# Category A: Duoprisms

Duoprisms (also called double prisms) are the cross products of two polygons, its as though we are multiplying one polygon by another one - this category has an infinite size, here we will list all of them up to the decagonal ones. Convex ones are cross products of two convex polygons, all others are starry. Some duoprisms are cross products of two identical polygons, these have a higher symmetry - since these are like multiplying a polygon to itself, they are actually the square of that polygon - these polygon squares are a subset of the 4-D powertopes as well as their duals. Their verfs are disphenoids and are therefore related to category 7. As N approaches infinity, the N x N (or N2) duoprism approaches the duocylinder which is the square of a circle.

## Polygonal Squares

These are the N2 duoprisms. Their cells are identical which makes these guys noble. The square duoprism is the tesseract which is mentioned in category 1. The first 7 are convex.

Triddip - (TRID dip) trigonal duoprism. Its cells are 6 trips. It has 9 vertices and has triddip symmetry. Symbol = ox ox = ox2 = oxo'x. It has 6 pieces with one type.

Pedip - (PEE dip) pentagonal duoprism. Its cells are 10 pips. It has 25 vertices and has pedip symmetry. Symbol = o^x o^x = o^x2. It has 10 pieces in one type.

Hiddip - (HID dip) hexagonal duoprism. Its cells are 12 hips. It has 36 vertices and has hiddip symmetry (usually has triddip symmetry when found in polytera). Symbol = o6x o6x = xx xx = xx2. It has 12 pieces of one type.

Hedip - (HEE dip) heptagonal duoprism. Its cells are 14 heps (heptagon prisms). It has 49 vertices and has hedip symmetry. Symbol = o7x o7x = o7x2. It has 14 pieces of one type.

Odip - (O dip) octagonal duoprism. It has 16 ops. It has 64 vertices and odip symmetry (in polytera, it usually has "squiddip" symmetry (or 42). Symbol = o8x o8x = x'x x'x = x'x2. It has 16 pieces with one type.

Edip - (E dip) enneagonal duoprism. It has 18 eps (enneagon prisms). It has 81 vertices and has edip symmetry. Symbol = o9x o9x = o9x2. It has 18 pieces with one type.

Dedip - (DEE dip) decagonal duoprism. It has 20 dips. It has 100 vertices and has dip symmetry. Symbol = o10x o10x = o10x2 = x^x x^x. It has 20 pieces in one type.

Stardip - (STAR dip) pentagrammic duoprism (or star duoprism). It has 10 stips. It has 25 vertices and has pedip symmetry. Symbol = o*x o*x = o*x2. It has 20 pieces in one type. This is the square of a star.

Shedip - (SHE dip) heptagrammic duoprism. The cells are 14 ships (7/2 prism). It has 49 vertices and has hedip symmetry. Symbol = o7/2x o7/2x = o7/2x2. It has 28 pieces with one type.

Gishdip - (GISH dip) great heptagrammic duoprism. The cells are 14 giships (7/3 prism). It has 49 vertices and has hedip symmetry. Symbol = o7/3x o7/3x = o7/3x2. It has 28 pieces with one type.

Stodip - (STO dip) octagrammic duoprism. The cells are 16 stops. It has 64 vertices and has odip symmetry. Symbol = o8/3x o8/3x = o8/3x2 = x"x x"x. It has 32 pieces with one type.

Stedip - (STE dip) enneagrammic duoprism. The cells are 18 steps (9/2 prism). It has 81 vertices and has edip symmetry. Symbol = o9/2x o9/2x = o9/2x2. It has 36 pieces with one type.

Gastedip - (ga STEE dip) great enneagrammic duoprism. The cells are 18 gisteps (9/4 prism). It has 81 vertices and has edip symmetry. Symbol = o9/4x o9/4x = o9/4x2. It has 36 pieces with one type.

Stadidip - (sta DID dip) decagrammic duoprism. The cells are 20 stiddips. It has 100 vertices and has dedip symmetry. Symbol = o10/3x o10/3x = o10/3x2 = x*'x x*'x. It has 40 pieces with one type.

N-Dip - (IN dip) N-gonal duoprism (N > 2, N is an integer). Cells are 2N N-gonal prisms. It has N2 vertices and has N-dip symmetry. Symbol = oNx oNx = oNx2. It has 2N pieces of one type. These are convex.

N/M-Dip - (NIM dip) M density N-grammic duoprism (N/M > 2, M and N relatively prime). Cells are 2N N/M-prisms. It has N2 vertices and has N-dip symmetry. Symbol = oN/Mx oN/Mx = oN/Mx2. It has 4N pieces of one type.

## Duoprisms based off of triangular and square symmetries.

These duoprisms may show up as facets for various non-prismattic uniform polytera which makes these a bit more significant. Triddip, hiddip, odip, and stodip could also fit here - but has been counted in the squared polygons above - as well as the tesseract from category 1. The symmetries and symbols reflect how they show up in higher polytopes. First six are convex.

Thiddip - (THID dip) trigonal-hexagonal duoprism. Its cells are 6 trips and 3 hips. It has 18 vertices and has trigon x trigon symmetry. Symbol = ox xx. It has nine pieces with two types.

Tisdip - (TISS dip) trigonal-square duoprism. Its cells are 4 trips and 3 cubes. It has 12 vertices and has tisdip symmetry. Symbol = ox o'x. It has seven pieces in two types.

Todip - (TOE dip) trigonal-octagonal duoprism. Its cells are 8 trips and 3 ops. It has 24 vertices and has tisdip symmetry. Symbol = ox x'x. It has 11 pieces with two types.

Shiddip - (SHID dip) square-hexagonal duoprism. Its cells are 6 cubes and 4 hips. It has 24 vertices and has tisdip symmetry. Symbol = xx o'x. It has 10 pieces with two types.

Hodip - (HO dip) hexagonal-octagonal duoprism. Its cells are 8 hips and 6 ops. It has 48 vertices and has tisdip symmetry. Symbol = xx x'x. It has 14 pieces with two types.

Sodip - (SO dip) square-octagonal duoprism. Its cells are 8 cubes and 4 ops. It has 32 vertices and has square x square (or sisdip) symmetry. Symbol = o'x x'x. It has 12 pieces in two types.

Tistodip - (tiss TOE dip) trigonal-octagrammic duoprism. Its cells are 8 trips and 3 stops. It has 24 vertices and has tisdip symmetry. Symbol = ox x"x. It has 19 pieces in two types.

Histodip - (hiss TOE dip) hexagonal-octagrammic duoprism. Its cells are 8 trips and 6 stops. It has 48 vertices and has tisdip symmetry. Symbol = xx x"x. It has 22 pieces with two types.

Sistodip - (sis TOE dip) square-octagrammic duoprism. Its cells are 8 cubes and 4 stops. It has 32 vertices and has sisdip symmetry. Symbol = o'x x"x. It has 20 pieces with two types.

Ostodip - (OSS toe dip) octagon-octagrammic duoprism. Its cells are 8 ops and 8 stops. It has 64 vertices and has sisdip symmetry. Symbol = x'x x"x. It has 24 pieces in two types.

## Other Convex Cases

These are the convex ones that don't fit in the above groups. They will be listed as M,N. Name - (pro noun see AYE shun) for the M,N-Dip. For an M,N-Dip the cells are N M-prisms and M N-prisms, they have M*N vertices with M,N-Dip symmetry. Symbol = oMx oNx. They have M+N pieces of two types.

3,5. Trapedip - (tra PEE dip) 3,7. Theddip - (THEH dip) 3,9. Tedip - (TEE dip) 3,10. Tradedip - (tra DEE dip)

4,5. Squipdip - (SQUIP dip) 4,7. Squahedip - (skwa HE dip) 4,9. Sendip - (SEND dip) 4,10. Squadedip - (skwa DEE dip)

5,6. Phiddip - (FID dip) 5,7. Pheddip - (FED dip) 5,8. Podip - (PO dip) 5,9. Peendip - (PEEN dip) 5,10. Padedip - (pa DEE dip)

6,7. Hahedip - (ha HEE dip) 6,9. Hendip - (HEND dip) 6,10. Hadedip - (ha DEE dip)

7,8. Heodip - (HYO dip) 7,9. Heendip - (HEEN dip) 7,10. Hedadip - (he DAY dip)

8,9. Oedip - (OIE dip) 8,10. Odedip - (o DEE dip) 9,10. Edidip - (e DID dip)

M,N. M,N-Dip - (em IN dip) - hmmm...we could call these the "emindips"

## Convex-Star Duoprisms

These are the duoprisms that are cross products of a convex and a star case, that has yet to be mentioned. For a P,N/M-Dip - the cells are P N/M-prisms and N P-prisms, they have N*P vertices with P,N-Dip symmetry. Symbol = oPx oN/Mx. They have P+2N pieces of two types.

3,5/2. Tistadip - (TISS ta dip) 3,10/3. Tistadedip - (TISS ta DEE dip)

4,5/2. Sistadip - (SIS ta dip) 4,10/3. Sistadedip - (SIS ta DEE dip) - Good name for a female rap star - but then again "Rapstar" would be a good name for the pentagram-rap duoprism

5,5/2. Starpedip - (STAR pe dip) 5,8/3. Pistodip - (PISS toe dip) 5,10/3. Pistadedip - (PISS ta DEE dip)

6,5/2. Stahdip - (STAH dip) 6,10/3. Histadedip - (HISS ta DEE dip)

7,5/2. Stahedip - (stah HE dip) 7,8/3. Hestodip - (HESS sto dip) 7,10/3. Hestadedip - (HESS ta DEE dip)

8,5/2. Starodip - (STARE o dip) 8,10/3. Ostadedip - (OSS ta DEE dip)

9,5/2. Starendip - (sta REN dip) 9,8/3. Stoendip - (STOIN dip) 9,10/3. Estadedip - (ES ta DEE dip)

10,5/2. Stardedip - (star DEE dip) 10,8/3. Stodedip - (sto DEE dip) 10,10/3. Distadedip - (DISS ta DEE dip)

P,N/M. P,N/M-Dip - (pa NIM dip) - I guess these can be called the "panimdips"

## Double Star Duoprisms

These are the duoprisms that are cross products of two star polygons. For an N/M,P/Q-Dip - the cells are P N/M-prisms and N P/Q-prisms, they have N*P vertices with P,N-Dip symmetry. Symbol = oN/Mx oP/Qx. They have 2P+2N pieces in two types.

5/2,8/3. Stastodip - (STAS toe dip) 5/2,10/3. Stastidedip - (stast IDE dee dip) 8/3,10/3. Stostidedip - (sto STIDE dee dip)

N/M,P/Q. N/M,P/Q-Dip - (NIM pik dip) - you guessed it, these are the "nimpiqdips"

Special thanks to Dr. Richard Klitzing for some of the heptagonal and enneagonal short names as well as helping me to keep these names from clashing with others.

## Conjugates

Below shows how the duoprisms pair up as conjugates, limited to the trigon and square symmetric ones.

Conjugates

The following are self-conjugate: triddip, thiddip, tisdip, shiddip, hiddip, and ostodip.

The following are conjugate pairs: odip-stodip, todip-tistodip, sodip-sistodip, and hodip-histodip.

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