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
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])
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
Marek14 wrote:I wonder if there are people who'd want unusual geometric games I still wonder about feasibility of hyperbolic Carcassone.
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).
[...]
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 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: [... snipped lots of good stuff...]
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?
[...]
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!
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
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.
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.
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.
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.
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.
Keiji wrote:[...] Surely with parametric equations there shouldn't be an issue writing a utility to generate all the hypercells in the figure?
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).
Marek14 wrote:Nice, especially the way stella displays the polychoral net Looks real fun.
It's really incredible that the whole thing has only two distinct types of cells!
quickfur wrote:Marek14 wrote:Nice, especially the way stella displays the polychoral net 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.
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.
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.)
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.
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