CRFP4DP/Augmentations (Meta, 14)

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Revision as of 16:56, 1 March 2014

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.

Pyromorphs

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

Stauromorphs

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 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 cantellated 16-cell can be augmented with eight octahedron||cuboctahedron (K4.29) to form 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.

Xylomorphs

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:

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. There are probably 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.