quickfur wrote:What about the Gosset polytopes?
quickfur wrote: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 wrote: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 wrote:quickfur wrote:What about the Gosset polytopes?
You mean things like E8? They're included in the xylochoric family.
[...]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.
quickfur wrote: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 wrote:Keiji wrote:quickfur wrote: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.
Fine. In any case, it would be nice to have some examples of how it works.
Keiji wrote:[...]
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 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.
[...]
<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.
In general, the n-cube is alternable into the demicube, and all omnitruncates and cartesian products of alternable polytopes are alternable.
Keiji wrote:[...]
Then perhaps the best solution would be to have a new class for alternated Cartesian products of zonotopes (= alternable polytopes).
[...]
Keiji wrote:[...]
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?
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