# The Catalan Polychora

The *Catalan Polychora* are the duals of the uniform
polychora. They are the generalization of the Catalan
solids to 4D. They are cell-transitive, but generally not vertex-, edge-,
or face-transitive, and have multiple edge lengths, non-regular faces, and
non-uniform cells.

Being duals of the uniform polychora, there are 47 Catalan non-prismatic polychora, and they follow the same classification according to the symmetry families of the 6 regular polychora.

Under a broader interpretation we may also include the duals of the duoprisms, the duotegums, in this category. They are an infinite family, being the duals of the infinite family of duoprisms.

Being the duals of the uniform polychora confers these polytopes the following properties:

- They are cell-transitive;
- All dichoral angles are equal;
- All dihedral angles around each edge are equal.

## The 5-cell family

The 5-cell family consists of 13 members. However, due to the self-duality
of the 5-cell, only 8 of these 13 are distinct. Furthermore, also due to
self-duality, 3 of them (marked with ^{*}) have a higher degree of
symmetry than the others.

(Dual rectified dual 5-cell: identical to the joined pentachoron.)

(Dual truncated dual 5-cell: identical to the tetrakis pentachoron.)

Joined pentachoron (dual of the rectified 5-cell): bounded by 10 triangular bipyramids with edge ratio 2 : 2 : 3.

(Dual cantellated dual 5-cell: identical to the notched triacontachoron.)

Bidecachoron

^{*}(dual of the bitruncated 5-cell): bounded by 30 tetragonal disphenoids with edge ratio √3 : √5.(Dual cantitruncated dual 5-cell: identical to the sphenoidal hexecontachoron.)

Triangular antitegmatic icosachoron

^{*}(dual of the runcinated 5-cell): bounded by 20 triangular trapezohedra with uniform edge length.Notched triacontachoron (dual of the cantellated 5-cell).

(Dual of runcitruncated dual 5-cell: identical to the rhombipyramidal hexecontachoron.)

Tetrakis pentachoron (dual of the truncated 5-cell).

Rhombipyramidal hexecontachoron (dual of the runcitruncated 5-cell).

Sphenoidal hexecontachoron (dual of the cantitruncated 5-cell).

Disphenoidal hecatonisocachoron

^{*}(dual of the omnitruncated 5-cell).

## The tesseract/16-cell family

The tesseract/16-cell family consists of 13 members having the symmetry of the tesseract and the 16-cell. Due to the coincidence of the 24-cell with the rectified 16-cell, three of these 13 members coincide with members of the 24-cell family, thus leaving 10 unique members in this family.

(Dual rectified 16-cell: regular.)

Hexakis tesseract (dual of the truncated 16-cell).

Joined 16-cell (dual of the rectified tesseract): bounded by 32 triangular bipyramids with edge length ratio √7 : 3√2.

(Dual of the cantellated 16-cell: identical to the joined 24-cell.)

Disphenoidal enneacontahexachoron (dual of the bitruncated tesseract / bitruncated 16-cell).

(Dual of the cantitruncated 16-cell: identical to the octakis 24-cell.)

Triangular antitegmatic hexacontatetrachoron (dual of the runcinated tesseract): bounded by 64 triangular trapezohedra with edge ratio (1/7)√(76+37√2) : 2 : 4−√2.

Notched enneacontahexachoron (dual of the cantellated tesseract).

Deltopyramidal hecatonenneacontadichoron (dual of the runcitruncated 16-cell).

Tetrakis 16-cell (dual of the truncated tesseract).

Rhombipyramidal hecatonenneacontadichoron (dual of the runcitruncated tesseract): bounded by 192 skew rhombic pyramids.

Sphenoidal hecatonenneacontadichoron (dual of the cantitruncated tesseract).

Tetrahedral triacosioctacontatetrachoron (dual of the omnitruncated tesseract / omnitruncated 16-cell).

## The 24-cell family

The 24-cell family consists of 14 members, 3 of which overlap with the
tesseract/16-cell family because of the coincidence of the
24-cell with the rectified 16-cell. Furthermore, since
the 24-cell is self-dual, only 9 of the members of this family are distinct.
The self-duality also causes some members (marked with ^{*}) of this
family to have a higher degree of symmetry than the 24-cell itself.

One special member of this family, the dual of the snub 24-cell, has a diminished 24-cell symmetry.

(Dual rectified dual 24-cell: identical to the joined 24-cell.)

(Dual truncated dual 24-cell: identical to the octakis 24-cell.)

Joined 24-cell (dual of the rectified 24-cell): bounded by 96 triangular bipyramids with edge ratio √5 : 3.

(Dual of cantellated dual 24-cell: identical to the notched enneacontahexachoron.)

Bitetracontoctachoron

^{*}(dual of the bitruncated 24-cell).(Dual of cantitruncated dual 24-cell: identical to the sphenoidal pentacosiheptacontahexachoron.)

Square antitegmatic hecatontetracontatetrachoron

^{*}(dual of the runcinated 24-cell): bounded by 144 square trapezohedra with edge ratio 0.658 : 1.Notched enneacontahexachoron (dual of the cantellated 24-cell).

(Dual runcitruncated dual 24-cell: identical to the deltopyramidal pentacosiheptacontahexachoron.)

Octakis 24-cell (dual of the truncated 24-cell).

Deltopyramidal pentacosiheptacontahexachoron (dual of the runcitruncated 24-cell).

Sphenoidal pentacosiheptacontahexachoron (dual of the cantitruncated 24-cell).

Disphenoidal chiliahecatonpentacontadichoron

^{*}(dual of the omnitruncated 24-cell).Quattro-icositetradiminished hexacosichoron (dual of the snub 24-cell).

## The 120-cell/600-cell family

The 120-cell/600-cell family comprises 14 duals of uniform truncates. All 14 are distinct. We include the pentagonal double antitegmoid (the dual of the grand antiprism) here, even though it does not have 120-cell symmetry.

Joined 120-cell (dual of the rectified 600-cell): bounded by 720 pentagonal bipyramids with edge ratio 3−φ : 2.

Dodecakis hecatonicosachoron (dual of the truncated 600-cell).

Joined 600-cell (dual of the rectified 120-cell).

Small notched trischiliahexacosichoron (dual of the cantellated 600-cell).

Disphenoidal trischiliahexacosichoron (dual of the bitruncated 120-cell).

Small sphenoidal heptachiliadiacosichoron (dual of the cantitruncated 600-cell).

Triangular antitegmatic dischiliatetracosichoron (dual of the runcinated 120-cell).

Great notched trischiliahexacosichoron (dual of the cantellated 120-cell).

Deltopyramidal heptachiliadiacosichoron (dual of the runcitruncated 600-cell).

Tetrakis hexacosichoron (dual of the truncated 120-cell).

Rhombipyramidal heptachiliadiacosichoron (dual of the runcitruncated 120-cell).

Great sphenoidal heptachiliadiacosichoron (dual of the cantitruncated 120-cell).

Tetrahedral myriatetrachiliatetracosichoron (dual of the omnitruncated 120-cell).

Pentagonal double antitegmoid (dual of the grand antiprism).

## Duotegums

The duotegums are the duals of the duoprisms. This is an infinite family, so we only list a few representative examples here.

## References

The naming of the 4D Catalans is borrowed from the Polytope wiki.