Pyroperihedral rotunda

Discussion of known convex regular-faced polytopes, including the Johnson solids in 3D, and higher dimensions; and the discovery of new ones.

Pyroperihedral rotunda

Postby Marek14 » Fri Nov 07, 2014 10:19 am

I noticed that http://hddb.teamikaria.com/wiki/CRFP4DP/Diminishings lists 7 rotundas. Two of these, stauroperihedral rotunda and rhodoperihedral rotunda are created by slicing of prismatotruncated tesseract/prismatotruncated 120-cell.

My research of tetrahedral vertices led me to vertex "346-646", consisting of triangular cupola, truncated tetrahedron, hexagonal prism and truncated octahedron. This is a part of "pyroperihedral rotunda", a diminishing of prismatotruncated pentachoron x3x3o3x.
Now, this wasn't considered a rotunda because it's bigger than the other part of the slice (truncated tetrahedron || truncated octahedron or tetrahedral canticupola); however, I'd argue otherwise. The key fact is that is we cut prismatotruncated pentachoron in truncated octahedral slice, we'll get two polychora that do not "bulge" -- i.e. their maximum 3D diameter is in the hyperplane of the cut. Or, in other words, two copies of a polychoron with this property can be glued together to form another convex polychoron. This means that prismatotruncated pentachoron can be considered a "cupolarotunda", a part of small family including tetrahedral canticupola, pyroperihedral rotunda and various bicupolas, birotundas, cupolarotundas and their elongated versions.

Therefore, I submit that this polychoron should be counted as a rotunda in its own right and not just as a diminishing.
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Re: Pyroperihedral rotunda

Postby Marek14 » Fri Nov 07, 2014 10:56 am

BTW, is there a deeper connection between great prismated tesseract x4x3x3x and prismatotruncated 24-cell x3x4o3x? The verf of x4x3x3x seems to be a part of verf of x3x4o3x, does that mean that the great cuboctahedra in x4x3x3x can be augmented with truncated octahedron || truncated cuboctahedron where octahedral prisms in x4x3x3x merge with square cupolas to various related polyhedra?
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Re: Pyroperihedral rotunda

Postby Klitzing » Fri Nov 07, 2014 2:44 pm

Marek14 wrote:BTW, is there a deeper connection between great prismated tesseract x4x3x3x and prismatotruncated 24-cell x3x4o3x? The verf of x4x3x3x seems to be a part of verf of x3x4o3x, does that mean that the great cuboctahedra in x4x3x3x can be augmented with truncated octahedron || truncated cuboctahedron where octahedral prisms in x4x3x3x merge with square cupolas to various related polyhedra?


Yes.
Just cf. gidpith (x3x3x4x) on my IncMats website, there this already is stated:
Augmenting toe || girco onto each girco of gidpith would lead to prico (which then would have an even larger symmetry)!

Resp. the other way round cf. prico (x3o4x3x), where it likewise is stated:
Diminishing the prico by toe-cupolae at hexadecachoral positions results in the gidpith.



Btw. this same relation occures at proh (x3o3x4x):
Augmenting co-cupolae onto the tic would lead to the srico (which would have an even larger symmetry)!

Resp. the other way round cf. srico (x3o4x3o):
Diminishing the srico by co-cupolae at hexadecachoral positions results in the proh.



Or a further such pairing is srit (o3x3o4x):
Augmenting oct-cupolae onto the sirco would lead to the spic (which would have an even larger symmetry)!

Resp. conversely cf. spic (x3o4o3x):
Diminishing spic by oct-cupolae at hexadecachoral positions results in the srit.



Or even simpler is tes (o3o3o4x):
Augmenting cubpy onto the cube would lead to the ico (which would have an even larger symmetry)!

Conversely cf. ico (x3o4o3o):
[...] the hex-diminished ico is nothing but the tes.



These 4 pairings then make up up a commutative diagram...

Various partial augmentations resp. diminishings also have been considered already.

--- rk
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