# Non-CRF facets of CRF duals (ConceptTopic, 3)

### From Hi.gher. Space

The set of **non-CRF facets of CRF duals** is currently being studied by Keiji. A proper name for the set has not yet been thought up. They are reminiscent of the Catalan solids. This page lists them in order of discovery.

- Triangulated triangular prism
- First discovered in the dual of the BXD and appeared again shortly after in the dual of the (4D analog of J37). It also appears in the dual of the J92 rhombochoron and so far it is the most commonly appearing polytope in this list. Self-dual and chiral.
- Hexagonal cupolawedge
- First discovered in the dual of the bilunabirotunda pseudopyramid and appeared later in the tetraaugmented J92 rhombochoron. Dual is the hexagonal pyrawedge which has not yet been seen in any other polytopes. Not chiral.
- Square cupolawedge
- First discovered in the dual of the triangular hebesphenorotunda pseudopyramid and appeared later at the same position in the tetraaugmented J92 rhombochoron. Dual is the pentagonal cupolawedge which has not yet been seen in any other polytopes. Not chiral.
- Rhombocupolawedge
- First discovered in the dual of the D4.7 and appeared again in the dual of the D4.11. Dual is the rhombopyrawedge which has not yet been seen in any other polytopes. Not chiral.
- Metabigyrotriangulated cube
- First discovered in the dual of the D4.7. Self-dual and chiral.
- Parabiorthotriangulated cube
- First discovered in the dual of the J92 rhombochoron. Self-dual, but not chiral. It is the first such known polytope (in this list, anyway) and is also the first to break an incorrect assumption made by the current version of the polytope explorer, which will thus not accept the polytope as legitimately self-dual.
- Triangulated pinched triangular cupola
- First discovered in the dual of the J92 rhombochoron. Self-dual. Has no symmetry at all, and is therefore chiral. (The parabiorthotriangulated metapinched triangular cupola displayed to the right was previously thought to be the dual by mistake.)
- Biunpinched cube
- First discovered in the dual of the J92 rhombochoron and appeared shortly after at the same position in the tetraaugmented J92 rhombochoron. Dual is the parabiorthotriangulated orthopinched triangular cupola which has not yet been seen in any other polytopes. Not chiral.
- Digon-unpinched pentagonal prism
- First discovered in the dual of the tetraaugmented J92 rhombochoron. Dual is the snub disphenoid, which makes it the first discovered non-CRF to be both the dual of a CRF, and the cell of a dual of a CRF, at the same time.
- Square transcupolawedge
- First discovered in the dual of the tetraaugmented J92 rhombochoron. Dual is the pentagonal transcupolawedge which has not yet been seen in any other polytopes. This pair is very similar to the square cupolawedge and pentagonal cupolawedge described above, with the only difference that a pair of edges is swapped.
- Transposed diminished cube
- First discovered in the dual of the D4.16. Dual is the digon-unpinched square pyramid which has not yet been seen in any other polytopes.