## Another New Group of Polytopes

Discussion of tapertopes, uniform polytopes, and other shapes with flat hypercells.

### Another New Group of Polytopes

I thought the symmetroform concept was too inclusive and didn't do its job so I made another group.
I have no idea what to name it but here are the rules:
1. vertex transitive
2. faces must be semiregular/regular and/or part of a vertex transitive faceting of a semiscaliform (vertex transitive with semiregular/regular faces) polytope (elements and the polytope itself).
This groups definition is a bit inelegant but it is very inclusive.

ubersketch
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### Re: Another New Group of Polytopes

If you mean 2D faces then you get regular polygons and therefore the polytope belongs to the class of scaliform polytopes (cells do not need to be uniform), such as the isogonal bidex with 48 teddis.

But if you want facets to be semiregular or regular, here is the following.

In 3D you get the family of uniform polytopes because a uniform polygon is a regular polygon or a star polygon.
In 4D you get the uniform polychora because a semiregular polyhedron has regular polygons and are the same as a uniform polyhedron.
In 5D you are limited to the 6 regular polychora, rectified 5-cell, rectified 600-cell, and snub 24-cell because the cells must be regular polyhedra. For 5D polytopes this means the 3 regular polytera alongside the 5-demicube, rectates of the 5-simplex (rectified 5-simplex, birectified 5-simplex), and rectates of the 5-orthoplex (rectified 5-orthoplex, birectified 5-cube/orthoplex)
In 6D, 7D, and 8D, you have the Gosset polytopes (up to 4_21) that contain semiregular facets alongside the 3 regular polytopes and the demihypercube.

Eventually you are restricted to the 3 regular polytopes and the demihypercube as the number of dimensions approaches infinity.
SárHwās Hnáras táygʲasi Hr̥táy kʲa waĉatás samHatás kʲa ĵn̥Hyántay. Táy kʲitˢtáH mánasaH kʲa dádHn̥tay Hántaray práti bʰraHtrásya pn̥tHáH kriHánt.

All human beings are born free and equal in dignity and rights. They are endowed with reason and conscience and should act towards one another in a spirit of brotherhood.
Mercurial, the Spectre
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### Re: Another New Group of Polytopes

Mercurial, the Spectre wrote:[...]
But if you want facets to be semiregular or regular, here is the following.

In 3D you get the family of uniform polytopes because a uniform polygon is a regular polygon or a star polygon.
In 4D you get the uniform polychora because a semiregular polyhedron has regular polygons and are the same as a uniform polyhedron.
In 5D you are limited to the 6 regular polychora, rectified 5-cell, rectified 600-cell, and snub 24-cell because the cells must be regular polyhedra. For 5D polytopes this means the 3 regular polytera alongside the 5-demicube, rectates of the 5-simplex (rectified 5-simplex, birectified 5-simplex), and rectates of the 5-orthoplex (rectified 5-orthoplex, birectified 5-cube/orthoplex)
In 6D, 7D, and 8D, you have the Gosset polytopes (up to 4_21) that contain semiregular facets alongside the 3 regular polytopes and the demihypercube.

Eventually you are restricted to the 3 regular polytopes and the demihypercube as the number of dimensions approaches infinity.

Mercurial, in 3D you were allowing non-convex figures. But at 5D you were swapping to convex ones only. I think those ought to be added there too.

In fact, uberscetch's definition looks like a direct extension of Gosset's definition of semiregularity:
1) vertex-uniform = vertex-transitive = isogonal
2) all facets (i.e. (D-1)-faces) are regular polytopes
1) vertex-transitive
2) all facets are semiregular polytopes
Thus we could re-write his definition into
1) all 0-faces are symmetry-equivalent
2) all (D-2)-faces are regular polytopes

That is, for D=4 you'd get that just the 2-faces are to be regular, which are the polygons. Therefore uberscetch here is quite near to the idea of CRFs. But in contrast, the setting of CRFs additionally restricts to convex shapes only, whereas OTOH it is more liberate in not asking for vertex-transitiveness at all.
Mercurial is right about the higher-dimensional restrictiveness of his definition. Within 3D and 4D the set of Wythoffians obviously is contained. But also figures beyond. But Wythoffians generally would allow for prisms as 3-faces. But for D >= 5 uberscetch's definition would disallow those prismatic 3-faces! That is from D=5 on not even all Wythoffians are contained within uberscetch's definition!

--- rk
Klitzing
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### Re: Another New Group of Polytopes

Klitzing wrote:
Mercurial, the Spectre wrote:[...]
But if you want facets to be semiregular or regular, here is the following.

In 3D you get the family of uniform polytopes because a uniform polygon is a regular polygon or a star polygon.
In 4D you get the uniform polychora because a semiregular polyhedron has regular polygons and are the same as a uniform polyhedron.
In 5D you are limited to the 6 regular polychora, rectified 5-cell, rectified 600-cell, and snub 24-cell because the cells must be regular polyhedra. For 5D polytopes this means the 3 regular polytera alongside the 5-demicube, rectates of the 5-simplex (rectified 5-simplex, birectified 5-simplex), and rectates of the 5-orthoplex (rectified 5-orthoplex, birectified 5-cube/orthoplex)
In 6D, 7D, and 8D, you have the Gosset polytopes (up to 4_21) that contain semiregular facets alongside the 3 regular polytopes and the demihypercube.

Eventually you are restricted to the 3 regular polytopes and the demihypercube as the number of dimensions approaches infinity.

Mercurial, in 3D you were allowing non-convex figures. But at 5D you were swapping to convex ones only. I think those ought to be added there too.

In fact, uberscetch's definition looks like a direct extension of Gosset's definition of semiregularity:
1) vertex-uniform = vertex-transitive = isogonal
2) all facets (i.e. (D-1)-faces) are regular polytopes
1) vertex-transitive
2) all facets are semiregular polytopes
Thus we could re-write his definition into
1) all 0-faces are symmetry-equivalent
2) all (D-2)-faces are regular polytopes

That is, for D=4 you'd get that just the 2-faces are to be regular, which are the polygons. Therefore uberscetch here is quite near to the idea of CRFs. But in contrast, the setting of CRFs additionally restricts to convex shapes only, whereas OTOH it is more liberate in not asking for vertex-transitiveness at all.
Mercurial is right about the higher-dimensional restrictiveness of his definition. Within 3D and 4D the set of Wythoffians obviously is contained. But also figures beyond. But Wythoffians generally would allow for prisms as 3-faces. But for D >= 5 uberscetch's definition would disallow those prismatic 3-faces! That is from D=5 on not even all Wythoffians are contained within uberscetch's definition!

--- rk

Oh, I may be biased towards convex figures but any semiregular star may work. Sorry

But then again, we could call these order-3 regular polytopes in the basis that the ridges must be regular (compare order-2 for semiregular polytopes and order-1 for regular polytopes).
SárHwās Hnáras táygʲasi Hr̥táy kʲa waĉatás samHatás kʲa ĵn̥Hyántay. Táy kʲitˢtáH mánasaH kʲa dádHn̥tay Hántaray práti bʰraHtrásya pn̥tHáH kriHánt.

All human beings are born free and equal in dignity and rights. They are endowed with reason and conscience and should act towards one another in a spirit of brotherhood.
Mercurial, the Spectre
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### Re: Another New Group of Polytopes

sounds sound.
(Oh, what a sound this sentence has. )

Thus the convex order-(D-1) regular polytopes would just be the isogonal subset of the CRFs.
--- rk
Klitzing
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### Re: Another New Group of Polytopes

Note: By semiregular, I mean, each face is vertex-transitive which allows for unequal edge lengths.
So this means:
1. isogonal element or part of an isogonal element/polytope.
2. isogonal
This means you may have stuff like [url="http://www.software3d.com/NobleSnub.php"]this[/url].

ubersketch
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### Re: Another New Group of Polytopes

You shouldn't begin to overload the meaning of an already defined notion!
"semiregular" has being defined by Gosset long ago, then meaning regularity of facets.
And because any edge would be contained in some facet, all edges are supposed to be of the same size, as this is an elementar part of regular polytopes.

Btw. the same holds true for order-k-regular polytopes, as being now coined by Mercurial.

--- rk
Klitzing
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