CRF polychora discovery project (Meta, 13)
From Hi.gher. Space
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*15 rhodomorphs, | *15 rhodomorphs, | ||
*17 prisms of convex uniform polyhedra (not 18, because one was already covered as the tesseract, a stauromorph), | *17 prisms of convex uniform polyhedra (not 18, because one was already covered as the tesseract, a stauromorph), | ||
| - | *the [[ | + | *the [[snub demitesseract]] and the [[grand antiprism]]. |
== Prisms of Johnson solids == | == Prisms of Johnson solids == | ||
Revision as of 02:23, 22 November 2011
This page documents an ongoing project to discover as many CRF polychora as possible, and perhaps as a long-term goal prove that every CRF polychoron has been found.
Convex uniform polychora
The first 64 CRF polychora are the convex uniform polychora, which can be divided up into:
- 9 pyromorphs,
- 9 xylomorphs,
- 12 stauromorphs (not 15, because three were already covered as xylomorphs),
- 15 rhodomorphs,
- 17 prisms of convex uniform polyhedra (not 18, because one was already covered as the tesseract, a stauromorph),
- the snub demitesseract and the grand antiprism.
Prisms of Johnson solids
There are 92 Johnson solids. Each one has a prism which is a CRF polychoron, bringing the running total to 156.
Prismatoid forms
wintersolstice has proposed that each of the following CRF polyhedra:
- tetrahedron
- octahedron
- icosahedron
- cube
- triangular prism
- pentagonal prism
- square antiprism
- pentagonal antiprism
- snub disphenoid
- augmented triangular prism
- biaugmented triangular prism
- triaugmented triangular prism
- gyroelongated square pyramid
- gyroelongated square bipyramid
- diminished icosahedon
- metabidiminished icosahedon
- tridiminished icosahedon
- square pyramid
- pentagonal pyramid
- pentagonal bipyramid
lead to each of the following forms as CRF polychora:
- pyramid
- bipyramid
- elongated pyramid
- elongated bipyramid
except for the tetrahedral pyramid and the octahedral bipyramid, which are already covered as the pyrochoron and the aerochoron respectively. This generates 82 shapes in total.
Can't we just have all possible combinations of (110 CRF polyhedra) × {pyramid, bipyramid, elongated pyramid, elongated bipyramid}, excluding shapes already covered? That would produce up to 440 shapes.
Cupolae of regular polyhedra
We can generate 21 CRF polychora from the possible combinations of {tetrahedral, cubic, dodecahedral} × {cupola, orthobicupola, gyrobicupola, elongated cupola, elongated orthobicupola, elongated gyrobicupola, antiprism}. There are an additional 8 forms constructed as {octahedral, icosahedral} × {cupola, orthobicupola, elongated cupola, elongated orthobicupola}, as these forms do not use both duals. This gives 29 shapes in total.
Each cupola is constructed as the spline from the base polyhedron to its extratruncate. In the case of gyrobicupolae, the "other end" of the polychoron is the dual of the base shape. In the case of antiprisms, the spline is directly from the base shape to its dual.
The ability to construct these shapes with regular faces needs to be checked.
Ring cupolae
Nine CRF polychora are available from the possible combinations of {triangle, square, pentagon} × {ortho, gyro, magna}. Keiji discovered the ortho- and gyro- forms, and quickfur discovered the magna- form. Keiji has (perhaps temporarily) dubbed these shapes ring cupolae in general, but specific naming patterns have not yet been derived ("orthoringcupola" as one word is unsightly, and separating them to "ring triangular orthocupola" or such does not transmit the meaning well).
The ortho- and gyro- forms are constructed as in this post. The magna- forms are constructed as in this post (second-to-last paragraph).
Mrrl discovered an infinite family of ringed forms, with a 3-membered ring consisting of two antiprisms and a prism, with various Johnson polyhedra filling in the gaps. The first member contains two square antiprisms, one cube, four tetrahedra and four square pyramids. Details can be found in this post. In general, members of this family consists of two n-gonal antiprisms and an n-gonal prism, forming a 3-membered ring, with n tetrahedra and n square pyramids filling in the lateral gaps, for all n>=3.
