I live!

Discussion of shapes with curves and holes in various dimensions.

Re: I live!

Postby Marek14 » Fri Nov 18, 2011 8:04 am

Another idea: a program that could display 5D shapes as 2D arrays of 3D slices.

And how many toruses are there if we allow other quadrics in their definition?

In 3D, it's easy to imagine parabola+circle or hyperbola+circle (basically a parabola/hyperbola-shaped tube). Replacing each point of a circle with a parabola would give a shape formed by rotating parabola around an axis parallel with its direct line, but not touching it. Replacing each point of parabola with hyperbola or vice versa would lead to something strange, I think.

Related to the idea of projective geometry torus, and various ways it works in Euclideam geometry if you put the point at infinity to various positions related to it.
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Re: I live!

Postby Marek14 » Fri Nov 18, 2011 8:12 am

Also, I seem to remember that there are some graphotopes that couldn't be expressed in the other notations... simplest of them is the longdome:

O--O--O--O

Three of planar sections of longdome (xy, yz and zw) are circles, the other three (xz, xw and yw) are squares. The xyz and yzw hyperplanar cuts are domes (crinds), while the xyw and xzw ones are cylinders.

If you put it on the floor in xyw or yzw orientation, it will make a 2D circular mark and can be rolled in the floor direction perpendicular to it; this will transform it into xzw, respectively xyw orientation.

If you put it on the floor in xyw or xzw orientation, the mark will be a line, parallel to a lateral direction of its crind "shadow". It can be rolled in two perpandicular directions; one will transform the line into a circle, and the shape into yzw, respectively xyz orientation, the other direction will transform the line into a perpendicular line, and the shape into xzw, respectively xyw orientation.

(That's one feature of graphotopes - you can show their cross-sections by erasing nodes, and "traces" (how they touch floor) by erasing a node together with all other connected nodes. And any rolling motion they can do in coordinate directions happens along some edge of the graph, transforming one "trace-shadow" combination into another).
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Re: I live!

Postby Keiji » Fri Nov 18, 2011 11:17 am

Marek14 wrote:Also, I seem to remember that there are some graphotopes that couldn't be expressed in the other notations... simplest of them is the longdome


It's not one of the following?:

Cubic crind ([xyz]w)
Cylindrical crind ([(xy)z]w)
Dicrind ([xy]zw)
Duocrind ([xy][zw])
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Re: I live!

Postby Marek14 » Fri Nov 18, 2011 1:44 pm

Keiji wrote:
Marek14 wrote:Also, I seem to remember that there are some graphotopes that couldn't be expressed in the other notations... simplest of them is the longdome


It's not one of the following?:

Cubic crind ([xyz]w)
Cylindrical crind ([(xy)z]w)
Dicrind ([xy]zw)
Duocrind ([xy][zw])


I was unable to find the definition of these operations so far, but I assume that [] signifies combining two or more shapes in orthogonal dimensions together in such way that all cross-sections with axes from different shapes are squares, while () combines them such that the cross-sections are circles.

Cubic crind, if I understand these shapes correctly, corresponds to what I call "tridome", a single node joined to three others. Its planar sections are three circles and three squares, yes, but the planes with circles all share a single axis. And its hyperplanar sections are completely off, being three crinds and one cube.

Cylindrical crind is called "spheridome" in graphotope system. It is a triangle with an extra node connected to one of its nodes. Its planar sections are four circles and two squares and hyperplanar sections are two crinds, a cylinder and a sphere.

Dicrind would be "hemiglome", I guess, a shape with five circular planar sections and only a single square. The graph is K4 minus one edge. Hyperplanar cross-sections are two crinds and two spheres.

Duocrind would be "cyclodome" - four nodes connected in a circular fashion. Planar sections are four circles and two squares, all hyperplanar sections are crinds.

So, if I get these correctly, nope, longdome is not in there. And for good reason, since these operations translate to graphs as follows:

[]: combine the graphs of the component shapes and put them into the same graph without adding any edges.
(): combine the graphs of the component shapes and put them into the same graph, then add all edges that join nodes from different components.

However, not all graphs can be formed by these two operations, and the longdome graph (chain of four nodes) is the simplest one that cannot.

So graphotopes would include all those shapes that can be build from [] and () operations, but also many other shapes.

The 4D graphotopes, complete list, would be (with my old names):

tesseract [xyzw]
cubinder [(xy)zw]
dominder [([xy]z)w]
duocylinder [(xy)(zw)]
tridome ([xyz]w)
spherinder [(xyz)w]
longdome - inexpressible in this notation
spheridome ([(xy)z]w)
cyclodome ([xy][zw])
hemiglome ([xy]zw)
glome (xyzw)

I can actually prove that longdome doesn't exist in this notation in another way: a graph can be transformed into complementary graph (with edges precisely in the locations where the original graph doesn't have them) by switching () and [] operations in its definition. But the 4-chain of longdome is a self-dual graph! So it should stay the same after exchanging the () and [] operations, which doesn't seem possible since there is always one main pair of brackets that encloses the whole shape (a rough division between those that "feel square" and those that "feel round").
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Re: I live!

Postby Keiji » Fri Nov 18, 2011 2:03 pm

You are completely right!

Maybe we should adopt a SMILES-esque notation for the graphotopes.

Using, say, "o" for a node and "." for a nonconnection we could have:

Code: Select all
tesseract                           [xyzw]            o.o.o.o
cubinder                            [(xy)zw]          oo.o.o
dominder     AKA crindal prism      [([xy]z)w]        ooo.o
duocylinder                         [(xy)(zw)]        oo.oo
tridome      AKA cubic crind        ([xyz]w)          oo(o)o
spherinder                          [(xyz)w]          o1oo1.o
longdome                            not a bracketope  oooo
spheridome   AKA cylindrical crind  ([(xy)z]w)        o1oo1o
cyclodome    AKA duocrind           ([xy][zw])        o1ooo1
hemiglome    AKA dicrind            ([xy]zw)          o1oo1o1
glome                               (xyzw)            o1o2o1o1,2


By the way, here's the old topic on graphotopes: viewtopic.php?f=3&t=350

Do you have a surface equation for the longdome? I couldn't find one in said topic. I don't think anyone has yet proved the longdome actually exists (i.e. is embeddable as a convex shape in Euclidean 4-space), and I would like to know it does before I go and cry over it not being a brick product... :P
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Re: I live!

Postby Marek14 » Fri Nov 18, 2011 2:31 pm

Someone DID derive an equation, I think, but the details are hazy.

You could build one by imagining a series of cross-sections. The longdome has two "major" directions (corresponding to degree-2 nodes) and two "minor" directions (corresponding to degree-1 nodes).

Slicing it along a major direction is simpler. So, let's say that the longdome is x-y-z-w, so major directions are y and z. I'll use z.

The slice for z = 0 is a cylinder, with equation max((x^2 + y^2), w^2) = 1. The slices for z = 1 and z = -1 are a line with equations (y = 0 & w = 0). Now the question is how do the slices in-between look... I haven't studied that part in detail, but I guess that they will be prisms whose w-dimension will vary from 0 for z = 1 to 1 for z = 0 (height will be, of course, twice this number as w will count the same in positive and negative direction) and whose base shape will be side slices of crind (which transform between line on one side and circle on the other). The variation of w dimension will be sinusoidal (so half-height^2 + z^2 = 1).

If you cut longdome along minor direction (let's say w), the w = 0 slice will be a dome (max(x^2, z^2) + y^2) = 1 and the w = 1 and w = -1 extreme slices will be circles (x^2 + y^2 = 1 & z = 0), but it's harder to imagine the shape of in-between slices in this case, at least for me.
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Re: I live!

Postby wendy » Sat Nov 19, 2011 9:23 am

The formulae for this are the 'radiant forms'. It's one of my inventions..

Suppose a polytope has a centre where r=0, and the surface gives r=1, in every direction.

Then, the prism product of X and Y gives max[x,y]. The tegum product of x,y gives sum<x,y>, the crind product gives rss(x,y), being the root-sum-squares.

A cylinder is [x,(y,z)], gives max[x,rss(y,z)].

For a given point x,y,z, one evaluates the scale by this rule. eg for the direction through 3,3,4 one gets max[3, rss(3,4)] = max[3,5] = 5. Since this is the working radius, the surface falls a 3/5, 3/5, 4/5.
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Re: I live!

Postby Keiji » Sat Nov 19, 2011 1:21 pm

Yes, I get all that (did so years ago), and that works great for the bracketopes. I was asking for an equation for the longdome, which is not a bracketope.
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Re: I live!

Postby Marek14 » Sun Nov 20, 2011 10:08 am

I wonder if there are people who'd want unusual geometric games :) I still wonder about feasibility of hyperbolic Carcassone.
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Re: I live!

Postby quickfur » Sun Nov 20, 2011 3:29 pm

Marek14 wrote:I wonder if there are people who'd want unusual geometric games :) I still wonder about feasibility of hyperbolic Carcassone.

I for one would love to have a game based on polyhedra/polychora. I'm seriously thinking about that polyhedral shooter we talked about. :) Though I will finally need to get off my lazy butt to actually install 3D drivers for my video card... haven't had the pressing need to so far.
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Re: I live!

Postby Mrrl » Sun Nov 20, 2011 4:45 pm

There are many hyperbolic Rubiks-like puzzles. They don't require special game programming, so it's more easy to start programming with them. Implementation of hyperbolic geometry (periodic patterns in plane/space) is difficult enough...
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Re: I live!

Postby Marek14 » Mon Nov 21, 2011 3:25 pm

I spent some time today cutting 5D toratopes in my mind. Got some good idea of how they work. I think the wiki could use some images - the (3+2)D technique should work well (selecting 3 axis to display and using the remaining 2 as x and y axis of a table that shows various cuts). It allows to get a good idea of the 4D cuts as well (which are rows and columns).

Not sure whether the toratope have robust nomenclature which could handle the higher-dimensional ones.

I'm thinking of classifying the dimensions of toruses by their radiuses. For example, a 3D torus has an outer radius and an inner radius, and we can also say that it has two outer dimensions (dimensions where the outer radius is measured) and one inner dimension (the one dimension that is added with inner radius). This allows for easy classification for groups of toruses (for example, we can say that one cut of ditorus are two toruses displaced in outer dimension, while a cut of tiger are two toruses displaced in inner dimension).

The numbers of dimensions associated with a particular radius can differ, and it can be even zero (for example, tiger has three radius, the outer radius A, outer radius B and inner radius. Its four dimensions are two outer dimensions A and two outer dimensions B -- it has no inner dimension, but its bigger brother, toraduocyldyinder has the extra dimension as its inner dimension.

A "bigger brother" of a toratope is defined thusly:

1. A toratope has a "skeleton". Each particular toratope is defined as a "set of all points which have the same distance from the skeleton". Skeleton of a 3D torus is a circle and skeleton of tiger is cartesian product of two circles.

2. A "bigger brother" of a toratope is a toratope in higher dimension that shares its skeleton. So a toracubinder is a bigger brother of torus because it's also based on a circle.

As an aside, I consider this definition more intuitive than "bending" and joining ends, which is hard to visualize.

Somewhat logically, then, a "youngest brother" is a toratope whose skeleton has only 1 less dimension than its surface. Tiger is a youngest brother because its surface is 3-dimensional, while circle x circle, its skeleton is 2-dimensional. Toraduocyldyinder has the same 2D skeleton, bud a 4D surface (and is a 5D toratope altogether).

Toratopes as rotations (stated here previously):

Toratopes can be also defined as rotations of lower-dimensional toratopes. Generally, a (n-1) dimensional toratope is rotated in a plane formed by 1 dimension of its own plus 1 additional new dimension. Alternately, we can say that it's rotated around a (n-2)-dimensional hyperplane formed by all of its dimensions except one. Either way, there are two ways to rotate it:

I. Around a coordinate hyperplane. Replaces a "I" in the toratopic notation with "II"
II. Around a remote hyperplane. This hyperplane is parallel to a coordinate hyperplane, but remote enough so it doesn't intersect the original toratope. This replaces a "I" in the toratopic notation with "(II)".

The "I" replaced in both cases is the one which represents the dimension NOT in the hyperplane.

Thus, all 4D and 5D toratopes are constructed as following:

Glome: Rotate a sphere around a coordinate plane.
Skeleton: Point. Bigger brother of sphere.
3D cuts: all 4 cuts are spheres.

Toracubinder: Rotate a torus around the outer coordinate plane, OR rotate a sphere around a remote plane.
Outer radius, inner radius; 2 outer dimensions, 2 inner dimensions.
Skeleton: Circle. Bigger brother of torus.
3D cuts: 2 cuts (inner) are toruses, 2 cuts (outer) are two displaced spheres.

Tiger: Rotate a torus around a remote outer plane.
Outer radius A, outer radius B, inner radius; 2 outer dimensions A, 2 outer dimensions B
Skeleton: Circle x circle. Youngest brother.
3D cuts: all 4 cuts are two toruses displaced through inner dimensions.

Toraspherinder: Rotate a torus around a outer-inner coordinate plane.
Outer radius, inner radius; 3 outer dimensions, 1 inner dimension
Skeleton: Sphere. Youngest brother.
3D cuts: 1 cut (inner) is two concentric spheres, 3 cuts (outer) are toruses.

Ditorus: Rotate a torus around a remote outer-inner plane.
Outer radius, mid-radius, inner radius; 2 outer dimensions, 1 mid-dimension, 1 inner dimension
Skeleton: Torus. Youngest brother.
3D cuts: 1 cut (inner) is two concentric toruses differing in their inner radii, 1 cut (mid) is two concentric toruses differint in their outer radii, 2 cuts (outer) are two toruses displaced through outer dimension.

Pentasphere: Rotate a glome around a coordinate hyperplane.
Skeleton: Point. Bigger brother of glome.
4D cuts: all 5 cuts are glomes.
3D cuts: all 10 cuts are spheres.

Toratesserinder: Rotate a toracubinder around not-inner coordinate hyperplane (i.e. hyperplane that is missing one of the two inner dimensions of toracubinder), OR rotate a glome around a remote hyperplane.
Outer radius, inner radius; 2 outer dimensions, 3 inner dimensions
Skeleton: Circle. Bigger brother of toracubinder.
4D cuts: 3 cuts (inner) are toracubinders; 2 cuts (outer) are 2 displaced glomes.
3D cuts: 3 cuts (inner-inner) are toruses; 6 cuts (outer-inner) are 2 displaced spheres; 1 cut (outer-outer) is empty.

Toraduocyldyinder: Rotate a toracubinder around a remote not-inner hyperplane.
Outer radius A, outer radius B, inner radius; 2 outer dimensions A, 2 outer dimensions B, 1 inner dimension
Skeleton: Circle x circle. Bigger brother of tiger.
4D cuts: 1 cut (inner) is tiger; 4 cuts (outer[A] and outer[B]) are 2 toracubinders displaced through inner dimension.
3D cuts: 4 cuts (outer[A]-inner and outer[B]-inner) are 2 toruses displaced through inner dimension; 2 cuts (outer[A]-outer[A] and outer[B]-outer[B]) are empty; 4 cuts (outer[A]-outer[B]) are four spheres arranged in rectangle.

Toracubspherinder: Rotate a toracubinder around a not-outer coordinate hyperplane, OR rotate a toraspherinder around a not-inner coordinate hyperplane.
Outer radius, inner radius; 3 outer dimensions, 2 inner dimensions
Skeleton: Sphere. Bigger brother of toraspherinder.
4D cuts: 2 cuts (inner) are toraspherinders; 3 cuts (outer) are toracubinders.
3D cuts: 1 cut (inner-inner) is two concentric spheres; 6 cuts (outer-inner) are toruses, 3 cuts (outer-outer) are two displaced spheres.

Toracubtorinder: Rotate a toracubinder around a remote not-outer hyperplane, OR rotate a ditorus around a not-inner coordinate hyperplane.
Outer radius, mid-radius, inner radius; 2 outer dimensions, 1 mid-dimension, 2 inner dimensions
Skeleton: Torus. Bigger brother of ditorus.
4D cuts: 2 cuts (inner) are ditoruses, 1 cut (mid) is two concentric toracubinders differing in their outer radii, 2 cuts (outer) are 2 toracubinders displaced through outer dimension.
3D cuts: 1 cut (inner-inner) is two concentric toruses differing in their inner radii, 2 cuts (mid-inner) are two concentric toruses differing in their outer radii, 4 cuts (outer-inner) are two toruses displaced through outer dimension, 2 cuts (outer-mid) are 4 spheres arranged in a line, 1 cut (outer-outer) is empty.

Cylspherintigroid: Rotate a tiger around a coordinate hyperplane, OR rotate a toraspherinder around a remote not-inner hyperplane.
Outer spherical radius, outer circular radius, inner radius; 3 spherical dimensions, 2 circular dimensions
Skeleton: Sphere x circle. Youngest brother.
4D cuts: 2 cuts (circular) are two toraspherinders displaced through inner dimension, 3 cuts (spherical) are tigers.
3D cuts: 1 cut (circular-circular) is empty, 6 cuts (spherical-circular) are two toruses displaced through inner dimension, 3 cuts (spherical-spherical) are, once again, two toruses displaced through inner dimension (but with different geometry of other parallel cuts).

Cyltorintigroid: Rotate a tiger around a remote hyperplane, OR rotate a ditorus around a remote not-inner hyperplane.
Outer torus radius, outer circular radius, mid-torus radius, inner radius; 2 outer torus dimensions, 2 outer circular dimensions, 1 mid-torus dimension
Skeleton: Torus x circle. Youngest brother.
4D cuts: 2 cuts (circular) are two ditoruses displaced through inner dimension, 1 cut (mid-torus) is two concentric tigers differing in one of their outer radii, 2 cuts (outer-torus) are two displaced tigers.
3D cuts: 1 cut (circular-circular) is empty, 2 cuts (mid-torus-circular) are two pairs of concentric toruses differing in their outer radii displaced in inner dimension, 4 cuts (outer-torus-circular) are four toruses arranger in two stacks of two, 2 cuts (outer-torus-mid-torus) are four toruses displaced through inner dimension, 1 cut (outer-torus-outer-torus) is empty, but with different geometry of other parallel cuts than the circular-circular one.

Toraglominder: Rotate a toraspherinder around a not-outer coordinate hyperplane.
Outer radius, inner radius; 4 outer dimensions, 1 inner dimension
Skeleton: Glome. Youngest brother.
4D cuts: 1 cut (inner) is two concentric glomes, 4 cuts (outer) are toraspherinders.
3D cuts: 4 cuts (outer-inner) are two concentric spheres, 6 cuts (outer-outer) are toruses.

Cylindrical ditorus: Rotate a toraspherinder around a remote not-outer hyperplane, OR rotate a ditorus around a not-mid coordinate hyperplane.
Outer radius, mid-radius, inner-radius; 2 outer dimensions, 2 mid-dimensions, 1 inner dimension
Skeleton: Toracubinder. Youngest brother.
4D cuts: 1 cut (inner) is two concentric toracubinders differing in their inner radii, 2 cuts (mid) are ditoruses, 2 cuts (outer) are two toraspherinders displaced through outer dimension.
3D cuts: 2 cuts (mid-inner) are two concentric toruses differing in their inner radii, 2 cuts (outer-inner) are two displaced pairs of concentric spheres, 1 cut (mid-mid) is two concentric toruses differing in their outer radii, 4 cuts (outer-mid) are two toruses displaced through outer dimension, 1 cut (outer-outer) is empty.

Tigric torus: Rotate a ditorus arount a remote not-mid hyperplane.
Outer radius A, outer radius B, mid-radius, inner radius; 2 outer dimensions A, 2 outer dimensions B, 1 inner dimension
Skeleton: Tiger. Youngest brother.
4D cuts: 1 cut (inner) is two concentric tigers differing in their inner radii, 4 cuts (outer[A] and outer[B]) are two ditoruses displaced through mid-dimension.
3D cuts: 4 cuts (outer[A]-inner and outer[B]-inner) are two pairs of concentric toruses differing in their inner radii displaced through inner dimension, 2 cuts (outer[A]-outer[A] and outer[B]-outer[B]) are empty, 4 cuts (outer[A]-outer[B]) are four toruses lying in the shape of a rectangle.

Spheric ditorus: Rotate a ditorus around a not-outer coordinate hyperplane.
Outer radius, mid-radius, inner radius; 3 outer dimension, 1 mid-dimension, 1 inner dimension
Skeleton: Toraspherinder. Youngest brother.
4D cuts: 1 cut (inner) is two concentric toraspherinders differing in their inner radii, 1 cut (mid) is two concentric toraspherinders differing in their outer radii, 3 cuts (outer) are ditoruses.
3D cuts: 1 cut (mid-inner) is four concentric spheres, 3 cuts (outer-inner) are two concentric toruses differing in their inner radii, 3 cuts (outer-mid) are two concentric toruses differing in their outer radii, 3 cuts (outer-outer) are two toruses displaced through outer dimension.

Tritorus: Rotate a ditorus around a remote not-outer hyperplane.
Outer radius, mid-1-radius, mid-2-radius, inner radius; 2 outer dimensions, 1 mid-1 dimension, 1 mid-2 dimension, 1 inner dimension
Skeleton: Ditorus. Youngest brother.
4D cuts: 1 cut (inner) is two concentric ditoruses differing in their inner radii, 1 cut (mid-2) is two concentric ditoruses differing in their mid-radii, 1 cut (mid-1) is two concentric ditoruses differing in their outer radii, 2 cuts (outer) are two ditoruses displaced through outer dimension.
3D cuts: 1 cut (mid-2-inner) is four concentric toruses differing in their inner radii, 1 cut (mid-1-inner) is four concentric toruses with one of each combination of two outer and two inner radii, 2 cuts (outer-inner) are two pairs of concentric toruses differing in their inner radii displaced through outer dimension, 1 cut (mid-1-mid-2) is four concentric toruses differing in their outer radii, 2 cuts (outer-mid-2) are two pairs of concentric toruses differing in their outer radii displaced through outer dimension, 2 cuts (outer-mid-1) are 4 toruses in a line displaced through outer dimension, 1 cut (outer-outer) is empty.
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Re: I live!

Postby quickfur » Mon Nov 21, 2011 9:54 pm

Marek14 wrote:[...]
I'm thinking of classifying the dimensions of toruses by their radiuses. For example, a 3D torus has an outer radius and an inner radius, and we can also say that it has two outer dimensions (dimensions where the outer radius is measured) and one inner dimension (the one dimension that is added with inner radius). This allows for easy classification for groups of toruses (for example, we can say that one cut of ditorus are two toruses displaced in outer dimension, while a cut of tiger are two toruses displaced in inner dimension).

I like this. This is a very useful way of thinking about toroidal shapes. So basically the inner dimension is the dimension orthogonal to the skeleton (as you defined below), right? And the outer dimension is the dimension that lies in the (hyper)plane of the skeleton?

[...]
A "bigger brother" of a toratope is defined thusly:

1. A toratope has a "skeleton". Each particular toratope is defined as a "set of all points which have the same distance from the skeleton". Skeleton of a 3D torus is a circle and skeleton of tiger is cartesian product of two circles.

2. A "bigger brother" of a toratope is a toratope in higher dimension that shares its skeleton. So a toracubinder is a bigger brother of torus because it's also based on a circle.

I like this!! Classifying toratopes by their skeleton is a brilliant idea! This makes them so much easier to visualize.

So the tiger is basically just the ridge of a duocylinder expanded into a 4-manifold, right? Heh... finally I can visualize the tiger, thanks to your definition. :)

As an aside, I consider this definition more intuitive than "bending" and joining ends, which is hard to visualize.

Yeah, no kidding.

Somewhat logically, then, a "youngest brother" is a toratope whose skeleton has only 1 less dimension than its surface. Tiger is a youngest brother because its surface is 3-dimensional, while circle x circle, its skeleton is 2-dimensional. Toraduocyldyinder has the same 2D skeleton, bud a 4D surface (and is a 5D toratope altogether).

So the youngest brother of a toracubinder is the torus, right?

I like this terminology. Older brothers are "fatter" (have more dimensions) but are "essentially the same" as their younger brothers (same skeleton). I think this is a much better way of classifying toratopes than trying to roll them up or fold them (which is easy to understand when you first start out, but when you get to shapes like the tiger, it's totally mind-bending to visualize what happens when you try to roll it up or fold it). Visualizing toratopes via their skeletons is so much easier!

[...]
Toratopes can be also defined as rotations of lower-dimensional toratopes. Generally, a (n-1) dimensional toratope is rotated in a plane formed by 1 dimension of its own plus 1 additional new dimension. Alternately, we can say that it's rotated around a (n-2)-dimensional hyperplane formed by all of its dimensions except one. Either way, there are two ways to rotate it:

I. Around a coordinate hyperplane. Replaces a "I" in the toratopic notation with "II"
II. Around a remote hyperplane. This hyperplane is parallel to a coordinate hyperplane, but remote enough so it doesn't intersect the original toratope. This replaces a "I" in the toratopic notation with "(II)".

The "I" replaced in both cases is the one which represents the dimension NOT in the hyperplane.

OK, so in terms of skeletons, rotating around the coordinate hyperplane doesn't change the skeleton, right? And rotating around a remote hyperplane makes a cartesian product of the skeleton with a circle?

Thus, all 4D and 5D toratopes are constructed as following: [... snipped lots of good stuff...]

I like it. I really like it. This is a much better way of classifying toratopes than what we had before. Yay!
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Re: I live!

Postby Marek14 » Mon Nov 21, 2011 10:42 pm

quickfur wrote:
Marek14 wrote:[...]
I'm thinking of classifying the dimensions of toruses by their radiuses. For example, a 3D torus has an outer radius and an inner radius, and we can also say that it has two outer dimensions (dimensions where the outer radius is measured) and one inner dimension (the one dimension that is added with inner radius). This allows for easy classification for groups of toruses (for example, we can say that one cut of ditorus are two toruses displaced in outer dimension, while a cut of tiger are two toruses displaced in inner dimension).

I like this. This is a very useful way of thinking about toroidal shapes. So basically the inner dimension is the dimension orthogonal to the skeleton (as you defined below), right? And the outer dimension is the dimension that lies in the (hyper)plane of the skeleton?


In case of torus, yes. But in general cases, it can be more complicated. Basically, the outer dimensions are dimensions of the skeleton if the skeleton is a circle, sphere or something like that. But the skeleton can be any toratope or cartesian product of toratopes. However, the general division into "skeleton dimensions" and "inflated dimensions" should be always possible (inflated since you can imagine, say, torus to be an inflated circle), but bear in mind that it's possible to have no skeleton dimensions (for spheres, whose skeleton is a point), and no inflated dimensions (for tiger and other beasts, which have skeleton bent in significantly more dimensions than its actual dimensionality (circle x circle is 2D surface, but you need 4D to contain it)).

[...]
A "bigger brother" of a toratope is defined thusly:

1. A toratope has a "skeleton". Each particular toratope is defined as a "set of all points which have the same distance from the skeleton". Skeleton of a 3D torus is a circle and skeleton of tiger is cartesian product of two circles.

2. A "bigger brother" of a toratope is a toratope in higher dimension that shares its skeleton. So a toracubinder is a bigger brother of torus because it's also based on a circle.

I like this!! Classifying toratopes by their skeleton is a brilliant idea! This makes them so much easier to visualize.

So the tiger is basically just the ridge of a duocylinder expanded into a 4-manifold, right? Heh... finally I can visualize the tiger, thanks to your definition. :)



Something like that. Imagine it with inflation, as if you had a swimmer's ring in the shape of duocylinder ridge and blew air into it.


As an aside, I consider this definition more intuitive than "bending" and joining ends, which is hard to visualize.

Yeah, no kidding.

Somewhat logically, then, a "youngest brother" is a toratope whose skeleton has only 1 less dimension than its surface. Tiger is a youngest brother because its surface is 3-dimensional, while circle x circle, its skeleton is 2-dimensional. Toraduocyldyinder has the same 2D skeleton, bud a 4D surface (and is a 5D toratope altogether).

So the youngest brother of a toracubinder is the torus, right?

I like this terminology. Older brothers are "fatter" (have more dimensions) but are "essentially the same" as their younger brothers (same skeleton). I think this is a much better way of classifying toratopes than trying to roll them up or fold them (which is easy to understand when you first start out, but when you get to shapes like the tiger, it's totally mind-bending to visualize what happens when you try to roll it up or fold it). Visualizing toratopes via their skeletons is so much easier!

[...]
Toratopes can be also defined as rotations of lower-dimensional toratopes. Generally, a (n-1) dimensional toratope is rotated in a plane formed by 1 dimension of its own plus 1 additional new dimension. Alternately, we can say that it's rotated around a (n-2)-dimensional hyperplane formed by all of its dimensions except one. Either way, there are two ways to rotate it:

I. Around a coordinate hyperplane. Replaces a "I" in the toratopic notation with "II"
II. Around a remote hyperplane. This hyperplane is parallel to a coordinate hyperplane, but remote enough so it doesn't intersect the original toratope. This replaces a "I" in the toratopic notation with "(II)".

The "I" replaced in both cases is the one which represents the dimension NOT in the hyperplane.

OK, so in terms of skeletons, rotating around the coordinate hyperplane doesn't change the skeleton, right? And rotating around a remote hyperplane makes a cartesian product of the skeleton with a circle?


Hm, rotation about coordinate hyperplane CAN change the skeleton. Toraspherinder is an example, as it's generated by rotating a torus, but its skeleton is a sphere. I guess the skeleton is also generated by the same rotation. Thus, the weird skeleton of tiger is generated by the fact that a circle is rotated in a plane that is completely orthogonal to the plane of the circle!

Thus, all 4D and 5D toratopes are constructed as following: [... snipped lots of good stuff...]

I like it. I really like it. This is a much better way of classifying toratopes than what we had before. Yay!


I also had my own names for those, which were long, but should give you better idea...

I think I used the ^ operator (read "sur" for some reason I don't exactly recall) for inflation, so circle^circle was torus, circle^sphere would be toracubinder and sphere^circle a toraspherinder. Tiger's bigger brothers would be called Tiger+1, Tiger+2, etc... the ((III)(II)) toratope would be "sphere tiger", ((II)I)(II)) would be "torus tiger" and ((III)((II)I)) would be "sphere-torus tiger".
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Re: I live!

Postby Marek14 » Tue Nov 22, 2011 1:13 pm

Some more notes:

Families and patriarchs:

A "family" of toratopes is a group with the same bracket structure. For example, ((II)II), ((III)I) and ((II)I) are all in the same family that can be written as "(())".
All toratopes within the same family have the same number and classification of radii. For example, this family (the Torus family) has an outer radius and an inner radius. The Ditorus family "((()))" has an outer radius, a mid-radius, and an inner radius. However, the number of dimensions for each radius may differ.
The patriarch of the family is the lowest-dimensional member. The family can be named after him.

We can differentiate "pure" and "mixed" families:

In a "pure" family, the skeleton of each member is a toratope from a specific family (its ancestor). For example, the Torus family's ancestor is the Sphere family since each toratope from the Torus family has a sphere of some dimension as its skeleton.

In a "mixed" family, the skeleton of each member is a cartesian product of toratopes from specific families. The Tiger family "(()())" has the Sphere family as its double ancestor.

Second thing:

I noticed some confusion on the page for construction of 4D toruses and cross-sections. To simplify it, we can simply extend the definition of torus notation to disconnected shapes.

(II) is a circle, (III) is a sphere, (IIII) is a glome, etc. So, what is (I)? (I) is a 1D circle, in other words, TWO POINTS.

If ((II)II) is a toracubinder and ((II)I) is a torus, what is ((II))? It's two concentric circles.

What's ()? That's an empty set.

In other words:

There are two kinds of entities, brackets "()" and slashes "I". The rule is as follows:

If a bracket is empty, it signifies an empty set. From now on, we'll assume that brackets are non-empty since any bracket that includes an empty bracket is an empty set as well.
If a bracket contains one slash, it's two points.
If a bracket contains two or more slashes, it's a sphere of some dimension.
If a bracket contains only a single bracket, it's two concentric copies of the object inside that bracket, differing only in the innermost radius.
If a bracket contains a single bracket and some slashes, it's a toratope with object inside the inner bracket as a skeleton and some extra inner dimensions.
If a bracket contains multiple brackets, it's the cartesian product of objects in all the brackets, inflated by an inner radius while preserving the dimensions.
If a bracket contains both multiple brackets AND slashes, it's the same thing, except that the inflation adds some dimensions as well.

We'll illustrate this on the torus tiger, or cyltorintigroid, (((II)I)(II)).

It's 4D cuts are obtained by erasing one slash:
(((I)I)(II)): (I) is two points, ((I)I) is two points inflated into second dimension, i.e. two separated circles. (II) is a circle, so (((I)I)(II)) is two separated circles x circle + inflated, meaning two separated tigers.
(((II))(II)): (II) is a circle, ((II)) are two concentric circles, so (((II))(II)) is a cartesian product of two concentric circles and another circle, then inflated. This is two tigers which differ in one of their outer radii.
(((II)I)(I)): ((II)I) is a torus, (I) is two points, so (((II)I)(I)) is two toruses displaced in fourth dimension, then inflated. This is two ditoruses displaced in their inner dimension.
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Re: I live!

Postby Marek14 » Thu Nov 24, 2011 11:43 am

I wonder, would it be possible to make some crude toratope models? Angular, perhaps like this:

Glome 1:
x = cos a cos b cos c
y = cos a cos b sin c
z = cos a sin b
w = sin a

a and b from -pi to pi, c from 0 to 2pi, all sampled by 8 points.

Glome 2:
x = cos a cos b
y = cos a sin b
z = sin a cos c
w = sin a sin c

a from -pi to pi, b and c from 0 to 2pi, all sampled by 8 points.

Toraspherinder:
x = 4 cos b cos c + cos a cos b cos c
y = 4 cos b sin c + cos a cos b sin c
z = 4 sin b + cos a sin b
w = sin a

I think that ratio of 4:1 between radii makes for a good picture.

a from 0 to 2pi, sampled by 8 points.
b from -pi to pi, sampled by 32 points.
c from 0 to 2pi, sampled by 32 points.

Tiger:
x = 4 cos b + cos a cos b
y = 4 sin b + cos a sin b
z = 4 cos c + sin a cos c
w = 4 sin c + sin a sin c

a from 0 to 2pi, sampled by 8 points.
b and c from 0 to 2pi, sampled by 32 points.

Ditorus:
x = 16 cos c + 4 cos b cos c + cos a cos b cos c
y = 16 sin c + 4 cos b sin c + cos a cos b sin c
z = 4 sin b + cos a sin b
w = sin a

a from 0 to 2pi, sampled by 8 points.
b from 0 to 2pi, sampled by 32 points.
c from 0 to 2pi, sampled by 256 points.

Toracubinder:
x = 4 cos c + cos a cos b cos c
y = 4 sin c + cos a cos b sin c
z = cos a sin b
w = sin a

a from -pi to pi, sampled by 8 points.
b from 0 to 2pi, sampled by 8 points.
c from 0 to 2pi, sampled by 32 points.
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Re: I live!

Postby quickfur » Fri Nov 25, 2011 3:05 am

Marek14 wrote:I wonder, would it be possible to make some crude toratope models? Angular, perhaps like this:

Glome 1:
x = cos a cos b cos c
y = cos a cos b sin c
z = cos a sin b
w = sin a

a and b from -pi to pi, c from 0 to 2pi, all sampled by 8 points.

Glome 2:
x = cos a cos b
y = cos a sin b
z = sin a cos c
w = sin a sin c

I've already done these two, in the wireframe spheres thread, including a few renders of them.

For the rest, it will take some work, because my viewer currently doesn't handle nonconvex shapes. It can probably be coerced to do something with a hand-made (i.e. not done by convex hull algo) polytope definition, but I doubt the results will be correct. :P
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Re: I live!

Postby quickfur » Fri Nov 25, 2011 3:08 am

quickfur wrote:[...] It can probably be coerced to do something with a hand-made (i.e. not done by convex hull algo) polytope definition, but I doubt the results will be correct. :P

Correction: the only thing that won't work correctly would be the visibility clipping, but the difficult part is generating the model without using a convex hull algo. If you have any good methods for doing this, i'm willing to give it a try.
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Re: I live!

Postby Marek14 » Fri Nov 25, 2011 7:44 am

quickfur wrote:
quickfur wrote:[...] It can probably be coerced to do something with a hand-made (i.e. not done by convex hull algo) polytope definition, but I doubt the results will be correct. :P

Correction: the only thing that won't work correctly would be the visibility clipping, but the difficult part is generating the model without using a convex hull algo. If you have any good methods for doing this, i'm willing to give it a try.


Well, I wanted to get an OFF file I could import to Stella. Stella handles nonconvex polychora without problems (it implements all the nonconvex uniforms).
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Re: I live!

Postby Keiji » Fri Nov 25, 2011 2:32 pm

quickfur wrote:
quickfur wrote:[...] It can probably be coerced to do something with a hand-made (i.e. not done by convex hull algo) polytope definition, but I doubt the results will be correct. :P

Correction: the only thing that won't work correctly would be the visibility clipping, but the difficult part is generating the model without using a convex hull algo. If you have any good methods for doing this, i'm willing to give it a try.


Surely with parametric equations there shouldn't be an issue writing a utility to generate all the hypercells in the figure?
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Re: I live!

Postby quickfur » Fri Nov 25, 2011 3:42 pm

Keiji wrote:[...] Surely with parametric equations there shouldn't be an issue writing a utility to generate all the hypercells in the figure?

That's what I thought, but when I tried to use the parametric equations to generate the tiger, I got some strange results. Maybe there are some bugs in my code. :P
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Re: I live!

Postby quickfur » Fri Nov 25, 2011 3:51 pm

Marek14 wrote:[...] Well, I wanted to get an OFF file I could import to Stella. Stella handles nonconvex polychora without problems (it implements all the nonconvex uniforms).

Here's the toroidal subdivision of the 3-sphere: 3-sphere (you may have to manually change the .txt extension to .off, apparently kawachan changed the .off into .txt.)
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Re: I live!

Postby Marek14 » Fri Nov 25, 2011 5:01 pm

Nice, especially the way stella displays the polychoral net :D Looks real fun.

It's really incredible that the whole thing has only two distinct types of cells!
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Re: I live!

Postby quickfur » Fri Nov 25, 2011 5:30 pm

Marek14 wrote:Nice, especially the way stella displays the polychoral net :D Looks real fun.

It's really incredible that the whole thing has only two distinct types of cells!

Well, that's not perfectly true... they are mostly square frustums with varying edge lengths (and different volumes). They are only "the same" topologically. :)
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Re: I live!

Postby Marek14 » Fri Nov 25, 2011 5:48 pm

quickfur wrote:
Marek14 wrote:Nice, especially the way stella displays the polychoral net :D Looks real fun.

It's really incredible that the whole thing has only two distinct types of cells!

Well, that's not perfectly true... they are mostly square frustums with varying edge lengths (and different volumes). They are only "the same" topologically. :)


I think that's wrong, actually. There are only four layers between two toroidal poles, and every cell around the poles is identical, and every cell in the middle layers seems to be identical as well, as there can be transformed into each other just by rotations and reflections.

Stella would recognize any differences, it only deems cells of the same type if there is a symmetry that maps one onto another.
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Re: I live!

Postby quickfur » Fri Nov 25, 2011 6:08 pm

Marek14 wrote:
quickfur wrote:[...] Well, that's not perfectly true... they are mostly square frustums with varying edge lengths (and different volumes). They are only "the same" topologically. :)


I think that's wrong, actually. There are only four layers between two toroidal poles, and every cell around the poles is identical, and every cell in the middle layers seems to be identical as well, as there can be transformed into each other just by rotations and reflections.

Stella would recognize any differences, it only deems cells of the same type if there is a symmetry that maps one onto another.

Hmm you're right, I did make the model with only 4 layers. :) I was thinking of the general case with many layers. That's the danger of thinking in general terms. :P
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Re: I live!

Postby Marek14 » Fri Nov 25, 2011 6:47 pm

Yes, but that's good since it means that a low-resolution toratopes will also be simple, cell-wise.
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Re: I live!

Postby Keiji » Fri Nov 25, 2011 7:42 pm

quickfur wrote:
Marek14 wrote:[...] Well, I wanted to get an OFF file I could import to Stella. Stella handles nonconvex polychora without problems (it implements all the nonconvex uniforms).

Here's the toroidal subdivision of the 3-sphere: 3-sphere (you may have to manually change the .txt extension to .off, apparently kawachan changed the .off into .txt.)


Actually you can link to http://teamikaria.com/hddb/dl/GQP10XZ3E ... J2NH7G.off or even http://teamikaria.com/hddb/dl/GQP10XZ3E ... sphere.off if you want. Kawachan doesn't care what you put after the first dot or slash, and I implemented it like this entirely so that you can hint filename or extension. The reason it changed it to .txt is because it inspects the contents of files to determine their type and doesn't know the OFF file format.
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Re: I live!

Postby quickfur » Fri Nov 25, 2011 8:12 pm

Keiji wrote:[...] Kawachan doesn't care what you put after the first dot or slash, and I implemented it like this entirely so that you can hint filename or extension. The reason it changed it to .txt is because it inspects the contents of files to determine their type and doesn't know the OFF file format.

I see. Cool!
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