CRF polychora discovery project (Meta, 13)

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== Convex Non-Uniform Scaliform ==
== Non-Uniform Scaliform ==
Outside the uniform polychora (uniformed-celled and vertex transitive) there are 4 '''known''' convex scaliform (equilateral and vertex transitive)  
Outside the uniform polychora (uniformed-celled and vertex transitive) there are 4 '''known''' convex scaliform (equilateral and vertex transitive)  

Revision as of 16:52, 31 March 2012

See also: List of convex regular-faced polyhedra

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 polychora9 + 9 + 12 + 15 + 17 + 26464
Prisms of Johnson solids9292156
Prismatoid forms13×4 - 250206
Cupolae of regular polyhedra3×7 + 2×4 - 1 + 634240
Bicupolic rings3×39249
Duoprisms augmented with pyramidsSee below16331894
Non-Uniform Scaliform441898

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.

Richard Klitzing's segmentotopes

Dr. Richard Klitzing enumerated the full set of segmentotopes: CRF polychora constructed by the convex hull of two lower-dimensional polytopes placed in parallel hyperplanes spaced appropriately so that the result will have equal-length edges.

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

We can generate 52 CRF polychora by all possible combinations of {tetrahedron, cube, octahedron, icosahedron, square antiprism, pentagonal antiprism, triangular prism, pentagonal prism, square pyramid, pentagonal pyramid, diminished icosahedron, metabidiminished icosahedron, tridiminished icosahedron} × {pyramid, bipyramid, elongated pyramid, elongated bipyramid}. However, two of these - the "tetrahedral pyramid" and the "octahedral bipyramid" - are already covered as the pyrochoron and the aerochoron respectively, leaving us with 50 new CRF polychora. This brings the running total to 206.

The remaining CRF polyhedra cannot generate pyramidal forms for one (or both) of the following reasons:

  • the polyhedron's vertices are further from its center than its edge length, thus any pyramid of it would require base-apex edge lengths longer than base-base edge lengths, and thus not be CRF;
    • note that this reason is implied if the polyhedron contains a contour with at least six edges, but the converse is not always true, e.g. in the case of the dodecahedron
  • the polyhedron cannot be inscribed in a sphere, thus there is no point equidistant from all base points, thus any pyramid of it would have at least two different base-apex edge lengths, and thus not be CRF.

wintersolstice originally proposed a list containing more polyhedra than those listed above, but this was incorrect due to the above reasons. He acknowledged that there was a mistake with the list some time ago, most likely realizing the same argument that has been written above, but did not give this explanation at the time.

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.

The elongated cubic orthobicupola is the same as the runcinated tesseract, leaving us with 28 new CRF polychora.

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.

Two copies of the icosahedron-dodecahedron antiprism can be fitted together by their dodecahedral bases; the relative sizes of the icosahedron and dodecahedron of equal edge length ensures that the result is convex, and therefore CRF. It consists of 2 icosahedral cells, 100 tetrahedra, and 24 pentagonal pyramids. An elongated form is obtained by inserting a dodecahedral prism. Both forms have augmented and biaugmented variants. The other polyhedron-dual antiprisms do not produce convex CRFs this way, so this is a unique combination giving 6 new CRF polychora in total.

Bicupolic rings

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 dubbed these shapes bicupolic rings in general, and the specific naming pattern is n-gonal formbicupolic ring, e.g. square orthobicupolic ring.

The ortho- and gyro- forms are constructed as in this post. The magna- forms are constructed as in this post (second-to-last paragraph).


So far, three CRF rotundae have been discovered.

Mrrl discovered that a CRF polychoron can be cut from the rectified 120-cell when diminishing the latter. This polychoron consists of 1 icosidodecahedron, 12 pentagonal rotundae, and 40 tetrahedra. It can be considered the 4D analogue of the 3D pentagonal rotunda.

A similar CRF rotunda can be obtained from the cantellated 600-cell by a similar cutting, producing a polychoron with 1 icosidodecahedron, 12 pentagonal rotundae, 42 pentagonal prisms, 20 cuboctahedra, and 20 triangular cupolae.

Mrrl also found that the top of the second rotunda can be diminished, to obtain another CRF rotunda with 1 truncated icosahedron, 12 pentagonal rotunda, 30 pentagonal prisms, and 40 triangular cupolae.

These rotundae have birotunda forms as well as their corresponding elongates, giving 12 new CRF polychora in total.

Augmented uniform polychora

Some of the uniform polychora can be augmented (have CRF pyramids erected on one or more of their cells) and still remain CRF. The criteria for such augmentations to be CRF are:

  • There must exist a CRF pyramid whose base is in the shape of the cell being augmented.
  • The sum of each dichoral angle of the pyramid's cells with its base and the dichoral angle of the cell being augmented and the corresponding neighbouring cell must be ≤ 180° in order to remain convex. If two adjacent cells are being augmented, then the sum of dichoral angles of two adjacent pyramid cells with their respective bases and the dichoral angle between the two augmented cells must be ≤ 180°.
  • In the case where the sum of dichoral angles is exactly 180°, adjacent cells in the augments will merge; in such a case, the merged cells must themselves be CRF.

The pyrochoron has a single augmentation, consisting of two pyrochora joined cell-to-cell.

The tesseract has 14 augmentations with cubical pyramids, one of which corresponds with the xylochoron. However, these augmentations are also a subset of the xylochoron's diminishings, so their count should not be included under the present category.

The xylochoron has 20 augmentations (forum post). This has been found to be invalid, due to it being based on a faulty computation of the xylochoron's dichoral angle.

There are probably many other augmentable uniform polychora, these have yet to be explored.

Augmented duoprisms

With pyramids

The duoprisms are a source of 1633 CRF polychora via augmentation with CRF pyramids, especially because the pentagonal prism pyramid is very shallow. This shallowness permits it to be fitted onto pentagonal prisms of n,5-duoprisms in various combinations up to n=20. The other prism pyramids (triangular and square) are less shallow, but still contribute a good number of CRF polychora.

The following lists the number of CRF polychora generated by augmenting duoprisms:

3,3-duoprism: 3
3,4-duoprism: 5
3,5-duoprism: 11
3,6-duoprism: 4
4,5-duoprism: 17
4,6-duoprism: 4
4,7-duoprism: 4
4,8-duoprism: 7
5,5-duoprism: 35
5,6-duoprism: 12
5,7-duoprism: 17
5,8-duoprism: 29
5,9-duoprism: 45
5,10-duoprism: 77
5,11-duoprism: 15
5,12-duoprism: 25
5,13-duoprism: 30
5,14-duoprism: 48
5,15-duoprism: 63
5,16-duoprism: 98
5,17-duoprism: 132
5,18-duoprism: 208
5,19-duoprism: 290
5,20-duoprism: 454
Total: 1633 augmentations

The 4,4-duoprism is omitted here, because it coincides with the tesseract, the augmentations of which are covered under another category.

The sharp drop in the number of augmentations between the 3,5-duoprism and the 3,6-duoprism, between the 4,5-duoprism and the 4,6-duoprism, and between the 5,5-duoprism and the 5,6-duoprism is because pyramids of hexagonal (or higher) prisms cannot be CRF, since equilateral triangles tile the hexagon and so no hexagonal (or higher) pyramid can be formed without breaking the regular-faced requirement. Thus, only one of the duoprism's two rings can be augmented.

The drop between the 5,10-duoprism and the 5,11-duoprism is caused by the fact that adjacent pentagonal prism pyramids erected on an n-membered duoprism ring are no longer convex after n=10, so from the 5,11-duoprism onwards only non-adjacent augmentations are permitted, thus reducing the number of possible combinations. Adjacent augments on the 5,10-duoprism have pentagonal pyramid cells that are coplanar, thus merging into a pentagonal bipyramid.

Augments of the 5,20-duoprism have pentagonal pyramids coplanar with the adjacent pentagonal prism, so they merge into elongated pentagonal pyramids. If the next prism in the ring is also augmented, then another pentagonal pyramid is added to the coplanar cell, turning it into an elongated pentagonal bipyramid.

No other duoprisms can be augmented with CRF pyramids and still remain convex.

With other segmentochora

Besides CRF pyramids, certain other segmentotopes can augment duoprisms to form CRFs. The full enumeration of such duoprism augmentations is currently in progress.

So far, it has been found that some of the n,6-duoprisms, n,8-duoprisms, and n,10-duoprisms can be augmented by the segmentotopes m-gon||2m-prism for m=3,4,5. These augments induce an orientation on both duoprism rings, reducing symmetry and increasing the number of distinct augmentations. These augmentations have been enumerated for the following duoprisms:

  • 3,6-duoprism: The first ring can be augmented with triangle||hexagonal_prism and the second ring with triangular prism pyramids, albeit not at the same time. Total: 9 augmentations.
  • 4,6-duoprism: The first ring augmentable with triangle||hexagonal_prism; second ring with cubical pyramid, albeit not at the same time. Total: 7 augmentations.
  • 5,6-duoprism: The first ring augmentable with triangle||hexagonal_prism; second ring with pentagonal prism pyramid. First ring augments reduce the symmetries of the second ring by half, thus increasing the number of distinct combinations. Total: 64 augmentations.
  • 6,6-duoprism: First ring augmentable with triangle||hexagonal_prism; only one ring augmentable at a time. Total: 4 augmentations.
  • 3,8-duoprism: First ring augmentable with square||octagonal_prism; second ring cannot be augmented. Total: 5 augmentations.
  • 4,8-duoprism: First ring augmentable with square||octagonal prism; second ring with cubical pyramid (non-adjacent only). Both rings can be augmented simultaneously, but only when all augments on the first ring are in ortho orientation. Total: 44 augmentations.
  • 5,8-duoprism: TBD.

The tesseract (considered as a 4,4-duoprism) can be augmented with 4 square pyramid prisms and 4 line||square segmentochora to form a CRF polychoron having 4 cubes, 16 triangular prisms, and 16 tetrahedra. This is somewhat different augmentation from what is considered above, in that the initial augmentation creates a non-convex polychoron, but the gaps can be filled in with CRF segmentochora to form a valid CRF polychoron. This shape can also be generated as the convex hull of a tesseract and an octagon. It was discovered by quickfur on 9 Jan 2012.

Diminished polychora

Some regular polychora can be diminished to give CRF polychora. The 24-cell can be diminished into the tesseract by removing 8 square pyramids. Removing less than 8 pyramids in various configurations generates a number of distinct diminished 24-cells. It is also possible to remove square pyramids that do not correspond with facets of the tesseract, this generates a few more CRF polychora not included in the tesseract construction.

The diminished 16-cell coincides with the octahedral pyramid (see Prismatoid forms above).

The 600-cell has a large number of diminishings, two of which are uniform (the snub 24-cell and the grand antiprism). Removing icosahedral pyramids from the 600-cell generates a large number of CRF polychora; removing 24 in 24-cell configuration generates the snub 24-cell. Removing two rings of 10 vertices each from mutually complementary 2-planes generates the grand antiprism; removing subsets of these vertices generates various intermediates (full exploration of the possibilities still in progress).

Infinite families

The obvious infinite family is that of the m,n-duoprisms (mn ≥ 3).

There is also an infinite family of prisms of the n-gonal antiprisms.

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. Keiji has devised a similar naming scheme to the one he used for the cupolic rings: the collective term is the family of biantiprismatic rings, and the specific term is the n-gonal biantiprismatic ring, e.g. square biantiprismatic ring. These ringed forms are included as an infinite subfamily in Klitzing's list of segmentotopes (they are known as wedges in Klitzing's terminology).

Non-Uniform Scaliform

Outside the uniform polychora (uniformed-celled and vertex transitive) there are 4 known convex scaliform (equilateral and vertex transitive)

Note: all the faces on a scaliform are regular

they are:

Bi-icositetradiminished 600-cell

The 600-cell can be made as the convex hull of 5 24-cells removing the vertices of one and taking the convex hull, creates the the snub 24-cell, removing another (and taking the convex hull) creates this shape. It cells are 48 tridimished icosahedra

Prismatorhombato snub 24-cell

This is made by shrinking the 24 icosahedral cells of a snub 24-cell(all the tetrahedra are removed) Where icosahedra used to be joined by triangles they are now seperated by triangular prisms(96 in total), the holes in the shape are filled with 24 truncated tetrahedra and 96 triangular cupola.

Truncated Tetrahedra Cupoliprism (one of Richard Klitzing segmentotopes)

this is made by taking two Truncated Tetrahedra with hexagons lined up to triangles and then putting 8 triangular cuplolae on the hexagons and triangles, then filling in the holes with 5 tetrahedra.

Swirlprismatodiminished Rectified 600-cell

the Rectified 600-cell can be made as the convex hull of 6 600-cells, removed the vertices of one and take the convex hull and you get this shape. It's cells are 600 square pyramids, 120 pentagonal prisms and 120 pentagonal antiprisms.

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