN5D » Sun Jun 01, 2014 11:02 pm

Behold 3D mortals: I bring you another nine dimensional toratope. I've tried to render this one before, only to render it's 8D cousin instead. But, this time I got it! It was a pesky and elusive pair of missing parentheses; they were the culprit. I couldn't get the concentric pairing in the 4x2x4 array of torii before. Found them today, so check it out:


(((((II)I)(II))I)((II)I)) : Tigritoratigritorus



Image



(sqrt(((sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2) - R2)^2 + (sqrt(w^2 + v^2) - R3)^2 - R4)^2 + u^2) - R5)^2 + (sqrt((sqrt(t^2 + s^2) - R6)^2 + r^2) - R7)^2 - R8^2 = 0



In this 3D midsection, I used the equation:

(sqrt(((sqrt((sqrt(x^2 + a^2) - 5.5)^2 + 0^2) - 2.75)^2 + (sqrt(y^2 + b^2) - 3.5)^2 - 3)^2 + c^2) - 1.9)^2 + (sqrt((sqrt(z^2 + d^2) - 2.5)^2 + 0^2) - 1.25)^2 - 0.75^2 = 0

And a bounding box of XYZ = -11 / 11


It's what we get when inflating the skin of a tigritorus (((II)I)(II)) with a whole other tigritorus. One is the minor shape, the other is the major shape that was inflated by the minor. This one being 1 out of the 3 ways to inflate a (((II)I)(II)) with another (((II)I)(II)) . The end result is this:


A 4x2x4 array of concentric pairs in the major diameter: (((((I))(I)))((I)))

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This is rendering in 130 cube resolution, trying to resolve the lumpiness of the outer torii. Those must be more difficult for the program or something.

Enjoy!
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Re: The Tiger Explained

Postby ICN5D » Tue Jun 03, 2014 7:02 am

Made a neat little construction tree of the shape above. All shapes in the table below have a crosscut present in that huge array.


(((((II)I)(II))I)((II)I))
can be created through rotation starting with...


Code: Select all
2D :                                      (II)

3D :                                     ((II)I)

4D :                              ((II)(II))    (((II)I)I)

5D :                   (((II)I)(II))    (((II)(II))I)    ((((II)I)I)I)

6D :      (((II)I)((II)I))    ((((II)I)(II))I)    ((((II)(II))I)I)    (((((II)I)I)I)I)

7D : (((((II)I)(II))I)I)    ((((II)(II))I)(II))    ((((II)I)I)((II)I))    (((((II)I)I)I)(II))

8D :     (((((II)I)(II))I)(II))    ((((II)(II))I)((II)I))    (((((II)I)I)I)((II)I))
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Re: The Tiger Explained

Postby ICN5D » Sun Jun 08, 2014 5:17 am

Here's a cool one. Found lots of neat stuff going on in the rotations, so I had to show you all. This shape is actually the first one I explored with rotation PLUS translation. Seven dimensions gives us a spare axis, after six have been used up with start-to-finish rotation points. Plus, as you can tell, I played around more with making rotation montages this time.


(((II)I)((II)I)I) - 7D Spheritigric Duotorus , sphere-->duoring-->duoring


Image



I call it the spheritigric duotorus, since it can be made by starting with an ordinary 3D sphere, (III), and inflating a duoring with it. This will make ((II)(II)I), the spheritiger. Then, we can take this shape and inflate a larger duoring, making a duotorus of the spheritiger, (((II)I)((II)I)I) .



I used three unique rotation equations for exploration, but I feel there could be four in total. I put the spare axis for translation in the notation, denoted as a letter.



(((I))((I))I)
(sqrt((sqrt(x^2 + a^2) - 2.35)^2 + b^2) -1.15)^2 + (sqrt((sqrt(y^2 + c^2) - 2.6)^2 + d^2) -1.25)^2 + z^2 - 0.75^2 = 0

(((Xz)y)((Yx)d)Z)
(sqrt((sqrt((x*sin(c))^2 + (z*cos(a))^2) - 2.35)^2 + (y*cos(b))^2) -1.15)^2 + (sqrt((sqrt((y*sin(b))^2 + (x*cos(c))^2) - 2.6)^2 + d^2) -1.25)^2 + (z*sin(a))^2 - 0.5^2 = 0

(((Xy)z)((Yc)x)Z)
(sqrt((sqrt((x*sin(d))^2 + (y*cos(a))^2) - 2.35)^2 + (z*cos(b))^2) -1.15)^2 + (sqrt((sqrt((y*sin(a))^2 + c^2) - 2.6)^2 + (x*cos(d))^2) -1.25)^2 + (z*sin(b))^2 - 0.5^2 = 0

(((Xa)x)((Yz)y)Z)
(sqrt((sqrt((x*sin(b))^2 + a^2) - 2.35)^2 + (x*cos(b))^2) -1.15)^2 + (sqrt((sqrt((y*sin(d))^2 + (z*cos(c))^2) - 2.6)^2 + (y*cos(d))^2) -1.25)^2 + (z*sin(c))^2 - 0.5^2 = 0



Axial Midcut of a 4x4 array of 16 spheres: (((I))((I))I) . One can interpret this cut as a 2x2 square of 2x2 arrays, illustrating the two duorings. A single 2x2 array of four spheres is a cut of spheritiger, sphere along duoring. This 7D one is then another, larger duoring inflated by a single 2x2 array of spheres.

Image



Slight merging of all 16 into a single structure, made by translation away from center

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Beautiful multitangent cut, based off 4x4 array. There are in fact twenty narrowed down touching points, most I've ever seen.

Image



Slight rotation to an empty 3D hole, away from the 2x4 vertical rectangle stacking of 8 torii. This 7D toratope shares all midcuts with (((II)I)((II)I)), with the addition of the 4x4 array of spheres. This makes for some really cool rotations.

Image



Rotating from the 4x4 array spheres to the 2 concentric pair stacked four high of torii

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Rotating from an empty 3D hole to the 4x4 array of spheres

Image




This axial cut lies between four tigers laid out in a 2x2 square. We are moving out of the hole, and scanning our 3D plane along two of the four tigers. Doing so will show a cut evolution of two tigers, hence the double tiger dance.

Image




This is the same cut as above. But, this time we are rotating from one empty to another. While doing so, our 3D plane ( line of sight ) scans past two of those four tigers, the diagonal opposites. Interestingly enough, we end up seeing an oblique cut evolution of two tigers, instead of tiger dances.

Image



This is another empty to empty rotation. But, this one slides our 3D plane between a 2x4 vertical column of spheritoruses : (((I)I)((I))I) . This rotation is from ((()I)((I))I) --> (((I)I)(())I) . During rotation, we scan our 3D plane along four out of the 8, two slightly ahead of the other two.

Image




A complex morphing single rotation, made by a combination of translated plus rotated planes positioned

Image
Last edited by ICN5D on Mon Jun 09, 2014 11:06 pm, edited 1 time in total.
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Re: The Tiger Explained

Postby Marek14 » Sun Jun 08, 2014 5:32 am

This looks very cool :)
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Re: The Tiger Explained

Postby ICN5D » Mon Jun 09, 2014 4:28 am

Yup. Need to make more montages. I made some new rotation equations with tigric ditorus ((((II)I)I)(II))) , and saw some amazing things. Also with ditoritiger ((((II)(II))I)I), saw new supercool empty to empty scans. I'm looking into making animated GIFs too.
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Re: The Tiger Explained

Postby ICN5D » Mon Jun 09, 2014 10:58 pm

experimenting .....

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Re: The Tiger Explained

Postby ICN5D » Tue Jun 10, 2014 12:25 am

this one is better! GIF Movie Gear, non-web based. Though, it reduces to 256 colors, making it appear blockier than normal.....


Image


ohhhhh, man! I'm feeling the potential here!! time for complete exploratory GIFs for all 4D toratopes!!


Colors, or no colors?
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Re: The Tiger Explained

Postby ICN5D » Tue Jun 10, 2014 2:11 am

Not before some cool 6D ones, though......



Not sure how to clean up the dark grey diagonal lines. Maybe use a darker gray background?



Tigric Duotorus : (((II)I)((II)I))


Rotating from (((II))((I))) to (((I))((II))) and back to (((II))((I)))



Image





Rotating from (((I)I)((I))) to (((II))((I))) and back to (((I)I)((I)))


Image
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Re: The Tiger Explained

Postby ICN5D » Tue Jun 10, 2014 4:23 am

Ummmmmmm, is that not the most mindblowingly awesome thing you've ever seen?



It's a double 90 degree rotation from oblique to oblique in tigric duotorus (((II)I)((II)I)).


Starting from a perfect 45 degree angle between cuts of (((I)I)(()I)) and ((()I)((I)I)), we see two tiger cages as diagonal opposites in the cut (((I)I)((I)I)). We then rotate on two independent axes to a perfect 45 degree angle between cuts of (((II))((I))) and (((I))((II))). This makes a structure which is a tiger cage from four tigers, concentric in both major diameters, (((II))((II))) .



Image
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Re: The Tiger Explained

Postby Marek14 » Tue Jun 10, 2014 5:02 am

Once again, extremely interesting :)

As for me, I'm still working on an encyclopedia of cuts for 7D toratopes. In particular, I can now visualize cut evolutions of empty cuts pretty well :)

50/90 toratopes are now processed.
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Re: The Tiger Explained

Postby ICN5D » Tue Jun 10, 2014 6:59 am

Heck yeah! The last ones are exactly what I was looking for. Nothing beats good ole' animated pictures, without having to go to youtube!


You've done 50/90 in 7D? Sweet, it'll be a good reference to pick shapes from.


Funny you should mention empty cuts ..... I formed the equation for tetratiger ((II)(II)(II)(II)), and made a rotation equation, too:



((II)(II)(II)(II)) - Tetratiger , (0000)0-Tiger // glome-->quattroring
----------------------------------------------------------------------------------------------------------------------
(sqrt(x^2 + y^2) - R1)^2 + (sqrt(z^2 + w^2) - R2)^2 + (sqrt(v^2 + u^2) - R3)^2 + (sqrt(t^2 + s^2) - R4)^2- R5^2 = 0

• ((I)(I)(I)(I)) - 2x2x2x2 array of 16 glomes
(sqrt(x^2 + a^2) - 2.5)^2 + (sqrt(y^2 + b^2) - 2.5)^2 + (sqrt(z^2 + c^2) - 2.5)^2 + (sqrt(d^2 + 0^2) - 2.5)^2 - 1^2 = 0

• ((Ia)(Ib)(I)(Cd))
(sqrt(x^2 + a^2) - 2.5)^2 + (sqrt(y^2 + b^2) - 2.5)^2 + (sqrt(z^2 + 0^2) - 2.5)^2 + (sqrt(c^2 + d^2) - 2.5)^2 - 1^2 = 0
-- Adjusting C translates away from center of 2x2x2x2 array of glomes, passing by either +W or -W 2x2x2 cube array
-- Adjusting D controls merging of +W,-W 2x2x2 arrays in 4-space

• ((Xc)(Yz)(Zx)(Cy))
(sqrt((x*sin(0))^2 + (c*cos(a))^2) - 2.5)^2 + (sqrt((y*sin(d))^2 + (z*cos(b))^2) - 2.5)^2 + (sqrt((z*sin(b))^2 + (x*cos(0))^2) - 2.5)^2 + (sqrt((c*sin(a))^2 + (y*cos(d))^2) - 2.5)^2 - 1^2 = 0
-- Set C=3.5 for only visible morphings.

• ((Xc)(Yx)(Zy)(Cz))
(sqrt((x*sin(b))^2 + (c*cos(a))^2) - 2.5)^2 + (sqrt((y*sin(0))^2 + (x*cos(b))^2) - 2.5)^2 + (sqrt((z*sin(d))^2 + (y*cos(0))^2) - 2.5)^2 + (sqrt((c*sin(a))^2 + (z*cos(d))^2) - 2.5)^2 - 1^2 = 0
-- Adjusting B scans along diagonal 2x2x2 array, max illum at 0.785/45deg
-- Adjusting C translates away fr center of 2x2x2x2 array, max illum at +2.5,-2.5 in 4-space



** If using the equation for ((Ia)(Ib)(I)(Cd)) , you can see what it looks like when slightly merging the 16 glomes trace of tetratiger in 4-space. Then, you can translate along 4-space to vary the size of the 8 spheres, while slicing through cassini oval deformed glomes. Plus, there's some strange things going on with the sudden appearance of a duoring that inflates to a tiger, then deflates and vanishes, during a rotation. You have to translate out from center to a 2x2x2 array, then rotate an axis. That 4D array does some very strange things!


Plus, it was really neat exploring an array with all empty 3D holes, I'd have to say. And, I noticed in my mind and experiment that no matter how big the array is, our 3D plane will only illuminate 1/2N of the lowest trace, where N is the number of dimensions above a 3D trace array. So, for example, if cutting a 10D shape like pentatiger ((II)(II)(II)(II)(II)), we get a 5D array of 32 pentaspheres. For both rotation and translation, we can only illuminate (or cut through) 1/4 of the total number of shapes in the 5D 2x2x2x2x2 array, showing up always as a 2x2x2 array of 8 spheres that inflate then deflate.



Lately, I've been working on this shape in 10D , and trying to imagine its evolution sequence. So far, I've gotten it down to its evolutions in 4D, which is kind of an interesting exercise. Now, I've got to parse it's 3D cuts, and their evolution sequence.


Evolution Sequence Terminology:

{inf} - inflate
{spl} - split
{mrp} - morph
{mrg} - merge
{dfl} - deflate




((((II)(II))(II))((II)(II))) - (((00)0,0)0,(00)0)0 -tiger
-------------------------------------------------------------
Lowest non-empty cut is a 5D array of 32 tigritorii:
((((I)(I))(I))((I)(I))) - 2x2x2x2x2 array of 32 (((II)I)(II))


4D Slices, all five axial cuts slide through gaps between shapes, as they are separated in 5-space
---------------------------------------------------------------------------------------------------
• (((()(I))(I))((I)(I)))
• ((((I)())(I))((I)(I))) - Empty zone, moving out reveals a 2x2x2x2 array of 16 locations in one evolution sequence of:

<point {inf} blob {mph} tiger {spl} 2 tigers along line {mrg} tiger {mrp} blob {dfl} point>



• ((((I)(I))())((I)(I))) - Empty zone, moving out reveals a 2x2x2x2 array of 16 locations in one evolution sequence of:

<duoring {inf} tiger {spl} 2 concentric(maj1) tigers {mrg} tiger {dfl} duoring>



• ((((I)(I))(I))(()(I)))
• ((((I)(I))(I))((I)())) - Empty zone, moving out reveals a 2x2x2x2 array of 16 locations in one evolution sequence of:

<ring {inf} torus {inf} ditorus {spl} 2 stacked(minor) ditoruses {mrg} ditorus {dfl} torus {dfl} ring>




And the of course the remaining 20 cuts in 3D.
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Re: The Tiger Explained

Postby Marek14 » Tue Jun 10, 2014 8:35 am

Let's have a look at toratope #42, torus double tiger:

42. Torus double tiger ((((II)I)(II))(II))
6D blobs:
Minor stack of two torus tiger toruses ((((II)I)(II))(I)) -> torus double tiger
Tertiary stack of two ditorus tigers ((((II)I)(I))(II)) -> torus double tiger
Major pair of double tigers ((((II))(II))(II)) -> torus double tiger
Two[tiger] double tigers ((((I)I)(II))(II)) -> torus double tiger
5D slabs:
Empty slab A ((((II)I)(II))()) -> minor stack of two torus tiger toruses (x2)
2x2 tertiary/minor stack of tritoruses ((((II)I)(I))(I)) -> minor stack of two torus tiger toruses, tertiary stack of two ditorus tigers
Minor stack of two major pairs of tiger toruses ((((II))(II))(I)) -> minor stack of two torus tiger toruses, major pair of double tigers
Two minor stacks of two tiger toruses ((((I)I)(II))(I)) -> minor stack of two torus tiger toruses, two[tiger] double tigers
Empty slab B ((((II)I)())(II)) -> tertiary stack of two ditorus tigers (x2)
Medium stack of two major[torus] pairs of torus tigers ((((II))(I))(II)) -> tertiary stack of two ditorus tigers, major pair of double tigers
Two[torus] medium stacks of two torus tigers ((((I)I)(I))(II)) -> tertiary stack of two ditorus tigers, two[tiger] double tigers
Medium stack of four torus tigers ((((I))(II))(II)) -> major pair of double tigers, two[tiger] double tigers
Empty slab C (((()I)(II))(II)) -> two[tiger] double tigers (x2)
4D slices:
Empty slice A ((((II)I)(I))()) -> empty slab A, 2x2 tertiary/minor stack of tritoruses (x2)
Empty slice B ((((II))(II))()) -> empty slab A, minor stack of two major pairs of tiger toruses (x2)
Empty slice C ((((I)I)(II))()) -> empty slab A, two minor stacks of two tiger toruses (x2)
Empty slice D ((((II)I)())(I)) -> 2x2 tertiary/minor stack of tritoruses (x2), empty slab B
2x2 medium/minor stack of major pairs of ditoruses ((((II))(I))(I)) -> 2x2 tertiary/minor stack of tritoruses, minor stack of two major pairs of tiger toruses, medium stack of two major[torus] pairs of torus tigers
Two 2x2 medium/minor stacks of ditoruses ((((I)I)(I))(I)) -> 2x2 tertiary/minor stack of tritoruses, two minor stacks of two tiger toruses, two[torus] medium stacks of two torus tigers
4x2 medium/minor stack of ditoruses ((((I))(II))(I)) -> minor stack of two major pairs of tiger toruses, two minor stacks of two tiger toruses, medium stack of four torus tigers
Empty slice E (((()I)(II))(I)) -> two minor stacks of two tiger toruses (x2), empty slab C
Empty slice F ((((II))())(II)) -> empty slab B, medium stack of two major[torus] pairs of torus tigers (x2)
Empty slice G ((((I)I)())(II)) -> empty slab B, two[torus] medium stacks of two torus tigers (x2)
4x2 [A/A] array of tigers ((((I))(I))(II)) -> medium stack of two major[torus] pairs of torus tigers, two[torus] medium stacks of two torus tigers, medium stack of four torus tigers
Empty slice H (((()I)(I))(II)) -> two[torus] medium stacks of two torus tigers (x2), empty slab C
Empty slice I (((())(II))(II)) -> medium stack of four torus tigers (x2), empty slab C
3D cuts:
Empty cut A ((((II)I)())()) -> empty slice A (x2), empty slice D (x2)
Empty cut B ((((II))(I))()) -> empty slice A, empty slice B, 2x2 medium/minor stack of major pairs of ditoruses (x2)
Empty cut C ((((I)I)(I))()) -> empty slice A, empty slice C, two 2x2 medium/minor stacks of ditoruses (x2)
Empty cut D ((((I))(II))()) -> empty slice B, empty slice C, 4x2 medium/minor stack of ditoruses (x2)
Empty cut E (((()I)(II))()) -> empty slice C (x2), empty slice E (x2)
Empty cut F ((((II))())(I)) -> empty slice D, 2x2 medium/minor stack of major pairs of ditoruses (x2), empty slice F
Empty cut G ((((I)I)())(I)) -> empty slice D, two 2x2 medium/minor stacks of ditoruses (x2), empty slice G
4x2 array of vertical stacks of two toruses ((((I))(I))(I)) -> 2x2 medium/minor stack of major pairs of ditoruses, two 2x2 medium/minor stacks of ditoruses, 4x2 medium/minor stack of ditoruses, 4x2 [A/A] array of tigers
Empty cut H (((()I)(I))(I)) -> two 2x2 medium/minor stacks of ditoruses (x2), empty slice E, empty slice H
Empty cut I (((())(II))(I)) -> 4x2 medium/minor stack of ditoruses (x2), empty slice E, empty slice I
Empty cut J ((((I))())(II)) -> empty slice F, empty slice G, 4x2 [A/A] array of tigers (x2)
Empty cut K (((()I)())(II)) -> empty slice G (x2), empty slice H (x2)
Empty cut L (((())(I))(II)) -> 4x2 [A/A] array of tigers (x2), empty slice H, empty slice I

We can look at the empty cuts that only have one () string (if there are multiples, all evolutions are empty and nonempty cuts are only found in non-coordinate directions):
For example, in cut B ((((II))(I))()), I can remove the empty parentheses to get ((((II))(I))) or replace them with I to get ((((II))(I))I). What does that mean? Well, the first result, a vertical stack of two major/minor quartets of toruses, is how the cut will look in the middle of "full" part. The second result, a medium stack of two major pairs of ditoruses, shows the total evolution of this part. So in this particular cut, you'll see vertical stack of two major pairs of toruses appear, then each of the 4 toruses separates in a minor pair, then they come back together and disappear... and then the whole thing is repeated. The whole slice corresponds to the only non-empty slice this empty cut evolves into, 2x2 medium/minor stack of major pairs of ditoruses.

In case like the empty cut L (((())(I))(II)), where there's a double pair of empty parentheses, the double pair means that the nonempty evolution we derive won't be repeated two times, but four times. In this case, you'll get two parallel circles separating in vertical stacks of two toruses (vertical stack of four toruses altogether), and then shrinking back in two circles and disappearing. And this repeats 4 times, since each of these are two tigers from the 4x2 array.
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Re: The Tiger Explained

Postby ICN5D » Wed Jun 11, 2014 2:50 am

Today, I explored Tiger ditorus ((((II)I)I)(II)). Made a bunch of double 90 degree rotation animations from axial to axial. Also watched a ton of stuff going on with double tiger, I'll have to make some of that one. It was really the triple 90 degree from the 2x2x2 array (((I)(I))(I)) to an identical array, but arranged differently.






This is a single 90 degree rotation from the vertical square of major pairs to an empty 3D hole. The scanning effect of our 3D plane is amazing to see in action. You can clearly see a great big hollow ditorus! What your eyes see before your brain notices is the way a 3D beam illuminates a very thin portion of the whole structure. We can scan this laserpointer-like beam to show its other invisible, extradimensional parts, as if made of ultra-transparent glass. Cutting a shape down to 3D loses information each time, and we get a thinner representation of the whole thing. One thing I'd like to try is to merge the 8 torii vertically, making them closer towards the center. Then scan our beam to the hole! We will see a lot more of the shape that way, stay tuned for that one.


Image








A double 90 degree rotation from ((((I)I))(I)) to ((((II)))(I)) , showing how our 3D laser pointer hits structure that's there, and passes through holes that are empty. The way the solid parts morph around the complex structure of holes is amazingly consistent. We are definitely tracing out invisible parts of an extradimensional shape!!! :D


Image







A double 90 degree rotation from ((((I)))(II)) to ((((I))I)(I)) . What really gets me is the orientation of all the axial cuts, and how they're related to the rest of the invisible structure. And, what a straight-line cut through the middle of an extradimensional shape looks like, when rotated around. Absolutely amazing.


Image







A double 90 degree rotation from ((((I)))(II)) to ((((I))I)(I)). Super cool to see how the central structure behaves like a common anchorpoint for both circles of rotation. Also neat to see two distinct linear pipe-like views when illuminating the large invisible disk .

Image
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Re: The Tiger Explained

Postby ICN5D » Wed Jun 11, 2014 3:00 am

Hmmm, well that's strange. The gifs disappeared, and when i click on it, it says " 509 Bandwidth limit exceeded" . Are these gifs too large or something? It also happened to a ton of pictures, too. Is this one of the net neutrality things?
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Re: The Tiger Explained

Postby Polyhedron Dude » Wed Jun 11, 2014 5:24 am

ICN5D, I got one thing to say about all of these toroidal pics and animations - AWESOME!!! :mrgreen:
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Re: The Tiger Explained

Postby Marek14 » Wed Jun 11, 2014 6:18 am

I see, so the last picture is rotation ((((Xy))x)(YI)), correct? So my ideas about these were incorrect and now I see why -- the main direction was in x axis and changed to y. This way, there are no real invariants in the rotation, unlike rotation ((((I))x)(XI)) where the trace of ditorus tiger ((((I)))(I)), 8x2 array of circles, would stay invariant throughout the rotation. Just as well that current 7D exploration no longer even attempts to explore rotations :)

As an inspiration, here are cut sequences for "base" 7D toratopes (new species) I've explored -- I have 9 so far:

Code: Select all
6. Pentatorus ((((((II)I)I)I)I)I)
6D blobs:
Minor pair of tetratoruses ((((((II)I)I)I)I)) -> pentatorus
Quaternary pair of tetratoruses ((((((II)I)I)I))I) -> pentatorus
Tertiary pair of tetratoruses ((((((II)I)I))I)I) -> pentatorus
Secondary pair of tetratoruses ((((((II)I))I)I)I) -> pentatorus
Major pair of tetratoruses ((((((II))I)I)I)I) -> pentatorus
Two tetratoruses (((((((I)I)I)I)I)I) -> pentatorus
5D slabs:
Minor quartet of tritoruses (((((((II)I)I)I))) -> minor pair of tetratoruses, quaternary pair of tetratoruses
Tertiary/minor quartet of tritoruses ((((((II)I)I))I)) -> minor pair of tetratoruses, tertiary pair of tetratoruses
Secondary/minor quartet of tritoruses ((((((II)I))I)I)) -> minor pair of tetratoruses, secondary pair of tetratoruses
Major/minor quartet of tritoruses ((((((II))I)I)I)) -> minor pair of tetratoruses, major pair of tetratoruses
Two minor pairs of tritoruses ((((((I)I)I)I)I)) -> minor pair of tetratoruses, two tetratoruses
Tertiary quartet of tritoruses ((((((II)I)I)))I) -> quaternary pair of tetratoruses, tertiary pair of tetratoruses
Secondary/tertiary quartet of tritoruses ((((((II)I))I))I) -> quaternary pair of tetratoruses, secondary pair of tetratoruses
Major/tertiary quartet of tritoruses ((((((II))I)I))I) -> quaternary pair of tetratoruses, major pair of tetratoruses
Two tertiary pairs of tritoruses ((((((I)I)I)I))I) -> quaternary pair of tetratoruses, two tetratoruses
Secondary quartet of tritoruses ((((((II)I)))I)I) -> tertiary pair of tetratoruses, secondary pair of tetratoruses
Major/secondary quartet of tritoruses ((((((II))I))I)I) -> tertiary pair of tetratoruses, major pair of tetratoruses
Two secondary pairs of tritoruses ((((((I)I)I))I)I) -> tertiary pair of tetratoruses, two tetratoruses
Major quartet of tritoruses ((((((II)))I)I)I) -> secondary pair of tetratoruses, major pair of tetratoruses
Two major pairs of tritoruses ((((((I)I))I)I)I) -> secondary pair of tetratoruses, two tetratoruses
Four tritoruses ((((((I))I)I)I)I) -> major pair of tetratoruses, two tetratoruses
Empty slab (((((()I)I)I)I)I) -> two tetratoruses (x2)
4D slices:
Minor octet of ditoruses ((((((II)I)I)))) -> minor quartet of tritoruses, tertiary/minor quartet of tritoruses, tertiary quartet of tritoruses
Medium/minor/minor octet of ditoruses ((((((II)I))I))) -> minor quartet of tritoruses, secondary/minor quartet of tritoruses, secondary/tertiary quartet of tritoruses
Major/minor/minor octet of ditoruses ((((((II))I)I))) -> minor quartet of tritoruses, major/minor quartet of tritoruses, major/tertiary quartet of tritoruses
Two minor quartets of ditoruses ((((((I)I)I)I))) -> minor quartet of tritoruses, two minor pairs of tritoruses, two tertiary pairs of tritoruses
Medium/medium/minor octet of ditoruses ((((((II)I)))I)) -> tertiary/minor quartet of tritoruses, secondary/minor quartet of tritoruses, secondary quartet of tritoruses
Major/medium/minor octet of ditoruses (((((II))I))I)) -> tertiary/minor quartet of tritoruses, major/minor quartet of tritoruses, major/secondary quartet of tritoruses
Two medium/minor quartets of ditoruses (((((I)I)I))I)) -> tertiary/minor quartet of tritoruses, two minor pairs of tritoruses, two secondary pairs of tritoruses
Major/major/minor octet of ditoruses ((((((II)))I)I)) -> secondary/minor quartet of tritoruses, major/minor quartet of tritoruses, major quartet of tritoruses
Two major/minor quartets of ditoruses ((((((I)I))I)I)) -> secondary/minor quartet of tritoruses, two minor pairs of tritoruses, two major pairs of tritoruses
Four minor pairs of ditoruses ((((((I))I)I)I)) -> major/minor quartet of tritoruses, two minor pairs of tritoruses, four tritoruses
Empty slice A (((((()I)I)I)I)) -> two minor pairs of tritoruses (x2), empty slab
Medium octet of ditoruses ((((((II)I))))I) -> tertiary quartet of tritoruses, secondary/tertiary quartet of tritoruses, secondary quartet of tritoruses
Major/medium/medium octet of ditoruses ((((((II))I)))I) -> tertiary quartet of tritoruses, major/tertiary quartet of tritoruses, major/secondary quartet of tritoruses
Two medium quartets of ditoruses ((((((I)I)I)))I) -> tertiary quartet of tritoruses, two tertiary pairs of tritoruses, two secondary pairs of tritoruses
Major/major/medium octet of ditoruses ((((((II)))I))I) -> secondary/tertiary quartet of tritoruses, major/tertiary quartet of tritoruses, major quartet of tritoruses
Two major/medium quartets of ditoruses ((((((I)I))I))I) -> secondary/tertiary quartet of tritoruses, two tertiary pairs of tritoruses, two major pairs of tritoruses
Four medium pairs of ditoruses ((((((I))I)I))I) -> major/tertiary quartet of tritoruses, two tertiary pairs of tritoruses, four tritoruses
Empty slice B (((((()I)I)I))I) -> two tertiary pairs of tritoruses (x2), empty slab
Major octet of ditoruses ((((((II))))I)I) -> secondary quartet of tritoruses, major/secondary quartet of tritoruses, major quartet of tritoruses
Two major quartets of ditoruses ((((((I)I)))I)I) -> secondary quartet of tritoruses, two secondary pairs of tritoruses, two major pairs of tritoruses
Four major pairs of ditoruses ((((((I))I))I)I) -> major/secondary quartet of tritoruses, two secondary pairs of tritoruses, four tritoruses
Empty slice C (((((()I)I))I)I) -> two secondary pairs of tritoruses (x2), empty slab
Eight ditoruses ((((((I)))I)I)I) -> major quartet of tritoruses, two major pairs of tritoruses, four tritoruses
Empty slice D (((((()I))I)I)I) -> two major pairs of tritoruses (x2), empty slab
Empty slice E (((((())I)I)I)I) -> four tritoruses (x2), empty slab
3D cuts:
Minor 16-plet of toruses ((((((II)I))))) -> minor octet of ditoruses, medium/minor/minor octet of ditoruses, medium/medium/minor octet of ditoruses, medium octet of ditoruses
Major/minor/minor/minor 16-plet of toruses ((((((II))I)))) -> minor octet of ditoruses, major/minor/minor octet of ditoruses, major/medium/minor octet of ditoruses, major/medium/medium octet of ditoruses
Two minor octets of toruses ((((((I)I)I)))) -> minor octet of ditoruses, two minor quartets of ditoruses, two medium/minor quartets of ditoruses, two medium quartets of ditoruses
Major/major/minor/minor 16-plet of toruses ((((((II)))I))) -> medium/minor/minor octet of ditoruses, major/minor/minor octet of ditoruses, major/major/minor octet of ditoruses, major/major/medium octet of ditoruses
Two major/minor/minor octets of toruses ((((((I)I))I))) -> medium/minor/minor octet of ditoruses, two minor quartets of ditoruses, two major/minor quartets of ditoruses, two major/medium quartets of ditoruses
Four minor quartets of toruses ((((((I))I)I))) -> major/minor/minor octet of ditoruses, two minor quartets of ditoruses, four minor pairs of ditoruses, four medium pairs of ditoruses
Empty cut A (((((()I)I)I))) -> two minor quartets of ditoruses (x2), empty slice A, empty slice B
Major/major/major/minor 16-plet of toruses ((((((II))))I)) -> medium/medium/minor octet of ditoruses, major/medium/minor octet of ditoruses, major/major/minor octet of ditoruses, major octet of ditoruses
Two major/major/minor octets of toruses ((((((I)I)))I)) -> medium/medium/minor octet of ditoruses, two medium/minor quartets of ditoruses, two major/minor quartets of ditoruses, two major quartets of ditoruses
Four major/minor quartets of toruses ((((((I))I))I)) -> major/medium/minor octet of ditoruses, two medium/minor quartets of ditoruses, four minor pairs of ditoruses, four major pairs of ditoruses
Empty cut B (((((()I)I))I)) -> two medium/minor quartets of ditoruses (x2), empty slice A, empty slice C
Eight minor pairs of toruses ((((((I)))I)I)) -> major/major/minor octet of ditoruses, two major/minor quartets of ditoruses, four minor pairs of ditoruses, eight ditoruses
Empty cut C (((((()I))I)I)) -> two major/minor quartets of ditoruses (x2), empty slice A, empty slice D
Empty cut D (((((())I)I)I)) -> four minor pairs of ditoruses (x2), empty slice A, empty slice E
Major 16-plet of toruses ((((((II)))))I) -> medium octet of ditoruses, major/medium/medium octet of ditoruses, major/major/medium octet of ditoruses, major octet of ditoruses
Two major octets of toruses ((((((I)I))))I) -> medium octet of ditoruses, two medium quartets of ditoruses, two major/medium quartets of ditoruses, two major quartets of ditoruses
Four major quartets of toruses ((((((I))I)))I) -> major/medium/medium octet of ditoruses, two medium quartets of ditoruses, four medium pairs of ditoruses, four major pairs of ditoruses
Empty cut E (((((()I)I)))I) -> two medium quartets of ditoruses (x2), empty slice B, empty slice C
Eight major pairs of toruses ((((((I)))I))I) -> major/major/medium octet of ditoruses, two major/medium quartets of ditoruses, four medium pairs of ditoruses, eight ditoruses
Empty cut F (((((()I))I))I) -> two major/medium quartets of ditoruses (x2), empty slice B, empty slice D
Empty cut G (((((())I)I))I) -> four medium pairs of ditoruses (x2), empty slice B, empty slice E
16 toruses ((((((I))))I)I) -> major octet of ditoruses, two major quartets of ditoruses, four major pairs of ditoruses, eight ditoruses
Empty cut H (((((()I)))I)I) -> two major quartets of ditoruses (x2), empty slice C, empty slice D
Empty cut I (((((())I))I)I) -> four major pairs of ditoruses (x2), empty slice C, empty slice E
Empty cut J (((((()))I)I)I) -> eight ditoruses (x2), empty slice D, empty slice E

7. Tiger tritorus (((((II)(II))I)I)I)
6D blobs:
Minor pair of tiger ditoruses (((((II)(II))I)I)) -> tiger tritorus
Tertiary pair of tiger ditoruses (((((II)(II))I))I) -> tiger tritorus
Secondary pair of tiger ditoruses (((((II)(II)))I)I) -> tiger ditorus
Secondary stack of two tetratoruses (((((II)(I))I)I)I) & (((((I)(II))I)I)I) -> tiger tritorus
5D slabs:
Minor quartet of tiger toruses (((((II)(II))I))) -> minor pair of tiger ditoruses, tertiary pair of tiger ditoruses, secondary pair of tiger ditoruses
Medium/minor quartet of tiger toruses (((((II)(II)))I)) -> minor pair of tiger ditoruses
Secondary stack of two minor pairs of tritoruses (((((II)(I))I)I)) & (((((I)(II))I)I)) -> minor pair of tiger ditoruses, secondary stack of two tetratoruses
Medium quartet of tiger toruses (((((II)(II))))I) -> tertiary pair of tiger ditoruses, secondary pair of tiger ditoruses
Secondary stack of two tertiary pairs of tritoruses (((((II)(I))I))I) & (((((I)(II))I))I) -> tertiary pair of tiger ditoruses, secondary stack of two tetratoruses
Secondary stack of two secondary pairs of tritoruses (((((II)(I)))I)I) & (((((I)(II)))I)I) -> secondary pair of tiger ditoruses, secondary stack of two tetratoruses
Empty slab (((((II)())I)I)I) & ((((()(II))I)I)I) -> secondary stack of two tetratoruses (x2)
2x2 array of tritoruses (((((I)(I))I)I)I) -> secondary stack of two tetratoruses (x2, in two different ways)
4D slices:
Minor octet of tigers (((((II)(II))))) -> minor quartet of tiger toruses, medium/minor quartet of tiger toruses, medium quartet of tiger toruses
Medium stack of two minor quartets of ditoruses (((((II)(I))I))) & (((((I)(II))I))) -> minor quartet of tiger toruses, secondary stack of two minor pairs of tritoruses, secondary stack of two tertiary pairs of tritoruses
Medium stack of two medium/minor quartets of ditoruses (((((II)(I)))I)) & (((((I)(II)))I)) -> medium/minor quartet of tiger toruses, secondary stack of two minor pairs of tritoruses, secondary stack of two secondary pairs of tritoruses
Empty slice A (((((II)())I)I)) & ((((()(II))I)I)) -> secondary stack of two minor pairs of tritoruses (x2), empty slab
2x2 array of minor pairs of ditoruses (((((I)(I))I)I)) -> secondary stack of two minor pairs of tritoruses (x2, in two different ways), 2x2 array of tritoruses
Medium stack of two medium quartets of ditoruses (((((II)(I))))I) & (((((I)(II))))I) -> medium quartet of tiger toruses, secondary stack of two tertiary pairs of tritoruses, secondary stack of two secondary pairs of tritoruses
Empty slice B (((((II)())I))I) & ((((()(II))I))I) -> secondary stack of two tertiary pairs of tritoruses (x2), empty slab
2x2 array of medium pairs of ditoruses (((((I)(I))I))I) -> secondary stack of two tertiary pairs of tritoruses (x2, in two different ways), 2x2 array of tritoruses
Empty slice C (((((II)()))I)I) & ((((()(II)))I)I) -> secondary stack of two secondary pairs of tritoruses (x2), empty slab
2x2 array of major pairs of ditoruses (((((I)(I)))I)I) -> secondary stack of two secondary pairs of tritoruses (x2, in two different ways), 2x2 array of tritoruses
Empty slice D (((((I)())I)I)I) & ((((()(I))I)I)I) -> empty slab, 2x2 array of tritoruses (x2)
3D cuts:
Vertical stack of two minor octets of toruses (((((II)(I))))) & (((((I)(II))))) -> minor octet of tigers, medium stack of two minor quartets of ditoruses, medium stack of two medium/minor quartets of ditoruses, medium stack of two medium quartets of ditoruses
Empty cut A (((((II)())I))) & ((((()(II))I))) -> medium stack of two minor quartets of ditoruses (x2), empty slice A, empty slice B
2x2 array of minor quartets of toruses (((((I)(I))I))) -> medium stack of two minor quartets of ditoruses (x2, in two different ways), 2x2 array of minor pairs of ditoruses, 2x2 array of medium pairs of ditoruses
Empty cut B (((((II)()))I)) & ((((()(II)))I)) -> medium stack of two medium/minor quartets of ditoruses (x2), empty slice A, empty slice C
2x2 array of major/minor quartets of toruses (((((I)(I)))I)) -> medium stack of two medium/minor quartets of ditoruses (x2, in two different ways), 2x2 array of minor pairs of ditoruses, 2x2 array of major pairs of ditoruses
Empty cut C (((((I)())I)I)) & ((((()(I))I)I)) -> empty slice A, 2x2 array of minor pairs of ditoruses (x2), empty slice D
Empty cut D (((((II)())))I) & ((((()(II))))I) -> medium stack of two medium quartets of ditoruses (x2), empty slice B, empty slice C
2x2 array of major quartets of toruses (((((I)(I))))I) -> medium stack of two medium quartets of ditoruses (x2, in two different ways), 2x2 array of medium pairs of ditoruses, 2x2 array of major pairs of ditoruses
Empty cut E (((((I)())I))I) & ((((()(I))I))I) -> empty slice B, 2x2 array of medium pairs of ditoruses (x2), empty slice D
Empty cut F (((((I)()))I)I) & ((((()(I)))I)I) -> empty slice C, 2x2 array of major pairs of ditoruses (x2), empty slice D
Empty cut G ((((()())I)I)I) -> empty slice D (x4, in two different ways)

10. Torus tiger ditorus (((((II)I)(II))I)I)
6D blobs:
Minor pair of torus tiger toruses (((((II)I)(II))I)) -> torus tiger ditorus
Tertiary pair of torus tiger toruses (((((II)I)(II)))I) -> torus tiger ditorus
Tertiary stack of two tetratoruses (((((II)I)(I))I)I) -> torus tiger ditorus
Major pair of tiger ditoruses (((((II))(II))I)I) -> torus tiger ditorus
Two tiger ditoruses (((((I)I)(II))I)I) -> torus tiger ditorus
5D slabs:
Minor quartet of torus tigers (((((II)I)(II)))) -> minor pair of torus tiger toruses, tertiary pair of torus tiger toruses
Tertiary stack of two minor pairs of tritoruses (((((II)I)(I))I)) -> minor pair of torus tiger toruses, tertiary stack of two tetratoruses
Major/minor quartet of tiger toruses (((((II))(II))I)) -> minor pair of torus tiger toruses, major pair of tiger ditoruses
Two minor pairs of tiger toruses (((((I)I)(II))I)) -> minor pair of torus tiger toruses, two tiger ditoruses
Tertiary stack of two tertiary pairs of tritoruses (((((II)I)(I)))I) -> tertiary pair of torus tiger toruses, tertiary stack of two tetratoruses
Major/medium quartet of tiger toruses (((((II))(II)))I) -> tertiary pair of torus tiger toruses, major pair of tiger ditoruses
Two medium pairs of tiger toruses (((((I)I)(II)))I) -> tertiary pair of torus tiger toruses, two tiger ditoruses
Empty slab A (((((II)I)())I)I) -> tertiary stack of two tetratoruses (x2)
Secondary stack of two major pairs of tritoruses (((((II))(I))I)I) -> tertiary stack of two tetratoruses, major pair of tiger ditoruses
Two secondary stacks of two tritoruses (((((I)I)(I))I)I) -> tertiary stack of two tetratoruses, two tiger ditoruses
Secondary stack of four tritoruses (((((I))(II))I)I) -> major pair of tiger ditoruses, two tiger ditoruses
Empty slab B ((((()I)(II))I)I) -> two tiger ditoruses (x2)
4D slices:
Minor stack of two minor quartets of ditoruses (((((II)I)(I)))) -> minor quartet of torus tigers, tertiary stack of two minor pairs of tritoruses, tertiary stack of two tertiary pairs of tritoruses
Major/minor/minor octet of tigers (((((II))(II)))) -> minor quartet of torus tigers, major/minor quartet of tiger toruses, major/medium quartet of tiger toruses
Two minor quartets of tigers (((((I)I)(II)))) -> minor quartet of torus tigers, two minor pairs of tiger toruses, two medium pairs of tiger toruses
Empty slice A (((((II)I)())I)) -> tertiary stack of two minor pairs of tritoruses (x2), empty slab A
Medium stack of two major/minor quartets of ditoruses (((((II))(I))I)) -> tertiary stack of two minor pairs of tritoruses, major/minor quartet of tiger toruses, secondary stack of two major pairs of tritoruses
Two medium stacks of two minor pairs of ditoruses (((((I)I)(I))I)) -> tertiary stack of two minor pairs of tritoruses, two minor pairs of tiger toruses, two secondary stacks of two tritoruses
Medium stack of four minor pairs of ditoruses (((((I))(II))I)) -> major/minor quartet of tiger toruses, two minor pairs of tiger toruses, secondary stack of four tritoruses
Empty slice B ((((()I)(II))I)) -> two minor pairs of tiger toruses (x2), empty slab B
Empty slice C (((((II)I)()))I) -> tertiary stack of two tertiary pairs of tritoruses (x2), empty slab A
Medium stack of two major/medium quartets of ditoruses (((((II))(I)))I) -> tertiary stack of two tertiary pairs of tritoruses, major/medium quartet of tiger toruses, secondary stack of two major pairs of tritoruses
Two medium stacks of two medium pairs of ditoruses (((((I)I)(I)))I) -> tertiary stack of two tertiary pairs of tritoruses, two medium pairs of tiger toruses, two secondary stacks of two tritoruses
Medium stack of four medium pairs of ditoruses (((((I))(II)))I) -> major/medium quartet of tiger toruses, two medium pairs of tiger toruses, secondary stack of four tritoruses
Empty slice D ((((()I)(II)))I) -> two medium pairs of tiger toruses (x2), empty slab B
Empty slice E (((((II))())I)I) -> empty slab A, secondary stack of two major pairs of tritoruses (x2)
Empty slice F (((((I)I)())I)I) -> empty slab A, two secondary stacks of two tritoruses (x2)
4x2 array of ditoruses (((((I))(I))I)I) -> secondary stack of two major pairs of tritoruses, two secondary stacks of two tritoruses, secondary stack of four tritoruses
Empty slice G ((((()I)(I))I)I) -> two secondary stacks of two tritoruses (x2), empty slab B
Empty slice H ((((())(II))I)I) -> secondary stack of four tritoruses (x2), empty slab B
3D cuts:
Empty cut A (((((II)I)()))) -> minor stack of two minor quartets of ditoruses (x2), empty slice A, empty slice C
Vertical stack of two major/minor/minor octets of toruses (((((II))(I)))) -> minor stack of two minor quartets of ditoruses, major/minor/minor octet of tigers, medium stack of two major/minor quartets of ditoruses, medium stack of two major/medium quartets of ditoruses
Two vertical stacks of two minor quartets of toruses (((((I)I)(I)))) -> minor stack of two minor quartets of ditoruses, two minor quartets of tigers, two medium stacks of two minor pairs of ditoruses, two medium stacks of two medium pairs of ditoruses
Vertical stack of four minor quartets of toruses (((((I))(II)))) -> major/minor/minor octet of tigers, two minor quartets of tigers, medium stack of four minor pairs of ditoruses, medium stack of four medium pairs of ditoruses
Empty cut B ((((()I)(II)))) -> two minor quartets of tigers (x2), empty slice B, empty slice D
Empty cut C (((((II))())I)) -> empty slice A, medium stack of two major/minor quartets of ditoruses (x2), empty slice E
Empty cut D (((((I)I)())I)) -> empty slice A, two medium stacks of two minor pairs of ditoruses (x2), empty slice F
4x2 array of minor pairs of toruses (((((I))(I))I)) -> medium stack of two major/minor quartets of ditoruses, two medium stacks of two minor pairs of ditoruses, medium stack of four minor pairs of ditoruses, 4x2 array of ditoruses
Empty cut E ((((()I)(I))I)) -> two medium stacks of two minor pairs of ditoruses (x2), empty slice B, empty slice G
Empty cut F ((((())(II))I)) -> medium stack of four minor pairs of ditoruses (x2), empty slice B, empty slice H
Empty cut G (((((II))()))I) -> empty slice C, medium stack of two major/medium quartets of ditoruses (x2), empty slice E
Empty cut H (((((I)I)()))I) -> empty slice C, two medium stacks of two medium pairs of ditoruses (x2), empty slice F
4x2 array of major pairs of toruses (((((I))(I)))I) -> medium stack of two major/medium quartets of ditoruses, two medium stacks of two medium pairs of ditoruses, medium stack of four medium pairs of ditoruses, 4x2 array of ditoruses
Empty cut I ((((()I)(I)))I) -> two medium stacks of two medium pairs of ditoruses (x2), empty slice D
Empty cut J ((((())(II)))I) -> medium stack of four medium pairs of ditoruses (x2), empty slice D, empty slice H
Empty cut K (((((I))())I)I) -> empty slice E, empty slice F, 4x2 array of ditoruses (x2)
Empty cut L ((((()I)())I)I) -> empty slice F (x2), empty slice G (x2)
Empty cut M ((((())(I))I)I) -> 4x2 array of ditoruses (x2), empty slice G, empty slice H

17. Ditorus tiger torus (((((II)I)I)(II))I)
6D blobs:
Minor pair of ditorus tigers (((((II)I)I)(II))) -> ditorus tiger torus
Quaternary stack of two tetratoruses (((((II)I)I)(I))I) -> ditorus tiger torus
Secondary pair of two torus tiger toruses (((((II)I))(II))I) -> ditorus tiger torus
Major[torus] pair of torus tiger toruses (((((II))I)(II))I) -> ditorus tiger torus
Two[torus] torus tiger toruses (((((I)I)I)(II))I) -> ditorus tiger torus
5D slabs:
Minor stack of two minor pairs of tritoruses (((((II)I)I)(I))) -> minor pair of ditorus tigers, quaternary stack of two tetratoruses
Medium/minor quartet of torus tigers (((((II)I))(II))) -> minor pair of ditorus tigers, secondary pair of two torus tiger toruses
Major[torus]/minor quartet of torus tigers (((((II))I)(II))) -> minor pair of ditorus tigers, major[torus] pair of torus tiger toruses
Two[torus] minor pairs of torus tigers (((((I)I)I)(II))) -> minor pair of ditorus tigers, two[torus] torus tiger toruses
Empty slab A (((((II)I)I)())I) -> quaternary stack of two tetratoruses (x2)
Tertiary stack of two secondary pairs of tritoruses (((((II)I))(I))I) -> quaternary stack of two tetratoruses, secondary pair of two torus tiger toruses
Tertiary stack of two major pairs of tritoruses (((((II))I)(I))I) -> quaternary stack of two tetratoruses, major[torus] pair of torus tiger toruses
Two tertiary stacks of two tritoruses (((((I)I)I)(I))I) -> quaternary stack of two tetratoruses, two[torus] torus tiger toruses
Major quartet of tiger toruses (((((II)))(II))I) -> secondary pair of two torus tiger toruses, major[torus] pair of torus tiger toruses
Two major[A] pairs of tiger toruses (((((I)I))(II))I) -> secondary pair of two torus tiger toruses, two[torus] torus tiger toruses
Four[torus] tiger toruses (((((I))I)(II))I) -> major[torus] pair of torus tiger toruses, two[torus] torus tiger toruses
Empty slab B ((((()I)I)(II))I) -> two[torus] torus tiger toruses (x2)
4D slices:
Empty slice A (((((II)I)I)())) -> minor stack of two minor pairs of tritoruses (x2), empty slab A
Minor stack of two medium/minor quartets of ditoruses (((((II)I))(I))) -> minor stack of two minor pairs of tritoruses, medium/minor quartet of torus tigers, tertiary stack of two secondary pairs of tritoruses
Minor stack of two major/minor quartets of ditoruses (((((II))I)(I))) -> minor stack of two minor pairs of tritoruses, major[torus]/minor quartet of torus tigers, tertiary stack of two major pairs of tritoruses
Two minor stacks of two minor pairs of ditoruses (((((I)I)I)(I))) -> minor stack of two minor pairs of tritoruses, two[torus] minor pairs of torus tigers, two tertiary stacks of two tritoruses
Major/major/minor octet of tigers (((((II)))(II))) -> medium/minor quartet of torus tigers, major[torus]/minor quartet of torus tigers, major quartet of tiger toruses
Two major[A]/minor quartets of tigers (((((I)I))(II))) -> medium/minor quartet of torus tigers, two[torus] minor pairs of torus tigers, two major[A] pairs of tiger toruses
Four minor pairs of tigers (((((I))I)(II))) -> major[torus]/minor quartet of torus tigers, two[torus] minor pairs of torus tigers, four[torus] tiger toruses
Empty slice B ((((()I)I)(II))) -> two[torus] minor pairs of torus tigers (x2), empty slab B
Empty slice C (((((II)I))())I) -> empty slab A, tertiary stack of two secondary pairs of tritoruses (x2)
Empty slice D (((((II))I)())I) -> empty slab A, tertiary stack of two major pairs of tritoruses (x2)
Empty slice E (((((I)I)I)())I) -> empty slab A, two tertiary stacks of two tritoruses (x2)
Medium stack of two major quartets of ditoruses (((((II)))(I))I) -> tertiary stack of two secondary pairs of tritoruses, tertiary stack of two major pairs of tritoruses, major quartet of tiger toruses
Two medium stacks of two major pairs of ditoruses (((((I)I))(I))I) -> tertiary stack of two secondary pairs of tritoruses, two tertiary stacks of two tritoruses, two major[A] pairs of tiger toruses
Four medium stacks of two ditoruses (((((I))I)(I))I) -> tertiary stack of two major pairs of tritoruses, two tertiary stacks of two tritoruses, four[torus] tiger toruses
Empty slice F ((((()I)I)(I))I) -> two tertiary stacks of two tritoruses (x2), empty slab B
Medium stack of eight ditoruses (((((I)))(II))I) -> major quartet of tiger toruses, two major[A] pairs of tiger toruses, four[torus] tiger toruses
Empty slice G (((()I))(II))I) -> two major[A] pairs of tiger toruses (x2), empty slab B
Empty slice H (((())I)(II))I) -> four[torus] tiger toruses (x2), empty slab B
3D cuts:
Empty cut A (((((II)I))())) -> empty slice A, minor stack of two medium/minor quartets of ditoruses (x2), empty slice C
Empty cut B (((((II))I)())) -> empty slice A, minor stack of two major/minor quartets of ditoruses (x2), empty slice D
Empty cut C (((((I)I)I)())) -> empty slice A, two minor stacks of two minor pairs of ditoruses (x2), empty slice E
Vertical stack of two major/major/minor octets of toruses (((((II)))(I))) -> minor stack of two medium/minor quartets of ditoruses, minor stack of two major/minor quartets of ditoruses, major/major/minor octet of tigers, medium stack of two major quartets of ditoruses
Two vertical stacks of two major/minor quartets of toruses (((((I)I))(I))) -> minor stack of two medium/minor quartets of ditoruses, two minor stacks of two minor pairs of ditoruses, two major[A]/minor quartets of tigers, two medium stacks of two major pairs of ditoruses
Four vertical stacks of two minor pairs of toruses (((((I))I)(I))) -> minor stack of two major/minor quartets of ditoruses, two minor stacks of two minor pairs of ditoruses, four minor pairs of tigers, four medium stacks of two ditoruses
Empty cut D ((((()I)I)(I))) -> two minor stacks of two minor pairs of ditoruses (x2), empty slice B, empty slice F
Vertical stack of eight minor pairs of toruses (((((I)))(II))) -> major/major/minor octet of tigers, two major[A]/minor quartets of tigers, four minor pairs of tigers, medium stack of eight ditoruses
Empty cut E ((((()I))(II))) -> two major[A]/minor quartets of tigers (x2), empty slice B, empty slice G
Empty cut F ((((())I)(II))) -> four minor pairs of tigers (x2), empty slice B, empty slice H
Empty cut G (((((II)))())I) -> empty slice C, empty slice D, medium stack of two major quartets of ditoruses (x2)
Empty cut H (((((I)I))())I) -> empty slice C, empty slice E, two medium stacks of two major pairs of ditoruses (x2)
Empty cut I (((((I))I)())I) -> empty slice D, empty slice E, four medium stacks of two ditoruses (x2)
Empty cut J ((((()I)I)())I) -> empty slice E (x2), empty slice F (x2)
8x2 array of toruses (((((I)))(I))I) -> medium stack of two major quartets of ditoruses, two medium stacks of two major pairs of ditoruses, four medium stacks of two ditoruses, medium stack of eight ditoruses
Empty cut K ((((()I))(I))I) -> two medium stacks of two major pairs of ditoruses (x2), empty slice F, empty slice G
Empty cut L ((((())I)(I))I) -> four medium stacks of two ditoruses (x2), empty slice F, empty slice H
Empty cut M ((((()))(II))I) -> medium stack of eight ditoruses (x2), empty slice G, empty slice H

18. Double tiger torus ((((II)(II))(II))I)
6D blobs:
Minor pair of double tigers ((((II)(II))(II))) -> double tiger torus
Tertiary stack of two tiger ditoruses ((((II)(II))(I))I) -> double tiger torus
Secondary stack of two torus tiger toruses ((((II)(I))(II))I) & ((((I)(II))(II))I) -> double tiger torus
5D slabs:
Minor stack of two minor pairs of tiger toruses ((((II)(II))(I))) -> minor pair of double tigers, tertiary stack of two tiger ditoruses
Secondary stack of two minor pairs of torus tigers ((((II)(I))(II))) & ((((I)(II))(II))) -> minor pair of double tigers, secondary stack of two torus tiger toruses
Empty slab A ((((II)(II))())I) -> tertiary stack of two tiger ditoruses (x2)
2x2 secondary/tertiary stack of tritoruses ((((II)(I))(I))I) & ((((I)(II))(I))I) -> tertiary stack of two tiger ditoruses, secondary stack of two torus tiger toruses
Empty slab B ((((II)())(II))I) & (((()(II))(II))I) -> secondary stack of two torus tiger toruses (x2)
2x2 array of torus tigers ((((I)(I))(II))I) -> secondary stack of two torus tiger toruses (x2, in two different ways)
4D slices:
Empty slice A ((((II)(II))())) -> minor stack of two minor pairs of tiger toruses (x2), empty slab A
2x2 medium/minor stack of minor pairs of ditoruses ((((II)(I))(I))) & ((((I)(II))(I))) -> minor stack of two minor pairs of tiger toruses, secondary stack of two minor pairs of torus tigers, 2x2 secondary/tertiary stack of tritoruses
Empty slice B ((((II)())(II))) & (((()(II))(II))) -> secondary stack of two minor pairs of torus tigers (x2), empty slab B
2x2 array of minor pairs of tigers ((((I)(I))(II))) -> secondary stack of two minor pairs of torus tigers (x2, in two different ways), 2x2 array of torus tigers
Empty slice C ((((II)(I))())I) & ((((I)(II))())I) -> empty slab A, 2x2 secondary/tertiary stack of tritoruses (x2)
Empty slice D ((((II)())(I))I) & (((()(II))(I))I) -> 2x2 secondary/tertiary stack of tritoruses (x2), empty slab B
2x2 array of medium stacks of two ditoruses ((((I)(I))(I))I) -> 2x2 secondary/tertiary stack of tritoruses (x2, in two different ways), 2x2 array of torus tigers
Empty slice E ((((I)())(II))I) & (((()(I))(II))I) -> empty slab B, 2x2 array of torus tigers (x2)
3D cuts:
Empty cut A ((((II)(I))())) & ((((I)(II))())) -> empty slice A, 2x2 medium/minor stack of minor pairs of ditoruses (x2), empty slice C
Empty cut B ((((II)())(I))) & ((()(II))(I))) -> 2x2 medium/minor stack of minor pairs of ditoruses (x2), empty slice B, empty slice D
2x2 array of vertical stacks of two minor pairs of toruses ((((I)(I))(I))) -> 2x2 medium/minor stack of minor pairs of ditoruses (x2, in two different ways), 2x2 array of minor pairs of tigers, 2x2 array of medium stacks of two ditoruses
Empty cut C ((((I)())(II))) & (((()(I))(II))) -> empty slice B, 2x2 array of minor pairs of tigers (x2), empty slice E
Empty cut D ((((II)())())I) & (((()(II))())I) -> empty slice C (x2), empty slice D (x2)
Empty cut E ((((I)(I))())I) -> empty slice C (x2, in two different ways), 2x2 array of medium stacks of two ditoruses (x2)
Empty cut F ((((I)())(I))I) & (((()(I))(I))I) -> empty slice D, 2x2 array of medium stacks of two ditoruses (x2), empty slice E
Empty cut G (((()())(II))I) -> empty slice E (x4, in two different ways)

27. Duotorus tiger torus ((((II)I)((II)I))I)
6D blobs:
Minor pair of duotorus tigers ((((II)I)((II)I))) -> duotorus tiger torus
Major[circle] pair of torus tiger toruses ((((II)I)((II)))I) & ((((II))((II)I))I) -> duotorus tiger torus
Two[circle] torus tiger toruses ((((II)I)((I)I))I) & ((((I)I)((II)I))I) -> duotorus tiger torus
5D slabs:
Major[circle]/minor quartet of torus tigers ((((II)I)((II)))) & ((((II))((II)I))) -> minor pair of duotorus tigers, major[circle] pair of torus tiger toruses
Two[circle] minor pairs of torus tigers ((((II)I)((I)I))) & ((((I)I)((II)I))) -> minor pair of duotorus tigers, two[circle] torus tiger toruses
Tertiary stack of four tritoruses ((((II)I)((I)))I) & ((((I))((II)I))I) -> major[circle] pair of torus tiger toruses, two[circle] torus tiger toruses
Major[A]/major[B] quartet of tiger toruses ((((II))((II)))I) -> major[circle] pair of torus tiger toruses (x2, in two different ways)
Two major[B] pairs of tiger toruses ((((I)I)((II)))I) & ((((II))((I)I))I) -> major[circle] pair of torus tiger toruses, two[circle] torus tiger toruses
Empty slab ((((II)I)(()I))I) & (((()I)((II)I))I) -> two[circle] torus tiger toruses (x2)
2x2 [A/B] array of torus tigers ((((I)I)((I)I))I) -> two[circle] torus tiger toruses (x2, in two different ways)
4D slices:
Minor stack of four minor pairs of ditoruses ((((II)I)((I)))) & ((((I))((II)I))) -> major[circle]/minor quartet of torus tigers, two[circle] minor pairs of torus tigers, tertiary stack of four tritoruses
Major[A]/major[B]/minor octet of tigers ((((II))((II)))) -> major[circle]/minor quartet of torus tigers (x2, in two different ways), major[A]/major[B] quartet of tiger toruses
Two major[B]/minor quartets of tigers ((((I)I)((II)))) & ((((II))((I)I))) -> major[circle]/minor quartet of torus tigers, two[circle] minor pairs of torus tigers, two major[B] pairs of tiger toruses
Empty slice A ((((II)I)(()I))) & (((()I)((II)I))) -> two[circle] minor pairs of torus tigers (x2), empty slab
2x2 [A/B] array of minor pairs of tigers ((((I)I)((I)I)) -> two[circle] minor pairs of torus tigers (x2, in two different ways), 2x2 [A/B] array of torus tigers
Empty slice B ((((II)I)(()))I) & (((())((II)I))I) -> tertiary stack of four tritoruses (x2), empty slab
Medium stack of four major pairs of ditoruses ((((II))((I)))I) & ((((I))((II)))I) -> tertiary stack of four tritoruses, major[A]/major[B] quartet of tiger toruses, two major[B] pairs of tiger toruses
Two medium stacks of four ditoruses ((((I)I)((I)))I) & ((((I))((I)I))I) -> tertiary stack of four tritoruses, two major[B] pairs of tiger toruses, 2x2 [A/B] array of torus tigers
Empty slice C (((()I)((II)))I) & ((((II))(()I))I) -> two major[B] pairs of tiger toruses (x2), empty slab
Empty slice D ((((I)I)(()I))I) & (((()I)((I)I))I) -> empty slab, 2x2 [A/B] array of torus tigers (x2)
3D cuts:
Empty cut A ((((II)I)(()))) & (((())((II)I))) -> minor stack of four minor pairs of ditoruses (x2), empty slice A, empty slice B
Vertical stack of four major/minor quartets of toruses ((((II))((I)))) & ((((I))((II)))) -> minor stack of four minor pairs of ditoruses, major[A]/major[B]/minor octet of tigers, two major[B]/minor quartets of tigers, medium stack of four major pairs of ditoruses
Two vertical stacks of four minor pairs of toruses ((((I)I)((I)))) & ((((I))((I)I))) -> minor stack of four minor pairs of ditoruses, two major[B]/minor quartets of tigers, 2x2 [A/B] array of minor pairs of tigers, two medium stacks of four ditoruses
Empty cut B (((()I)((II)))) & ((((II))(()I))) -> two major[B]/minor quartets of tigers (x2), empty slice A, empty slice C
Empty cut C ((((I)I)(()I))) & (((()I)((I)I))) -> empty slice A, 2x2 [A/B] array of minor pairs of tigers (x2), empty slice D
Empty cut D ((((II))(()))I) & (((())((II)))I) -> empty slice B, medium stack of four major pairs of ditoruses (x2), empty slice C
Empty cut E ((((I)I)(()))I) & (((())((I)I))I) -> empty slice B, two medium stacks of four ditoruses (x2), empty slice D
4x4 array of toruses ((((I))((I)))I) -> medium stack of four major pairs of ditoruses (x2, in two different ways), two medium stacks of four ditoruses (x2, in two different ways)
Empty cut F (((()I)((I)))I) & ((((I))(()I))I) -> two medium stacks of four ditoruses (x2), empty slice C, empty slice D
Empty cut G (((()I)(()I))I) -> empty slice D (x4, in two different ways)

32. Triger torus (((II)(II)(II))I)
6D blobs:
Minor pair of trigers (((II)(II)(II))) -> triger torus
Medium stack of 221-tiger 1-toruses (((II)(II)(I))I) & (((II)(I)(II))I) & (((I)(II)(II))I) -> triger torus
5D slabs:
Minor stack of two minor pairs of 221-tigers (((II)(II)(I))) & (((II)(I)(II))) & (((I)(II)(II))) -> minor pair of trigers, medium stack of 221-tiger 1-toruses
Empty slab (((II)(II)())I) & (((II)()(II))I) & ((()(II)(II))I) -> medium stack of 221-tiger 1-toruses (x2)
2x2 medium stack of 221-ditoruses (((II)(I)(I))I) & (((I)(II)(I))I) & (((I)(I)(II))I) -> medium stack of 221-tiger 1-toruses (x2, in two different ways)
4D slices:
Empty slice A ((((II)(II)())) & (((II)()(II))) & ((()(II)(II))) -> minor stack of two minor pairs of 221-tigers (x2), empty slab
2x2 vertical stack of minor pairs of spheritoruses (((II)(I)(I))) & (((I)(II)(I))) & (((I)(I)(II))) -> minor stack of two minor pairs of 221-tigers (x2, in two different ways), 2x2 medium stack of 221-ditoruses
Empty slice B (((II)(I)())I) & (((I)(II)())I) & (((II)()(I))I) & (((I)()(II))I) & ((()(II)(I))I) & ((()(I)(II))I) -> empty slab, 2x2 medium stack of 221-ditoruses (x2)
2x2x2 array of torispheres (((I)(I)(I))I) -> 2x2 medium stack of 221-ditoruses (x3, in three different ways)
3D cuts:
Empty cut A ((((II)(I)())) & ((((I)(II)())) & (((II)()(I))) & (((I)()(II))) & ((()(II)(I))) & ((()(I)(II))) -> empty slice A, 2x2 vertical stack of minor pairs of spheritoruses (x2), empty slice B
2x2x2 array of pairs of spheres (((I)(I)(I))) -> 2x2 vertical stack of minor pairs of spheritoruses (x3, in three different ways), 2x2x2 array of torispheres
Empty cut B ((((II)()())I) & (((()(II)())I) & (((()()(II))I) -> empty slice B (x4, in two different ways)
Empty cut C ((((I)(I)())I) & ((((I)()(I))I) & (((()(I)(I))I) -> empty slice B (x2, in two different ways), 2x2x2 array of torispheres (x2)

39. Tiger torus tiger ((((II)(II))I)(II))
6D blobs:
Minor stack of two tiger ditoruses ((((II)(II))I)(I)) -> tiger torus tiger
Medium pair of double tigers ((((II)(II)))(II)) -> tiger torus tiger
Secondary stack of two ditorus tigers ((((II)(I))I)(II)) & ((((I)(II))I)(II)) -> tiger torus tiger
5D slabs:
Empty slab A ((((II)(II))I)()) -> minor stack of two tiger ditoruses (x2)
Minor stack of two medium pairs of tiger ditoruses ((((II)(II)))(I)) -> minor stack of two tiger ditoruses, medium pair of double tigers
2x2 secondary/minor stack of tritoruses ((((II)(I))I)(I)) & ((((I)(II))I)(I)) -> minor stack of two tiger ditoruses, secondary stack of two ditorus tigers
Medium stack of two medium pairs of torus tigers ((((II)(I)))(II)) & ((((I)(II)))(II)) -> medium pair of double tigers, secondary stack of two ditorus tigers
Empty slab B ((((II)())I)(II)) & (((()(II))I)(II)) -> secondary stack of two ditorus tigers (x2)
2x2 [torus] array of torus tigers ((((I)(I))I)(II)) -> secondary stack of two ditorus tigers (x2, in two different ways)
4D slices:
Empty slice A ((((II)(II)))()) -> empty slab A, minor stack of two medium pairs of tiger ditoruses (x2)
Empty slice B ((((II)(I))I)()) & ((((I)(II))I)()) -> empty slab A, 2x2 secondary/minor stack of tritoruses (x2)
2x2 medium/minor stack of medium pairs of ditoruses ((((II)(I)))(I)) & ((((I)(II)))(I)) -> minor stack of two medium pairs of tiger ditoruses, 2x2 secondary/minor stack of tritoruses, medium stack of two medium pairs of torus tigers
Empty slice C ((((II)())I)(I)) & (((()(II))I)(I)) -> 2x2 secondary/minor stack of tritoruses (x2), empty slab B
2x2 array of minor stacks of two ditoruses ((((I)(I))I)(I)) -> 2x2 secondary/minor stack of tritoruses (x2, in two different ways), 2x2 [torus] array of torus tigers
Empty slice D ((((II)()))(II)) & (((()(II)))(II)) -> medium stack of two medium pairs of torus tigers (x2), empty slab B
2x2 [A/A] array of major[A] pairs of tigers ((((I)(I)))(II)) -> medium stack of two medium pairs of torus tigers (x2, in two different ways), 2x2 [torus] array of torus tigers
Empty slice E ((((I)())I)(II)) & (((()(I))I)(II)) -> empty slab B, 2x2 [torus] array of torus tigers (x2)
3D cuts:
Empty cut A ((((II)(I)))()) & ((((I)(II)))()) -> empty slice A, empty slice B, 2x2 medium/minor stack of medium pairs of ditoruses (x2)
Empty cut B ((((II)())I)()) & (((()(II))I)()) -> empty slice B (x2), empty slice C (x2)
Empty cut C ((((I)(I))I)()) -> empty slice B (x2, in two different ways), 2x2 array of minor stacks of two ditoruses (x2)
Empty cut D ((((II)()))(I)) & (((()(II)))(I)) -> 2x2 medium/minor stack of medium pairs of ditoruses (x2), empty slice C, empty slice D
2x2 array of two vertical stacks of two major pairs of toruses ((((I)(I)))(I)) -> 2x2 medium/minor stack of medium pairs of ditoruses (x2, in two different ways), 2x2 array of minor stacks of two ditoruses, 2x2 [A/A] array of major[A] pairs of tigers
Empty cut E ((((I)())I)(I)) & (((()(I))I)(I)) -> empty slice C, 2x2 array of minor stacks of two ditoruses (x2), empty slice E
Empty cut F ((((I)()))(II)) & (((()(I)))(II)) -> empty slice D, 2x2 [A/A] array of major[A] pairs of tigers (x2), empty slice E
Empty cut G (((()())I)(II)) -> empty slice E (x4, in two different ways)

42. Torus double tiger ((((II)I)(II))(II))
6D blobs:
Minor stack of two torus tiger toruses ((((II)I)(II))(I)) -> torus double tiger
Tertiary stack of two ditorus tigers ((((II)I)(I))(II)) -> torus double tiger
Major pair of double tigers ((((II))(II))(II)) -> torus double tiger
Two[tiger] double tigers ((((I)I)(II))(II)) -> torus double tiger
5D slabs:
Empty slab A ((((II)I)(II))()) -> minor stack of two torus tiger toruses (x2)
2x2 tertiary/minor stack of tritoruses ((((II)I)(I))(I)) -> minor stack of two torus tiger toruses, tertiary stack of two ditorus tigers
Minor stack of two major pairs of tiger toruses ((((II))(II))(I)) -> minor stack of two torus tiger toruses, major pair of double tigers
Two minor stacks of two tiger toruses ((((I)I)(II))(I)) -> minor stack of two torus tiger toruses, two[tiger] double tigers
Empty slab B ((((II)I)())(II)) -> tertiary stack of two ditorus tigers (x2)
Medium stack of two major[torus] pairs of torus tigers ((((II))(I))(II)) -> tertiary stack of two ditorus tigers, major pair of double tigers
Two[torus] medium stacks of two torus tigers ((((I)I)(I))(II)) -> tertiary stack of two ditorus tigers, two[tiger] double tigers
Medium stack of four torus tigers ((((I))(II))(II)) -> major pair of double tigers, two[tiger] double tigers
Empty slab C (((()I)(II))(II)) -> two[tiger] double tigers (x2)
4D slices:
Empty slice A ((((II)I)(I))()) -> empty slab A, 2x2 tertiary/minor stack of tritoruses (x2)
Empty slice B ((((II))(II))()) -> empty slab A, minor stack of two major pairs of tiger toruses (x2)
Empty slice C ((((I)I)(II))()) -> empty slab A, two minor stacks of two tiger toruses (x2)
Empty slice D ((((II)I)())(I)) -> 2x2 tertiary/minor stack of tritoruses (x2), empty slab B
2x2 medium/minor stack of major pairs of ditoruses ((((II))(I))(I)) -> 2x2 tertiary/minor stack of tritoruses, minor stack of two major pairs of tiger toruses, medium stack of two major[torus] pairs of torus tigers
Two 2x2 medium/minor stacks of ditoruses ((((I)I)(I))(I)) -> 2x2 tertiary/minor stack of tritoruses, two minor stacks of two tiger toruses, two[torus] medium stacks of two torus tigers
4x2 medium/minor stack of ditoruses ((((I))(II))(I)) -> minor stack of two major pairs of tiger toruses, two minor stacks of two tiger toruses, medium stack of four torus tigers
Empty slice E (((()I)(II))(I)) -> two minor stacks of two tiger toruses (x2), empty slab C
Empty slice F ((((II))())(II)) -> empty slab B, medium stack of two major[torus] pairs of torus tigers (x2)
Empty slice G ((((I)I)())(II)) -> empty slab B, two[torus] medium stacks of two torus tigers (x2)
4x2 [A/A] array of tigers ((((I))(I))(II)) -> medium stack of two major[torus] pairs of torus tigers, two[torus] medium stacks of two torus tigers, medium stack of four torus tigers
Empty slice H (((()I)(I))(II)) -> two[torus] medium stacks of two torus tigers (x2), empty slab C
Empty slice I (((())(II))(II)) -> medium stack of four torus tigers (x2), empty slab C
3D cuts:
Empty cut A ((((II)I)())()) -> empty slice A (x2), empty slice D (x2)
Empty cut B ((((II))(I))()) -> empty slice A, empty slice B, 2x2 medium/minor stack of major pairs of ditoruses (x2)
Empty cut C ((((I)I)(I))()) -> empty slice A, empty slice C, two 2x2 medium/minor stacks of ditoruses (x2)
Empty cut D ((((I))(II))()) -> empty slice B, empty slice C, 4x2 medium/minor stack of ditoruses (x2)
Empty cut E (((()I)(II))()) -> empty slice C (x2), empty slice E (x2)
Empty cut F ((((II))())(I)) -> empty slice D, 2x2 medium/minor stack of major pairs of ditoruses (x2), empty slice F
Empty cut G ((((I)I)())(I)) -> empty slice D, two 2x2 medium/minor stacks of ditoruses (x2), empty slice G
4x2 array of vertical stacks of two toruses ((((I))(I))(I)) -> 2x2 medium/minor stack of major pairs of ditoruses, two 2x2 medium/minor stacks of ditoruses, 4x2 medium/minor stack of ditoruses, 4x2 [A/A] array of tigers
Empty cut H (((()I)(I))(I)) -> two 2x2 medium/minor stacks of ditoruses (x2), empty slice E, empty slice H
Empty cut I (((())(II))(I)) -> 4x2 medium/minor stack of ditoruses (x2), empty slice E, empty slice I
Empty cut J ((((I))())(II)) -> empty slice F, empty slice G, 4x2 [A/A] array of tigers (x2)
Empty cut K (((()I)())(II)) -> empty slice G (x2), empty slice H (x2)
Empty cut L (((())(I))(II)) -> 4x2 [A/A] array of tigers (x2), empty slice H, empty slice I
Marek14
Pentonian
 
Posts: 1102
Joined: Sat Jul 16, 2005 6:40 pm

Re: The Tiger Explained

Postby Marek14 » Wed Jun 11, 2014 7:10 am

Here's two more ideas:

1. Put your images and gifs on wiki page of the particular toratopes to have a permanent storage for them.
2. You could try to combine the "main" toratope you're exploring with second, thinner, that goes through the hole to accentuate it better. In case of ditorus tiger, there could be a torus tiger inserted, for example.
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Re: The Tiger Explained

Postby quickfur » Wed Jun 11, 2014 5:31 pm

Polyhedron Dude wrote:ICN5D, I got one thing to say about all of these toroidal pics and animations - AWESOME!!! :mrgreen:

I totally agree!!! These are some truly mindblowing animations. You should make a website dedicated to them. :lol:

A thought just occurred to me while looking at these crazy animations... what if there was an alternate universe where electron orbitals followed toratopic rules instead of the usual Schroedinger equation? Then you'd have crazy chemistries going on with orbitals shaped like these toratopes, where a state change would trigger a shift in the shape of the orbital akin to the transformations shown in your animations. So you'd have molecules where atoms are linked by concentric toroidal rings, sequences of toruses, etc.. You could get the most bizarre-looking molecules and the craziest chemical reactions. :D
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Re: The Tiger Explained

Postby quickfur » Wed Jun 11, 2014 5:33 pm

Actually, since you already have a wiki account for uploading these images/animations, you might as well just start making wiki pages for their corresponding toratopes, and link to the images/animations from there.
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Re: The Tiger Explained

Postby Marek14 » Wed Jun 11, 2014 8:27 pm

I updated my "basic toratopes" file, correcting some numbers and extending it to 10D, which should contain 2058 toratopes, including 145 basic ones (there are probably some mistakes, though).
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Re: The Tiger Explained

Postby ICN5D » Wed Jun 11, 2014 8:28 pm

Polyhedron Dude wrote:ICN5D, I got one thing to say about all of these toroidal pics and animations - AWESOME!!! :mrgreen:


Thanks Polyhedron Dude! You can imagine my excitement when I see a complex morphing, and jump on the chance to bring it to life. It's a synergistic marriage of mundane math and sterile technology, producing something more beautiful than the starting ingredients.





quickfur wrote:I totally agree!!! These are some truly mindblowing animations. You should make a website dedicated to them. :lol:

Actually, since you already have a wiki account for uploading these images/animations, you might as well just start making wiki pages for their corresponding toratopes, and link to the images/animations from there.


I've been thinking about a dedicated website. Like a GIF animation explorer gallery. And, of course making wiki pages here as well. None exist for the ones I've imaged, except for a few 5D's. I'd have to admit 6D is very interesting, more-so than 5! It only takes about 10-30 minutes from taking the first screenshot, to compiling the GIF, to uploading, to submitting the post on here, for each one.

A thought just occurred to me while looking at these crazy animations... what if there was an alternate universe where electron orbitals followed toratopic rules instead of the usual Schroedinger equation? Then you'd have crazy chemistries going on with orbitals shaped like these toratopes, where a state change would trigger a shift in the shape of the orbital akin to the transformations shown in your animations. So you'd have molecules where atoms are linked by concentric toroidal rings, sequences of toruses, etc.. You could get the most bizarre-looking molecules and the craziest chemical reactions. :D


Now you're thinking! That's been my suspicion of how it actually is, since learning the toratopes. Before rendering them, I used the notation and cut algorithm to figure it all out. But, I would have never guessed how amazing they really are with multirotations, and empty to empty scans. Maybe that's what the electron's complex space really is, just normal physical clouds, but existing above our 3-plane in higher dimensions. We don't have the ability to rotate our physical 3D plane to make them visible. But, the 3D graphing program does, and when we do rotate past diagonally opposed shapes, we see them appear suddenly out of nowhere, then disappear just as quickly. Those parts are indeed there, but they're displaced above and below XYZ. So, maybe when we see two electrons occupying two opposing real orbitals, there are actually four in total, with an identical pair in the opposing complex orbitals (which could be the case anyways, I'm just rambling .... )





Marek14 wrote:I see, so the last picture is rotation ((((Xy))x)(YI)), correct? So my ideas about these were incorrect and now I see why -- the main direction was in x axis and changed to y. This way, there are no real invariants in the rotation, unlike rotation ((((I))x)(XI)) where the trace of ditorus tiger ((((I)))(I)), 8x2 array of circles, would stay invariant throughout the rotation. Just as well that current 7D exploration no longer even attempts to explore rotations :)


Yes it's ((((Xy))x(YI)). I guess this is the first empirical evidence of such a thing. Single rotations in ((((II)I)I)(II)) always have the invariant 8x2 trace of circles. But double rotations flip the orientation around to another invariant plane. I suppose that's probably the best way to describe it. Take whatever the trace would be, establish the dimensionality of its hyperplane it lies in, and extrapolate the other(s). It seems to be related to how many orthogonal n-planes the shape can have, according to its lowest trace.

In this case, it's a 2D hyperplane that contains the 8x2 trace of circles ((((I)))(I)), and there seems to be three of them. If the trace is 2D, there are three such orthogonal 2-planes in 6D. Considering what ((((II)I)I)(II)) looks like as a whole, it's a large flat biscuit with the majority of it's structure spread out along a 2-plane. Two of the 8x2 trace hyperplanes become a vertical knife cut through the top of the biscuit, at 90 degree orientations, like making quarter slices. The third trace array hyperplane slices the biscuit horizontally like an english muffin.

The 8x2 trace array hyperplane can be rotated to create all 3D cuts. It's a matter of how you orient and spin the plane, and there are some really complex ways to do so. That's worth investigating. How to manipulate the lowest trace to produce all 3D axials? Bisecting rotations are simple, but non-intersectings are more complex. Non-intersecting rotations seem to isolate the trace into separate groups, then do regular bisecting rotations of those individual groups. Let's have a look:

Code: Select all
((((I)))(I)) - 8x2 array of circles

   O O O O O O O O

   O O O O O O O O



((((II)))(I)) - simple rotation around bisecting line, divides trace into two separate 4x2 groups

          |
   O O O O|O O O O
          |
   O O O O|O O O O
          |



((((I)))(II)) - simple rotation around bisecting line, divides trace into two separate 8x1 groups

   O O O O O O O O
----------------------
   O O O O O O O O



((((I)I))(I)) - complex rotation around non-intersecting line, isolates trace into two separate 4x2 groups, then rotate in opposing directions around two bisecting lines, dividing each into 2x2 groups

      |         |
   O O|O O   O O|O O
      |         |
   O O|O O   O O|O O
      |         |

     CW        CCW


((((I))I)(I)) - complex rotation around non-intersecting line, isolates trace into four separate 2x2 groups, then rotate around four bisecting lines, dividing each into 1x2 group

    |    |    |    |
   O|O  O|O  O|O  O|O
    |    |    |    |
   O|O  O|O  O|O  O|O
    |    |    |    |

   CW   CCW  CW   CCW





1. Put your images and gifs on wiki page of the particular toratopes to have a permanent storage for them.
2. You could try to combine the "main" toratope you're exploring with second, thinner, that goes through the hole to accentuate it better. In case of ditorus tiger, there could be a torus tiger inserted, for example.




1) yeah, that sounds like a better idea. Right now, they're in the unboxed supercluster

2) How would that work? What should I expect to see? I remember you mentioned rendering chained toratopes, is that the same? Show how they link together through higher directions?
Last edited by ICN5D on Wed Jun 11, 2014 9:01 pm, edited 1 time in total.
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Re: The Tiger Explained

Postby Marek14 » Wed Jun 11, 2014 8:50 pm

ICN5D wrote:
Marek14 wrote:1. Put your images and gifs on wiki page of the particular toratopes to have a permanent storage for them.
2. You could try to combine the "main" toratope you're exploring with second, thinner, that goes through the hole to accentuate it better. In case of ditorus tiger, there could be a torus tiger inserted, for example.




1) yeah, that sounds like a better idea. Right now, they're in the unboxed supercluster

2) How would that work? What should I expect to see? I remember you mentioned rendering chained toratopes, is that the same? Show how they link together through higher directions?


Well, basically, the holes in toratopes can be imagined as open (or infinitely long) toratopes.

A hole in torus is basically a cylinder -- you can put a long, thin cylinder through the hole.

In 4D, the hole in torisphere is a spherinder (or, better, sphere x line) and you could render a torisphere with spherinder-shaped pole stuck through it. In torus cut, the pole would shrink into zero diameter as the hole in torus closes, in pair of spheres cut it would appear in spherical cross-section as a sphere remaining in the pair's center.

The hole in spheritorus is circle x plane, you could show it as a cubinder with long square edges. In two spheres cut, this would look like a board between the spheres, thinning as they come closer and disappearing before they merge. In torus view, it would be a cylindrical pole that persists unchanged throughout the cut.

The ditorus is more interesting: it has two separate holes. One of them (the "outer hole") is torus x line, while the "inner hole" is the hole in the torus, circle x plane, same as spheritorus. In two torus cut, the outer hole could be displayed as two poles through the torus holes. As the toruses merge, the poles would show the Cassini ovals crosscuts. The inner hole would be a board between the toruses, thinning and disappearing, just like in spheritorus case.

Alternate image here would be ditorus/torisphere lock. Instead of making two poles, make a torus going through the two ditoruses and evolve it like a torisphere -- the results should be spectacular.

In ditorus's "major pair of toruses" cut, outer hole would be a hollow cylinder between both toruses and inner hole would be a stable pole through the center of the figure. In "minor pair of toruses" cut, the outer hole would be a stable torus in the inner region of the figure and inner hole would be a pole through the figure. Note that the ditorus/torisphere lock can be displayed in these orientations as well -- how it would look is left as an exercise.

As for tiger, it has two spheritorus-type holes (circle x plane), which cross in a finite region in the middle of tiger. In the standard cut of tiger, one of them would look as a thinning board separating two toruses and the other as a stable pole going through the middle of toruses.
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Re: The Tiger Explained

Postby ICN5D » Wed Jun 11, 2014 9:30 pm

Ahhh! I see how that works! You're right, it would look super cool. Especially cutting it along 4-space. That brings me to my next question I've had for some time: how do I form implicit surface equations for open toratopes? Maybe I want to animate the cuts of a duocylinder, or a cyltrianglinder! I like your thinking on representing the dimensionality of holes, even better animating them. If I can render it, I can animate it, no problem.

Plus, thank you so much for making that toratope list to 10D, it's awesome! There's so many amazing complex ones in there. There comes a time, after so much has been spent reading the notation, that you can immediately understand and see the shape, at first glance of its symbols. It seems that a lot of investigation and rendering has gone into CRFs, where the toratopes have been pretty much left alone. Are they scarier being the beasts of high-D?

CRFs are dimensionally simple, yet structurally complex. Conversely, toratopes are structurally simple, yet dimensionally complex. No matter, I'll fill in the vacancy for such exploration and illustration. Since finding CalcPlot3D, I've explored 25 toratopes using about 80 equations, made 2200 screenshots, and 15 movies. And it's just the beginning!
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