CRFP4DP/Augmentations (Meta, 14)

From Hi.gher. Space

Augmented uniform polychora

Some of the uniform polychora can be augmented (have CRF pyramids or other prismatoids 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 prismatoid whose base is in the shape of the cell being augmented.
  • The sum of each dichoral angle of the prismatoid'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 prismatoid 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 cantitruncated pyrochoron (x3x3x3o) can be augmented with the segmentotope x3x3o||x3x3x (truncated tetrahedron atop truncated octahedron) to produce a CRF augmentation that contains 4 gyrobifastigium (J26) cells (the latter being produced by the merging of pairs of corealmar triangular prisms).


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 tesseract (considered as a 4,4-duoprism) can be augmented with 4 square pyramid prisms and 4 square||orthogonal line 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.

The truncated tesseract can be augmented with eight cuboctahedron||truncated_cube's to form the octaaugmented truncated tesseract, a CRF polychoron with 24 square orthobicupolae, 16 tetrahedra, 64 triangular prisms, and 8 cuboctahedra.

The runcinated tesseract can be augmented with eight cubical pyramids to form a polychoron with 24 elongated square bipyramids, 24 triangular prisms, and 16 tetrahedra.

The runcitruncated 16-cell (x4o3x3x) can be augmented with eight octahedron||cuboctahedron (K4.29) to form the octaaugmented runcitruncated 16-cell, a polychoron with 8 octahedra, 24 elongated square bipyramids, 16 truncated tetrahedra, 32 hexagonal prisms, and 160 triangular prisms.

The runcitruncated tesseract (x4x3o3x) can be augmented with eight cuboctahedon||truncated_cube to produce the cantellated 24-cell (o3x4o3x), which is uniform.


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 with n-prism pyramids:

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. Ten non-adjacent pentagonal prisms can be augmented in this way, producing the omniaugmented 5,20-duoprism.

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.
  • 8,8-duoprism: augmentable with square magnabicupolic rings; these are included as a subset of the diminishings and gyrations of the cantellated tesseract.

Since the n-gonal cupola may be constructed by radially expanding the triangular faces of the n-gonal pyramid, so the segmentochoron 2n-prism||n-gon (n-gonal magnabicupolic ring) may be constructed by radially expanding the pyramid cells of the n-prism pyramid. The dihedral angles of the triangles with the base polygon remain the same, implying that for every n-gonal prism pyramid, the n-gonal magnabicupolic ring exhibits the same dichoral angles between the base prism and the cupola and tetrahedral cells above it as the respective pyramids and tetrahedra in the n-gonal prism pyramid do. Therefore, the bicupolic rings may be fitted on m,2n-duoprisms in the corresponding positions as the prism pyramid augments occur on the m,n-duoprism.

Therefore, every augmentation of an m,n-duoprism by prism pyramids has a corresponding augmentation of the m,2n-duoprism with n-gonal magnabicupolic rings. So there are at least 1633 augmentations of duoprisms by these bicupolic rings. However, there are many more, because these segmentochora may be rotated relative to each other in some of the m,2n-duoprisms, and for m=3,4,5, the other ring of prisms may be augmentable by m-gonal prism pyramids, leading to many more possible combinations. A computer search found 11,956,959 duoprism augmentations with n-gonal prism pyramids and magnabicupolic rings (this includes the 1633 augmentations in the previous section), 11,921,273 of which are due to the augmentations of the 10,10-duoprism, which exhibited combinatorial explosion due to the augmentability and orientability of both rings. Note that these counts are not verified yet, as they are still pending independent review.

Augmented polyhedral prisms

Some of the polyhedral prisms have prism side-cells that can be augmented with prism pyramids, due to the shallow dichoral angles of certain augments such as the pentagonal prism pyramid. These include: