## The Tiger Explained

Discussion of shapes with curves and holes in various dimensions.
You can continuously deform a tetratorus into a tiger.

The tetratorus has successively parallel axies, viz

torus(torus(torus(w,x),y)z).

The tiger is nested pairs, ie torus(torus(w,x),torus(y,z))

One can visualise the circle produced by the final fold, projected into a realm (plane), as a line. Rotating the torus around is simply freely moving this line around. If it is perpendicular to the plane of the circle of the torus, then one gets a tiger. If it goes through the torus like a puncture, then it's a tetratorus.

W
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A continuous transformation isn't enough to show that they're the same object. There is a continuous map between duocylinder and torus, and between circle and square.

You need a linear transformation i.e. rotations, reflections and scaling. I'm pretty sure there would be a linear map from one tiger to any other tiger, and from a 3-torus to any other 3-torus (except degenerate cases), so the parameters don't matter. All we need is a single linear map from any tiger to any 3-torus.

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I am anyway more interested in the topological differences.
So (surface of) coffee cup and torus is the same for me *g*.
Though I hadnt time yet verifying Wendys assertion. Is it really true?
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Though I hadnt time yet verifying Wendys assertion. Is it really true?

I very much doubt it. There's too much difference between the equations. I can accept that they might be topologically equivalent, but I don't know how to prove it.

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wendy wrote:
torus(torus(torus(w,x),y)z).

what does that mean? are the toruses in those coodinates?
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wendy wrote:torus(torus(torus(w,x),y)z).

what does that mean? are the toruses in those coodinates?

It's a notation Wendy just made up. Torus() essentially means the torus product. I don't think we'll be using it much, it seems too cumbersome.

In general, torus(A, B) = sqrt( A<sup>2</sup> + B<sup>2</sup>) - r<sub>i</sub>

torus(torus(x,y),z) is the standard 3D torus (21), with the major circle in the xy plane.

torus(torus(torus(w,x),y),z) is the 3-torus, ((21)1)

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### Re: The Tiger Explained

In my understanding, the Duocylinder has the surface of two torii. Duiring the 4-D spin, a cylinder's flat circle ends are joined together into a continuous rolling surface of a circle torus. The line torus part of a cylinder undergoes a different kind of spin, where the line-part of the torus is isolated and spun, creating another circle torus. Having two circle torii with their surfaces connected entirely to each other makes for no flat surface panels, only two curved with a hole. The contact patch of a duocylinder resting on our 3-D plane takes the shape of a circle. It can roll along two planes simultaneously, much like the topology of a sphere. Cross sections from both angles are a circle expanding to a cylinder, collapsing back to a circle.

So, how does one spherate a dual-torus surface with no apparent flat and open ends? The spheration of a cylinder serves to join the open flat circle ends into a continuous closed loop. Spheration of prisms is very straightforward, a simple joining of the ends. However, if starting with a non-prism shape with no flat ends to join, how does the motion apply? Do both surface circle torii join to close another unanticipated loop?

Or is the tiger something more like a double-torus ( not ditorus ), taking the shape of a 4-D figure eight, with two closed loops and two holes? Using the two cylinders of various heights in the cross section, perhaps this is the key?

-Philip
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### Re: The Tiger Explained

You're confusing surface as both 2 right angles, and one equal sign, which they are in 3d.

An equal sign is like x=0, divides space of any dimension into two, gives a space of n-1 dimensions. Two equal signs (eg y=1) gives a space of n-2, and so forth. Right-angles gives spaces of m dimensions, if there are m lines at right angles to each other. equal signs + right angles add to the space in question.

A 'surface' should always be read as one equal sign, because it covers. You need a different word for the 2-space.

In the duocylinder, there are two faces (cylinder-circles) [ie 1 equal sign], which are separated by a something topologically equal to a torus-surface [two equal signs, which makes two right angles in 4D].

Spheration of this hedrix is pretty much the same as the spheration of a circle in 3d, but done in 4d on a bi-circular prism margin. So you have for example, the hedrix (2-space), given by w^2+x^2 = r^2 and y^2 + z^2 = r^2 (2 equal signs - intersection), and then you pick points that are some smaller s distant from one of these places.
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### Re: The Tiger Explained

@ wendy : Was that an answer to my question? Forgive my crudeness, but I didn't understand some of the things you mentioned about equal signs, right angles and the like. I do see the duocylinder as having two surface panels, and I see than you mentioned it.

It's possible that we use different words when referring to certain shapes. What exactly do you mean with the " cylinder-circle"? If the duocylinder has two rolling surfaces, how does one "spherate", or join the ends together, if there are no flat ends?

I saw that you call the cyltrianglinder a "trangle-cirlce", or something of the like. If this is the case, wouldn't the "cylinder-circle" be a prism of the duocylinder in 5-D? By embedding a circle into every point on and within a cylinder, this syntax is identical to the embedding of a circle into every point of a triangle. The duocylinder can also be understood as a circle embedded into every point of a circle. Can the tiger be defined in a similar fashion? Can something be embedded into every point of another to make the tiger?
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### Re: The Tiger Explained

Spheration is placing a sphere at each point of a subspace. This has the effect of turning it into a round figure over the shape. A torus in 3d is a spherated circle, the circle runs through the centre of the torus (inside the tube, so to speak). Something like the atomium ( http://www.atomium.be ) is an example of spheration, where the edges have been turned into cylinders, and the vertices into larger spheres. Zome-tools make spherations of polyhedra.

The tiger is a spheration of the margin space between the two faces of the bi-circular prism (duo-cylinder). You keep only the bit between the two faces, and turn this thing into a solid by replacing points with a sphere, in the same way that a circle becomes a torus.

The thing with equal signs and right angles.

The equation for a dividing plane in N dimensions, is an equation like 'x=0'. This has N-1 dimensions. In 1D, a point divides the line, in 2D, a line divides space, in 3D, a 2d plane divides it, in 4D, a 3d plane divides space. The more equal signs, the lesser the defined space. In 2d, you need two equal signs to set a point, in 3D, you need 3 equal signs.

A right angle here refers to lines co-perpendicular to each other. A 2d space is defined by two perpendicular lines, a 3D space is defined by 3 perpendicular lines.

What happens is that people think that because a 2d surface is a 1 equal-sign in 3D (and hence divides space), that a 2d space divides 4d too. This is not much helped by calling a 2d space a 'plane'. It's best to call it a 2-flat or hedrix, and leaving plane to mean 'one equal sign'. Most of the words like this, like 'face' (which is commonly used to refer to a 2-flat part of the surface), giving 'facet' as a 1-equal sign of the surface. In 3d, they are the same, but in 6d, there are three dimensions between the two: a facet is a 5-flat surface part.
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### temp toratope topic 2

Some time ago, I found a relationship between a square and a torus. After examining the sequence of the tiger using the spherate motion ((||)(||)), I noticed that it may be related to a cylinder the same way. So if ((||)|) is a torus, (||)| is a cylinder, then is

|| : ((||)|) == (||)| : ((||)(||)) ??

That is, the process that turns a square into torus as process X. Does process X also turn a cylinder into a tiger?

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### Re: Understanding the Cyltrianglinder

((II)(II)) is just a notation. The fact that II alone is a square and ((II)I) is a torus does not mean that a torus is constructed from a square by spheration.

Spheration is a difficult product to wrap your head around, which is why I generally don't use it and just stick to the toratope notation. I managed to find the list though (which I should really add to the wiki):

((11)1) torus = circle # circle
((11)11) toracubinder = circle # sphere
((111)1) toraspherinder = sphere # circle
(((11)1)1) ditorus = (circle # circle) # circle
((11)(11)) tiger = (circle x circle) # circle

Notice that spheration (#) does not give you the expected result that the number of dimensions in the product is the sum of the dimensions of the operands, and it is not even consistent! In fact, in the case of the tiger, it only gives Σn-2, while in the other cases, it gives Σn-1. The Cartesian product (x) for comparison always gives Σn.

circle x circle is the duocylinder, so if you are trying to make analogies, the tiger is to the duocylinder as the torus is to the circle, and as the ditorus is to the torus. Yet the latter two cases there increase the dimension by one, while in the case of the tiger the dimension does not increase at all.

It looks like your "process X" is "replace an outer I with an (II), then put the whole lot in parentheses". In this case, yes, it turns a square into a torus and a cylinder into a tiger. However, if the operand of X does not have an outer I the result is not defined, e.g. X(square) = torus and X(cylinder) = tiger, but X(tiger) does not exist!

Also remember that process X combined with a finite, constant number of other linear processes, cannot possibly create all toratopes. If it could, then the number of toratopes in dimension n would be O(na) where a is the number of processes. However the number of toratopes per dimension is actually O(en) (see the graph of A000669), which grows faster than O(na).

This is what I was saying about considering products in your studies not just linear processes - linear processes can get you so far, but they can only ever give you a subset of the objects available by using products.

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### Re: Understanding the Cyltrianglinder

|| : ((||)|) == (||)| : ((||)(||)) ??

That is, the process that turns a square into torus as process X. Does process X also turn a cylinder into a tiger?

I guess what I meant to say was that process X is "lathe then spherate". If lathing a square into a cylinder, then spherating into a torus is process X. Does lathing a cylinder into a duocylinder, then spherating into a "torus" create the tiger?

This tiger has been a tough one. Spherating a cylinder joins the flat ends into a closed loop. The duocylinder doesn't seem to have any flat ends though, only two curved rolling surfaces. Unless they are flat but circular. In this case, I might actually get it.

((11)1) torus = circle # circle
((11)11) toracubinder = circle # sphere
((111)1) toraspherinder = sphere # circle
(((11)1)1) ditorus = (circle # circle) # circle
((11)(11)) tiger = (circle x circle) # circle

By this notation, it looks like the tiger is the duocylinder-torus, ( (||)(||)|) but it shouldn't be, right?

((11)1) torus = circle # circle ---->> circle extruded along the path of a circle

((11)11) toracubinder = circle # sphere ----->> circle extruded along the plane of a sphere

((111)1) toraspherinder = sphere # circle ---->> sphere extruded along the path of a circle

(((11)1)1) ditorus = (circle # circle) # circle --->> torus extruded along the path of a circle

((11)(11)) tiger = (circle x circle) # circle -->> duocylinder extruded along the path of a circle ???

The 'tiger' is a "spherated bi-glomohedrix prism". A glomo-hedr-ix is a round-2d-cloth, that is, the surface of a 3d sphere. The bi- bit means there are two of them, perpendicular to each other. A prism here may be read as a cartesian product. Spheraation is a surface finish, like a paint job. Here, it replaces thin things like lines and points and 2d fabric with solid things. Because the prism here is only 2d in 4d, we replace each point on the surface with a circle orthogonal to the 2d surface. That's what you get by replacing each point with a sphere.

You know, that is starting to make some sense. I'm going to dwell on this one for a while. It will come to me eventually, it always does. What you call the glomohedrix I have been referring to as the dot-globus, represented as *(00), meaning "dot extruded along the plane of a sphere". It's cool to see the real name for it, I like it better! This "spherated bi-glomohedrix prism" is the spheration of a hollow-sphere prism? Like the 2-D surface of a circle-torus, is this the surface of a sphere-torus?

-Philip
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### Re: Understanding the Cyltrianglinder

ICN5D wrote:I guess what I meant to say was that process X is "lathe then spherate". If lathing a square into a cylinder, then spherating into a torus is process X. Does lathing a cylinder into a duocylinder, then spherating into a "torus" create the tiger?

Spheration certainly does not mean "join the ends into a loop". Spheration is a product and is applied to two objects, and torus is circle spherated by circle.

Putting parentheses on the outside of something in toratopic notation is a partial linear operation, which is not spheration (and doesn't have a name, in fact). This operation joins the ends of a cylinder into a loop, but it does it by passing those ends through the original cylinder, like rolling up the hem of a sock, not by stretching those ends around in a big loop outside of the original cylinder like connecting the two ends of a hose to each other. On a torus, this does not matter, because they result in the same object, but when you put parentheses around (111)1 the spherinder, if you used the "hose" analogy, you'd end up with ((11)11) the toracubinder. Whereas if you use the "sock" analogy, you correctly get ((111)1) the toraspherinder. It is also worth noting neither of these analogies work unless there is one and only one "1" at the outermost level. So it works for (11)1, (111)1 and ((11)1)1, but it does not work for (11)11 and (11)(11).

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### Re: Understanding the Cyltrianglinder

Spheration in my definition is applied as a surface finish. Keiji is prolly thinking of the comb product, which produces toruses, or the crind product which produces spheres.
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### Re: Understanding the Cyltrianglinder

Okay, I might be getting it now! See, I thought if (||)| is the cylinder and ((||)|) is the torus, then the extra parentheses around the cylinder meant the linear extrusion of the circle turned into a circular extrusion.

The name " spherated bi-glomohedrix prism " is beginning to make sense the more I think about it. If the cartesian product of two circles means a prism of two perpendicular circles ( bi-circular prism), then the bi-glomohedrix prism is the cartesian product of two perpendicular glomohedrices. Then take that rigid prism and round it out during the spherate process. That last part I'm still having trouble with. I'm good with the bi-glomohedrix prism. Or, is the bi-glomohedrix prism by itself the tiger, and the word spherate isn't an additional process, just an addition to the name?

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### Re: Understanding the Cyltrianglinder

The torus is actually Comb{circle ø circle}. The comb product multiplies two polygons to give a polyhedron. The tiger is Comb{circle ø circle ø circle}, which means you can multiply three polygons together to get a 4d solid.
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### temp toratope topic

Wendy, a question for you -

Would the 3D interior of a torus be written as disc # circle or circle # disc?

3D interior of a torus is "disc at every point of a circle", while "circle at every point of a disc" doesn't make any sense. But I'm not sure whether "A at every point of B" is written A # B or B # A.

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### Re: Understanding the Cyltrianglinder

I understand it now. Thank you very much for that Wendy. You have no idea how long I've been trying to figure out the tiger. It was the name spherated bi-glomohedric prism that really did it. I understand the duocylinder to be the bi-circular prism. The spheration process turns the circles into glomohedrices before the cartesian product is applied. And because the shape was spherated, this word has to be attached to the bi-glomohedric prism. Wonderful!
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### Re: Alternative Methods for the Same Goal

Is glomohedrix a general term for hollow n-spheres, or a specific one for a specific hollow n-sphere?
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### temp toratope topic 4

ICN5D wrote:I understand it now. Thank you very much for that Wendy. You have no idea how long I've been trying to figure out the tiger. It was the name spherated bi-glomohedric prism that really did it. I understand the duocylinder to be the bi-circular prism. The spheration process turns the circles into glomohedrices before the cartesian product is applied. And because the shape was spherated, this word has to be attached to the bi-glomohedric prism. Wonderful!

Trying to use these instruction to draw the tiger again, found that I got something really weird
It seems I still haaving trouble visulising a comb between a 2D and a 3D shape
Attachments
ddd4s.PNG
Trying to draw the tiger again...
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### Re: Understanding the Cyltrianglinder

Wow, your diagram makes it seem so easy!

Thanks

I think the hole is like the rightmost diagram you drew, under the word "middle"... except that the left-to-right hole can go in a second path as well, on the "outside" of the figure (like the largest cube in a "cube-in-a-cube" projection of a tesseract)

So that when you enter one hole, you can get to the other side by going left or right around the other hole, but you can't actually enter the other hole without going outside again.

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### Re: Understanding the Cyltrianglinder

Regards Keiji's question.

The repetition of A at each point of B, is identical to the repetition of B at each point of A.

Consider this: Suppose you had a pile of coins. This makes a disk at each point of the height. You can put a tape beside it and bend the coins into a torus, i suppose. Now, consider a single coin. You can stand a pile of matches on top of it, so you have a match for each point of the coin. You could bend the pile of matches into a circle and make a torus.

So it really doesn't matter for prism products or cartesian products, whether you multiply A#B or B#A.

It does matter for the comb product, though. It's only the surface that is repeated copies of two surfaces, not the body. The comb product does not involve the body of the figure. In 3d, it does not matter, because topologically, inside and outside is the same shape. You can for example, put a circle inside a torus, so that it can't vanish without crossing the surface, or you can put a circle outside with the same effect.

In 4D, you have eg Comb{circle, sphere} != Comb{sphere, circle}. The comb product works, that if you 'prefix' a measure, like circle in the first, it makes a hose-linkage. If you put it at the end, it makes a sock-linkage. So if you start off with a line-sphere product, the hose-link connects the line so the body is inside the spherinder. This means that you have a non-vanishing circle inside, and a non-vansihing sphere outside. (All circles outside can vanish). If you roll the spherinder wall down like a sock, then the non-vanishing circle is outside, and the non-vanishing sphere is inside. They're different shapes with the same surface.

The non-vanishing shape inside a tiger is not a circle or a sphere, but a torus. I have not figured out what lives outside yet.
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### Re: Understanding the Cyltrianglinder

Regarding Names of spheres etc.

The names i use for various shapes like spheres and circles etc, is based on the 'fabric / patch' idiom.

A fabric is a thing to itself, and is generally taken to be (indefinitely) unbounded. It allows us to talk about open spaces of N dimensions with a word.

A patch is a thing used to do other things, like make polyhedra. It's usually the attached prefix that tells you what is actually being done. Polyhedron is the marker word here. Poly is many (with closure), Hedron is 2d patches. That is, we take a mob of patches and sew them together in a way that we can't add any more, and that's it.

Cloth (fabric) comes in many dimensions. Teelix (0D) = button, Latrix (1D) = string or thread. Hedrix (2D), is a 2d sheet, Chorix (3D) is a 3d thing, Terix (4D) is a 4d fabric. Petix (5D), Ectix (6D), Zettix (7D), Yottix (8D). The plural is to replace -ix with -ices. The adjective is to replace -ix with -ic.

Patches are derived from cloth by replacing -ix with -on, to teelon, latron, hedron, choron. The plural is -a, the adjective is -al

We have a word for 'solid in N dimensions' (eg like a solid area of read is still 2D), by replacing -ix with -id. eg Hedrid.

We have a word for 'approximately that shape', (-ous) like saying a snake is a latrous animal (it's string-shaped).

The bit you put in front, makes what you do with it. Without it, it just means 'patch' or 'cloth'.

GLOMO means round, or positive curvature. It's used of spheres. a glomohedrix is then a "round 2d fabric" (by itself), is used to refer both to the surface of a 3d sphere, and the spheric 2d geometry, commonly designated S2. We live on a glomohedrix. gravity keeps us on the ground, and for the most part, it's spherical geometry that navigates us around. Glomohedron is a 'round 2d patch', (which does something). Here the something is that it contains volume. A glomohedron is a 3d-disk. HORO- deals with horizon-centred (euclidean geometry), and BOLLO- is hyperbolic. PLANO- makes flat things. A planolatron on a sphere is a great circle, not a eucliedan straight line.

SUR- refers to surface. A surtope is a surface polytope, like a vertex, edge etc. A surhedron is a "surface 2d patch". You might think of a jacket as having surteela (buttons), and several surhedra (panels) sewn together to make sleeves and brest and back and so forth. When you list the surtopes of a figure, there is no need to prefix sur- to each element.

And so on. Most of this is in the Polygloss.
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### Re: Understanding the Cyltrianglinder

So - if you wanted to extend the comb product so that it dealt with interiors as well, would the most logical thing be:

circle # sphere -> surface of toracubinder
circle # ball -> interior of toracubinder
disc # sphere -> exterior of toracubinder
sphere # circle -> surface of toraspherinder
sphere # disc -> interior of toraspherinder
ball # circle -> exterior of toraspherinder
?

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### Re: Understanding the Cyltrianglinder

One of your parts is going to have its interior wholy inside the torus. That's the one you make solid. So your list is correct.

Note that if you multiply 'disk # sphere', it's kind of like making a dodecahedron or something out of disk*pentagon prisms. You put the pentagons together to make a 'sphere', because you want the sphere to remain hollow. But because you made it out of disk-shaped prisms, you get solid disks.

You can swap inside and outside of these figures by 'central inversion' at an inside point. This replaces (r, angle) with (1/r, angle). The 0 of the inversion goes to infinity, and infinity goes to zero. Circles remain circles, and straight lines pass are circles through the 0 point, parallel lines are cotangent at 0.
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### Re: Understanding the Cyltrianglinder

Are you saying disc # sphere will not produce the exterior of a torus and I need to do circle # (complement of ball) instead?

Or are you saying yes it will produce the exterior and the dodecahedron analogy was in agreement?

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### temp toratope topic 3

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.
Attachments
Tiger, spheration and glomohedrix
tiger.jpg (51.47 KiB) Viewed 10892 times
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### Re: Understanding the Cyltrianglinder

Doesn't that create a ditorus? I'm not clear on the last process. Before the last process, it was a torus*line-prism. Did it turn into a torus*circle -prism?
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### Re: Understanding the Cyltrianglinder

It does, but then a tiger is an instance of a 'duo-torus'.
The dream you dream alone is only a dream
the dream we dream together is reality.

\(\LaTeX\ \) at https://greasyfork.org/en/users/188714-wendy-krieger

wendy
Pentonian

Posts: 1964
Joined: Tue Jan 18, 2005 12:42 pm
Location: Brisbane, Australia

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