The Tiger Explained

Discussion of shapes with curves and holes in various dimensions.

Re: The Tiger Explained

Postby ICN5D » Mon Dec 02, 2013 6:23 am

If it has just one hole, then it's the weirdest one ever. I can see how two individuals would intersect around each other, but not one that does two things. Did you say the hole is torus shaped? Are the two "tunnels" different? As in different shapes? I'm imagining taking the minor radius and elongating it into an oval. This effect on a 3D torus would have the appearance of an engine nacelle with the turbine blades removed. This is what I mean by tunnel. If the same was done to a tiger, then what do the tunnels look like?
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Re: The Tiger Explained

Postby wendy » Mon Dec 02, 2013 8:02 am

Have a look at the duocylinder. The tiger is essentially a fattened version of the margin between the faces.

If you now think of the duocylinder as just the surface, with the round torus-shaped faces as glass and the tiger as the thing the glass lives in.

Remove one of the glass panels, and you have a hole. It's complex enough that you can pick it up with a stick or with a 2d stick. The two holes are evidently identical, so it does not matter which one of the two glasses you remove. If you remove both, you can pick it up with a stick, but not a hedrix (2d stick), if you go in one window and out the other.
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Re: The Tiger Explained

Postby Keiji » Mon Dec 02, 2013 8:40 am

You can think of it like a cross shaped hole. Say you have a piece of card (or whatever) and cut out a + shape, then you can fit other cards through the hole in 2 ways - horizontally or vertically. Now, it's like that, except that the "horizontal" opening is the entire xy plane, and the "vertical" opening is the entire zw plane. The center point is where the 2 inserted planes would hit each other if you tried to put them both in at once.

In the xyz cross section of the tiger, if you put something in the xy plane, then it goes between the 2 parallel torii. If you put something in the zw plane, it appears as a line going through the holes in the torii. That's because you removed the w axis for the cross section. If you look at each of the cross sections as you move along the w axis, you'll find the torii moving closer together or further apart, or the minor radius of the torii shrinking or growing, but it will never occupy the z axis in any such cross section, meaning that all the points in the zw axis are unoccupied by the tiger. And that's where your zw plane goes.

Now, for another question... Alternating toracubinders and toraspherinders can form a chain in 4D, just like a chain of torii can exist in 3D. But can a chain of tigers exist in 4D? I don't think so, as you could imagine in the cross section, the torii of two tigers linked together, but then you can just move one of the tigers through w and its torus will disappear without intersecting the other tiger. But perhaps there is something else along the w axis that prevents it being moved that way?
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Re: The Tiger Explained

Postby wendy » Mon Dec 02, 2013 9:57 am

Tigers do chain up quite nicely. Because it can be lifted by both a one- and two- dimensional hook, it can function as a two-D and three-D hole, and therefore make a chain. It's quite good at making chain-mail too.

If you look down any of the axies, you get a pair of disjoint torii. It is easy to imagine that two tigers might intersect because you can connect two torii. Even though there is noting in the w-axis when y=z=0, there is something further out. Specifically, the cross-section of the two torii at the radius of the torus, forms another torus, and it is this that stops the second tiger escaping the 'grip' of the first.

Let's look at w=0. Tiger A appears as two torii, of radius R, at +/- Z, and parallel to the xy axis. A is lathed in the wz axis. We want to construct a second tiger B so that it does not intersect A, but is linked to it.

We first put B, so that one of the two torii is set around x=0, y=R, z=R. It is lathed in the w-axis too. The other half is at x=2R. It's fairly clear that in this section, it is linked.

Let's now replace say, z with w. tiger A does not change. It still has an XY section, exposed in fill, but the other axis is lathed from Z to ZX.

Tiger B turns the torus-section y-z into two toruses, the full section now appears in XW. We still have two linked torii (XW and YX are still linked), but the vacant height of B is now in the Y axis. We see they're still linked, so the attempt to separate them by moving torus B in the W axis will intersect tiger A.

There is no intersection, so tiger A and B really are linked together.
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Re: The Tiger Explained

Postby ICN5D » Mon Dec 02, 2013 9:39 pm

The + shaped hole is also what I am feeling. And, you're right, it would be two planes, not lines. The holes along Z, while being stretched along W will create the second plane. So, the tiger can accept two boards of infinite LxW, and two axles of infinite length. What is the other cross section of the tiger? Other than the two parallel torii?

If the torus has a circular-shaped hole, the toraspherinder has a spherical-shaped hole, then it looks like the tiger has two independent circle-shaped holes, crossing at right angles in the same place, as projected into 3D.
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Re: The Tiger Explained

Postby ICN5D » Tue Dec 03, 2013 3:48 am

I just had an interesting idea....If we agree that the tiger is the inflated margin of a duocylinder, that is, a circle embedded into the manifold of the cartesian product of two hollow circles, circle bi-glomolatrix prism,

What I am wondering, is if a shape exists that is an inflated margin of a cyltrianglinder. A spherated cyltrianglinder. This would be a circle embedded into the manifold, of the cartesian product, of a hollow circle and a hollow triangle. The circle-(glomolatrix,triangulatrix)-prism, circle-glomotriangulatric prism, circle-cyltriangulatric prism, just to play with the words.

This shape would have tiger-like cross sections, in a triangular pattern, instead of a circular one. It's cross sections would be:

Ring
Inflates to Torus
Splits into two Torii, parallel, one above another
Moves away
Torinder appears suddenly

Duocylinder == |O|O , cartesian product of two solid disks
Tiger == |O((O)(O)) , circle X cartesian product of two hollow circles, spherated duocylinder

Cyltrianglinder == |>|O , cartesian product of solid triangle and disk
Spherated Cyltriang == |O((>)(O)) , circle X cartesian product of hollow triangle and circle

(O) - hollow circle
(>) - hollow triangle
|O - solid disk
|> - solid triangle
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Re: The Tiger Explained

Postby Keiji » Tue Dec 03, 2013 6:56 am

wendy wrote:Tigers do chain up quite nicely. Because it can be lifted by both a one- and two- dimensional hook, it can function as a two-D and three-D hole, and therefore make a chain. It's quite good at making chain-mail too.

If you look down any of the axies, you get a pair of disjoint torii. It is easy to imagine that two tigers might intersect because you can connect two torii. Even though there is noting in the w-axis when y=z=0, there is something further out. Specifically, the cross-section of the two torii at the radius of the torus, forms another torus, and it is this that stops the second tiger escaping the 'grip' of the first.


You're right. If you pull away the second tiger along the w axis, the disjoint torii will move together. Eventually they will squash the second tiger so you can pull it no further.

What about a ditorus? Let's see...

With one in its "two concentric torii" cross-section, and the other in its "two disjoint torii" cross section, you could hook the second between the two concentric torii of the first. If you attempted to pull the second along w, the concentric torii would close up and lock it. If you attempted to pull the first, the second ditorus would spin around, but the torus it was hooked to would never disappear. So they would be chained together. Is that correct?
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Re: The Tiger Explained

Postby Keiji » Tue Dec 03, 2013 7:22 am

I've also noticed that some of what I've previously said about the toracubinder and toraspherinder is actually wrong.

The toracubinder is still your standard "inner tube" torus. One of its cross sections is a 3D torus; if you rotate it into 4D, then it morphs into two disjoint spheres. I previously said that the toraspherinder had a two-disjoint-sphere cross-section, but actually this is the one that you can fit a plane through.

The toraspherinder also has a 3D torus as a cross section. However if you rotate this one, it morphs into two concentric spheres. You can only fit a pole through this one.

So in essence, if you are traversing the hole in the torus, then the toraspherinder is the more direct analogue to the 3D torus. However, if you are looking at it from the outside, then the toracubinder is the more direct analogue. It all depends on perspective!
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Re: The Tiger Explained

Postby wendy » Tue Dec 03, 2013 8:17 am

You can get a fairly good model of a tiger from looking at just the squares of a duoprism. These essentially form a wire-frame for the tiger to grow on, and the polygon faces that become removed give some kind of notion of the hole.

But the projection of the tiger as two parallel torii, is also its cross-section, the elevation of the horizontal hole looks like the vertical one, and vice versa.

One should be wary about the question of the 'shape of the hole', when the parent figure is a hollowed-out convex figure (as the tiger derives from the duo-cylinder). Think of the faces of such a polytope as 'windows', which one can remove. For your ordinary figure, like a tesseract, removing a cube-face will make the thing with an indent (but not a hole, since by hole, something might be threaded through it). It is like taking the lid off a box. You need to remove two of these.

The tiger derives from the duocylinder. It has two faces, which makes two windows. Removing one of these windows makes a hole, which supports two types of hook (latral and hedral). You can remove both as well. There's only one cavity, but you need two patches to close the cavity in, one by itself will still leave a hole.
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Re: The Tiger Explained

Postby Secret » Tue Dec 03, 2013 1:31 pm

Spent some time on folding ditorus and tiger

Found that the tiger is sort of like a ditorus, it is also look something like a "duo-torus" (not to be confused with ditorus) in a way a cylinder is to a duocylinder,except (if all my foldings and spherations are correct) none of the 2-torii involved is a clifford torus. It does have a bi-circular symmetry as evidenced by the fact that you can inscribe a duocylinder into the hole of a tiger thus blocking it up save for the two torii portions that extend out from the cross shaped hole

I also finally work out how to fold the final step of the ditorus (which corresponds to the "rolling up a sock" type of folding) (also if you fold it by making a cublinder first, you end up with a different type of ditorus where one of the cross section is a clifford torus)

I also work out the full appearance of the minor hole of the ditorus (which in the projection of the complete object looks as if it is only a torus shaped cavity inside a torus) and found that the minor hole is indeed more complicated than the major hole

In the attached image, I messed up with the drawing of the hole of the tiger, so ignore that for now (i'll fix that along with illustration of tiger and ditorii chains in a later post). What I found is that, the hole has some characteristic of a duocylinder (the + cross like bit is actually the centre of this hole). Roughly speaking, the hole looks like as if you replace every circle you see in the duocylinder projection with the wedge like shape that is the shape of the major hole of the ditorus (i.e. the 3D hole of the ordinary 2-torus)
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Re: The Tiger Explained

Postby ICN5D » Tue Dec 03, 2013 5:03 pm

The hole inside the tiger is pretty much what I am seeing as well. Two independent pathways intersecting in the middle. It is curious how the projection of the duocylinder comes out. More mysterious phenomena with extra directions of space.
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Re: The Tiger Explained

Postby wendy » Wed Dec 04, 2013 7:20 am

And the tiger is the simplest of these swirl-torii. There are literally thousands of them, with any number of holes.

The octagonny gives rise to a simple one with six holes. The twelftychoron gives rise to one that has twelve holes. If you count the left annd right turns and all their tracks, the octagonny gives rise to eighteen, all crossing at right angles, the twelftychoron gives rise to seventytwo.
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Re: The Tiger Explained

Postby ICN5D » Wed Dec 04, 2013 5:11 pm

So, is there such a thing as a spherated cyltrianglinder? Or is it as trivial as a triangle-shaped torus?
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Re: The Tiger Explained

Postby Secret » Thu Dec 05, 2013 7:39 am

Hole of tiger

Btw what are swirl-torii? Are they related to the CRF polytopes swirl prisms?
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Re: The Tiger Explained

Postby wendy » Thu Dec 05, 2013 7:47 am

@Philip: Spheration is a surface finish roughly equating to turning a stick-man into something with flesh and blood. So we flesh out the clifford torus to make a tiger. If your cyltriangle thing already has a volume, it is little point to spherate it. None the same you can do things to polytopes (like poke out the faces and particular margins), and get something that keenly would do with a bit of flesh, which 'spherate' would do quite nicely.

@Secret: You're actually looking at the wrong hole when you claim it is not a spherated clifford torus. It's the body of the tiger itself that is wrapped around a clifford torus. It is inside the tiger's tummy, not outside of it.
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Re: The Tiger Explained

Postby Secret » Thu Dec 05, 2013 1:16 pm

@Wendy (+ others)

It seems I might have made a lot of misconceptions, so I think I should check to make sure I understand the operations and concepts, before going back to trying to visualise the hole of the tiger (or more generally, its interior and exterior)

To elaborate the questions in the pics:
Q1. A clifford torus is locally flat everywhere (just like a cylinder hose, which imagine you fold them using a paper grid, none of the squares were stretched). Are the orientation in space of all the circular cross sections the same throughout as shown?

Q2. In contrast, an ordinary torus (which I will refer hereafter as "donut" so as not to mix up with other torii) is intrinsically curved especially near the hole (because the paper grid is stretched along the lateral direction). Does that mean the orientation of the circular cross section are only the same for opposite pairs ony as shown?

Q3. About 2 pages back, it is mentioned that spheration puff up a point into spheres, and on this page it is mentioend that spheration make a skeleton puffy so that it is fleshy. Exactly when we puff it up into 2-spheres (glomohedrix) and when we puff it up into 1-spheres (circle/glomolatrix). What are the orientation of the spheres in the object when it is being spherated. It is also mentioned that spheration is meaningless if your object is already bulky (i.e. has a n-volume e.g. cylintriangulinder), yet back in 1-3 pages when keiji mentioned about whether disk # circle = circle # disk, it is said they are equivalent as the former is like placing matches all over the disk and then bend this pile of matches in a circle which also give you a bulky donut (forgot formal term to describe objects with its interior filled/has a volume), thus making an example where spheration of a bulky object is meaningful. So my questions are
a. Given an n dimensional object A, do we always expand every point of A into circles (Example A to D)?
b. How do overlapping circles join together, if any?
c. What determines how the circles (or spheres) are oriented as A is being spherated (Example E vs F)?
d. How to spherate a given object mathematically. e.g. for a parabloid x^2+y^2=4. I know that a circle is x^2+y^2=r^2 and a sphere is x^2+y^2+z^2=r^2, but how to spherate the parabloid?

Q4. Another thing that confuses me is the comb product. I recall in another thread a few months ago we discussed that comb (n-gon,m-gon) means "place n and m gons so that they share one vertice, chop open the gons and unravel them so that they form two perpendicular lines that intersect each other. We then apply the cartesian product (where cartesian product means given set A and B, AxB is a set that contains a permultations of one element from A and B each) so that they span a hedrix grid, we then roll up the grid into a donut". So is comb(n-hedron,m-gon) will analogously form a grid of cubes which can then be folded into a ditorus.

In terms of maths e.g. given equations A ad B, how to compute comb(A,B)?

(To aid in the discussion, I have quoted the relevant part of the polygloss)
polygloss wrote:comb *
A product derivable from the regular tiling of measure polytopes.
In Euclidean space, this is the Cartesian product applied to tilings, but it also applies in spaces where the cartesian product does not exist, such as hyperbolic space.
The word is a backform from honeycomb.
The surtope polynomial is the product of the polynomials of the bases, ignoring both the bulk and nulloid terms.
The comb product of polytopes is the cartesian products of their surfaces. For example, the comb product of 2 pentagons gives a connected sheet in 4d of 25 squares. This is also hight hotel.
comb product *
A surtope product of two figures, excluding both the nulloid and bulk. It reduces dimension by every application.
The comb-product of tilings gives a tiling, but because it is really only meaningful to take cartesian products in horic space, the comb product is noted there. The comb product of a horogon gives rise to the infinite family of cubics in every dimension.


I tend to rely a lot on oblique projections with a projection angle chosen such that the extent of overlap is the least (I tend to call that the bipyramid projection angle since the object often resemble a bypyramid in this particular angle). If my understanding of the basic concepts are wrong, it means my diagrams are also wrong. I must make sure I got the diagrams right before I can visualise the various features in a higher dimension object
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Re: The Tiger Explained

Postby Klitzing » Thu Dec 05, 2013 2:30 pm

The comb product is not too complicated.

Consider 2 polytopal objects A and B. Polytopal here includes all geometries: sherical, euclidean, and hyperbolic.
Each polytope has a set of to be differentiated subsets of vertices (e.g. by symmetry or by different surroundings, etc.): 0(A)k, i.e. the 0-dimensional elements of A of type k. The same for edges: 1(A)k (generally a different number of indices k), for faces: 2(A)k, etc. But there are single objects called nulloid (empty set: -1(A)) and the (full-dimensional) body: d(A), which belong to a polytope as well. Those 2 boardering cases won't contribute in the product process of consideration.

Thus the elements of comb(A,B) are:
0(A)k x 0(B)m (for all k,m)
1(A)k x 0(B)m, 0(A)k x 1(B)m
2(A)k x 0(B)m, 1(A)k x 1(B)m, 0(A)k x 2(B)m
etc. giving one additional product type per dimension. But as soon as you come to n(A)k with n>=d (same for B and its dimension D), then those either don't exist (n>d) or are to be ignored (n=d). - This is why the number of products finally decreases again, getting finally down to
(d-1)(A)k x (D-1)(B)m
and then, being a polytope again, you've to add furthermore:
-1(comb(A,B)) (its nulloid) resp.
((d-1)(D-1)+1)(comb(A,B)) (its full-dimensional body).

If both A and B are spherical, comb(A,B) comes out to be equivalent to the surface of the A,B-torus.

Most natural the comb product applies when both A and B are both euclidean. Then d-1 resp. D-1 would be the tiling dimension each. (In fact: d resp. D then would be the dimensions of either half-space below the tilings each, their "bodies".) And by application you'd get a further tiling with tiling dimension (d-1)(D-1). (For sure: comb(A,B) then adds the half-sapce below again, in order to become a valid polytope too.)

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The Comb Product

Postby wendy » Fri Dec 06, 2013 8:08 am

There are five regular solids in every dimension, and these are all power-topes, of some base form to some Nth power.

The fifth of these five products is the comb product, which applied to the horogon, gives rise to the various n-cubics. These are tilings in euclidean space, but exist as polytopes in hyperbolic space. This product is the 'repetition of surface', which means that only the surface is used in the product, and the surface of the product is the (prism) product of the surfaces. So if we multiply polygons (bounded by 1d segments), we get a figure bounded by 2d segments - a polyhedron.

One can think of the number-line as a polygon with lots of sides. The interior would be a half-plane. Now if you take the cartesian product of the surface, you get a square grid, and give it an interior of a half-space, you get a very large polyhedron. And so on.

One of the features of tilings, is that one might find periodic areas, and mark off a single instance. For example, the number-line wraps up into clock-face arithmetic, and likewise, many games use a double-cyclic board so there are no artificial edges. (Some use seas and deserts to create an edge, like KQ3 and KQ4).

Mathematicians call the two-dimensional double-cylinder thing a 'torus'. One can find a similar topology on the surface of a doughnut, but it's the topology of the surface that matters.

In four dimensions, you can do a cartesian product of two circles without distortion. Such are very important things, and 'torii of lattitude' are the 4d version of 'lines of lattitude'. A planet under uniform rotation, and at an oblique angle to the sun, will produce climate-related lattitude by the same way lines of lattitude do in 3d.

You pretty much can fold any N-1 dimensional surface into an N-dimensional polytope, with an assortment of holes. This means that the comb product can be applied to polytopes as well as tilings, but the nature of the folding means that it does not have a neat mathematical expression. In practice, it is only what Keiji calls the 'brick' products that do have such a thing. So pyramids don't either.

Another way the comb product differs from the other products is that it is not 'associative' (ie ab != ba). You can see this in three dimensions, when you multiply a pentagon by a heptagon. You get something that has 35 square faces, but the product can yield a wheel with five or seven segments around the perimeter.

In four dimensions, the product of say, the dodecahedron by the decagon, does yield 100 faces, but there are two versions of this too, and they're topologically different. You can either have a hollow dodecahedron where the surface is puffed up by decagons (the pentagon-faces come to be pentagon-decagon prisms), or you can have a decagon, where the edges are puffed up to be dodecahedral in section. The surface is identical, but you can chain the two bits together!

The "tiger" is a comb of three polygons, but done in a way that is not by sock-and-hose wrapping. You can still make it by wrapping xy into a cylinder in the wx, a cylinder yz, and then wrap the third axis into a little ball to cover what has already been made.

Drawing circles in prospective is very hard anyway, but you can use what i call 'surtope paint'. This is another kind of surface transform. You spray it onto a smooth surface, and it 'cracks' into polytopes. For example a light application to a sphere might give a cube. More agressive painting might give dodecahedra or truncated icosahedra.

Applying the surtope paint to the tiger will give in various stages, something like a tesseract or something like a bi-dodecagon prism. This gives you a wire-frame to freely rotate a clifford frame around. It should be noted that when they build road tunnels over here, the make the thing as faces of a curved prism, which are transported to site.

Let's look at a the tiger. You can make (dec) 144 of these shapes, which will replace the 144 squares of the bi-dodecahedron prism. So it's in one way a square. The thing is actually a square-circle prism, but the square at one end of the section is tapered inwards and at the other end, expanded. It doesn't matter what order you place them, eventually.
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Re: The Tiger Explained

Postby wendy » Fri Dec 06, 2013 8:56 am

Spheration, comb, and crind products are not the same thing, even though they are connected to round objects.

Spheration is a paint-job that makes thin things thick. It does this by expanding the thing perpendicular to the space the thing is in.

Comb product is kind of surtope product, formed by "repeating the surface". This means that the surface of the product is the product of the surfaces.

Crind is a brick product, like prism and tegum, where the dimension of the product is the sum of its elements. In terms of the radiant formula, it is 'rss'. It does not form comb products.

A1. When you make a clifford-torus out of squares, it's much the same as making a circle out of lines. The lines are straight, and the sort of follow the circle. Likewise, the squares are straight and undistorted, and follow a clifford torus, but the vertices are only on the a torus. None the same, as polygons are useful for drawing circles, polygon-polygon prisms make useful ways of making clifford-torii.

A2. The torus is curved differently according to direction you go, but these are identical to whatever place you start. If we consider the bi- dodecagon prism, you get a single circle of radius R by following the edges of the square, and R sqrt(2) if you follow the diagonals of the square.

A3. Spheration does not actually intersect. It's a process that is done 'around' a thin thing. For example, if you use a very fine pen to write large letters on a peice of cardboard, they would not be seen from afar. But if you puff them up, by using a broad brush, the thin lines are replaced by the brush-strokes, and the characters can be seen from afar. This does not change the nature of what is written, but makes it solid for people to see.

Likewise, a circle in 3d is not something we can have as a loose object. But if you turn the thin line into a pipe, the circle becomes a torus, and you can do things with it: it is a solid. The section is solid. look at the points and lines of the http:\\www.atonium.be or simliar stick-and-ball models to see the inspiration for spheration.

A3b The idea behind spheration is to make thin things thick, not to make new shapes. So a disk (filled circle) is thin in 3d. but we can make it thick by spheration (here we make each point of the disk into a line perpendicular to the disk), and the disk becomes a coin. Spheration will turn a circle into a tyre-tube, but there is a cross-section to the original circle, that is perpendicular to the original circle, and consists of a disk that is centred on where the circle crosses the cross-section.

A3c You can't make a tiger from spheration of a torus in any dimension. Spheration is a paint-job like painting things red or whatever. You can't paint a new room onto the house, you need a builder to do that. Likewise, we need building operators to make a torus, not a paint job. This is why it's a bi-latric prism (the product of two circle-surfaces). You then 'spherate' this by puffing the 2d sections of it into four dimensions, the new dimensions are all perpendicular to the original 2d section. So if your clifford torus is divided into 2d sections, the bits you need to make it solid are 4d bits that have match the 2d sections, and have a circular section. It's kind of like making a circle or torus out of cut-out cylinders.

A4. The actual description of a tiger in terms of the comb product, is a 'comb(polygon, polygon, polygon)'. Circles are a kind of round polygon. Since comb removes one dimension from each of the elements, (to get the surface), and then adds one back at the end, you see that the polygons give rise to 1D, and 1D+1D+1D = 3D, and the figure that has a 3D surface is a 4D thing.

A4 When you multiply a polygon-surface by a polyhedron-surface, you are taking in effect the cartesian products of the nets (the unfolded surface that people cut out to fold up to make the models). If you multiply a dodecahedron by a dodecagon, you replace the 12 pentagons of the dodecahedron, with columns of 10 pentagonal prism. You can fold the dodecahedron together, and you get a tube in 4D, that looks like the sides (but not the ends), of a long dodecagonal prism. You can turn this into a comb product by either the hose or sock method. The shapes are different. So we write: comb(dodecahedron, decagon) vs comb(decagon, dodecahedron). We think of the elephant, who has a hose at the front. So the first one takes a decagon section, and joins it hose-fashion into a dodecahedron. You get something like a puffed up dodecahedron, with a decagonal cross-section. The second one is like a puffed up decagon, made with dodecahedral rods.

Central inversion, where the point is inside the figure, changes one to the other.

As to the mathematical description. I don't have any. There are too many variables in the way the thing folds down to lesser dimensions to do this. If you look through the early parts of this discussion, you would find that i was alarmed to map the tiger onto the ditorus. There is still much for me to learn here.
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Re: The Tiger Explained

Postby ICN5D » Fri Dec 06, 2013 10:04 pm

This could be just a fluke of interpretation, but by referring to the projection of the Tiger:

Image
Magically resized it for you! ~Keiji

It sort of resembles a cube in some way. I think by now, we have determined that it has two separate entry ways, giving two separate exits. According to the similarity to a cube's 6 sides, it should have two more, non-pathway panels to it, as in closed sides.

If this is not so, what are the boundary zones between the two torii? Or is the tiger a complex wireframe of inflated innertubes?

The duocylinder does not have any holes, but by the spheration process the tiger has two. I think I see what the sock-rolling method does.I have learned that spheration does not mean to join the ends of the cylinder, it just has the same effect. Correlation is not causation here. So, it must mean to snip the circles out of a cylinder, leaving a hollow tube, then roll one end over and back along the length of the tube, and join the empty ends together. This will make the torus. If this is the correct analogy, then spherating a duocylinder will apply the same sock rolling method two-fold. This will make for a single shape made of two torii, superimposed at right angles.

-Philip
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Re: The Tiger Explained

Postby Keiji » Fri Dec 06, 2013 10:50 pm

I regret posting the inaccurate information previously that stated the tiger has two holes. It doesn't. It has one hole, through which a plane can be inserted in two perpendicular orientations, e.g. xy and zw.

This diagram should help you understand how the tiger works. In each, I'm inserting a long, thin cubinder into the tiger. In the top row, I'm inserting it oriented in the xy plane. In the bottom row, I'm inserting it oriented in the zw plane. The tiger is in the same position in all four projections. The red-blue gradiented lines do not appear in the cross-section, but are parts of the tiger which are located in the fourth dimension.

Image
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Re: The Tiger Explained

Postby ICN5D » Fri Dec 06, 2013 11:13 pm

Thank Keiji! Those pics help a lot. I can see how 2 planes fit in it, now. That Tiger is the weirdest shape ever. It does some amazing things. I get excited and want to tell my FB crowd, but it's way beyond most of them. Probably because it's not a pic of a cute kitten. Though, there is a striking similarity in the naming.....


BTW, those diagrams are majorly wiki-worthy. I want to help you complete it with the rest of the surface elements and cross sections, if you want.

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Re: The Tiger Explained

Postby wendy » Sat Dec 07, 2013 9:41 am

By the way of holes, a "tiger" might be two "clifford holes" let through the a hollow glome's surface. It really does have two holes, and the holes are reproducable in any number. However, the way one counts holes mathematically is consistant, but gives an answer that is less obvious.

Let's look at holes, and how the tiger is different.

Suppose you start with a model of a cube (in 3d), in which the vertices and edges might be puffed up. How many holes would you think there were in this? Six? You might be supprised to find it has only five holes, although you really can't pick them out. If you fold this down into a Schlegel diagram, you get a pretzel with five holes in it. You can think of the removal of one face as removing the interior too (ie removing the lid off the box), and the remaining five walls make true holes.

The way you can detect a hole is to pass some kind of sphere-surface that can not disappear. This means that you can change the size without crossing the surface, and some of it will always be seen. The actual nature of the hole is then by putting up a screen to prevent any loops going through the part. The number and type of screens needed to stop any hollow sphere linking is then the "number of holes".

Since a surface divides the space into two, and holes are represented by hollow spheres and their intersecting screens in one space, we can measure the number of holes inside as well as outside. In our model cube-frame, there are loops that can form along any path, that might not disappear. How many breaks do we need to make so that you still have one peice, but no loops: It's five! In three dimensions, the number of holes outside is the same as the number of holes inside. The number of holes is called the 'genus' of the figure.

Four dimensions is different, even for simple holes.

The complement of a hole needing a 2d screen, is one needing 3d. For example, in the spherinder bent to a loop, you have a 2d screen outside, and a 3d screen inside (the join), which when inserted, will stop non-vanishing hollow-spheres.

The 'tiger' has a different kind of hole, which the simple-hole notion can not describe. This hole is probably best described as a puffed-up circle, that is concentric with the glome, and cuts into the surface of the hollow glome. These are (for reasons that shall become apparent soon), "clifford holes". The tiger has two of them.

If you take a duo-cylinder, which we agree is the convex prototype of the tiger, you can turn it into something different by doing this. ( 1 ), replace the two faces by glass windows. (2), replace the margin (clifford torus), by a tiger. (3) drain the interior. You now have something that is a tiger + windows, topologically equal to a hollow glome.

If you just remove one of the windows, you will see that you can pass a 2d plane through it, and a 1d stick, and both of these can be used to 'lift' the tiger. In short, the hole we have created has two different kinds of closure. You can prevent the 2d planes by inserting a 2d screen across the torus. This means, "perpendicular to the height of the prism". In essence, this turns the doughnut-shaped face of the duocylinder back into a bent cylinder, and the 2d plane equates to a non-vanishing 2d circle inside this space.

To stop one poking sticks through it (which corresponds to "points of entry"), you have to fill the thing right up. (i think).

Each hole acts separately, so it has two holes.

Clifford Parallels

We call these holes "clifford holes", because the hole follows a 'straight line' in spheric space (ie a great circle on a glome), and puffing them up, makes them into the same shape as the holes in the tiger (ie what you get if you bend a long cylinder into a face of a duocylinder).

Straight lines in spheric space (glomochorix = E3), can run equidistant to each other, but are not coplanar. You can for a given great circle, draw an equidistant great circle through any given point.

For those who understand maths, here are some equations to ponder over.

The equation for a straight line is "y = ax + b". When we want it to pass through the origin, we put "y = ax", or "a = y/x". The actual space representing all kinds of 'a' is a semicircle, from y=1, to x=1, to y=-1.

The same holds true for complex numbers, where x, y, a, and b are all complex. We see that x => x+iX, and y => y+iY, maps without distortion onto a four-point x, X, y, Y. So the complex line becomes a 2-space (argand diagram), and the complex plane (x,y), becomes 4-space.

A line is still defined by three points: one at (0,0), one at (x,y), and one at (cx, cy), where 'c' is any complex number. If we write c = r cis(wt), where w = speed of angle, and t = time. We put then that over t fron 0 to 2pi, will make every point circle (0,0), especially since pi itself represents a simple change of sign. This is 'clifford rotation'.

The space of 'clifford rotations' (ie the half-circle), now comes to a sphere whose diameter is x=0, y=0, and x=0, y=1, and y real. This is a 3-sphere, and every point comes to be a non-intersecting clifford-circle.
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Re: The Tiger Explained

Postby Polyhedron Dude » Sun Dec 08, 2013 9:58 am

Here we might need to cue the "Eye of the Tiger" song:

♪ Dun Dundaa Dundun Daaa Dundun Tadaaaaaa - ♪ Cross Sections of the Tiger ♪ - Dundundaa Dundun Tadaaa....♪ :mrgreen:

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http://pages.suddenlink.net/hedrondude/tiger.png - full pic
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Re: The Tiger Explained

Postby wendy » Sun Dec 08, 2013 10:22 am

Polyhedrondude: That's quite good, but is there a chance for an obique view, for Secret and others?

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Re: The Tiger Explained

Postby Polyhedron Dude » Sun Dec 08, 2013 1:55 pm

wendy wrote:Polyhedrondude: That's quite good, but is there a chance for an obique view, for Secret and others?

Wendy


Something like this?

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http://pages.suddenlink.net/hedrondude/tiger2.png - full pic
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Re: The Tiger Explained

Postby Keiji » Sun Dec 08, 2013 6:30 pm

Those are some excellent cross section renderings! Can I add them to the wiki; are they released under a CC license at all?
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Re: The Tiger Explained

Postby ICN5D » Sun Dec 08, 2013 7:57 pm

The seventh cross section from the left in the oblique view is what I've been seeing. Nice to see the full transformation! I also saw that it had four openings and a closed end, kind of like a cube with four squares punched out. That's a powerful flow graph, for those to better see it. Finally after seven years and +25,000 views, the tiger is being exposed, in visual form and layman's terms.
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Re: The Tiger Explained

Postby Polyhedron Dude » Sun Dec 08, 2013 10:30 pm

Keiji wrote:Those are some excellent cross section renderings! Can I add them to the wiki; are they released under a CC license at all?


Go for it. Consider them CC licensed. I'll be sending the ditorus next :mrgreen: .

ICN5D wrote:The seventh cross section from the left in the oblique view is what I've been seeing. Nice to see the full transformation! I also saw that it had four openings and a closed end, kind of like a cube with four squares punched out. That's a powerful flow graph, for those to better see it. Finally after seven years and +25,000 views, the tiger is being exposed, in visual form and layman's terms.


Its an amazing looking object. I sometimes feel like I'm like the Voyager probe sent to take pictures of multidimensional shapes revealing them for the first time :mrgreen: .
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Re: The Tiger Explained

Postby wendy » Mon Dec 09, 2013 7:49 am

In some way you are playing 'voyager' for us. The holes in the "tiger" (torus swirl-prism), are profoundly different to anything we know of in 3d.

But the pictures are very good, and very enlightening. The first one we have discussed for some time, but the second one shows the tiger creating two lunes, which become holes in the middle section.

Very good!!!!
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