## The Tiger Explained

Discussion of shapes with curves and holes in various dimensions.

### Re: Understanding the Cyltrianglinder

wendy wrote:Here goes. A picture of a tiger being folded from its net. The last stage before the tiger is a torus-prism (ie a 3d torus * line prism). This is joined top to bottom as one makes a torus from a prism.

From what I can see in that picture, is the Tiger look something like a duocylinder except the circular bits commonly drawn in the duocylinder projection becomes 2-torii themselves?

I still need to sharpen my visualisation skills more
in particular, my folding result of the tiger look something like what I expect for a ditorus, except that its crossection is a clifford torus rather than an ordinary 2-torus, while my ditorus folding is quite screwed since it is very difficult to stretch the 3D hypersurface and make sense on the actual shape of the "paper stripe in between" (As a result, the hidden surface culling may not be accurate)

P.S. The two mathematica diagrams are attempt to render a 3D projection of the Tiger and ditorus using the equations provided in the Wiki of this forum, the ditorus came out nicely but the tiger looks completely screwed. Seems I cannot make a (orthographic) projection simply by replacing w with a constant...
Attachments
ddds.PNG
Some folding attempt
ddd4s.PNG
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Secret
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### Re: Understanding the Cyltrianglinder

If I understand the net of a tiger to be a cube, then I see how the Comb Product works. In Comb{circle,circle,circle}, the three circles represent the dimensions and nature of the net, as a 3D cube. Then, it's a linear process of hollowing out, and filling with circles in each step:

-The first circle is applied by hollowing out the cube, reducing by N-1 dimensions, and filling it with circles. This makes a cubinder, {square,circle}, not the same as {cube, circle}, tesserinder.

-The second circle is applied by hollowing out the cubinder, reducing by N-1, and filling it with circles. This somehow makes a torus-prism, though I can't visualize it yet. Torus prism = {line-torus,circle} / {glomolatrix-prism,circle} or {torus,line}.

-The third circle, again, is applied by hollowing out the torus-prism, reducing by N-1, and filling it with circles. This will finally make the tiger.

In Secret's drawings, I see a two different ways the cube is being transformed. The bottom sequence is:

-Cube rotated into 4D makes Cubinder
-Cyl ends of cubinder joined together makes cylinder-torus, or torus-prism
-Torus-prism ends joined together makes di-torus

This is different than the duo-torus definition Wendy gave. By comparing the definitions of:

- duocylinder as a bi-circular prism == {circle,circle}-prism

Spherating the duocylinder gives:

- tiger as a spherated bi-glomohedric prism == {glomohedrix,glomohedrix}-prism

Then this means the tiger is a cartesian product of two glomohedrices, or hollow spheres. Is this right? I think the tiger has two holes, I'm not sure though.

I think I may have seen the tiger long ago, and just called it something different. I was trying to lathe a torus into 4D, by rotating its major circle around a bisecting line to its manifold. This created a spherical version of the circular manifold, and thus a circle embedded into the 2-manifold of a sphere, {circle,glomohedrix}-prism. But, this is different than the {glomohedrix,glomohedrix}-prism. Or are they the same?

-Philip
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### Re: Understanding the Cyltrianglinder

If you take a 3D torus ((II)I), make a prism of it and join the ends together (in a hose linkage), you get a ditorus (((II)I)I). The same happens if you roll the ends inside (in a sock linkage). Neither makes a tiger ((II)(II)).

I think perhaps Wendy has confused what the definition of tiger is as she has also written it as circle # circle # circle which is not what I was expecting:

I wrote:(((11)1)1) ditorus = (circle # circle) # circle
((11)(11)) tiger = (circle x circle) # circle

Keiji

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### Re: The Tiger Explained

This was in an off-topic thread, it should be here:

If I understand the net of a tiger to be a cube, then I see how the Comb Product works. In Comb{circle,circle,circle}, the three circles represent the dimensions and nature of the net, as a 3D cube. Then, it's a linear process of hollowing out, and filling with circles in each step:

-The first circle is applied by hollowing out the cube, reducing by N-1 dimensions, and filling it with circles. This makes a cubinder, {square,circle}, not the same as {cube, circle}, tesserinder.

-The second circle is applied by hollowing out the cubinder, reducing by N-1, and filling it with circles. This somehow makes a torus-prism, though I can't visualize it yet. Torus prism = {glomolatrix-prism,circle} or {torus,line}.

-The third circle, again, is applied by hollowing out the torus-prism, reducing by N-1, and filling it with circles. This will finally make the tiger.

By comparing the definitions of:

- duocylinder, as a bi-circular prism == {circle,circle}-prism

Spherating the duocylinder gives:

- tiger, as a spherated bi-glomohedric prism == {glomohedrix,glomohedrix}-prism

Then this means the tiger is a cartesian product of two glomohedrices, (hollow spheres). Is this right? I think the tiger has two holes, I'm not sure though.

I think I may have seen the tiger long ago, and just called it something different. I was trying to lathe a torus into 4D, by rotating its major circle around a bisecting line through its manifold. This created a spherical version of the circular manifold, and thus a circle embedded into the 2-manifold of a sphere, {circle,glomohedrix}-prism. But, this is different than the {glomohedrix,glomohedrix}-prism. Or are they the same?

-Philip
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ICN5D
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### Re: The Tiger Explained

ICN5D wrote:If I understand the net of a tiger to be a cube, then I see how the Comb Product works. In Comb{circle,circle,circle}, the three circles represent the dimensions and nature of the net, as a 3D cube. Then, it's a linear process of hollowing out, and filling with circles in each step:

-The first circle is applied by hollowing out the cube, reducing by N-1 dimensions, and filling it with circles. This makes a cubinder, {square,circle}, not the same as {cube, circle}, tesserinder.

-The second circle is applied by hollowing out the cubinder, reducing by N-1, and filling it with circles. This somehow makes a torus-prism, though I can't visualize it yet. Torus prism = {glomolatrix-prism,circle} or {torus,line}.

-The third circle, again, is applied by hollowing out the torus-prism, reducing by N-1, and filling it with circles. This will finally make the tiger.

Except that's not the tiger, that's the ditorus.

Keiji

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### Re: The Tiger Explained

Yeah, still having a tough time with it. I found Mirek14's explanation of the four different ways to rotate a torus. One of which makes a tiger. And I have no idea how in the world it works. But, I also found the surface volume formula, and I see the two torii formulas in it. So, maybe the tiger is some kind of duo-torus, similar to a duocylinder.

(√(x2 + y2) − a)2 + (√(z2 + w2) − b)2 = r2

I see the duocyl as having only two curved surfaces at right angles, but I'm not sure how to spherate it. Does it have something to do with the sharp edges between the two? As in, a square has sharp corners, spherating it smooths out to a circle. So then, spherating a duocyl should remove the sharp edges uniformly? But spherating a cylinder makes a torus. Then, wouldn't this mean that a duocylinder, having two cylinder-like attributes, turns into something like two torii in one? If there is one minor radius, and two perpendicular major radii, it seems like there is a single attachment between two independent torii, like a 4D figure-8, with two holes.
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### Re: Understanding the Cyltrianglinder

I'm imagining the tiger's cross sections go something like this:

Start with a horizontal ring that thickens into a torus that fisses into two torii - one above the other. The torii move away from each other, slow down, and then comes back together fusing together, then shrinking to a ring and vanishing. It would have two orthogonal chasms - like the uniform polychoron Ondip.
Last edited by Polyhedron Dude on Wed Nov 27, 2013 8:24 am, edited 1 time in total.
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### Re: Understanding the Cyltrianglinder

Holy crap, that is an amazing way to describe the darn thing! Probably the best I've ever seen. I think I'm beginning to get it. It does have two torii attributes, but I wasn't sure in which way. That is an incredibly complicated freaking shape, man. No wonder aliens don't bother with us. It'd be like trying to teach calculus to a dog.
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### Re: The Tiger Explained

Yes, the tiger can be seen as having two perpendicular torus-like properties.

I think I mentioned earlier, in response to Secret's diagram, that there would be two axes with holes that you could enter, and once you enter one of them, you could go either left or right to continue through that hole and around the perpendicular one to come out the other side, and that you can't enter the other hole without leaving the first one first.

I will try and explain it using my frontal/lateral approach. Here, frontal is the direction you go in if you want to go through a hole in a torus, and lateral is the direction you could look to see the "wall" (surface) of the torus.

3D torus has 2 laterals, 1 frontal. (Where you have 2 laterals, they form a circle, or ring)
Toracubinder has 3 laterals, 1 frontal. It's the closest analog to the 3D torus.
Toraspherinder has 2 laterals, 2 combined frontals. The "combined" means that rather than being able to fit a linear object, like an arbitrarily long pole through the torus you could fit an arbitrarily large planar object, like a large board of wood through - if you remove one of the dimensions and visualize it in 3D, you have the "top" and "bottom" of the toraspherinder above and below the board, while the "sides" of the toraspherinder are out in the 4th dimension not intersecting with the board.
Tiger has 2 laterals, 2 split frontals. The "split" means that things can only be long in one dimension to fit through again, like poles, so you can't fit your wooden board through this one - however, instead of only having one hole that it can go through, there are two perpendicular holes. As mentioned above, whichever you choose, you then go left or right around the other one before coming out the other side.
This is incorrect, see later in the topic.
Ditorus I'm not sure how to describe in numbers yet and actually this is the hardest to explain (even though it is easier to draw than the tiger!). The easiest way to visualise a ditorus is as a normal 3D torus, but then the surface of the 3D torus is "puffed up" to make a 3D torus cross section as well. This torus also has two holes, the "major" and the "minor" ones. If you look at it from this 3D perspective it's impossible to get to the "minor" hole, but you can go through the "major" one just as you would go through a 3D torus. However, if you go half way into the "major" hole so you're now sitting dead center - at the origin, if you use the parametric equations as such - you can "hop" into the minor hole by going a short distance into the fourth dimension. You're then stuck "inside" the 3D projection envelope until you go into the fourth dimension again to "hop" back out of it.

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### Re: Understanding the Cyltrianglinder

I had a look at the 'tiger' on the wiki at http://teamikaria.com/hddb/wiki/Tiger . It is quite interesting, although little is revealed there.

The equation given is " (sqrt(w^2 + x^2) - a)^2 + (sqrt(y^2 + z^2) -b)^2 = r^2 ". It looks quite frightening. But it disects quite nicely into bi-polar coordinates, where we write (w,x) as (x, \phi), and (y,z) as (y, \psi). The angular coordinates play no part, and we are left with a different equation:

(x-a)^2 + (y-b)^2 = r^2. This is the equation for a circle of radius r, centred on x=a, y=b.

What happens now. The coordinates are valid for x>0, y>0, and there is no dependance on the angular coordinates. So you have this quadrant, and you rotate the y-axis around the wx pland, and you rotate the x axis around the yz plane. Every point on the xy quadrant becomes a 'clifford torus'. A rectangle whose vertices lie at (a,b), (a,0), (0,0), (0,b), gives the duo-cylinder. The line from (a,0) to (0,b) gives the bi-circular tegum. The ellipse quarter from (a,0) to (0,b), gives a bi-circular crind: a kind of equi-ellipsoid, xOoOxOo, which becomes a glome when a=b.

A tiger is then a circle centred at a, b, both greater than r. When the circle is centred at a=0, b or a, b=0, the resulting figure is a spherated circle, or a torus of spherical cross section and circular hole.

Because the tiger is centred on a, b in this scheme, and this is both a "clifford torus", and the margin between the two faces of a duocylinder, we see that there are many descriptions that invoke the duocylinder, simply to effect the necessary clifford torus at the centre of the tiger.

The Clifford Torus

In maths speak, a torus is a rectangle-region, where the sides wrap around like a cylinder at each end. So if you're at the top in column x, the next step upwards is to end at the bottom of the same column x. A particular instance of this is the surface of a doughnut-shape thing in 3d, which is also called a torus. In 4D, the margin between the faces of a duocylinedr is a reasonably undeformed instance of such a rectangle: it shoud be, because it is the prism-product of two circle-surfaces.

The name 'clifford', is usually attached to things that have a relation to 'clifford parallels'. This means that one can divide all of 4-space (or any even number-space), into circles that neither intersect each other, but are none the less concentric on a common centre. For those of you familiar with "complex numbers", it is very easy to verify this, because a 2N space maps onto a product of N argand diagrams, and the multiplication of every point by a value cis(w t), does not change the gradient, and makes every point orbit the centre in one cycle.

The construction of the glome from rotating a xy quadrant over the wx and yz spaces, will sweep the quadrant from 0 to 90 degrees into the climate lattitudes of a 4D planet. 90 is cold, 0 is hot (as in lattitude), and the trace of the point at x degrees (say 27.5), will give all points whose lattitued is 27.5 degrees - a clifford torus.

A clifford torus at a, b gives a rectangle of size 2pi a * 2pi b, the diagonal in either direction, and its parallels (we cut the rectangle from a torus: this can be done anywhere), gives circles, and if the diagonals in all rectangles run the same way, all space is divided into parallel equal-distant circles. Every point on our xy quadrant is mapped onto a set of these: this means any figure that you construct by rotating around the wx and then yz axies, is bounded by a set of concentric circles.

The tiger

The tiger is then a 'spherated bi-circular prism'. I used the term 'glomohedrix' here, explaining that this word meant the surface of a sphere. Glomohedrix means sphere-surface, but spheres are not used here: circles are.

A tiger = spherated bi-circular prism = spherated bi-latrix prism.

Spheration turns the point (a,b) into a circle centred thereon, the symmetry by rotations around the wx and yz carry the circle (orthogonally), first into a torus, and then into a bi-torus or tiger. It's just that in the tiger, two of the circles are orthogonal.

Bi-glomohedrix prism are indeed used in relation to clifford-parallels, but not here. The phase space of all of the great circles + directions in 4D, map onto individual points on the 2glmhdx prism. But it's not part of the tiger.

The 'realmic sections' and Cassini ovals, are a bit of a false lead here. People think of toruses in terms of the cross section, not the equation that it might make on the surface of the water as it is being submerged. This is what is implied by being a rototope. This is also what Keiji is hinting at at things like circle # disk etc. The disk is rotated to keep orthogonal to the circle as its orbit sweeps out the torus.

Products

Keiji is coming to the conclusion that the tiger is ((II)(II)), while the duotorus is (((II)I)I), are different things. In practice, i suspect that it's more due to a mis-understanding on how the products actually work. One suspects by the issues he has with the assorted products what's being implied.

The duotorus and the tiger are both Comb{circle, circle, circle}, in much the same way that the rectangle and square are Prism{line, line}. The tiger is a special instance of a duo-torus, which has bi-cylinderic symmetry.

A prism product is in euclidean geometry, a cartesian product. A Comb product isn't.

Unlike the other products (crind, prism, tegum, pyramid), the comb product, applied to polytopes (rather than euclidean tilings, where it is perfectly regular), is not always applied at the centre. Also, if one fills the resulting surface, one can not suppose that the figures are topologically equal. A surface gives two different topological shapes, not the same. This is why we have sock and hose closures of surfaces.

The reason we have the 'spherated' part is that unless you get down to fixing the order of the comb product, the spherated (which is a kind of thick paint applied to thin things like points and lines), allows us to turn things like circles and bi-circular prisms, into solid things that you can wave about.

The three circles in the tiger, two make the clifford torus, the third runs around the clifford torus, as an outcome of spheration. You can see the spheration circle in the circle we rotate around the torus. Generally, the process of hose and sock places each subsequent circle around the centre of the previous one: so you start with a circle, make it to a 3d torus, and then turn the skin of the 3d torus into a bi-torus.
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### Re: The Tiger Explained

The tiger is indeed a duo-torus.

Consider the 1*12*12 prism, which is the prism-products of the nets.

The first step is to roll the inch axis into a circle. This is the spheration. We then have a rectangle in the yz plane.

The next step is to connect the rectangle into a circle in the xy space. This turns the surface into a 3d torus * line prism, with the z axis still unwrapped. You then turn the z line into a circle onto a zw space. That gives a tiger.

The equation given for the tiger in the wiki, http://teamikaria.com/hddb/wiki/Tiger , when you convert the xy and wz axis into polar coordinates, becomes the circle centred on a,b and of radius r.

(x-a)^2 + (y^b)^2 = r^2.

We then rotate this around xy and wz axies, one after the other, (which represents the trace of making the angular coordinates constant).

The idea of converting a point (a, b) [ie r=0 gives a clifford torus], into r>0, equates to spheration. But the circumference of the circle is one of the three involved in the product, the other two are are the xy and wz sweeps.
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### Re: Understanding the Cyltrianglinder

wendy wrote:Keiji is coming to the conclusion that the tiger is ((II)(II)), while the duotorus is (((II)I)I), are different things. In practice, i suspect that it's more due to a mis-understanding on how the products actually work. One suspects by the issues he has with the assorted products what's being implied.

That's not the conclusion, that's the definition.

tiger is defined as ((II)(II)) and ditorus is defined as (((II)I)I). There is an algorithm described on the rotope page which tells you how to convert this notation to a surface equation, which is where the equation you found on the wiki comes from. That equation clearly defines the surface, interior and exterior of both objects, and it's quite obvious from the equations that they are two completely different things.

The word duotorus makes me think of the exact same thing as the tiger - it's a duocylinder in torus form, so duotorus is ((II)(II)), the same as the tiger. But this is not the conventional name for the object (tiger is). The ditorus on the other hand should not be confused with any such "duotorus" as the di- prefix means "do an operation twice", the operation here being add I and enclose in parentheses. You will notice a convention with my object naming that di-, tri-, tetra-, penta- etc. mean to repeat a linear operation a number of times, whereas duo-, trio-, quadro-, quinto- etc. mean to repeat a sub-object or sub-grouping a number of times.

Keiji

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### Re: The Tiger Explained

Yep, I get it now! Finally after 6 years since the first introduction to it. I didn't believe it was a real shape, because I could never see it. I couldn't see how "torusing" the torus could make anything else other than the ditorus. But, the tiger definitely is a strange, valid way to do it.

Since the tiger has one minor and two major radii, the minor and first major are in the base-torus. The SECOND major radius is applied in a really bizarre way:

* It's the torusing of the first major radius, while remaining parallel. It really is an amazing shape.
Cross sections:
- Thin ring
- Inflates to torus
- Divides into 2 torii, parallel, one above another
- Separates to distance of second major diameter
- Moves toward each other
- Fuses into 1 torus
- Deflates to thin ring

* It is NOT the torusing of the base torus, where the plane of the first major radius changes along the way, in the path of a circle. This makes a ditorus.

One hole is through the "top", through the original torus, the second is through the "side", so to speak, through the second torus. These two holes have a similar orientation to the two curved rolling surfaces on a duocylinder. It was PolyhedronDudes's description of the cross sections that did it. That is key for normal people. All of the " R satisfies the margin of the bisecting planes through the such and such ", is too abstract for most. I would never have come the correct understanding from all other descriptions. Comparing the two parallel torii analogously with a torus' two-circle cross-section is the best way. Now, I will spread the word of the Tiger, to the farthest corners of the world. I almost want to call the tiger a toric bi-glomolatric prism. A torus that follows a pathway of the cartesian procuct of two hollow circles.

----So, what is the net of a tiger? I don't see how it can be a cube. I don't see how it could be anything recognizable, either.

Toraspherinder has 2 laterals, 2 combined frontals. The "combined" means that rather than being able to fit a linear object, like an arbitrarily long pole through the torus you could fit an arbitrarily large planar object, like a large board of wood through - if you remove one of the dimensions and visualize it in 3D, you have the "top" and "bottom" of the toraspherinder above and below the board, while the "sides" of the toraspherinder are out in the 4th dimension not intersecting with the board

That is weird. I didn't know that about it. I thought it was just another innertube, made out of a sphere-prism. Things like a cylconinder are very easy for me, but shapes with holes have some strange properties. I suppose it's just a matter of deriving basic features of a 3D circle-torus, then extrapolating analogously into higher dimensions. Just like with what I do with the classic rotopes. Just haven't examined in greater detail, the intricate nature of torii.

Ditorus I'm not sure how to describe in numbers yet and actually this is the hardest to explain (even though it is easier to draw than the tiger!). The easiest way to visualise a ditorus is as a normal 3D torus, but then the surface of the 3D torus is "puffed up" to make a 3D torus cross section as well. This torus also has two holes, the "major" and the "minor" ones. If you look at it from this 3D perspective it's impossible to get to the "minor" hole, but you can go through the "major" one just as you would go through a 3D torus.

Wouldn't the holes be the middle and major? The ditorus comes pretty easily to me. It's a torus embedded into the 1-manifold of circle, and a circle embedded into the 2-manifold of a torus, |O(O) x (O) and |O x (O)(O). The thick section of a ditorus actually feels kind of familiar, and looks like a connection between the surface of two 3D torii. If a torus prism has two torus ends, connected by an infinite number of lines, then lathing a torus prism into 5D, would put circles in place of those lines. This looks awfully familiar to a ditorus, but they have different dimensions.

---- If the ditorus is 4D, could it also be something like, bear with me on the word play here, a spherated torohedric prism? A spherated prism of a hollow torus?

However, if you go half way into the "major" hole so you're now sitting dead center - at the origin, if you use the parametric equations as such - you can "hop" into the minor hole by going a short distance into the fourth dimension. You're then stuck "inside" the 3D projection envelope until you go into the fourth dimension again to "hop" back out of it.

I can see how that would work. Never thought about that either! But, you're right, one could start in the big hole, walk along the curved surface of the smaller torus, around in 4D, and end up inside the small hole. That, also, is amazing! Thanks for that

-Philip
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### Re: The Tiger Explained

ICN5D wrote:
Toraspherinder has 2 laterals, 2 combined frontals. The "combined" means that rather than being able to fit a linear object, like an arbitrarily long pole through the torus you could fit an arbitrarily large planar object, like a large board of wood through - if you remove one of the dimensions and visualize it in 3D, you have the "top" and "bottom" of the toraspherinder above and below the board, while the "sides" of the toraspherinder are out in the 4th dimension not intersecting with the board

That is weird. I didn't know that about it. I thought it was just another innertube, made out of a sphere-prism. Things like a cylconinder are very easy for me, but shapes with holes have some strange properties. I suppose it's just a matter of deriving basic features of a 3D circle-torus, then extrapolating analogously into higher dimensions. Just like with what I do with the classic rotopes. Just haven't examined in greater detail, the intricate nature of torii.

The toracubinder is the "inner tube" made of a sphere-prism. It confused me for a long time when I started studying toratopes too. One could say that the names "toracubinder" and "toraspherinder" are reversed, because they make more intuitive sense that way, however the names were chosen the "unintuitive" way round so that the notations match up:

(II)II cubinder <--> ((II)II) toracubinder
(III)I spherinder <--> ((III)I) toraspherinder

If we decided to call the 4D "inner tube" by the name "toraspherinder", then we'd end up getting (III)I spherinder <--> ((II)II) toraspherinder and vice versa for cubinder, which would be very confusing.

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### Re: The Tiger Explained

The toracubinder is the "inner tube" made of a sphere-prism.

(II)II cubinder <--> ((II)II) toracubinder
(III)I spherinder <--> ((III)I) toraspherinder

If we decided to call the 4D "inner tube" by the name "toraspherinder", then we'd end up getting (III)I spherinder <--> ((II)II) toraspherinder and vice versa for cubinder, which would be very confusing.

Wouldn't it make more sense to call ((III)I) the innertube of sphere-prism? I know the name and notation has been in place for a long time, but there seems to be a break in the logical flow.

Comparing the sequence of circle, cylinder, torus as:

(II) - circle
(II)I - cylinder, circle-prism
((II)I) - torus, innertube of circle-prism
----------------------------------------------
(III) - sphere
(III)I - spherinder, sphere-prism
((III)I) - toraspherinder? , innertube of sphere-prism

1) This way we can interpret how (II) is a circle, and ((III)I) is a sphere embedded into the 1-manifold of a circle.

2) The (III) takes the place of the first " I " in a (II).

3) Much like how ((II)I) is a circle embedded into the 1-manifold of a circle, to make the torus. A (II) replaces the first " I " in (II), making ((II)I)

I don't know, it makes a little more sense to me that way. But, once again, the names and notation have been in place for a long time, and I'm just the new guy!

So, it must be the definition of spherate that does it. The cubinder has two flat panels, and one curved. Spherating will smooth out the square-like attribute, into a circular-like attribute. Perhaps to be interpreted in this case, as a spherical extrusion of the circle. In effect, the curved part of the cubinder has a planar smoothing into a sphere, if that makes any sense.

III - cube
(III) - sphere
(II)II - square-like cylinder, circle embedded into 2-plane of square
((II)II) - circle-like torus,sphere embedded into 1-plane of circle

Which kind of violates the logic behind:

II - square
(II) - circle
(II)I - line-like cylinder, circle embedded into 1-plane of line
((II)I) - circle-like torus, circle embedded into 1-plane of circle

BTW, I found out how to describe the tiger with my notation: |O((O)(O)). It reads " circle embedded into the manifold of the cartesian product of two hollow circles ". (O) is a hollow circle, or glomolatrix. The |O(O)(O) will make a ditorus, so by encasing the two (O)(O) with extra parentheses will separate it into its own product, before embedding the circle. Finally!

-Philip
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### Re: The Tiger Explained

No, you still don't understand. The notation defines the equations, which define the object. We can't just redefine the notation so that ((II)II) and ((III)I) swap for our convenience - ((II)II) will always define the equation "(√(x2 + y2) − R)2 + z2 + w2 = r2" because that is what happens if you apply the algorithm for notation to equation to ((II)II). And that equation will always mean an "inner tube" torus, one that can be constructed by taking a torus, extending it into the fourth dimension and bending the ends round in a hose linkage.

The only thing that could possibly be swapped are the names toracubinder and toraspherinder, but they were chosen to match the notation of the cubinder and spherinder. And after nearly a decade of using the names in this way, switching them now would only cause endless confusion.

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### Re: The Tiger Explained

Got it. It doesn't represent the visual, but formulaic definition. Okay.

(III) - sphere
((II)II) - sphere-torus, because the (II) is the manifold of the circle.
((III)I) - (circle,glomohedrix) prism, because the (III) is the 2-manifold of the sphere
((III)II) - (sphere, glomohedrix) prism, toracubspherinder
((II)III) - glome-torus
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### Re: The Tiger Explained

So, I guess in a quick summation, the Tiger is a four dimensional torus that has two independent passageways going through the middle. Both cross at a right angle to each other. Going through the middle, there would be a wall around you, where you couldn't get into the other hole. You would have to go all the way through to the outside. Then you could turn right or left, and go through the other hole. It's the craziest thing ever.
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### Re: The Tiger Explained

Tiger has only one hole in it, but that hole is torus-shaped. You can do as Philip suggests, go at right angles, and meet at the same point, but you can do that anywhere on both a torus and inside the tiger. The tiger has eaten a doughnut, ant that is not going to disappear.

It can rotate in clifford style around the centre, without any wobble. That is, the tiger is comprised entirely of circles with a common centre.

I've been fixing on trying to work out its area, and i have a summation that gives it, but not the wit to make it happen. Basically, for a circle in the xy plane, you sum(xy) over a circle of radius r centred on the points a, b. I was hoping something would pop into my head, but it hasnt happened to this moment.

The measurement of the dimensionality of a hole or cavity, is the notion of a "non-vanishing sphere". If you put a surface of a shape inside another, and then allow it to continiously deform but not cross the boundary of the cavity, it can either vanish by going to zero or infinity, or be stuck. A torus in 3d has an interior where a circle can be non-vasishing, and outside, one can put a circle (around the tyre), that also is non vanishing. All spheres vanish.

In 4D, the spheric torus, one can put a circle inside it, or a sphere around it, so that these do not vanish.

The maximum figure one can have in a tiger is a non-vanishing torus. That is, one can take a (clifford) torus, and it can not vanish by distortions to a point. It is the prism product of the two lines that Philip was talking about. I suppose outside, it's a bit more complex. A tiger will work, but i was kind of hoping for something simpler. I need to think this one through.
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### Re: The Tiger Explained

By saying the hole in the tiger is torus-shaped, are you referring only to the non-vanishing torus or something else?

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### Re: The Tiger Explained

I'm seeing the two holes as tunnels, closed off from each other, crossing at right angles in the very middle of the tiger. They cross around each other in the 4th extra direction of 4D. This allows two axles of infinite length, at right angles, to cross through the middle.

Standing on the outside of a 3D torus, one could go through the hole, or turn R or L and travel around the ring. For the Tiger, one could stand on the edge facing a hole, go straight into one tunnel, turn R or L and go through the other tunnel, or turn up or down, and travel along its complex ring structure.
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### Re: The Tiger Explained

Let's look at the tiger in terms of the duo-cylinder. The actual tiger appears around the interface between the two faces of the duocylinder.

The shape inside the tiger is then this margin between the faces of the duocylinder. This is the surface of the torus-faces of the duo-cylinder, which means that this is the surface which can not be made to disappear.

The two 'tunnels' at right angles hold full 2d planes, not lines. The middle section of a tiger is two toruses, one above the other, rather like the top and bottom rims of a cylinder. The full lathe-figure for producing a tiger from a plane figure, is a plane with four circles, placed at equal distances from the x axis, and equal distances from the y axis. But the x-axis becomes the w-x 2-pland (hedrix), and the y-axis becomes the y-z hedrix.

There are no tunnels inside the tiger.

The net of the tiger is a rectangular prism.
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### Re: The Tiger Explained

Oh! Suddenly that makes a lot more sense.

So if I rewrite what you just said in terms of the 'wooden board' example for the toraspherinder...

Instead of there being a sphere above the board and a sphere below the board, in the 3D cross section, those spheres are replaced with torii?

The non-vanishing torus you speak of, this torus is not embedded into 3D, but instead is a 'true' torus, congruent to the ridge of a duocylinder?

I just looked up Clifford torus on Wikipedia, I should have done that before, then it would have made sense already!

What is the cross section of a Clifford torus in 3D? A circle?

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### Re: The Tiger Explained

It should be noted that the tiger is a kind of 'fattened cilfford torus', in much the same way that the torus is a 'fattened circle'. This is what spherated means. You can get clifford torii from the margin or boundary between the faces of a duocylinder. A tiger is then a solid model of a clifford torus.

The middle section of a clifford circle is two circles parallel to the board in the xy plane, one at z=+a, the other at z=-a. These are circles of radius b. The tiger fattens these circles to torii.

Clifford torii have some use on the 4D planet. If you intersect a cylinder with a sphere, you get lines of lattitude north and south of the equator. For example, eg 27.5 degrees N and S. This is a kind of 'climate' level. If you do the same thing with a duo-cylinder, you get a 27.5 climate level on the 4d planet. That is it has the same kind of climate, but it's a kind of single space.

The space that contains say, 27 to 28 degrees on the climata, gives a tiger centered on 27.5 deg, and r = 0.5 deg.
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### Re: The Tiger Explained

Haha, so to say:
The tiger is the habitat of tigers on a 4D planet!
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### Re: The Tiger Explained

wendy wrote:It should be noted that the tiger is a kind of 'fattened cilfford torus', in much the same way that the torus is a 'fattened circle'. This is what spherated means. You can get clifford torii from the margin or boundary between the faces of a duocylinder. A tiger is then a solid model of a clifford torus.

Now I get it!

The Clifford torus, seen as a 3D cross section, starts as a circle in xy plane, then the circle moves in the z axis as you move along the w axis. Two circles appear in each section with one at +z and one at -z. Seeing the tiger as a fattened Clifford torus perfectly explains the sequence of cross sections of the tiger, and I can see how you can fit a plane between the two parallel torii, which then ends up going through the middle of the tiger when considered from an overall 4D perspective instead of a cross section. You can also insert the plane either in the xy orientation, or in the zw orientation, so there are not two holes, but rather two ways to put something in the same hole.

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### Re: The Tiger Explained

So the Tiger can accept two boards of infinite length and width? That makes sense, according to one cross section. The two parallel cross-section torii, when separated, will allow a board like this through. But, what about through the holes of the torii? It looks like only an axle on infinite length can get through both. As in, one infinite board and one infinite axle.
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### Re: The Tiger Explained

The stick that philip is seeing in the cross-section, is actually a cross-section of the second board. The thing is rotated in the extra dimension and through top-to-bottom, so the stick becomes a 2d board.
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### Re: The Tiger Explained

The full lathe-figure for producing a tiger from a plane figure, is a plane with four circles, placed at equal distances from the x axis, and equal distances from the y axis. But the x-axis becomes the w-x 2-pland (hedrix), and the y-axis becomes the y-z hedrix.

This is starting to look like two 3D torii, bisecting at right angles through their major radius. The wireframe of the two major radii, before embedding the minor radius circle, are two hollow circles bisecting at right angles. This is the cartesian product of the two hollow circles. It seems like this is related to crind, by the way it could roll along its circular edges. That is, its two innertube circumferences bisect at right angles, crossing at two points on the minor radius. This would allow one hole, but two ways to go through it.
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### Re: The Tiger Explained

The toruses do not intersect. Remember, there are four dimensions, so each of the two circles occupy two of the dimensions each. The circles don't intersect in the manner of a crind. It's like a cylinder. One circle runs around the wall, and the other circle joins the top and bottom through a fourth dimension. The tiger, is here a hollowed out margin of a duocylinder, but the shape that covers the tiger will cover the duocylinder.

The nature of the hole in the tiger is considerably more complex than anything that you see in three dimensions, even allowing for 1D holes (pairs of spheres held at a common distance).

The simple linkage hole consists of a spherated sphere, this is the surface of an m sphere fattened up. For example, the torus is a 2d circle, fattened up to 3D or 4D. A hook is a part of a sphere: in 3D, apart from the hook (part of a circle), you have a spoon (part of a sphere). Hooks are good for picking up rings, and a spoon can pick up a sphere of a sphere-pair (the other would dangle underneath). You still have these kinds of things in 4D, the dimensions of the intersecting spheres add to 5 here.

The tiger can be picked up by two different devices. You can use a line-hook or a hedrid hook. These are hooks of one or two dimensions. The spoon in 4D would be three dimensions, since it still has to hold a peasecorn or liquid. A plant tenderil, growing in the shape of a hedrid or flat 2space, would be able to wrap around a line or hedrix by curling up into a tiger. A tendril in the xy axis would curl up around the wx and yz axis to give a tiger, in much the same way that you see tendrils twist around lines in3d.
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