CRF polychora discovery project (Meta, 13)

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(→Prismatoid forms: not all johnson solids can be made pyramids)
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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.
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.
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<span style='color: red;'>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.</span>
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<div style='background-color: #fee; border: 1px solid #e00; padding: 0.5em 1em; margin: 1em 0;'>
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:<span style='color: red;'>Probably not: the larger johnson solids cannot be pyramidized without stretching the new faces into non-regular polygons. It's basically the 4D equivalent of how polygons with degree >=6 cannot have pyramids made from them with equilateral triangles because the angle defect is negative, so you have to stretch the triangles to make them meet at the apex.&mdash;[[User:Quickfur|quickfur]] 01:26, 22 November 2011 (UTC)</span>
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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.
 +
:Probably not: the larger johnson solids cannot be pyramidized without stretching the new faces into non-regular polygons. It's basically the 4D equivalent of how polygons with degree >=6 cannot have pyramids made from them with equilateral triangles because the angle defect is negative, so you have to stretch the triangles to make them meet at the apex.&mdash;[[User:Quickfur|quickfur]] 01:26, 22 November 2011 (UTC)
 +
::In that case, this calls for a case-by-case consideration of each CRF polyhedron (110 + 4 from infinite families). Let's fill in the table below. I've put in "OK" for the shapes in wintersolstice's list, "No" for shapes including a polygon with 6 or more sides, and "??" for all others. So far there are 32 No's and 20 OK's, leaving 62 to be checked. [[User:Hayate|Hayate]] 01:53, 22 November 2011 (UTC)
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{| style='width: 45em;'
 +
|Kt1||OK
 +
|Ko1||OK
 +
|Ko4||OK
 +
|Ki1||??
 +
|Ki4||OK
 +
|Kt3||??
 +
|Ko2||??
 +
|Ko3||??
 +
|Ko6||??
 +
|Ko5||??
 +
|-
 +
|Ko7||??
 +
|Ko0||??
 +
|Ki2||??
 +
|Ki3||??
 +
|Ki6||??
 +
|Ki5||??
 +
|Ki7||??
 +
|Ki0||??
 +
|J₁||OK
 +
|J₂||OK
 +
|-
 +
|J₃||No
 +
|J₄||No
 +
|J₅||No
 +
|J₆||No
 +
|J₇||??
 +
|J₈||??
 +
|J₉||??
 +
|J₁₀||OK
 +
|J₁₁||OK
 +
|J₁₂||??
 +
|-
 +
|J₁₃||OK
 +
|J₁₄||??
 +
|J₁₅||??
 +
|J₁₆||??
 +
|J₁₇||OK
 +
|J₁₈||No
 +
|J₁₉||No
 +
|J₂₀||No
 +
|J₂₁||No
 +
|J₂₂||No
 +
|-
 +
|J₂₃||No
 +
|J₂₄||No
 +
|J₂₅||No
 +
|J₂₆||??
 +
|J₂₇||??
 +
|J₂₈||??
 +
|J₂₉||??
 +
|J₃₀||??
 +
|J₃₁||??
 +
|J₃₂||??
 +
|-
 +
|J₃₃||??
 +
|J₃₄||??
 +
|J₃₅||??
 +
|J₃₆||??
 +
|J₃₇||??
 +
|J₃₈||??
 +
|J₃₉||??
 +
|J₄₀||??
 +
|J₄₁||??
 +
|J₄₂||??
 +
|-
 +
|J₄₃||??
 +
|J₄₄||??
 +
|J₄₅||??
 +
|J₄₆||??
 +
|J₄₇||??
 +
|J₄₈||??
 +
|J₄₉||OK
 +
|J₅₀||OK
 +
|J₅₁||OK
 +
|J₅₂||??
 +
|-
 +
|J₅₃||??
 +
|J₅₄||No
 +
|J₅₅||No
 +
|J₅₆||No
 +
|J₅₇||No
 +
|J₅₈||??
 +
|J₅₉||??
 +
|J₆₀||??
 +
|J₆₁||??
 +
|J₆₂||OK
 +
|-
 +
|J₆₃||OK
 +
|J₆₄||??
 +
|J₆₅||No
 +
|J₆₆||No
 +
|J₆₇||No
 +
|J₆₈||No
 +
|J₆₉||No
 +
|J₇₀||No
 +
|J₇₁||No
 +
|J₇₂||??
 +
|-
 +
|J₇₃||??
 +
|J₇₄||??
 +
|J₇₅||??
 +
|J₇₆||No
 +
|J₇₇||No
 +
|J₇₈||No
 +
|J₇₉||No
 +
|J₈₀||No
 +
|J₈₁||No
 +
|J₈₂||No
 +
|-
 +
|J₈₃||No
 +
|J₈₄||OK
 +
|J₈₅||??
 +
|J₈₆||??
 +
|J₈₇||??
 +
|J₈₈||??
 +
|J₈₉||??
 +
|J₉₀||??
 +
|J₉₁||??
 +
|J₉₂||No
 +
|}
 +
 
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Triangular prism OK &nbsp; &nbsp; Square antiprism OK &nbsp; &nbsp; Pentagonal prism OK &nbsp; &nbsp; Pentagonal antiprism OK
== Cupolae of regular polyhedra ==
== Cupolae of regular polyhedra ==

Revision as of 02:53, 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.

Probably not: the larger johnson solids cannot be pyramidized without stretching the new faces into non-regular polygons. It's basically the 4D equivalent of how polygons with degree >=6 cannot have pyramids made from them with equilateral triangles because the angle defect is negative, so you have to stretch the triangles to make them meet at the apex.—quickfur 01:26, 22 November 2011 (UTC)
In that case, this calls for a case-by-case consideration of each CRF polyhedron (110 + 4 from infinite families). Let's fill in the table below. I've put in "OK" for the shapes in wintersolstice's list, "No" for shapes including a polygon with 6 or more sides, and "??" for all others. So far there are 32 No's and 20 OK's, leaving 62 to be checked. Hayate 01:53, 22 November 2011 (UTC)
Kt1OK Ko1OK Ko4OK Ki1?? Ki4OK Kt3?? Ko2?? Ko3?? Ko6?? Ko5??
Ko7?? Ko0?? Ki2?? Ki3?? Ki6?? Ki5?? Ki7?? Ki0?? J₁OK J₂OK
J₃No J₄No J₅No J₆No J₇?? J₈?? J₉?? J₁₀OK J₁₁OK J₁₂??
J₁₃OK J₁₄?? J₁₅?? J₁₆?? J₁₇OK J₁₈No J₁₉No J₂₀No J₂₁No J₂₂No
J₂₃No J₂₄No J₂₅No J₂₆?? J₂₇?? J₂₈?? J₂₉?? J₃₀?? J₃₁?? J₃₂??
J₃₃?? J₃₄?? J₃₅?? J₃₆?? J₃₇?? J₃₈?? J₃₉?? J₄₀?? J₄₁?? J₄₂??
J₄₃?? J₄₄?? J₄₅?? J₄₆?? J₄₇?? J₄₈?? J₄₉OK J₅₀OK J₅₁OK J₅₂??
J₅₃?? J₅₄No J₅₅No J₅₆No J₅₇No J₅₈?? J₅₉?? J₆₀?? J₆₁?? J₆₂OK
J₆₃OK J₆₄?? J₆₅No J₆₆No J₆₇No J₆₈No J₆₉No J₇₀No J₇₁No J₇₂??
J₇₃?? J₇₄?? J₇₅?? J₇₆No J₇₇No J₇₈No J₇₉No J₈₀No J₈₁No J₈₂No
J₈₃No J₈₄OK J₈₅?? J₈₆?? J₈₇?? J₈₈?? J₈₉?? J₉₀?? J₉₁?? J₉₂No

Triangular prism OK     Square antiprism OK     Pentagonal prism OK     Pentagonal antiprism OK

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.

Retrieved from "http://hi.gher.space/wiki/CRF_polychora_discovery_project"

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