SSC3 draft

[Keiji]

I've just "published" a draft version of SSC3 on the wiki.

Feedback would be most appreciated. Comments on how SSC3 isn't as readable as SSC2 is will be ignored; I feel it is worth it since it allowed the notation to be far simpler syntactically.

[quickfur]

What about the Gosset polytopes?

[quickfur]

Also, I was thinking that you might be able to include alternated shapes using a /2 notation, because alternation requires even shapes (2-faces must be even polygons). Alternation will give you all the antiprisms for free, as well as the snubs (including the snub 24-cell) and many other shapes not currently representable. This operation can be (mostly) restricted to a few select kanitopes and prismatoids of even shapes. (To be alternable, both the base shape and the operands must be even, so you could even suffix "/2" at the end of the notation to indicate alternation.)

[quickfur]

IMHO i feel that store and recall is unnecessary... it's really just syntactic sugar or a conventional abbreviation; it need not be part of the base specification, but just an addendum to define "commonly-accepted" convention. (Sorta like how we use ellipses (...) to indicate omitted or unspecified portions of a string -- it's not part of string syntax, just a convention we use to denote omission or wildcarding.) This will simplify the base spec, which is always a good thing.

[Keiji]

What about the Gosset polytopes?

You mean things like E8? They're included in the xylochoric family.

Also, I was thinking that you might be able to include alternated shapes using a /2 notation, because alternation requires even shapes (2-faces must be even polygons). Alternation will give you all the antiprisms for free, as well as the snubs (including the snub 24-cell) and many other shapes not currently representable. This operation can be (mostly) restricted to a few select kanitopes and prismatoids of even shapes. (To be alternable, both the base shape and the operands must be even, so you could even suffix "/2" at the end of the notation to indicate alternation.)

I don't see any point having alternation. There are few enough shapes where alternation is worthwhile it's better just to include them in a different fashion like I have.

IMHO i feel that store and recall is unnecessary... it's really just syntactic sugar or a conventional abbreviation; it need not be part of the base specification, but just an addendum to define "commonly-accepted" convention. (Sorta like how we use ellipses (...) to indicate omitted or unspecified portions of a string -- it's not part of string syntax, just a convention we use to denote omission or wildcarding.) This will simplify the base spec, which is always a good thing.

Yes, they are unnecessary. I include them just to avoid stupidly long expressions.

[quickfur]

What about the Gosset polytopes?

You mean things like E8? They're included in the xylochoric family.

Xylochoric? Really? *squirms*

OK, fine. They are completely unrelated symmetries; you realize that, right?

[...]I don't see any point having alternation. There are few enough shapes where alternation is worthwhile it's better just to include them in a different fashion like I have.

I don't know what's your definition of worthwhile, but there are 7 alternations per regular family in 3D (not all distinct, so <21), and 15 alternations per regular family, which is 60 (but in reality a little less because alternated tesseract = 16-cell). Very few of them are actually uniform, but since you're including johnson polytopes, these ones are probably worth your attention too.

IMHO i feel that store and recall is unnecessary... it's really just syntactic sugar or a conventional abbreviation; it need not be part of the base specification, but just an addendum to define "commonly-accepted" convention. (Sorta like how we use ellipses (...) to indicate omitted or unspecified portions of a string -- it's not part of string syntax, just a convention we use to denote omission or wildcarding.) This will simplify the base spec, which is always a good thing.

Yes, they are unnecessary. I include them just to avoid stupidly long expressions.

Fine. In any case, it would be nice to have some examples of how it works.

[Keiji]

What about the Gosset polytopes?

You mean things like E8? They're included in the xylochoric family.

Xylochoric? Really? *squirms*

OK, fine. They are completely unrelated symmetries; you realize that, right?

Wikipedia's table lists E, F and G together, and because those are all special case families and do not overlap each other in dimensions, I've just put them all into family number 4. The symmetry being different doesn't bother me - I just didn't see any point reserving extra numbers needlessly.

[...]I don't see any point having alternation. There are few enough shapes where alternation is worthwhile it's better just to include them in a different fashion like I have.

I don't know what's your definition of worthwhile, but there are 7 alternations per regular family in 3D (not all distinct, so <21), and 15 alternations per regular family, which is 60 (but in reality a little less because alternated tesseract = 16-cell). Very few of them are actually uniform, but since you're including johnson polytopes, these ones are probably worth your attention too.

Well, if you would care to list the ones I haven't given expressions for, then I'll see if they can be represented or not. In any case, SSC2 doesn't have an alternation operation either, so it's not like I'm removing things from what can be represented.

Fine. In any case, it would be nice to have some examples of how it works.

An example would be the cubic truncate of cubic truncates. This can be written as:

<A=<3, 2, 3>, A, A, A>

This would expand to:

<<3, 2, 3>, <3, 2, 3>, <3, 2, 3>, <3, 2, 3>>

and the result would be the desired powertope.

[quickfur]

[...]
Wikipedia's table lists E, F and G together, and because those are all special case families and do not overlap each other in dimensions, I've just put them all into family number 4. The symmetry being different doesn't bother me - I just didn't see any point reserving extra numbers needlessly.

Fine. I was just offering some feedback as you asked for.

[...]
I don't know what's your definition of worthwhile, but there are 7 alternations per regular family in 3D (not all distinct, so <21), and 15 alternations per regular family, which is 60 (but in reality a little less because alternated tesseract = 16-cell). Very few of them are actually uniform, but since you're including johnson polytopes, these ones are probably worth your attention too.

Well, if you would care to list the ones I haven't given expressions for, then I'll see if they can be represented or not. In any case, SSC2 doesn't have an alternation operation either, so it's not like I'm removing things from what can be represented.

OK, for some reason this week has been make-a-big-fool-of-myself-by-saying-things-without-thinking week.

What I said was completely wrong. The only alternable polytopes are those whose 2-faces are all even. In 3D, the only such polytopes are the cube (alternates into tetrahedron--nothing new), the truncated octahedron (alternates into the icosahedron--nothing new), the great rhombicuboctahedron (alternates into the snub cube), and the great rhombicosidodecahedron (alternates into the snub dodecahedron). None of which isn't already covered.

In 4D, there are some interesting cases, though rather few in number: the omnitruncated 5-cell, the tesseract (alternates into the 16-cell), the omnitruncated tesseract, the truncated 24-cell (alternates into the snub 24-cell), the omnitruncated 24-cell, and the omnitruncated 120-cell. I'm not sure what the omnitruncates alternate into, looks like something interesting to explore. Besides these, all duoprisms of even polygons are alternable, though the result is non-uniform except for tesseract -> 16-cell.

In 5D and above, the 5-cube alternates into the demicube, which is already covered; there are also the omnitruncated hexateron and the omnitruncated 5-cube. I'm not sure if there are any others. In general, the n-cube is alternable into the demicube, and all omnitruncates and cartesian products of alternable polytopes are alternable. (Though I expect most of them to be non-uniform.) Although the number of alternable simplices/cubes are very limited, the number of cartesian products do grow quite quickly with increasing dimension, since they inherit from combinations of alternable polytopes from lower dimensions.

[...]
<A=<3, 2, 3>, A, A, A>

This would expand to:

<<3, 2, 3>, <3, 2, 3>, <3, 2, 3>, <3, 2, 3>>

and the result would be the desired powertope.

Ah, I see. Looks reasonable.

[Keiji]

In general, the n-cube is alternable into the demicube, and all omnitruncates and cartesian products of alternable polytopes are alternable.

Then perhaps the best solution would be to have a new class for alternated Cartesian products of zonotopes (= alternable polytopes).

The available zonotopes would be 2k-gons, hypercubes, omnitruncated simplices, omnitruncated hypercubes, omnitruncated dodecahedron and omnitruncated 120-cell.

This would generate the three snubs and the demihypercubes, but I'd rather leave those with the current definition.

The sequence could go <0, f₁, k₁, f₂, k₂, ...>
where f_i is the family (1 for a 2k-gon, 2 for a hypercube, 3 for an omnitruncated simplex, 4 for an omnitruncated hypercube, and 5 for an omnitruncated rhodomorph) and k_i is the corresponding index, which would be half the number of sides for a 2k-gon, and the dimensionality (at least 3) for the others. Each pair of numbers would define a zonotope operand of the Cartesian product, which would then be alternated.

The initial zero is obviously there to differentiate the class from the others, but it also resembles Dx 0 already used for snubs.

[quickfur]

[...]
Then perhaps the best solution would be to have a new class for alternated Cartesian products of zonotopes (= alternable polytopes).
[...]

Thanks for reminding me that zonotopes = alternable polytopes. There is actually a theorem that all zonotopes arise from projections of higher-dimensional cubes. (Conversely, all eutactic stars arise from projections of higher-dimensional crosses.) Unfortunately, exactly which n-cube projects to which zonotopes is not always obvious, and the relationship is not straightforward.

Oh, and BTW, did you include the alternated omnitruncated 24-cell somewhere?

[Keiji]

Not as a special case, no.

That theorem sounds interesting. Is that just for uniform zonotopes, or all zonotopes?

[Keiji]

After re-reading the Wikipedia page on zonotopes, I notice there are some that cannot be constructed as Cartesian products of the ones we've mentioned so far. For example, the rhombic dodecahedron.

However, I did get a new idea from the fact that zonotopes can be constructed from Minkowski sums of line segments. The line segment to +x is implied in every such construction. For each additional line segment we need, we start at +x and travel a certain number of degrees in each possible plane containing the x axis and one other coordinate axis. Trailing zero elements are left off.

I'm thinking we should enumerate the possible "useful" degree numbers, and my first thought was to use <0, ±45, ±72, ±90, ±120 or ±144>. This can generate the square, hexagon, octagon and decagon fine, but I'm not sure if it's enough for 3D or higher - combined angles aren't obvious when your trig is rusty

[quickfur]

Credit where credit is due: I learned what I did about zonotopes by reading this page:

http://home.inreach.com/rtowle/Polytopes/Chapter2/Polytopes2.html

[Keiji]

Conjecture: No finite, convex zonohedron is self-dual

The minimum number of edges each face of a zonohedron can have is 4, so to be self-dual all vertices would have to have order 4. But this can only happen if it is some rhombic tiling of the plane, thus infinite. As for zonohedra with faces of 6 or more sides, that would fail the theorem that states a planar graph must have at least one vertex of order 5.

Further conjecture: No finite, convex zonotope of at least three dimensions is self-dual.

I would quite like to see a formal proof (or disproof) of these...

[Keiji]

I think the rhombic dodecahedron has two possible alternations - the cube and the octahedron

So alternation is not necessarily unique? Darn.

[Mrrl]

Rhombic dodecahedron is not vertex-transitive. So there are other rules

[Keiji]

Oh, and a stronger conjecture about zonotopes:

Every finite, convex zonotope of at least three dimensions contains at least one vertex of order 3. Therefore, no such zonotope has a dual which is also a zonotope.

[quickfur]

Hmm. I was thinking of using the fact that zonotopes of projections of n-cubes, and trying to relate their duals to projections of n-crosses, but I realized that this line of thought may not be fruitful. For example, the 3-cube is a projection of the 4-cross, but its dual, the octahedron, is not a projection of the 4-cube.

And actually, now that I think of it... aren't all cubes projections of higher-dimensional crosses? In every dimension, the n-cube always has 2^n vertices half of which are antipodal to the other half. So the n-cube is a projection of a 2^{n-1}-cross. And since all zonotopes are projections of n-cubes, they are ultimately also projections of m-crosses! Even better yet, an n-cross always has 2n vertices, which means they are projections of (2n+1)-simplices. So ultimately, every convex polytope is a projection of an n-simplex.

Yeah, this line of thought is not useful. The self-duality of the higher dimensional object tells you nothing about the self-duality of its projections.

[Keiji]

I've put together a wiki page containing a summary of zonotope concepts so far: Zonotope

Any suggestions for a list of zonochora to include?

[quickfur]

You want to be careful about your "even stronger conjecture": in higher dimensions pretty much no vertex is order 3 because you need a lot of facets to meet at a vertex! Even a 5-cell has order 4 vertices. I think you want to reword that to "contains at least one vertex of order n".

EDIT: either that, or "contains at least one peak of order 3".

[Keiji]

Okay, I've adjusted the conjecture.

I take it a "peak" is an (n-3)-facet of an n-dimensional shape?
What was the (n-2)-facet again? Ridge, or margin?

[quickfur]

[...]
I take it a "peak" is an (n-3)-facet of an n-dimensional shape?
What was the (n-2)-facet again? Ridge, or margin?

The terminology I use is as follows:
0D surtope = vertex
1D surtope = edge
(2D surtope = face) -- I don't really like this because "face" is too often conflated with "facet" -- maybe wendy's term "surhedron" is best here
...
(n-3)D surtope = peak
(n-2)D surtope = ridge
(n-1)D surtope = facet
nD surtope = polytope itself

[wendy]

The terminology for space is by the -on stem

0d vertex, or teelon, 1d edge, or latron, 2d hedron, 3d choron, 4d teron,
5d peton, 6d ecton, 7d zetton, 8d yotton

A space of this dimension is refered to as an -ix, eg latrix, hedrix, chorix &c.
Measures of a specific dimension is by -age, eg hedrage, chorage. latrage would refer to the sum of lengths of edges, rather than a single edge. teelage is the count of vertices
Words in -ous, and in -id, refer to approximately and exactly of that feature. A snake is a 'fat line', becomes a latrous thing, while a page is a thin hedron, is hedrid.
A solid of n dimensions might be expressed by -id to, eg the dodecahedron is a chorid. When this is used as a patch on a 120ch, it becomes a patch (choron).

The terms relative to all-space are
N-1 dimensions: all terms with /face/, eg surface, face, facet, facing, terms with /plane/ or /plain/
N-2 dimensions: all terms with /mark/, eg margin, mark, demark (give surface boundaries to)
all terms with 'ring' refer to a hollow enclosure of marks or margins.

The terms relative to any subspace or any tiling (eg refering to parts of a hexagon in 4d)
M dimensions: cell
M-1 dimensions: wall
M-2 dimensions: sill.

Terms involving /flat/ or /plate/ hold to being the same curvature as all-space, a straight line is a /flat/ line.

The normal discussion of a polytope is to suppose it is solid in its all-space: that is a dodecahedron is discussed in terms of 3d space, not a 4d plate.