SharkRetriver wrote:the 24 is still strange because of the analogue problem, but it's a cool polychoron.
Well that's what makes it cool.
Analogue-wise, it's essentially a rhombic dodecahedron that just happens to be regular. It's in a series of "stellated" hypercubes, which are constructed by cutting one n-cube into (n-1)-cubical pyramids and attaching them onto the facets of another n-cube. In 1D, it is identical to the line segment (well, everything is, so that says nothing). In 2D, it is identical to the 2-cross (diamond, which happens to be the same as the square). In 3D, it starts to become distinct, as the rhombic dodecahedron. In 4D, it's the 24-cell, which, due to a set of interesting coincidences (such as the fact that 2^2 = 1+1+1+1) just happens to be regular. Then in 5D, it starts to become distinct from the maximal projection of the (n+1)-cube into n-space.
About the simplexes, they're odd as well. I don't like
But their mesotruncates and omnitruncates are even.
I looked up E polytopes on the wiki. I think that the trirectified 4_21 should be the next polytope of the month. There are only 4.8 million edges, lol
Oh don't worry about that one... I do have the omnitruncated 120-cell lined up. Among the 4D polytopes, it's the grand-daddy of them all. It has 14400 vertices, 28800 edges, 17040 faces, and 2640 cells. One of these days, it's gonna be up. Of course, that's nothing compared to the trirectified 4_21.
That's still a long way off. After I finish with 4D, I'm going to 5D, which has a lot more truncates, even though it only has 3 regular polytopes. If I'm still alive after that, it's 6D time, and if I outlive that
, then I'll get to the 4_21 in 7D.
Also, why is there only the snub 24-cell, and no other snub polychora?
Well, the concept of "snub", in the strict sense of surrounding the faces of the base polyhedron by triangles, really only applies to 3D. A more generally applicable operation is alternation. The 3D snubs are topologically equivalent to the alternations of the omnitruncate of their respective platonic solid. In a sense, you may think of the icosahedron as the "snub tetrahedron", since the "omnitruncated tetrahedron" is A111, which is equivalent to C011 (truncated octahedron), the alternation of which is topologically equivalent to the icosahedron. So in a sense, "snub" in 3D means "alternated omnitruncate".
However, the snub 24-cell in 4D is not
the alternation of an omnitruncate. It is topologically equivalent to the alternation of the truncated
24-cell (alternation of F0011), and its icosahedral facets are not completely
surrounded by tetrahedra, unlike the analogous 3D snubs. Four of each icosahedron's faces actually touch adjacent icosahedral cells. So in a sense, the snub 24-cell isn't really
a snub. It's just that there are no other uniform snubs in 4D, and it does
after all consist of facets of the same kind with the gaps filled in by regular tetrahedra.
Now, the key here is that alternation as an operation does not
, in general, give you a uniform polytope. The snub cube cannot be produced by alternating a uniform great rhombicuboctahedron, because even though the result is topologically
equivalent to a snub cube, its edge lengths are not equal and its faces are not regular. You need some unspecified amount of distortion in order to get the "real" snub cube. So nothing stops you from, say, taking the alternation of F1111 instead. You'll end up with a polytope with 48 snub cubes for facets plus a whole bunch of octahedra and probably some irregular shapes. However, it would not be possible to make the polytope uniform. Only the alternation of F0011 can be made uniform.
Alternation in 3D also gives rise to the infinite family of antiprisms: alternating a (2n)-gonal prism gives you an n-gonal antiprism. (Again, modulo some arbitrary distortion to make the result uniform.) Given this fact, it seems reasonable to suppose that alternating 2n,2n-duoprisms in 4D should give us some kind of analogue of the antiprisms. But alas, none of them can be made uniform except one: the alternation of the 4,4-duoprism (aka tesseract), which is the 16-cell, which also happens to be regular.
Given the large amount of polytopes that can be alternated in 4D, you could say that actually there are quite a good number of snubs; but only the snub 24-cell is uniform.
I'm also considering changing w to 10 before "mass-rendering" for various reasons...
Sounds good to me.