# Category B: Antiduoprisms

Antiduoprisms (also called antiprism prisms (or appips for short)) are the prisms of the antiprisms - this category has an infinite size, here we will list all of them up to the decagonal ones. Their verfs are pyramids of trapezoids or crossed trapezoids. As N approaches infinity, the N antiprism prism (or N appip) approaches a disk. There's also a group of polychora called duoantiprisms, but only the 2,2-dap and the 5,5/3-dap are uniform, these are hex (from cat. 1) and gudap (from cat. 20).

## Convex Cases

These are the convex cases, notice that the 2-appip and the 3-appip are missing, these are tepe and ope (tetrahedral prism and octahedral prism) respectively, both of which are in category 19.

Squapip - (SKWA pip) square antiprism prism (4-appip). Its cells are 2 squaps, 2 cubes, and 8 trips. It has 16 vertices and has squapip symmetry. Symbol = x s s's. It has 12 pieces with three types.

Pappip - (PAP pip) pentagonal antiprism prism (5-appip). Its cells are 2 paps, 2 pips, and 10 trips. It has 20 vertices and has pappip symmetry. Symbol = x s s^s. It has 14 pieces in three types.

Happip - (HAP pip) hexagonal antiprism prism (6-appip). Its cells are 2 haps (6 AP), 2 hips, and 12 trips. It has 24 vertices and has happip symmetry. Symbol = x s s6s. It has 16 pieces of three types.

Heappip - (he AP pip) heptagonal antiprism prism (7-appip). Its cells are 2 heaps (7 AP), 2 heps (7 P), and 14 trips. It has 28 vertices and has heappip symmetry. Symbol = x s s7s. It has 18 pieces of three types.

Oappip - (o AP pip) octagonal antiprism prism (8-appip). It has 2 oaps (8 AP), 2 ops, and 16 trips. It has 32 vertices and oappip symmetry. Symbol = x s s8s. It has 20 pieces of three types.

Eappip - (e AP pip) enneagonal antiprism prism (9-appip). It has 2 eaps (9 AP), 2 eps (9 P), and 18 trips. It has 36 vertices and has eappip symmetry. Symbol = x s s9s. It has 22 pieces of three types.

Dappip - (DAP pip) decagonal antiprism prism (10-appip). It has 2 daps (10 AP), 2 dips, and 20 trips. It has 40 vertices and has dappip symmetry. Symbol = x s s10s. It has 24 pieces of three types.

N-Appip - (in AP pip) N-gonal antiprism prism (N >= 2, N is an integer). Cells are 2 N-gonal antiprisms, 2 N-gonal prisms, and 2N trips. It has 4N vertices and has N-appip symmetry. Symbol = x s sNs. It has 2N+4 pieces of three types.

## Star Cases

These are the star cases with trapezoid pyramid verfs. For an N/M-appip, N/M must be greater than 2 to be in this set.

Stappip - (STA pip) pentagrammic (or star) antiprism prism (5/2-appip). Its cells are 2 staps, 2 stips, and 10 trips. It has 20 vertices and has pappip symmetry. Symbol = x s s*s. It has 34 pieces with four types.

Shappip - (SHAP pip) heptagrammic antiprism prism (7/2-appip). Its cells are 2 shaps (7/2 AP), 2 ships (7/2 P), and 14 trips. It has 28 vertices and has heappip symmetry. Symbol = x s s7/2s. It has 46 pieces in four types.

Gishappip - (gih SHAP pip) great heptagrammic antiprism prism (7/3-appip). Its cells are 2 gishaps (7/3 AP), 2 giships (7/3 P), and 14 trips. It has 28 vertices and has heappip symmetry. Symbol = x s s7/3s. It has 74 pieces of five types.

Stoappip - (sto AP pip) octagrammic antiprism prism (8/3-appip). Its cells are 2 stoaps (8/3 AP), 2 stops, and 16 trips. It has 32 vertices and has oappip symmetry. Symbol = x s s8/3s. It has 84 pieces of five types.

Steappip - (ste AP pip) enneagrammic antiprism prism (9/2-appip). It has 2 steaps (9/2 AP), 2 steps, and 18 trips. It has 36 vertices and eappip symmetry. Symbol = x s s9/2s. It has 58 pieces of four types.

Gisteappip - (GIS tee AP pip) great enneagrammic antiprism prism (9/4-appip). It has 2 gisteaps (9/4 AP), 2 gisteps (9 P), and 18 trips. It has 36 vertices and has eappip symmetry. Symbol = x s s9/4s. It has 130 pieces of six types.

Stidappip - (stid DAP pip) decagrammic antiprism prism (10/3-appip). It has 2 stidaps (10/3 AP), 2 stiddips, and 20 trips. It has 40 vertices and has dappip symmetry. Symbol = x s s10/3s. It has 104 pieces of five types.

N/M-Appip - (nim AP pip) M density N-grammic antiprism prism (N/M > 2). Cells are 2 N/M-gonal antiprisms, 2 N/M-gonal prisms, and 2N trips. It has 4N vertices and has N-appip symmetry. Symbol = x s sN/Ms. It has 2N(2M-1)+4 pieces of M+2 types.

## Crossed Cases

These are the star cases with crossed trapezoid pyramid verfs. For an N/M-appip, N/M must be less than 2 and greater than 1.5 to be in this set.

Starpip - (STAR pip) pentagrammic (or star) retroprism prism (5/3-appip). Its cells are 2 starps, 2 stips, and 10 trips. It has 20 vertices and has pappip symmetry. Symbol = x s s*'s. It has 78 pieces with seven types.

Gisharpip - (gih SHAR pip) great heptagrammic retroprism prism (7/4-appip). Its cells are 2 gisharps (7/4 AP), 2 giships (7/3 P), and 14 trips. It has 28 vertices and has heappip symmetry. Symbol = x s s7/4s. It has 134 pieces of eight types.

Storpip - (STORE pip) octagrammic retroprism prism (8/5-appip). Its cells are 2 stoarps (8/5 AP), 2 stops, and 16 trips. It has 32 vertices and has oappip symmetry. Symbol = x s s8/5s. It has 184 pieces of nine types.

Gisterpip - (gis TERP pip) great enneagrammic retroprism prism (9/5-appip). It has 2 gistearps (9/5 AP), 2 gisteps (9 P), and 18 trips. It has 36 vertices and has eappip symmetry. Symbol = x s s9/5s. It has 206 pieces of nine types.

N/(N-M)-Rappip - (NI nim RAP pip) N-M density N-grammic retroprism prism (2 > N/M > 1.5) (also called N/M-appip). Cells are 2 N/M-gonal antiprisms, 2 N/M-gonal prisms, and 2N trips. It has 4N vertices and has N-appip symmetry. Symbol = x s sN/Ms. It has 2N(2M+1)+8 pieces of M+4 types.

## Conjugates

Below shows how the antiduoprisms pair up as conjugates, limited to the square, pentagonal, and hexagonal symmetric ones.

Conjugates

The following are self-conjugate: squapip, stappip, and happip.

The following are conjugate pairs: pappip-starpip.

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