List of scaliform polychora

A polychoron is scaliform if it is vertex-transitive and all edges have the same length. All uniform polytopes are scaliform, but starting in the 4th dimension, not all scaliform polytopes are uniform. Besides the infinite family of hemiantiprisms, there are at least 1001 non-uniform scaliform polychora. Note that Bowers' site (the main source for this list) includes pseudo-scaliform, fissary, and even some exotic polytopes. Those will not be listed here.

Category A: Hemiantiprisms (infinite)
The hemiantiprisms are a type of alterprism. Two true antiprisms and two retroprisms are joined together at the bases in a rectangular manner, and bowtie tegum s passing through the center complete the polychoron. The skew octahemioctachoron (category 1) is a degenerate case.

Category 1: Simple Scaliforms (S1 - S9)
This category is for the simplest cases. Six of these are alterprisms while three are hexadecachoron facetings.

Category 2: Podary Scaliforms (S10 - S21)
The inverted quasiprismatodishexadecachoron (iquipadah) can be obtained as a blend of sidpith and odop. Blending odop with other sidpith members results in six scaliforms. They are nonuniform because they contain square cupolae and related polyhedra. They each have conjugates related to gaquipadah.

Category 3: Special Scaliforms (S22 - S91, 3 unnumbered)
These are the scaliforms that do not fit into any other category. There are many more of these than were previously thought, divided into the subcategories: noble diminishings, snub containing scaliforms (containing sirsid and seside cells), ike snubs (related to sadi), duoprismatic blends, truncate blends, blended tessic snubs (all blends of uniform polychora), ico scaliforms, gudap scaliforms, gap scaliforms, padiap scaliforms, ipe blends, greater ipe blends, ixitic ipe blends, tesapdid regiment, diminishings & friends (formerly a category of its own), and tispy regiment.

Category 4: Ex and Gishi Scaliforms (S92 - S117)
These are scaliforms from the ex and gishi regiments, related to sisp and gisp.

Category 5: Ondip Family (20 members)
Ondip can be obtained as a blend of 4 sidpith. Doing this with other members of the sidpith regiment results in ten scaliforms. They all have conjugates related to gondip.

Category 6: Hexadecagonal Scaliforms (8 members)
Unexpectedly, some scaliform polychora contain hexadecagons as faces.

Category 7: Scaliform Sidtaps and Gidtaps (24 members)
These are scaliform members of the sadsadox and gadsadox regiments. In them, cells from the rox and rigfix regiments fuse to form nonuniform cells.

Category 8: Typical Scaliform Idcossids (310 members)
These are scaliform members of the sadros daskydox regiment from blending uniform members of the padohi regiment.

Category 9: Typical Scaliform Dircospids (310 members)
These are scaliform members of the gadros daskydox regiment from blending uniform members of the gidipthi regiment.

Category 10: Sishi Scaliforms
This category contains various facetings of the small stellated hecatonicosachoron. This category is very large because one of the possible cell types is a small ditrigonary icosidodecahedron, which has many nonuniform facetings.

Category 11: Sidtaxhi Swirlprisms (11 members)
Facetings of sidtaxhi with swirlprismatic symmetry.

Category 12: Dattady Swirlprisms (17 members)
Facetings of dattady with swirlprismatic symmetry.

Category 13: Gadtaxady Swirlprisms (11 members)
Facetings of gadtaxady with swirlprismatic symmetry.

Category 14: Sidtaxhi Comboblends (66 members)
This regiment is formed from blending in the 12-sidtaxhi compound.

Category 15: Gadtaxady Comboblends (66 members)
This regiment is formed from blending in the 12-gadtaxady compound.

Category 16: Tetrasidpiths
This regiment is formed from blending in the 12-sidpith compound.

Category 17: Tetragittiths
This regiment is formed from blending in the 12-gittith compound.

Category 18: Ixitic Padohis (18 members)
The prismatorhombisnub icositetrachoron can be embedded in padohi. Blending five of them results in an ixitic-symmetric polychoron (same symmetry as setut).

Category 19: Ixitic Gidipthis (18 members)
The prismatorhombiretrosnub icositetrachoron can be embedded in gidipthi. Blending five of them results in an ixitic-symmetric polychoron.

Category 20: Prissi-Blended Idcossids
These are scaliform members of the sadros daskydox regiment from blending members of category 18.

Category 21: Prarsi-Blended Dircospids
These are scaliform members of the gadros daskydox regiment from blending members of category 19.

Category 22: Pecuexdap Regiment
Scaliform members of the pecuexdap regiment, having the vertices of gap and edge length of sishi.

Category 23: Disdi Super Regiment
These are scaliform members of the disdi regiment that are not ico-blended. Disdi itself is scaliform (in category 3) but is not included here due to its higher symmetry.

Category 24: Scaliform Idtessids
These are further scaliform members of the disdi regiment that, like the uniform idtessids, contain facets of an ico compound.

Category 25: Decasishi Scaliforms
These are in the sisdex (10-sishi compound) regiment. Due to blending, they become true polychora.

Category 26: Bhidhi Army
These are derived from sishi but have the same vertices as bhidhi.

Category 27: Octicoes and Hexadecicoes
The octicoes are formed from blending 8 ico regiment members. The hexadecicoes are formed from blending 16 ico regiment members.

Category 28: Monster Ipe Blends
The much more complex versions of the ipe blends, greater ipe blends and ixitic ipe blends from category 3.

Category 29: Hexasidpiths
A regiment formed by blending sidpith such that six sidpith verfs combine at each verf.

Category 30: Hexagittiths
A regiment formed by blending gittith such that six gittith verfs combine at each verf.