A long overdue discussion about the future of rotopes

Higher-dimensional geometry (previously "Polyshapes").

A long overdue discussion about the future of rotopes

Postby quickfur » Thu Nov 06, 2008 6:02 pm

Hayate wrote:[...]
You can blame me for all these problems, by the way, since I at-the-time-unknowingly introduced these problems when adding the tapering operation to rotopes, and I regret doing that.
[...]

Hmm, I'm not sure I see what the problem with the tapering operation is. I think it is relatively safe to assume that any tapered object does include the point at the apex, since even if the shape is of non-zero genus, the apex can be attained as the limit of the tapering process. Of course, the resulting object may acquire unusual properties in the process, but hey, that's what makes things more interesting. :)

The way I see it is, we can define the taper of any set S of n-dimensional points to be taper(S,h) = {(1-t/h)*<x_1, x_2, ... x_n, 0> + <0,0,0,...,t> | <x_1, x_2, ... x_n> in S, 0<=t<=h}, where h is the height of the (n+1)-dimensional pyramid thus produced. This definition ensures that as long as S is non-empty, the apex <0,0,0,...,h> is always in taper(S,h). Note that this works for any set S, including pathological fractal sets and non-constructive sets. So if the apex is well-defined for those things, it should be well-defined for "normal" objects too!

Split from "Defining the new surfaces when performing an extrusion". ~Hayate
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Re: Defining the new surfaces when performing an extrusion

Postby Keiji » Thu Nov 06, 2008 7:13 pm

Well, I guess that takes care of the ambiguous rotopes, but those are the least inconvenient of the lot. Would you mind explaining the pyramidal tigroids? :P
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Re: Defining the new surfaces when performing an extrusion

Postby quickfur » Thu Nov 06, 2008 7:16 pm

Hayate wrote:Well, I guess that takes care of the ambiguous rotopes, but those are the least inconvenient of the lot. Would you mind explaining the pyramidal tigroids? :P

Well, first, I gotta understand what tigroids are in the first place. :P

I remember those long threads about tigers in the forest, and perhaps I had even looked at some of the equations, but I've never actually gotten to the point where I understand just what they were. Maybe you could refresh my memory, if you don't mind. ;)
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Re: Defining the new surfaces when performing an extrusion

Postby Keiji » Thu Nov 06, 2008 7:18 pm

A tigroid is basically any strange rotope with a single outermost pair of parentheses.

A pyramidal tigroid is a tigroid with at least one superscript inside it, not counting any superscripts that don't have any sibling groups after them.

There is only one example in 4D, which is the cyltrianglintigroid.
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Re: Defining the new surfaces when performing an extrusion

Postby quickfur » Thu Nov 06, 2008 7:44 pm

Hayate wrote:A tigroid is basically any strange rotope with a single outermost pair of parentheses.

A pyramidal tigroid is a tigroid with at least one superscript inside it, not counting any superscripts that don't have any sibling groups after them.

There is only one example in 4D, which is the cyltrianglintigroid.

OK, before we even get to strange rotopes, I need to understand how exactly group notation works. In the "rotopic group notation" page, it is explained what individual elements mean, but it's unclear what is meant when you string multiple elements together. For example, I'm assuming that (xy) means spherated in the XY plane (i.e., a circle), and (xyz) means spherated in the XYZ plane (i.e., a sphere). But what does (wx)(yz) mean? By itself, (wx) means a circle in the WX plane, and (yz) means a circle in the YZ plane, but what does it mean when you juxtapose these elements? How are they combined?
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Re: Defining the new surfaces when performing an extrusion

Postby Keiji » Thu Nov 06, 2008 9:26 pm

Juxtaposition is the Cartesian product, however the taper operation applies to everything before it and takes precedence. So the cyltrianglinder, the non-spherated form of the cyltrianglintigroid, is the Cartesian product of a circle and a triangle.

Obviously since rotopic notation doesn't have any precedence brackets the Cartesian product of two triangles is not a rotope. This is another annoying thing that I don't like about the notation after tapering was added.
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Re: Defining the new surfaces when performing an extrusion

Postby quickfur » Thu Nov 06, 2008 9:49 pm

Hayate wrote:Juxtaposition is the Cartesian product, however the taper operation applies to everything before it and takes precedence. So the cyltrianglinder, the non-spherated form of the cyltrianglintigroid, is the Cartesian product of a circle and a triangle.

OK, so let me unpack the notation for the cyltrianglinder to see if I've understood it right: 1 is a line segment, so 1¹ is a tapered line segment, i.e., triangle. 2 is a circle, so 1¹2 is the Cartesian product of the triangle and a circle (what I call a triangular prismic cylinder, or 3-prismic cylinder). Or alternatively, x is a line segment along the X axis, so x^y is this line segment tapered along Y, i.e., a triangle in the XY plane. (zw) is the spheration in Z and W, i.e., a circle in the ZW plane, so x^y(zw) is the Cartesian product of the triangle and the circle (same as before). Do I understand it correctly so far?

Alright, now for the tigroid: (1¹2). To understand this properly, I need to know how the spheration operator works. When you write (abc), for example, I assume you are spherating along the axis a, b, and c, right? Or does it mean you take the Cartesian product of a, b, and c (i.e. a cube) and then the spheration operator does something to it that turns it into a sphere?

Obviously since rotopic notation doesn't have any precedence brackets the Cartesian product of two triangles is not a rotope. This is another annoying thing that I don't like about the notation after tapering was added.

Maybe we should adopt reverse Polish notation for rotopes, then. :lol:
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Re: Defining the new surfaces when performing an extrusion

Postby Keiji » Fri Nov 07, 2008 5:46 am

Do I understand it correctly so far?


Yes

I need to know how the spheration operator works. When you write (abc), for example, I assume you are spherating along the axis a, b, and c, right? Or does it mean you take the Cartesian product of a, b, and c (i.e. a cube) and then the spheration operator does something to it that turns it into a sphere?


I believe the spheration operator works on the result of the Cartesian product. However, (ab...n) is not spheration, that's just the definition of an n-sphere. The spheration operator is only invoked when there are more groups or superscripts inside the group. But I still don't know precisely how it works.

You know, as soon as I find a way to enumerate SSC2 without leaving enormous gaps, I can just obsolete rotopes and that will solve the whole problem :P
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Re: Defining the new surfaces when performing an extrusion

Postby quickfur » Fri Nov 07, 2008 6:30 pm

Hayate wrote:
quickfur wrote:[...]I need to know how the spheration operator works. When you write (abc), for example, I assume you are spherating along the axis a, b, and c, right? Or does it mean you take the Cartesian product of a, b, and c (i.e. a cube) and then the spheration operator does something to it that turns it into a sphere?


I believe the spheration operator works on the result of the Cartesian product. However, (ab...n) is not spheration, that's just the definition of an n-sphere. The spheration operator is only invoked when there are more groups or superscripts inside the group. But I still don't know precisely how it works.

Well then, that explains why we have so much trouble with it. We don't even know what it is in the first place! :D

You know, as soon as I find a way to enumerate SSC2 without leaving enormous gaps, I can just obsolete rotopes and that will solve the whole problem :P

This exercise has piqued my interest in enumerating shapes... Yesterday, I thought about it a bit, and came up with the following system for enumerating simple shapes that covers all the rotatopes, simplices, cross polytopes, and hypercubes, among other things, using only tapering operations(!).

The basic object is a point, denoted, unsurprisingly, by a dot (.). From this, everything else is derived. Derivations work via the following operators, which are written following an object, and are applied from left to right (they are non-associative and non-commutative). Each operator increases the dimension by 1.

| (vertical bar) - extrude the object along the new dimension. We don't really care about the length of extrusion if we only care about the topology of the resulting object. Formal definition: x| = { <x1,x2,...xn,t> | <x1,x2,...xn> is in x; 0<=t<=h }, for any object x and some chosen length h. For example, .| is a line segment, .|| is a square, .||| is a cube, and .|||| is a tesseract.

A (or delta) - taper the object to a point along the new dimension. Formal definition: xA = { (1-t/h)*<x1,x2,...xn,0> + <0,0,0,...t> | <x1,x2,...xn> is in x; 0<=t<=h }, where x is any object, and h is the height of the pyramid (which can be any arbitrary positive real, if we only care about the topology of the resulting object). For example, .|A is a triangle, .|AA is a tetrahedron, .|AAA is a pentachoron.

X (or diamond) - taper the object to two points along opposite directions along the new dimension. In other words, make a tegum based on the current object. (Formal definition is left as an exercise for the reader.) For example, .||X is an octahedron, .||XX is a 16-cell.

O (circle) - just like X, except that the object is not linearly tapered, but varies according to a circular curve. Formal definition: xO = { sqrt(R^2 - K*(1-t/h)^2)*<x1,x2,...xn,0> +/- <0,0,0,...t> | <x1,x2,...xn> is in x; 0<=t<=R }, where x is any object, and R is the radius of the circular path, usually taken to be 1, and K is some constant chosen so that the resulting object has circular cross-sections. So, .|O is a circle, and .|OO is a sphere; and .||O is a crind.

Note that these operations are indistinguishable in 1D, since any of them applied to a point gives a line segment. Also, since every object always begins with the point '.', we can abbreviate the notation by omitting the '.', and so we write:

| = line segment
|| = square
||| = cube
|||| = tesseract
||||| = 5-cube
A = line segment
AA = triangle
AAA = tetrahedron
AAAA = 5-cell
AAAAA = hexateron
X = line segment
XX = square (diamond)
XXX = octahedron
XXXX = 16-cell
XXXXX = 5-cross
O = line segment
OO = circle
OOO = sphere
OOOO = 3-sphere
OOOOO = 4-sphere

These generate the 4 families of basic shapes. Technically, we could omit the first symbol as well, since it is always the line segment, but I decided to leave it in so that we have the nice correspondence that the number of dimensions equals the number of symbols.

Here are the notations for Garrett Jones' rotatopes:

|O| = cylinder
|O|| = cubinder
|O|O = duocylinder
|OO| = spherinder

We can also make other interesting shapes:

|OA = cone (circular pyramid)
|OA| = cone prism (coninder)
|O|A = cylindrical pyramid (cylindrone)
||O = crind
||O| = crind prism
||OA = crind pyramid
|AO = triangular crind (made of 3 lunes)
|A| = triangular prism
|A|O = triangular prismic crind (made of 2 triangular lunar chorages and 3 square lunar chorages)
|AA| = tetrahedral prism
|A|| = 3,4-duoprism

I find it quite interesting that such a deficient notation (which doesn't even let you specify rotation except implicitly using the O operator) is capable of expressing such a great variety of shapes. :) I can, of course, think of several extensions that will add more shapes to the palette, but haven't decided on which ones to include yet.
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Re: Defining the new surfaces when performing an extrusion

Postby Keiji » Fri Nov 07, 2008 9:45 pm

quickfur wrote:Well then, that explains why we have so much trouble with it. We don't even know what it is in the first place! :D


Well, the spherator has been defined algebraically, but of course that only works if what is inside it can also be defined algebraically. The algorithm for converting toratopic group notation into an implicit equation can be found on HDDB. If we found a way to extend this algorithm to the complete rotopic group notation, that'd define all operations for us, but I don't think that's possible due to non-toratopes having vertices. :P

I thought about it a bit, and came up with the following system for enumerating simple shapes...


This looks like you just took the bracketopic products, added the taper operation and removed the whole idea of grouping things. Sure, it gives you the full set of classic rotopes without any definition problems and also contains the entire cross polytope set, but it doesn't have room for any toratopes other than hyperspheres or some of the more interesting (especially higher-dimensional) bracketopes such as the 3,3-duoprism.

I guess it is a more diverse set than bracketopes and rotopes individually, though. I definitely like the introduction of the trigonal crind, a shape which I've never thought about before!

Also, why do you claim it uses only the taper operation? Clearly only one of your four operators actually tapers.
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Re: Defining the new surfaces when performing an extrusion

Postby quickfur » Fri Nov 07, 2008 10:33 pm

Hayate wrote:
quickfur wrote:Well then, that explains why we have so much trouble with it. We don't even know what it is in the first place! :D


Well, the spherator has been defined algebraically, but of course that only works if what is inside it can also be defined algebraically. The algorithm for converting toratopic group notation into an implicit equation can be found on HDDB. If we found a way to extend this algorithm to the complete rotopic group notation, that'd define all operations for us, but I don't think that's possible due to non-toratopes having vertices. :P

Ah, so then the task is to define an algebraic function such that when applied to an equation specifying a shape will turn it into another equation that specifies the tapering of that shape. Once we have such a function, the spheration of tapered objects will be well-defined.

I thought about it a bit, and came up with the following system for enumerating simple shapes...


This looks like you just took the bracketopic products, added the taper operation and removed the whole idea of grouping things. Sure, it gives you the full set of classic rotopes without any definition problems and also contains the entire cross polytope set, but it doesn't have room for any toratopes other than hyperspheres or some of the more interesting (especially higher-dimensional) bracketopes such as the 3,3-duoprism.

I guess it is a more diverse set than bracketopes and rotopes individually, though. I definitely like the introduction of the trigonal crind, a shape which I've never thought about before!

Also, why do you claim it uses only the taper operation? Clearly only one of your four operators actually tapers.

All these operations are based on the idea of scaling and displacement, which is the fundamental idea behind the original taper operation (take an object, scale it down by the distance it is displaced along the new dimension). The original tapering operation is to translate an object along a new dimension, and linearly reducing its scale until it has vanished into a point. The extrusion operation may be thought of a "tapering" where the object is constantly scaled by 1 (so it's some kind of "degenerate" tapering). The tegum operation is basically tapering in two opposite directions simultaneously, and the O-operator is like the tegum operation, except that the scale of the object does not diminish linearly, but according to a square-root-of-square-sums function. In other words, both a diamond and a disc can be thought of as "stacking" line segments of diminishing length, the only difference being how the length is diminished, linearly or according to an rss function. (This is not to be confused with algebraically applying an rss function to a shape: the results are not the same.)

Any object produced by applying these operations to another object has the property that all its intersections with hyperplanes perpendicular to the last added dimension are scaled versions of the original object. The four operators I listed aren't the only possible ones, of course, since you could apply any function to the scaling factor as the object is displaced along the new dimension. For example, you could create paraboloids by applying an operator with a square root function on the scaling factor to a circle or sphere, say.

These functions don't have to be smooth, of course; you can, for example, create any convex polygon by using a function that varies according to the secant lengths of the polygon, and applying this function to a line segment. In fact, certain non-convex polygons can be generated this way as well, using a function that varies in a zig-zag fashion. (This breaks down with star polygons that require multi-valued functions.)

However, this notation only goes so far: it is unable to generate shapes for which there is no sequence of cross-sections that are scaled copies of each other. For example, you can't generate a dodecahedron, because there is no series of cross-sections of a dodecahedron that are scaled versions of a single polygon. (We were lucky with 2D polygons because 1D line segments are all isomorphic to each other. This is no longer true with 2D cross-sections.) Nonetheless, the shapes that can be generated this way are quite fascinating, given the limitations of the method.
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Re: A long overdue discussion about the future of rotopes

Postby quickfur » Fri Nov 07, 2008 11:17 pm

I should also add that these operations do generate an entire family tree of crind-like shapes, each with lune-like bounding manifolds.

To help with clarity in the following discussion, I'll introduce a naming scheme for crind-like objects. First, a "crind" without any additional qualifiers will refer to the square crind (||O in the notation system I described above, which I will call Taper Notation, or TN). If a modifier precedes it, it will refer to the result of applying the O operator to the object specified by the modifier; for example, a square crind is a square (||) with the O operator applied to it (||O). If we're speaking of a general crind-like shape in n-dimensions, we may denote it as an n-crind (with the caveat that it may refer to more than one kind of crind shape in that dimension). Note that a crind will always be 1 dimension higher than the modifying object, so it is sufficient to say, e.g., a cubical crind, which implies it is a 4-crind. Further, if we refer to a single n-crind without specifying its base shape, it will be understood that the base shape is an (n-1)-crind.

Now, crinds are bounded by lunes, for which we shall adopt a similar naming convention. A "lune" without any additional qualifiers will refer to the 2-lune, one of the 4 curved faces of the (square) crind. A triangular crind also has 2-lunes for its faces, albeit narrower (for our purposes, the term "2-lune" will apply to both). If preceded by an adjective specifying a shape (e.g., "square lune"), it will refer to the corresponding surfold of the crind-shape generated by applying the O operator to a shape having the adjectival shape as one of its surfolds (e.g., the square lune is a surfold of the cubical crind, |||O). An n-lune is a lune-shaped n-manifold. As with crinds, the dimension of a lune is always 1 higher than the dimension of the modifying adjective, so it suffices to say, e.g., square lune, which implies it is a 3-lune. If we refer to a single n-lune without specifying its base shape, it will be understood that the base shape is an (n-1)-lune.

If a crind or a lune appears as the base shape of a higher object, we will use the adjectives "crindal" and "lunar" to describe it. E.g., a crindal prism = extrusion of a crind; a lunar prism = extrusion of a lune.

Now, on to the shapes themselves. The 3D shapes involving crinds are ||O (the "standard" crind) and |AO (the triangular crind). The "circular crind" is identical to the sphere.

In 4D, we get a lot more variety:

|||O is the cubical crind, bounded by 6 square lunes
||O| is the crindal prism, bounded by 2 crinds and 4 lunar prisms
||OO is a crindal crind, bounded by 4 3-lunes
|AO is a triangular crind, bounded by 3 lunes
|AO| is a triangular crind prism, bounded by 3 3-lunes and 2 triangular crinds
|AOO is a triangular 3-crind, bounded by 3 3-lunes
|OAO is a conical crind, bounded by 1 circular lune and a nappe-lune
|XXO is an octahedral crind, bounded by 8 triangular lunes.
||AO is a square pyramidal crind, bounded by 4 triangular lunes and 1 square lune
||OA is a crind pyramid, bounded by a crind and 4 lunar pyramids
|A|O is a triangular prism crind, bounded by 2 triangular lunes and 3 square lunes
|AAO is a tetrahedral crind, bounded by 4 triangular lunes
(this list is not exhaustive)

Things get tastier in 5 dimensions:
OOO|O is a spherindrical crind, bounded by 2 spherical lunes and a spherindrical surfold lune
|AAOO is a tetrahedral crindal crind, bounded by 4 triangular lunar lunes
||||O is a tesseractic crind, bounded by 8 cubical lunes
||AAO is a square pyramidal pyramidal crind, bounded by 2 square pyramidal lunes, and 4 tetrahedral lunes
|O||O is a cubindrical crind, bounded by 4 cylindrical lunes and a square toroidal lune.

Well, you get the idea. By applying the O operator to any 4D object, we get a crindal object in 5D, with each surfold/surcell giving rise to new and bizarre lune shapes.

P.S. I think I made a mistake about the duocylinder: you cannot actually obtain the duocylinder with |O|O; that actually is a cylindrical crind bounded by 2 circular lunes and a lunar torus. You need a different operator to get the duocylinder from the cylinder, one which varies the height of the cylinder but does not change its radius.
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Re: A long overdue discussion about the future of rotopes

Postby Keiji » Sat Nov 08, 2008 3:41 pm

Right, I was actually wondering how you ended up with the duocylinder in there without any products. =p

But wow, that's quite some work you've done there. I'd gladly add the set to HDDB, if you could come up with an name for it! :D
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Re: A long overdue discussion about the future of rotopes

Postby quickfur » Mon Nov 10, 2008 4:11 am

Hayate wrote:Right, I was actually wondering how you ended up with the duocylinder in there without any products. =p

But wow, that's quite some work you've done there. I'd gladly add the set to HDDB, if you could come up with an name for it! :D

Quite some work? Not really! I came up with it over a period of, oh, 2 days? :P I question the utility of this notation for general use, because it really only covers a small set of possibilities. Further, actual, research would be needed to come up with a more general scheme. Consider it as my first attempt, my first dipping into the water. :)
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Re: A long overdue discussion about the future of rotopes

Postby Keiji » Mon Nov 10, 2008 6:01 am

I question the utility of this notation for general use, because it really only covers a small set of possibilities.


As I mentioned, it's more diverse than rotopes and bracketopes individually, doesn't have any annoying screwy thing about it, and doesn't actually miss out too much of the rotope/bracketope sets (only more complex products and tigroids).
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Re: A long overdue discussion about the future of rotopes

Postby quickfur » Mon Nov 10, 2008 5:50 pm

Hayate wrote:
I question the utility of this notation for general use, because it really only covers a small set of possibilities.


As I mentioned, it's more diverse than rotopes and bracketopes individually, doesn't have any annoying screwy thing about it, and doesn't actually miss out too much of the rotope/bracketope sets (only more complex products and tigroids).

With a few extensions, it could cover stuff like duocylinders as well, e.g., introduce an operator that scales the base object unevenly, so you could start with a cylinder with height tapered in both directions along a half-circular curve, but with constant radius, resulting in a duocylinder. Alternatively, if you introduce Cartesian products, then a greater variety of objects are available (I've already worked out a possible syntax to denote Cartesian products).

The main limitation, though, is that it misses out on many common interesting objects such as the uniform polytopes and icosahedral polytopes (icosahedron, dodecahedron, 120-cell, 600-cell, and their derivatives). I suppose if you shoehorn the Conway operators into the mix, you'll be able to get these, but things get less pretty with unusual shapes such as the grand antiprism.

But about tigroids, are there any tigroids which does not have an ill-defined operation (such as spheration on a complex object)?
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Re: A long overdue discussion about the future of rotopes

Postby quickfur » Mon Nov 10, 2008 7:41 pm

Just a little update: I've thought a bit more about some possible extensions to this notation system (which may or may not be of much interest).

Cartesian products

Since Cartesian products are so useful in generating interesting higher-dimensional shapes, it makes sense to extend the notation to include them. The extension is very simple: the Cartesian product of two shapes with notations X and Y are written simply as XY. Note that Y cannot be in abbreviated form (the base object "." must be present, since otherwise the string XY may be misinterpreted as applying the operations Y to the object X). So, the duocylinder would be written .|O.|O, or simply |O.|O (note that the "." here is NOT a Cartesian product operator; it is the base object, ".", the point).

Polygonal operators

Since polygons are so useful, it's nice to be able to denote them. For this, it is useful to add new operators that take the line segment .| and makes polygons out of them. For consistency with the rest of the operators, we still restrict ourselves to "tapering" operations: we simply scale the base object appropriately as we translate it along the new dimension. Since all (convex) polygons can be decomposed into a "stack of lines", it is possible to obtain them all this way.

Furthermore, we are really only interested in regular polygons currently, and it so happens that regular polygons allow us to uniformly scale the object at each offset along the new dimension (we don't need to translate it). To achieve this, we simply divide the desired polygon into two halves along any of its lines of symmetry, and the length of each "secant" orthogonal to the selected line of symmetry gives us the amount by which we need to scale the base object at that displacement.

For odd polygons, all lines of symmetry pass through exactly 1 vertex and bisect 1 edge. So, there is a unique way to derive the polygon from the line segment. This gives rise to a unique "tapering" operator that derives each odd polygon from the line segment. We may denote this operator by the degree of the polygon; hence, .|5 is the pentagon, and .|7 is the heptagon. This operator is general; we can apply it to higher-dimensional shapes to get objects made out of adjoined frustums and pyramids. For example, .||5 is the "pentagonization" of the square, which is a 3D solid made of a frustum adjoined with a square pyramid, such that it has a pentagonal cross-section along its axis of symmetry. To avoid ambiguity with multiple polygon operators in a sequence, we may delimit these numerical operators by writing (.|5)5 as .|[5>5. (The reason for writing [5> instead of <5> or [5] will be explained below.)

For even polygons, there are two different lines of symmetry: one which bisects two edges, and one which intersects with two vertices. So there are two ways of deriving the same polygon from the line segment. When the desired polygon is the square, we see that this corresponds with the difference between the cross polytope and the measure polytope: .|| is the measure, obtained by extruding .|, and .|X is the cross, obtained by bi-tapering .|. In 2D, these two classes of operators are equivalent; however, in 3D and above, they diverge. For example, the hexagon may be obtained either as a truncated bi-tapering (taper the object either way but stop before it reaches a point, resulting in 6 edges), or by an extrusion followed by a bi-tapering (extrude the object partway to obtain 2 of the hexagon's edges, then taper either side to a point to obtain 2 more edges on either side). We will denote the former as [6], and the latter as <6>. So, .|<6> and .|[6] are both the hexagon, but .||<6> is a solid made of a cube with square pyramids adjoined on two opposite faces, and .||[6] is a solid made of two frustums adjoined at their bases. Both have a hexagonal cross-section along their axis of symmetry.

Now the reason for writing the odd polygonal operators as <5] or [5> should be clear: the [ indicates edge bisection, and > corresponds with vertex intersection.

Examples

With these extensions to the notation, we can now create whole new families of objects.

All regular polygons are now available.

All the uniform prisms are also now available: .|A| (triangular prism, same as .|3 or .|[3> or .|<3]), .||| (square prism, same as cube, or diamond prism, .|X|), pentagonal prism (.|5|, or .|[5>|, or .|<5]|), hexagonal prism (.|6|, which refers to either .|[6]| or |<6>|), heptagonal prism (.|7|), etc.. Note that where unambiguous, we may dispense with the cumbersome bracketed notation and just write the degree of the polygon.

Polygonal pyramids are also now available: e.g., pentagonal pyramid, .|5A; hexagonal pyramid, .|6A. Polygonal bipyramids are also now available: e.g., heptagonal bipyramid, .|7X.

We can now have the general polygonal crind: e.g., heptagonal crind, .|7O or .|[7>O (to avoid confusion with .|70, a 70-gon).

Besides this, higher dimensional "polygonizations" are also now available: e.g., .||<6>, a cube (cuboid) with two square pyramids adjoined to two opposite ends; .|<5><6> (also written .|5<6>), a pentagonal prism with two adjoined pentagonal pyramids, .||[6], two adjoined square frustums (the shape of the face-first projection of the tetracube into 3D), .|5[6] (two adjoined pentagonal frustums), and so forth.

In 4D, we have things like .|||<6>: a tetracuboid with two adjoined cubical pyramids, .|||[6], two adjoined cubical hyper-frustums, .OOO5: a spherindrical frustum adjoined with a spherical cone, and many other such shapes.

The corresponding Cartesian products of these shapes are also now available, so all the duoprisms can now be generated, e.g., .|5.|7: the 5,7-duoprism; .|8.|3: the 8,3-duoprism, or .|O.|7: the heptagonal prismic cylinder (heptagon-circle Cartesian product). And what about .|<8>O.|| - the 5D (octagonal crind)-square Cartesian product? And what about (.|<8>O.||)[6], the edge-aligned hexagonalization of this object? (That's a 6D object consisting of two adjoined ((octagonal crind)-square Cartesian product)-al frustums.) The possibilities are endless. :)

(And we're still only dealing with objects derived from "tapering", plus the Cartesian products. Still no way to derive the uniform polytopes or icosahedral polytopes.)
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Re: A long overdue discussion about the future of rotopes

Postby Keiji » Mon Nov 10, 2008 11:31 pm

All that looks rather cumbersome and ugly if you ask me.

But, | corresponds to [4], X to <4> and A to <3] or [3>. O becomes [0] and <0> (both the same). I don't really like the inclusion of the initial . and |, so let's forget those. I will now shorten ][ and >< (positive continuation) to | and >[ and ]< (negative continuation) to X. This is not ambiguous as all evens use same brackets and odds use alternating brackets.

Therefore we have these series:
[4] = square
[4|4] = cube
[4|4|4] = tesseract

<4> = diamond
<4|4> = octahedron
<4|4|4> = hexadecachoron

[0] = circle
[0|0] = sphere
[0|0|0] = glome

<3> = triangle
<3|3> = tetrahedron
<3|3|3> = pentachoron

Crinds are, for example:

[4|0] = crind
[4|4|0] = cubic crind
<3|0] = trigonal crind
<0|4|0> = biconic crind

Elongated bipyramids would be:

[6] = hexagon
[6|6] = elongated hexagonal bipyramid
[4|6] = elongated square bipyramid

etc.

Cartesian products will be shown by separated brackets:

[0][0] = duocylinder
<3][4] = 3,4-duoprism
[6][0|0] = hexagon x sphere
<5][6]<7] = pentagon x hexagon x heptagon

So, how do you like my suggestions?
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Re: A long overdue discussion about the future of rotopes

Postby quickfur » Tue Nov 11, 2008 1:07 am

Hayate wrote:All that looks rather cumbersome and ugly if you ask me.

But, | corresponds to [4], X to <4> and A to <3] or [3>. O becomes [0] and <0> (both the same). I don't really like the inclusion of the initial . and |, so let's forget those. I will now shorten ][ and >< (positive continuation) to | and >[ and ]< (negative continuation) to X. This is not ambiguous as all evens use same brackets and odds use alternating brackets.

Therefore we have these series:
[4] = square
[4|4] = cube
[4|4|4] = tesseract

<4> = diamond
<4|4> = octahedron
<4|4|4> = hexadecachoron

[0] = circle
[0|0] = sphere
[0|0|0] = glome

<3> = triangle
<3|3> = tetrahedron
<3|3|3> = pentachoron

Mmmm, I like this much better. :) Using non-matching brackets are rather ugly, I admit.

Crinds are, for example:

[4|0] = crind
[4|4|0] = cubic crind
<3|0] = trigonal crind
<0|4|0> = biconic crind

Hmm, is there any way of getting rid of the non-matching brackets? I repent for introducing them. :P A different way of denoting them would be much nicer.

[...]Cartesian products will be shown by separated brackets:

[0][0] = duocylinder
<3][4] = 3,4-duoprism
[6][0|0] = hexagon x sphere
<5][6]<7] = pentagon x hexagon x heptagon

So, how do you like my suggestions?

I'm thinking that perhaps we should just stick to <> for odd numbers instead of the inexcusably ugly <] or [>. Since odd number operators are exactly the same, we only have visual ugliness to lose.
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Re: A long overdue discussion about the future of rotopes

Postby wendy » Tue Nov 11, 2008 9:59 am

At the moment, i recognise five different products that are freely applyable to solids (with surfaces)

1. prism = repetition of content = radial: max()
2. tegum = drawing of surface = radial: sum()
3. crind = () = radial: rss()
4. prism = drawing of content (not coherent, adds a dimension)
5. comb = repetition of surface (not coherent, removes a dimension)

The first three are coherent, in that a unit of volume is formed by the product of unit lengths in perpendiculars. So one has

P2 = square, P3 = cubic, P4 = tesseractic, P4= prismatoteric
T2 = rhombic, T3 = octahedral, T4 = tegmatoteric
C2 = circular, C3 = spheric, C4 = glomic = glomatoteric

The last C4 generalises to all dimensions, eg glomatoyottic for C8.

Pyramids add 'altitude' over the bases. The pyramid product of p,q,r is over p+q+r+2 dimensions, 2 being the dimension of the simplex with 3 vertices. The general formula for this is derived from the space in an extra dimension, as

pP.qQ.rR, that p+q+r = 1, p,q,r positive.

This is the general description of a tegum-face, where the elements have faces P, Q, R. Since this can be represented in, say 3 dimensions, with the point x,y,z represents the prism-product of xP.yQ.zR, the pyramid forms in the plane x+y+z = 1. [the CO product forms in x+y+z=2, is a general instance of the lace figures).

The comb product generally leads to products of tilings (in the case where a complete tiling is regarded as the surface of a polytope having a body of half-space, eg {6,3} is the hexagonal tiling, and all the ground under it. So 2d##2d = 3d. For ordinary polytopes, it leads to torii.

Lace figures are a more general form of product, in as far as it admits 'progression' of the bases. The antiprism, for example, is a lace-figure over one base.
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Re: A long overdue discussion about the future of rotopes

Postby Keiji » Tue Nov 11, 2008 2:48 pm

...Mmmm, I like this much better. :) Using non-matching brackets are rather ugly, I admit... Hmm, is there any way of getting rid of the non-matching brackets? I repent for introducing them. :P A different way of denoting them would be much nicer... I'm thinking that perhaps we should just stick to <> for odd numbers instead of the inexcusably ugly <] or [>. Since odd number operators are exactly the same, we only have visual ugliness to lose.


Then, <n> = [n] where n = 0 or n is odd.

Thus [5][6][7] = pentagon x hexagon x heptagon but the biconic crind must stay as <0|4|0> (as [0|4|0] would mean the cylindrical crind).
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Re: A long overdue discussion about the future of rotopes

Postby quickfur » Fri Nov 14, 2008 12:22 am

Hayate wrote:
...Mmmm, I like this much better. :) Using non-matching brackets are rather ugly, I admit... Hmm, is there any way of getting rid of the non-matching brackets? I repent for introducing them. :P A different way of denoting them would be much nicer... I'm thinking that perhaps we should just stick to <> for odd numbers instead of the inexcusably ugly <] or [>. Since odd number operators are exactly the same, we only have visual ugliness to lose.


Then, <n> = [n] where n = 0 or n is odd.

Thus [5][6][7] = pentagon x hexagon x heptagon but the biconic crind must stay as <0|4|0> (as [0|4|0] would mean the cylindrical crind).


OK, just to summarize so that we have an idea of where we are:

- Instead of starting with a point and a line segment, both of which are trivially derived, we could just give them special symbols that do not occur elsewhere: a dot (.) for the point, and a vertical bar (|) for the line segment.

- For 2D and higher, we have <n> or [n] as the n-polygon, where n is either 0, or 3, or greater. In 2D, the choice of angle brackets or square brackets does not matter, since polygons are self-dual. The special case of n=0 will denote the circular taper, so the circle is written either as <0> or [0].

- For 3D and higher, if n is 0 or odd, then <n> and [n] are equivalent, and denotes the circular/polygonal taper operation. If n is even, then <n> denotes the polygonal operation that produces two vertices, and [n] denotes the polygonal operation that produces two facets.

- Operations with the same bracket type performed in a sequence is denoted by delimiting with vertical bars: <4|4|4> denotes the operation <4> applied 3 times to the line segment; i.e., an octahedron. [4|4|4] denotes the operation [4] applied 3 times to the line segment: i.e., a cube.

- If a sequence of operations has different bracket type, the transition is denoted by delimiting with X: <3X4> denotes the application of <3> followed by [4]; i.e., a triangular prism; whereas <3|4> denotes a trigonal bipyramid. [4X4] denotes [4] followed by <4>; i.e., an octahedron. (I hope I got this one right.)

- The Cartesian product of two objects is denoted by concatenation: [4][5] is a 4,5-duoprism, and [0][0] is a duocylinder.

Are we on the same page so far? :)
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Re: A long overdue discussion about the future of rotopes

Postby quickfur » Fri Nov 14, 2008 12:31 am

wendy wrote:At the moment, i recognise five different products that are freely applyable to solids (with surfaces)

1. prism = repetition of content = radial: max()
2. tegum = drawing of surface = radial: sum()
3. crind = () = radial: rss()
4. prism = drawing of content (not coherent, adds a dimension)
5. comb = repetition of surface (not coherent, removes a dimension)

Is there a typo here? Prism occurs twice?

[...]Lace figures are a more general form of product, in as far as it admits 'progression' of the bases. The antiprism, for example, is a lace-figure over one base.

What are lace figures? How is the base progression specified? I can see how one could, for example, derive a dodecahedron by expanding a pentagon, then truncating it progressively until it becomes a dual pentagon, and then shrink that to some non-zero size; this would trace out a dodecahedron if varied over the third dimension. The icosahedron could also be generated in a similar way.

The corresponding sequences for the 4D regular polychora are much more complex.
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Re: A long overdue discussion about the future of rotopes

Postby Keiji » Fri Nov 14, 2008 6:32 am

quickfur wrote:- Instead of starting with a point and a line segment, both of which are trivially derived, we could just give them special symbols that do not occur elsewhere: a dot (.) for the point, and a vertical bar (|) for the line segment.


Well, | is used elsewhere, so how about - (hyphen)?

Rest of post


Yes, everything else is correct :D
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Re: A long overdue discussion about the future of rotopes

Postby wendy » Fri Nov 14, 2008 8:58 am

quickfur wrote:
wendy wrote:At the moment, i recognise five different products that are freely applyable to solids (with surfaces)

1. prism = repetition of content = radial: max()
2. tegum = drawing of surface = radial: sum()
3. crind = () = radial: rss()
4. prism = drawing of content (not coherent, adds a dimension)
5. comb = repetition of surface (not coherent, removes a dimension)

Is there a typo here? Prism occurs twice?

It appeared twice by accident, 4 ought read 'pyramid'.

quickfur wrote:
[...]Lace figures are a more general form of product, in as far as it admits 'progression' of the bases. The antiprism, for example, is a lace-figure over one base.

What are lace figures? How is the base progression specified? I can see how one could, for example, derive a dodecahedron by expanding a pentagon, then truncating it progressively until it becomes a dual pentagon, and then shrink that to some non-zero size; this would trace out a dodecahedron if varied over the third dimension. The icosahedron could also be generated in a similar way.

The corresponding sequences for the 4D regular polychora are much more complex.


The usual sequence of "presenting a polytope down an axis" is pretty much what is called a 'lace-tower'. You have the coordinate of the tower (height), and a progression produced by different dynkin-symbols for the sections. So the examples given above would be,

dodecahedron x5o - f5o - o5f - o5x
icosahedron o5o - x5o - o5x - o5o

These are read as oPo = point (zero edge P-gon), x5o = pentagon o5x = dual pentagon, f = x of size 1.61803398875.

The next step is to generalise over an axis of two or more dimensions, eg write a graphic with labled points.

Code: Select all
                                  o5o
                         o5o  x5o    o5x   o5o

                           o5x  o5f f5o x5o
                       o5o   f5o      o5f   o5o

                        x5o  o5f  x5x f5o o5x

                       o5o   f5o      o5f   o5o
                          o5x  o5f  f5o   x5o

                         o5o  x5o     o5x  o5o
                                 o5o


This is the 500ch or 3,3,5. presented as a lace city. The points are scattered over decagons, the outer rim appears first as ten o5o, and then internally as x5x. There are rings of o5x /x5o and o5f / f5o.

The complete coordinate system could be written as a sum of prisms. eg

o5o . x5x
o5x . o5f
x5o . f5o
o5f . x5o
f5o . o5x
x5x . o5o

You see then that there is rows representing the assorted sections in pentagonal sections. The second and third rows correspond to the icosahedron and dodecahedron of unit edge. Were this done on a proper decagon, you would see all sorts of interesting things, like different sections interacting with each other.

Removing the o5o / x5x points, one gets the grand antiprism!

Lace prisms go on further.

In essence, they can be regarded as a coordinate system, including a non-mirror-axis, where each axis has its own dimensions. The simplex or pyramid product can be presented as an 'altitude' that is a simplex, and in each column, a point, except for the applied dimension. eg
Code: Select all
     x5o    o4o    x--.
     o5o    x4o    .--x


This is the prism product of a pentagon and a square. In the wx axis at v=-1, lies a pentagon x5o (here pentagon[1] * square[0]). At v=+1 lies a square x4o.

The first two axies each have two dimensions (wx, yz), while the third axis has one (v).

Since we can freely add things to the axies, we could write:
Code: Select all
     x5o    o4x    x--.
     o5x    x4o    .--x


This is a lace prism, formed by pentagon-square prism "progressing" into a rotated form. We see among its chora x5o || o5x (pentagon on pentagon), which gives an pentagonal antiprism. The actual determination of faces and other surtope consist has been developed for these.

There are interesting presentations of the 2_21, easily derived from a coordinate system, that takes the form of three intersecting bi-triangle prisms, in much the same way that the icosahedron is made of three intersecting golden rectangles. The interesting thing here is that the triangles are reversed on re-use.

Code: Select all
    x3o o3x o3o       x f o
    o3o x3o o3x       o x f
    o3x o3o x3o       f o x
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Re: A long overdue discussion about the future of rotopes

Postby quickfur » Fri Nov 14, 2008 6:03 pm

wendy wrote:
quickfur wrote:
wendy wrote:At the moment, i recognise five different products that are freely applyable to solids (with surfaces)

1. prism = repetition of content = radial: max()
2. tegum = drawing of surface = radial: sum()
3. crind = () = radial: rss()
4. prism = drawing of content (not coherent, adds a dimension)
5. comb = repetition of surface (not coherent, removes a dimension)

Is there a typo here? Prism occurs twice?

It appeared twice by accident, 4 ought read 'pyramid'.

Ahhh, that makes more sense. :)

[...]The usual sequence of "presenting a polytope down an axis" is pretty much what is called a 'lace-tower'. You have the coordinate of the tower (height), and a progression produced by different dynkin-symbols for the sections. So the examples given above would be,

dodecahedron x5o - f5o - o5f - o5x
icosahedron o5o - x5o - o5x - o5o

These are read as oPo = point (zero edge P-gon), x5o = pentagon o5x = dual pentagon, f = x of size 1.61803398875.

The next step is to generalise over an axis of two or more dimensions, eg write a graphic with labled points.

Code: Select all
                                  o5o
                         o5o  x5o    o5x   o5o

                           o5x  o5f f5o x5o
                       o5o   f5o      o5f   o5o

                        x5o  o5f  x5x f5o o5x

                       o5o   f5o      o5f   o5o
                          o5x  o5f  f5o   x5o

                         o5o  x5o     o5x  o5o
                                 o5o


This is the 500ch or 3,3,5. presented as a lace city. The points are scattered over decagons, the outer rim appears first as ten o5o, and then internally as x5x. There are rings of o5x /x5o and o5f / f5o.

To me, once we get to this point, it seems more expedient to express these objects using another, more concise system, e.g., the Schläfli symbol {3,3,5}. The limitation of the "tapering" or "stacking" approach, as I have developed above, is that it deals with polytopes section-wise. For simple progressions of bases, this lends well to geometric intuition; but once we get to figures such as {3,3,5} or {5,3,3}, the main point of interest is not so much the cross-sections as the symmetry group. So, for regular or uniform polytopes, where the main point of interest is symmetry (esp. elaborate symmetry, as in the {3,3,5}), a notation based on symmetry seems more appropriate.

The complete coordinate system could be written as a sum of prisms. eg

o5o . x5x
o5x . o5f
x5o . f5o
o5f . x5o
f5o . o5x
x5x . o5o

You see then that there is rows representing the assorted sections in pentagonal sections. The second and third rows correspond to the icosahedron and dodecahedron of unit edge. Were this done on a proper decagon, you would see all sorts of interesting things, like different sections interacting with each other.

Removing the o5o / x5x points, one gets the grand antiprism!

Nice.

[...]There are interesting presentations of the 2_21, easily derived from a coordinate system, that takes the form of three intersecting bi-triangle prisms, in much the same way that the icosahedron is made of three intersecting golden rectangles. The interesting thing here is that the triangles are reversed on re-use.

Code: Select all
    x3o o3x o3o       x f o
    o3o x3o o3x       o x f
    o3x o3o x3o       f o x

Hmm, this sounds like an interesting object to render on my polytope viewer. All I need is to compute the Cartesian coordinates of the vertices. ;)
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Re: A long overdue discussion about the future of rotopes

Postby Keiji » Fri Nov 14, 2008 10:57 pm

So, anyway, have we finalized this set?

I'd like to name it, enumerate it, and add it to HDDB. :)
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Re: A long overdue discussion about the future of rotopes

Postby quickfur » Sat Nov 15, 2008 12:09 am

Hayate wrote:So, anyway, have we finalized this set?

I'd like to name it, enumerate it, and add it to HDDB. :)

Finalized?? I don't think my notation, as it stands, is anywhere close to being finalized. There are so many more ways to expand the notation so that it can name more objects of interest. If you're looking for completeness, Wendy's lace figures notation is much closer to "complete" than mine.

As far as naming is concerned... I was tempted to call it "tapertopes" except that the name is already used; I thought of "stackotopes" (being a stack of cross-sections) but it is sorely lacking in euphony.
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Re: A long overdue discussion about the future of rotopes

Postby Keiji » Sat Nov 15, 2008 11:05 pm

Well, the problem is that I hardly ever understand anything Wendy writes :XP:

Anyway, I think this notation is quite nice in itself.

I present from experience the following three reasons to not overcomplicate something:
1. It makes it very difficult to enumerate without duplicates or gaps.
2. It usually results in uglier notation.
3. It almost always produces badly defined shapes.

For all three of the above, compare: the full set of rotopes to the classic set; the newest extension of CSG to the original; SSC1 to bracketopes.

Your notation has many shapes not in the set of rotopes or bracketopes and some which are difficult to define in SSC2 (!), but is still nice and simple and non-ugly. That's why I think we should call it complete, before we start mutilating it :P
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Re: A long overdue discussion about the future of rotopes

Postby quickfur » Sun Nov 16, 2008 12:04 am

Hayate wrote:[...]I present from experience the following three reasons to not overcomplicate something:
1. It makes it very difficult to enumerate without duplicates or gaps.
2. It usually results in uglier notation.
3. It almost always produces badly defined shapes.

For all three of the above, compare: the full set of rotopes to the classic set; the newest extension of CSG to the original; SSC1 to bracketopes.

Your notation has many shapes not in the set of rotopes or bracketopes and some which are difficult to define in SSC2 (!), but is still nice and simple and non-ugly. That's why I think we should call it complete, before we start mutilating it :P

Well, my notation does have several duplicates, though, caused by the coincidence of [4] and <4>, for example, and anything derived from them. As for gaps, you can always pull a Gödel and enumerate all strings over the set of symbols and retain only the valid ones. As for badly defined shapes, it's just a matter of working out exactly what each symbol or syntactic construct means, and apply the rules consistently. This is often easier said than done, though, 'cos precision doesn't always coincide with intuition, and needs some extra care.

My main complaint about the current notation is that it is inherently only symmetric about one hyperplane (or not even, if you use the pyramid taper operation a lot), the others being coincidental. (Well, except for the Cartesian product, which is a nice extension, but it doesn't buy us very much in terms of symmetric shapes.) Many higher-dimensional shapes of interest have a higher inherent degree of symmetry, which would be nice to capture. However, symmetry is a tricky thing, in spite of its intuitive simplicity. It needs group theory to be described; no trivial enumeration is possible. Regardless, the nice thing about symmetry is that it would allow us to construct such objects as roundish shapes having, for example, icosahedral symmetry, etc..

On the other hand, perhaps highly-symmetric shapes deserve their own notation. As a first shot at it, I observe that one has the equivalent of "prime numbers" in the world of symmetric groups: the Platonic solids, for example, embody the "primes" in 3 dimensions, from which other highly symmetric objects may be derived. The 6 convex regular polychora embody the "primes" in 4 dimensions, from which other highly-symmetric objects may be derived. The interrelationships between these "primes" are quite intricate; for example, consider the tetrahedron, which at first glance seems to be an innocuous object with only a 4-fold symmetry. Not so; its edges exhibit an alternating symmetry which may be "factored" into the cube/octahedron. Upon deriving the cube, say, we notice that its edges form another symmetric group, which is fully expressed in the rhombic dodecahedron. An analogous relationship occurs in 4D between the cube and the 24-cell: if we take the set of vectors from the origin to the vertices and face centers of the tetracube and scale them to equal length, we find that we have the facet normals (or, dually, the vertices) of a 24-cell. The icosahedron is related to the octahedron via a Golden Ratio snubification process, and in 4D, the 24-cell is related to the 600-cell by a analogous relationship, with the snub 24-cell as an intermediary.

I'm not sure how this generalizes to 5D: we know that there are no "pentagonal" polytopes in 5D, but the alternated 5-cube does produce a uniform polytope, which makes one wonder what is obtained if one performs an alternation with the Golden Ratio on either polytope.
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