Polytwisters

Polytwisters
Above: a montage of cross sections of five different polytwisters.

Polytwisters are curvaceous four dimensional polytope like objects with an interesting twist, instead of having vertices as their simplest (or basic) elements, they have equatorial circles (also called rings). These rings actually swirl around each other in a twisted like arrangement. In place of edges, they have "strips" which look sort of like a full twist Mobius strip. Finally, for their facets, they have cylindrically curved "twisters". A twister resembles a polygonal rod (which is somewhat bloated) which is twisted 360 degrees and then curved back in on itself into a cylindrical ring. Polytwisters are related to Hopf Fibration. Hopf fibration represents the fact that a 4-D sphere (glome) can be split into an infinity of disjoint equators all swirling around each other. There is a one to one function that takes each point of a sphere to one of the equators of the glome - so in other words, a glome is a sphere of equators. So what would happen if we started with a dodecahedron instead of a sphere and took the "Hopf function" of it? - you would get the "Hopf dodecahedron" which looks a lot like the dodecatwister, except that the Hopf version's facets bow inwards in a concave way, while the dodecatwister remains convex. I suspect that Hopf polyhedra were known since Hopf discovered his fibration, however - their relatives, the polytwisters, may be something new. Polytwisters have symmetries like polyhedra, but also have the capacity to roll smoothly like cylinders.

Regular Polytwisters

There are 9 regular Hopf polyhedra (one for each regular polyhedron), but their are 36 regular polytwisters, as well as an infinity of regular dyadic polytwisters (called dysters for short). A polytwister is regular if all of it's rings are identical in a transitive way, same thing for its strips and twisters, also it's twisters must have a cross section which is topologically and symmetrically congruent to a regular polygon - in other words, an orthogonal twister section looks like a regular polygon with all of it's edges bowed out to the same degree. Below is a list of the regular polytwisters. Pictures can be seen by clicking on the names.

1. Tetter - (TET tur) also called tet twister, or tetratwister. It is the {3,3} twister. It has 4 rings, 6 strips, and 4 trigonal twisters. It's ring figure (rinf) is a bowed triangle. It is related to the Hopf tetrahedron. It would make an interesting 4-sided die where each of it's sides can roll like a cylinder. Tetteric symmetry is an continuous form of tetrahedral swirlprism symmetry which is connected to icoic symmetry.

2. Quitter - (QUIT tur) quasitetratwister (also stellated tetratwister), or quit twister. It is the {3/2,3} twister. It has 4 rings, 6 strips, and 4 retrotrigonal twisters. Its rinf is a bowed triangle. It's sections resemble strange drill bits. It has tetteric symmetry. It is formed by stellating the tetratwister.

3. Blitter - (BLIT tur) bloated tetratwister, or blit twister. It is the {3,3/2} twister. It has 4 rings, 6 strips, and 4 tripodic twisters. In the bloated twisters, the rings are no longer extroverted, they are now introverted. The rinf is a bloated retrotriangle. Its cross sections would make some bizarre lipstick designs. It has tetteric symmetry.

4. Itter - (IT tur) inverted tetratwister, or it twister. It is the {3/2,3/2} twister (I first thought this one and other inverted twisters didn't exist, but Mason Green found out that they did, however they seem to copycat existing bloated twisters). It has 4 rings, 6 strips, and 4 retrotripodic twisters. Like the bloated twisters, the rings are no longer extroverted, they are now introverted. The rinf is a bloated retrotriangle. It copy-cats blitter. It has tetteric symmetry.

5. Cubiter - (CUBE it tur) cube twister, also called hexatwister. It is the {4,3} twister and is related to the Hopf cube. It has 8 rings, 12 strips, and 6 square twisters. Its rinf is a bloated triangle. It has cubiteric symmetry which is a continuous form of cubic swirlprism symmetry which is related to contic symmetry. It would make a neat 6 sided die, which can roll on each side. I would love to see a Rubik's cubiter.

6. Quicter - (QUIC tur) quasicube twister, or quic twister. This is the {4/3,3} twister. It has 8 rings, 12 strips, and 6 retrosquare twisters. Its rinf is a bloated triangle. It has cubiteric symmetry.

7. Blicter - (BLIC tur) bloated cube twister, or blic twister. This is the {4,3/2} twister. It has 8 introverted rings, 12 strips, and 6 flower-like bloated square twisters. Its rinf is a bloated retrotriangle. It has cubiteric symmetry.

8. Icter - (IC tur) inverted cube twister, or ic twister. This is the {4/3,3/2} twister. It has 8 introverted rings, 12 strips, and 6 flower-like bloated retrosquare twisters. Its rinf is a bloated retrotriangle. It has cubiteric symmetry and it copycats blicter.

9. Octer - (OK tur) oct twister or octatwister. This is the {3,4} twister and is related to the Hopf octahedron. It has 6 rings, 12 strips, and 8 trigonal twisters. Its rinf is a bloated square. It has cubiteric symmetry. It would make a great 8 sided die, where each side rolls like a cylinder - if the octer rolls onto another side - it will roll off at an angle.

10. Quoter - (QUO tur) quasioctatwister or quo twister. This is the {3/2,4} twister. It has 6 rings, 12 strips, and 8 retrotrigonal twisters. Its rinf is a bloated square. It has cubiteric symmetry.

11. Bloter - (BLO tur) bloated octatwister or blo twister. This is the {3,4/3} twister. It has 6 introverted rings, 12 strips, and 8 tripodic twisters. Its rinf is a bloated retrosquare. It has cubiteric symmetry. Its sections sort of remind me of those strange looking shadows that result from small objects floating in a pool.

12. Ioter - (i O tur) inverted octatwister or Io twister (not related to Jupiter's volcanic moon). This is the {3/2,4/3} twister. It has 6 introverted rings, 12 strips, and 8 retrotripodic twisters. Its rinf is a bloated retrosquare. It has cubiteric symmetry. It copy cats bloter.

These have doter (dodecatwister) symmetry which is a continuous version of swirlprism symmetry which is related to hyic symmetry. Each of these have 30 strips. There are two pages of sections for these polytwisters, for the second page, click the dash after it's name.

13. Doter - (DOE tur) dodecatwister, or doe twister. This is the {5,3} twister, it is related to the Hopf dodecahedron. It has 20 rings, and 12 pentagonal twisters. Its rinf is a bloated triangle. It would make an interesting 12 sided die, where each of its sides roll.

14. Quadoter - (qua DOE tur) quasidodecatwister or quado twister (once called quiditer and quid twister - names that now goes to the quasi-icosidodecatwister). This is the {5/4,3} twister. It has 20 spiky rings and 12 retropentagonal twisters. Its rinf is a bloated triangle. This one looks very spiky, but what looks like spikes in the sections are actually parts of ring-like structures - this thing can roll smoothly as though it were on wheels (just like the other polytwisters).

15. Bladoter - (bla DOE tur) bloated dodecatwister or blado twister (once called bliditer and blid twister, names that now goes to the bloated icosidodecatwister). This is the {5,3/2} twister. It has 20 introverted rings, and 12 flower-like pentapodic twisters. Its rinf is a bloated retrotriangle.

16. Idoter - (i DOE tur) inverted dodecatwister or ido twister. This is the {5/4,3/2} twister. It has 20 introverted rings, and 12 flower-like retropentapodic twisters. Its rinf is a bloated retrotriangle.

17. Iketer - (IKE it tur) icosatwister or ike twister. This is the {3,5} twister and is related to the Hopf icosahedron. It has 12 rings and 20 trigonal twisters. Its rinf is a bloated pentagon. It would make a cool 20 sided die, with all sides rollable.

18. Quiketer - (QUIKE it tur) quasicosatwister or quike twister. This is the {3/2,5} twister. It has 12 rings, and 20 retrotrigonal twisters. Its rinf is a bloated pentagon.

19. Bliketer - (BLIKE it tur) bloated icosatwister or blike twister. This is the {3,5/4} twister. It has 12 introverted rings, and 20 tripodic twisters. Its rinf is a bloated retropentagon.

20. Iyiketer - (i YIKE it tur) inverted icosatwister or Iyike twister. This is the {3/2,5/4} twister. It has 12 introverted rings, and 20 retrotripodic twisters. Its rinf is a bloated retropentagon.

21. Gaditer - (GAD it er) great dodecatwister or gad twister. This is the {5,5/2} twister, it is related to the Hopf gad. It has 12 rings and 12 pentagonal twisters. Its rinf is a bloated star.

22. Gaquiditer - (ga KWID it tur) great quasidodecatwister or gaquid twister. This is the {5/4,5/2} twister. It has 12 rings and 12 bloated retropentagonal twisters. Its rinf is a bloated star.

23. Gabliditer - (ga BLID it tur) great bloated dodecatwister or gablid twister. This is the {5,5/3} twister. It has 12 introverted rings and 12 flower-like bloated pentagonal twisters. Its rinf is a bloated retrostar.

24. Giaditer - (ge AD it tur) great inverted dodecatwister or giad twister. This is the {5/4,5/3} twister. It has 12 introverted rings and 12 flower-like bloated retropentagonal twisters. Its rinf is a bloated retrostar.

25. Sissiditer - (sis SID it tur) small stellated dodecatwister, sissid twister. This is the {5/2,5} twister, it is similar to the Hopf sissid. It has 12 rings and 12 star twisters. Its rinf is a bloated pentagon.

26. Siquissiditer - (SIK kwiss SID it tur) small quasistellated dodecatwister, siquissid twister (sik KWISS id). This is the {5/3,5}. It has 12 rings and 12 retrostar twisters. Its rinf is a bloated pentagon.

27. Soblessiditer - (SAH bleh SID it tur) small bloatostellated dodecatwister, soblessid twister (sah BLESS id). This is the {5/2,5/4} twister, it is a copy cat of (looks identical to) bladoter. It has 12 introverted rings (inside pentagonal ring-like cups) and 12 flower-like star twisters (which look like pentapod twisters). Its rinf is a bloated retropentagon.

28. Sansiditer - (san SID it tur) small invertostellated dodecatwister, sansid twister (SAN sid). This is the {5/3,5/4} twister, it is a copy cat of (looks identical to) bladoter. It has 12 introverted rings (inside pentagonal ring-like cups) and 12 flower-like retrostar twisters. Its rinf is a bloated retropentagon.

29. Giketer - (GIKE it tur) great icosatwister or gike twister. This is the {3,5/2} twister and is related to the Hopf great icosahedron. It has 12 rings and 20 trigonal twisters. Its rinf is a bloated star.

30. Gaquiter - (ga QUIT tur) great quasicosatwister or gaqui twister. This is the {3/2,5/2} twister. It has 12 rings, and 20 retrotrigonal twisters. Its rinf is a bloated star.

31. Gabliter - (ga BLIT tur) great bloated icosatwister or gabli twister. This is the {3,5/3} twister. It has 12 introverted rings, and 20 tripodic twisters. Its rinf is a bloated retrostar.

32. Giyiter - (ge YI tur) great inverted icosatwister or giyi twister. This is the {3/2,5/3} twister. It has 12 introverted rings, and 20 retrotripodic twisters. Its rinf is a bloated retrostar.

33. Gissiditer - (gis SID it tur) great stellated dodecatwister, gissid twister. This is the {5/2,3} twister, it is similar to the Hopf gissid. It has 20 rings and 12 bloated star twisters. Its rinf is a bloated triangle. It is not near as spiky as one might first guess.

34. Gaquissiditer - (GAH quiss SID it tur) great quasistellated dodecatwister, gaquissid twister (gah KWISS sid). This is the {5/3,3} twister, which is quite interesting in appearance. It has 20 rings and 12 bloated retrostar twisters. Its rinf is a bloated triangle.

35. Goblessiditer - (GAH bleh SID it tur) great bloatostellated dodecatwister, goblessid twister (gah BLESS id). This is the {5/2,3/2} twister, it is a copy cat of gabliditer. It has 20 introverted rings and 12 flower-like bloated star twisters. Its rinf is a bloated retrotriangle.

36. Gansiditer - (gan SID it tur) great invertostellated dodecatwister, gansid twister (GAN sid). This is the {5/3,3/2} twister, it is a copy cat of gabliditer. It has 20 introverted rings and 12 flower-like bloated retrostar twisters. Its rinf is a bloated retrotriangle.

Dysters (dyadic twisters) are actually {2,n/d} polytwisters, where n>2, d<n, and where n and d are relatively prime - check out some sample sections which shows a section of the 27, 27/13, 27/26, 7, 7/2, 7/3, 7/4, 7/5, and 7/6 dysters. {n/d,2} polytwisters seem to look identical to the duocylinder (circle duoprism}.

Uniform Polytwisters

The 194 known uniform polytwisters can be divided up in the following way: there are 36 regular cases (already counted), 34 rectified cases, 72 sheaved cases, and 52 special cases. There are also the infinite group of dysters as well as their rectified counterparts, the redysters. A polytwister is uniform if all of it's rings are identical in a transitive way, and the twisters are all regular, in otherwords it's twisters' rings and strips are congruent. Not all uniform polyhedra has a uniform polytwister analougue - for example, the non-orientable hemis like thah, sidhei, and gidhid can't even form polytwister versions. Polytwisters based off of truncates, small and great rhombates, and snubs exist but they can't be made uniform - Why? well - consider the tut twister, it would have 4 trigon twisters and 4 hexagon twisters for sides, but the hexagon twisters' sides dont all bow out the same amount, those connected to hexagon twisters bow out a bit more than the sides connected to the trigon twisters, so the hexagon twister is more like a ditrigon twister instead of a regular hexagon twister - there are even cases where half the sides bow out and the other half bow in, so they are not quite uniform. Each of the 36 regular polytwisters has a sheaved case and a plated (antisheaved) case. Each dual pair of regulars has a rectified and a blotorectified case. Below are the non-regular uniform polytwisters. Pictures can be seen by clicking on the names.

37. Tatetter - (tuh TET tur) also called tatet twister, or tetratetratwister. It is the r{3,3/2} twister. It has 6 rings, 12 strips, 4 trigonal twisters and 4 retrotrigonal twisters. It's ring figure (rinf) is a very skinny bowed rectangle. Maybe its just me, but this thing has a sort of strange feminine beauty to it. It has tetteric symmetry. The r{3,3} and r{3/2,3/2} are octer and quoter respectively and isn't counted here.

38. Blatatetter - (BLAD uh TET tur) also called blatatet twister (BLAD uh tet), or bloated tetratetratwister. It is the br{3,3/2} twister. It has 6 rings (which blends in with the surface), 12 strips, 4 tripodic twisters and 4 retrotripodic twisters. It's rinf is some sort of bloated X-like invertibowed rectangle (lets just call it the "bloated X"). It has tetteric symmetry. The br{3,3} and br{3/2,3/2} are bloter and ioter respectively and isn't counted here.

39. Coter - (CO tur) also called co twister, or cuboctatwister. It is the r{3,4} twister. It has 12 rings, 24 strips, 8 trigonal twisters and 6 square twisters. It's rinf is a bowed rectangle. It has cubiteric symmetry.

40. Oquicter - (o QUIC tur) also called oquic twister, or octaquasicubitwister. It is the r{3,4/3} twister. It has 12 rings, 24 strips, 8 trigonal twisters and 6 retrosquare twisters. It's rinf is a skinny bowed rectangle. It has cubiteric symmetry.

41. Caquoter - (ka QUO tur) also called caquo twister, or cubiquasioctatwister. It is the r{3/2,4} twister. It has 12 rings, 24 strips, 8 retrotrigonal twisters and 6 square twisters. It's rinf is a skinny bowed rectangle. It has cubiteric symmetry.

42. Quicoter - (quic O tur) also called quico twister, or quasicuboctatwister. It is the r{3/2,4/3} twister. It has 12 rings, 24 strips, 8 retrotrigonal twisters and 6 retrosquare twisters. It's rinf is a bowed rectangle. It has cubiteric symmetry.

43. Blicoter - (blic CO tur) also called blico twister, or bloated cuboctatwister. It is the br{3,4} twister. It has 12 rings, 24 strips, 8 tripodic twisters and 6 floral tetrapodic twisters. It's rinf is a bloated X. It has cubiteric symmetry.

44. Oblicter - (o BLIK tur) also called oblic twister, or octabloated cube twister. It is the br{3,4/3} twister. It has 12 rings, 24 strips, 8 tripodic twisters and 6 retrotetrapodic twisters. It's rinf is a bloated X. It has cubiteric symmetry.

45. Cabloter - (ka BLO tur) also called cablo twister, or cubibloated octatwister. It is the br{3/2,4} twister. It has 12 rings, 24 strips, 8 retrotripodic twisters and 6 floral tetrapodic twisters. It's rinf is a bloated X. It has cubiteric symmetry.

46. Icoter - (i CO tur) also called ico twister (not to be confused with ico the polychoron), or inverticuboctatwister. It is the br{3/2,4/3} twister. It has 12 rings, 24 strips, 8 retrotripodic twisters and 6 retrotetrapodic twisters. It's rinf is a bloated X. It has cubiteric symmetry.

47. Iditer - (ID it tur) also called id twister, or icosidodecatwister. It is the r{3,5} twister. All polytwisters from #47 to #70 have 30 rings and 60 strips and they all have doteric symmetry. Iditer has 20 trigonal twisters and 12 pentagonal twisters. It's rinf is a bowed rectangle.

48. Iquiditer - (i QUID it tur) also called iquid twister, or icosiquasidodecatwister. It is the r{3,5/4} twister. It has 20 trigonal twisters and 12 retropentagonal twisters. It's rinf is a skinny bowed rectangle.

49. Daquiter - (da QUIT tur) also called daqui twister, or dodecaquasiicosatwister. It is the r{3/2,5} twister. It has 20 retrotrigonal twisters and 12 pentagonal twisters. It's rinf is a skinny bowed rectangle. This one looks like some sort of alien artifact.

50. Quiditer - (QUID it tur) also called quid twister, or quasiicosidodecatwister. It is the r{3/2,5/4} twister. It has 20 retrotrigonal twisters and 12 retropentagonal twisters. It's rinf is a bowed rectangle. This one has an attractive spiky look (but remember what looks like spikes on the sections are really sharp disc-like structures that swirl about each other.)

51. Bliditer - (BLID it tur) also called blid twister, or bloated icosidodecatwister. It is the br{3,5} twister. It has 20 tripodic twisters and 12 floral pentapodic twisters. It's rinf is a bloated X.

52. Ibliditer - (ih BLID it tur) also called iblid twister, or icosibloated dodecatwister. It is the br{3,5/4} twister. It has 20 tripodic twisters and 12 retropentapodic twisters. It's rinf is a bloated X.

53. Dabliter - (da BLI tur) also called dabli twister, or dodecabloated icosatwister. It is the br{3/2,5} twister. It has 20 retrotripodic twisters and 12 floral pentapodic twisters. It's rinf is a bloated X.

54. Iyiditer - (i YID dit tur) also called iyid twister, or inverticosidodecatwister. It is the br{3/2,5/4} twister. It has 20 retrotripodic twisters and 12 retropentapodic twisters. It's rinf is a bloated X.

55. Diditer - (DID it tur) also called did twister, or dodecadodecatwister. It is the r{5/2,5} twister. It has 12 pentagram twisters and 12 pentagonal twisters. It's rinf is a bowed rectangle.

56. Gidquiditer - (gid QUID it tur) also called gidquid twister, or great dodecaquasidodecatwister. It is the r{5/2,5/4} twister. It has 12 star twisters and 12 retropentagonal twisters. It's rinf is a skinny bowed rectangle.

57. Sidquiditer - (sid QUID it tur) also called sidquid twister, or small dodecaquasidodecatwister. It is the r{5/3,5} twister. It has 12 retrostar twisters and 12 pentagonal twisters. It's rinf is a bowed rectangle.

58. Quadiditer - (qua DID it tur) also called quadid twister, or quasidodecadodecatwister. It is the r{5/3,5/4} twister. It has 12 retrostar twisters and 12 retropentagonal twisters. It's rinf is a bowed rectangle.

59. Bladiditer - (bluh DID it tur) also called bladid twister, or bloated dodecadodecatwister. It is the br{5/2,5} twister. It has 12 starpod twisters and 12 floral pentapodic twisters. It's rinf is a bloated X.

60. Gidbliditer - (gid BLID it tur) also called gidblid twister, or great dodecabloated dodecatwister. It is the br{5/2,5/4} twister. It has 12 starpod twisters and 12 retropentapodic twisters. It's rinf is a bloated X.

61. Sidbliditer - (sid BLID it tur) also called sidblid twister, or small dodecabloated dodecatwister. It is the br{5/3,5} twister. It has 12 retrostarpod twisters and 12 floral pentapodic twisters. It's rinf is a bloated X.

62. Idoditer - (i DOE dit tur) also called idod twister, or invertidodecadodecatwister. It is the br{5/3,5/4} twister. It has 12 retrostarpod twisters and 12 retropentapodic twisters. It's rinf is a bloated X.

63. Giditer - (GID it tur) also called gid twister, or great icosidodecatwister. It is the r{5/2,3} twister. It has 12 pentagram twisters and 20 trigonal twisters. It's rinf is a bowed rectangle.

64. Geiquiditer - (gi QUID it tur) also called geiquid twister, or great icosiquasidodecatwister. It is the r{5/3,3} twister. It has 12 retrostar twisters and 20 trigonal twisters. It's rinf is a bowed rectangle.

65. Gidquiter - (gid QUITE tur) also called gidqui twister, or great dodecaquasiicosatwister. It is the r{5/2,3/2} twister. It has 12 star twisters and 20 retrotrigonal twisters. It's rinf is a bowed rectangle.

66. Gequiditer - (gee QUID it tur) also called gequid twister, or great quasiicosidodecatwister. It is the r{5/3,3/2} twister. It has 12 retrostar twisters and 20 retrotrigonal twisters. It's rinf is a bowed rectangle.

67. Gobliditer - (go BLID it tur) also called goblid twister, or great bloated icosidodecatwister. It is the br{5/2,3} twister. It has 12 starpod twisters and 20 floral tripodic twisters. It's rinf is a bloated X.

68. Geibliditer - (gi BLID it tur) also called geiblid twister, or great icosibloated dodecatwister. It is the br{5/3,3} twister. It has 12 retrostarpod twisters and 20 floral tripodic twisters. It's rinf is a bloated X.

69. Gidbliter - (gid BLITE tur) also called gidbli twister, or great dodecabloated icosatwister. It is the br{5/2,3/2} twister. It has 12 starpod twisters and 20 retrotripodic twisters. It's rinf is a bloated X.

70. Giyiditer - (gi YID dit tur) also called giyid twister, or great invertiicosidodecatwister. It is the br{5/3,3/2} twister. It has 12 retrostarpod twisters and 20 retrotripodic twisters. It's rinf is a bloated X.

Here are the sheaved cases

71. Vitter - (VIT tur) also called vit twister, or sheaved tetratwister. It is the 2{3,3} twister (where the 2 represents the dyads). It has 4 rings, 12 strips, 4 trigonal twisters, and 6 dyad twisters. Its rinf is a ditrigon. It has tetteric symmetry.

72. Quivitter - (QUIV it tur) also called quivit twister, or quasisheaved tetratwister. It is the 2{3/2,3} twister. It has 4 rings, 12 strips, 4 retrotrigonal twisters, and 6 dyad twisters. Its rinf is a ditrigon. It has tetteric symmetry.

73. Blivitter - (BLIV it tur) also called blivit twister, or bloatosheaved tetratwister. It is the 2{3,3/2} twister. It has 4 rings, 12 strips, 4 trigonal twisters, and 6 dyad twisters. Its rinf is a retroditrigon. It has tetteric symmetry.

74. Ivitter - (IV it tur) also called ivit twister, or invertisheaved tetratwister. It is the 2{3/2,3/2} twister. It has 4 rings, 12 strips, 4 retrotrigonal twisters, and 6 dyad twisters. Its rinf is a retroditrigon. It has tetteric symmetry.

75. Victer - (VIC tur) also called vic twister, or sheaved cube twister. It is the 2{4,3} twister. It has 8 rings, 24 strips, 6 tetragonal twisters, and 12 dyad twisters. Its rinf is a ditrigon. It has cubiteric symmetry.

76. Quivicter - (QUIV ic tur) also called quivic twister, or quasisheaved cube twister. It is the 2{4/3,3} twister. It has 8 rings, 24 strips, 6 retrotetragonal twisters, and 12 dyad twisters. Its rinf is a ditrigon. This one looks quite unusual. It has cubiteric symmetry.

77. Blivicter - (BLIV ic tur) also called blivic twister, or bloatosheaved cube twister. It is the 2{4,3/2} twister. It has 8 rings, 24 strips, 6 tetragonal twisters, and 12 dyad twisters. Its rinf is a retroditrigon. It has cubiteric symmetry.

78. Ivicter - (IV ic tur) also called ivic twister, or invertisheaved cube twister. It is the 2{4/3,3/2} twister. It has 8 rings, 24 strips, 6 retrotetragonal twisters, and 12 dyad twisters. Its rinf is a retroditrigon. It has cubiteric symmetry.

79. Voter - (VO tur) also called vo twister, or sheaved octatwister. It is the 2{3,4} twister. It has 8 rings, 24 strips, 8 trigonal twisters, and 12 dyad twisters. Its rinf is a ditetragon. It has cubiteric symmetry.

80. Quivoter - (quuh VO tur) also called quivo twister, or quasisheaved octatwister. It is the 2{3/2,4} twister. It has 8 rings, 24 strips, 8 retrotrigonal twisters, and 12 dyad twisters. Its rinf is a ditetragon. It has cubiteric symmetry.

81. Blivoter - (bluh VO tur) also called blivo twister, or bloatosheaved octatwister. It is the 2{3,4/3} twister. It has 8 rings, 24 strips, 8 trigonal twisters, and 12 dyad twisters. Its rinf is a crowned square (a semiuniform octagram which resembles a square with crowned vertices). It has cubiteric symmetry.

82. Ivoter - (ih VO tur) also called ivo twister, or invertisheaved octatwister. It is the 2{3/2,4/3} twister. It has 8 rings, 24 strips, 8 retrotrigonal twisters, and 12 dyad twisters. Its rinf is a crowned square. It has cubiteric symmetry.

83. Viditer - (VID it tur) also called vid twister, or sheaved dodecatwister. It is the 2{5,3} twister. This and the remaining sheaved twisters have 60 strips and have doteric symmetry. This one also has 20 rings, 12 pentagonal twisters, and 30 dyad twisters. Its rinf is a ditrigon.

84. Quividiter - (quiv VID it tur) also called quivid twister, or quasisheaved dodecatwister. It is the 2{5/4,3} twister. It has 20 rings, 12 retropentagonal twisters, and 30 dyad twisters. Its rinf is a ditrigon.

85. Blividiter - (bliv VID it tur) also called blivid twister, or bloatosheaved dodecatwister. It is the 2{5,3/2} twister. It has 20 rings, 12 pentagonal twisters, and 30 dyad twisters. Its rinf is a retroditrigon.

86. Ividiter - (iv VID it tur) also called ivid twister, or invertisheaved dodecatwister. It is the 2{5/4,3/2} twister. It has 20 rings, 12 retropentagonal twisters, and 30 dyad twisters. Its rinf is a retroditrigon.

87. Viketer - (VIKE e tur) also called vike twister, or sheaved icosatwister. It is the 2{3,5} twister. It has 12 rings, 20 trigonal twisters, and 30 dyad twisters. Its rinf is a dipentagon.

88. Quiviketer - (quiv VIKE e tur) also called quivike twister, or quasisheaved icosatwister. It is the 2{3/2,5} twister. It has 12 rings, 20 retrotrigonal twisters, and 30 dyad twisters. Its rinf is a dipentagon.

89. Bliviketer - (bliv VIKE e tur) also called blivike twister, or bloatosheaved icosatwister. It is the 2{3,5/4} twister. It has 12 rings, 20 trigonal twisters, and 30 dyad twisters. Its rinf is a retrodipentagon.

90. Iviketer - (iv VIKE e tur) also called ivike twister, or invertisheaved icosatwister. It is the 2{3/2,5/4} twister. It has 12 rings, 20 retrotrigonal twisters, and 30 dyad twisters. Its rinf is a retrodipentagon.

91. Goviditer - (go VID it tur) also called govid twister, or great sheaved dodecatwister. It is the 2{5,5/2} twister. It has 12 rings, 12 pentagonal twisters, and 30 dyad twisters. Its rinf is a truncated star.

92. Gaquaviditer - (GA qua VID it tur) also called gaquavid twister, or great quasisheaved dodecatwister. It is the 2{5/4,5/2} twister. It has 12 rings, 12 retropentagonal twisters, and 30 dyad twisters. Its rinf is a truncated star.

93. Gablaviditer - (GA bla VID it tur) also called gablavid twister, or great bloatosheaved dodecatwister. It is the 2{5,5/3} twister. It has 12 rings, 12 pentagonal twisters, and 30 dyad twisters. Its rinf is a quasitruncated star (resembles decagram).

94. Gividiter - (giv VID it tur) also called givid twister, or great invertisheaved dodecatwister. It is the 2{5/4,5/3} twister. It has 12 rings, 12 retropentagonal twisters, and 30 dyad twisters. Its rinf is a quasitruncated star.

95. Sividiter - (siv VID it tur) also called sivid twister, or stellisheaved dodecatwister. It is the 2{5/2,5} twister. It has 12 rings, 12 star twisters, and 30 dyad twisters. Its rinf is a dipentagon.

96. Quisviditer - (quis VID it tur) also called quisvid twister, or quasistellisheaved dodecatwister. It is the 2{5/3,5} twister. It has 12 rings, 12 retrostar twisters, and 30 dyad twisters. Its rinf is a dipentagon.

97. Blisviditer - (blis VID it tur) also called blisvid twister, or bloatostellisheaved dodecatwister. It is the 2{5/2,5/4} twister. It has 12 rings, 12 star twisters, and 30 dyad twisters. Its rinf is a retrodipentagon.

98. Isviditer - (iss VID it tur) also called isvid twister, or invertistellisheaved dodecatwister. It is the 2{5/3,5/4} twister. It has 12 rings, 12 retrostar twisters, and 30 dyad twisters. Its rinf is a retrodipentagon.

99. Goviter - (GO vit tur) also called govi twister, or great sheaved icosatwister. It is the 2{3,5/2} twister. It has 12 rings, 20 trigonal twisters, and 30 dyad twisters. Its rinf is a truncated star.

100. Gaquiviter - (ga QUIV it tur) also called gaquivi twister, or great quasisheaved icosatwister. It is the 2{3/2,5/2} twister. It has 12 rings, 20 retrotrigonal twisters, and 30 dyad twisters. Its rinf is a truncated star.

101. Gabliviter - (ga BLIV it tur) also called gablivi twister, or great bloatosheaved icosatwister. It is the 2{3,5/3} twister. It has 12 rings, 20 trigonal twisters, and 30 dyad twisters. Its rinf is a quasitruncated star.

102. Giviter - (GI vit tur) also called givi twister, or great invertisheaved icosatwister. It is the 2{3/2,5/3} twister. It has 12 rings, 20 retrotrigonal twisters, and 30 dyad twisters. Its rinf is a quasitruncated star.

103. Gisviditer - (gis VID it tur) also called gisvid twister, or great stellisheaved dodecatwister. It is the 2{5/2,3} twister. It has 12 rings, 12 star twisters, and 30 dyad twisters. Its rinf is a ditrigon.

104. Gaqsviditer - (goks VID it tur) also called gaqsvid twister, or great quasistellisheaved dodecatwister. It is the 2{5/3,3} twister. It has 12 rings, 12 retrostar twisters, and 30 dyad twisters. Its rinf is a ditrigon.

105. Gablisviditer - (GA blis VID it tur) also called gablisvid twister, or great bloatostellisheaved dodecatwister. It is the 2{5/2,3/2} twister. It has 12 rings, 12 star twisters, and 30 dyad twisters. Its rinf is a retroditrigon.

106. Gise Viditer - (gise VID it tur) also called gise vid twister, or great invertistellisheaved dodecatwister. It is the 2{5/3,3/2} twister. It has 12 rings, 12 retrostar twisters, and 30 dyad twisters. Its rinf is a retroditrigon.

Now for the "plated" or antisheaved cases

107. Potter - (POT tur) also called pot twister, or plated tetratwister. It is the p{3,3} twister (where the p represents the bloated dyads doing a plating effect). It has 4 rings, 12 strips, 4 floral tripod twisters, and 6 bloated dyad twisters. Its rinf is an introverted ditrigon. It has tetteric symmetry.

108. Quipiter - (QUIP it tur) also called quipit twister, or quasiplated tetratwister. It is the p{3/2,3} twister. It has 4 rings, 12 strips, 4 retrotripod twisters, and 6 bloated dyad twisters. Its rinf is an introverted ditrigon. It has tetteric symmetry.

109. Blipiter - (BLIP it tur) also called blipit twister, or bloatoplated tetratwister. It is the p{3,3/2} twister. It has 4 rings, 12 strips, 4 floral tripod twisters, and 6 bloated dyad twisters. Its rinf is an introverted retroditrigon. It has tetteric symmetry.

110. Ipiter - (IPE it tur) also called ipit twister, or invertiplated tetratwister. It is the p{3/2,3/2} twister. It has 4 rings, 12 strips, 4 retrotripod twisters, and 6 bloated dyad twisters. Its rinf is an introverted retroditrigon. It has tetteric symmetry.

111. Picter - (PIC tur) also called pic twister, or plated cube twister. It is the p{4,3} twister. It has 8 rings, 24 strips, 6 bloated tetragonal twisters, and 12 bloated dyad twisters. Its rinf is an introverted ditrigon. It has cubiteric symmetry.

112. Quipicter - (QUIP ic tur) also called quipic twister, or quasiplated cube twister. It is the p{4/3,3} twister. It has 8 rings, 24 strips, 6 bloated retrotetragonal twisters, and 12 bloated dyad twisters. Its rinf is an introverted ditrigon. It has cubiteric symmetry.

113. Blipicter - (BLIP ic tur) also called blipic twister, or bloatoplated cube twister. It is the p{4,3/2} twister. It has 8 rings, 24 strips, 6 bloated tetragonal twisters, and 12 bloated dyad twisters. Its rinf is an introverted retroditrigon. This one looks cool. It has cubiteric symmetry.

114. Ipicter - (IP ic tur) also called ipic twister, or invertiplated cube twister. It is the p{4/3,3/2} twister. It has 8 rings, 24 strips, 6 bloated retrotetragonal twisters, and 12 bloated dyad twisters. Its rinf is an introverted retroditrigon. It has cubiteric symmetry.

115. Poter - (PO tur) also called po twister, or plated octatwister. It is the p{3,4} twister. It has 8 rings, 24 strips, 8 bloated trigonal twisters, and 12 bloated dyad twisters. Its rinf is an introverted ditetragon. It has cubiteric symmetry.

116. Quipoter - (quih PO tur) also called quipo twister, or quasiplated octatwister. It is the p{3/2,4} twister. It has 8 rings, 24 strips, 8 bloated retrotrigonal twisters, and 12 bloated dyad twisters. Its rinf is an introverted ditetragon. It has cubiteric symmetry.

117. Blipoter - (blih PO tur) also called blipo twister, or bloatoplated octatwister. It is the p{3,4/3} twister. It has 8 rings, 24 strips, 8 bloated trigonal twisters, and 12 bloated dyad twisters. Its rinf is an introverted retroditetragon. It has cubiteric symmetry.

118. Ipoter - (ih PO tur) also called ipo twister, or invertiplated octatwister. It is the p{3/2,4/3} twister. It has 8 rings, 24 strips, 8 bloated retrotrigonal twisters, and 12 bloated dyad twisters. Its rinf is an introverted retroditetragon. It has cubiteric symmetry.

119. Piditer - (PID it tur) also called pid twister, or plated dodecatwister. It is the p{5,3} twister. This and the remaining plated twisters have 60 strips and have doteric symmetry. This one also has 20 rings, 12 bloated pentagonal twisters, and 30 bloated dyad twisters. Its rinf is an introverted ditrigon.

120. Quipiditer - (quip PID it tur) also called quipid twister, or quasiplated dodecatwister. It is the p{5/4,3} twister. It has 20 rings, 12 bloated retropentagonal twisters, and 30 bloated dyad twisters. Its rinf is an introverted ditrigon.

121. Blipiditer - (blip PID it tur) also called blipid twister, or bloatoplated dodecatwister. It is the p{5,3/2} twister. It has 20 rings, 12 bloated pentagonal twisters, and 30 bloated dyad twisters. Its rinf is an introverted retroditrigon.

122. Ipiditer - (ip PID it tur) also called ipid twister, or invertiplated dodecatwister. It is the p{5/4,3/2} twister. It has 20 rings, 12 bloated retropentagonal twisters, and 30 bloated dyad twisters. Its rinf is an introverted retroditrigon.

123. Piketer - (PIKE e tur) also called pike twister, or plated icosatwister. It is the p{3,5} twister. It has 12 rings, 20 bloated trigonal twisters, and 30 bloated dyad twisters. Its rinf is an introverted dipentagon.

124. Quipiketer - (quip PIKE e tur) also called quipike twister, or quasiplated icosatwister. It is the p{3/2,5} twister. It has 12 rings, 20 bloated retrotrigonal twisters, and 30 bloated dyad twisters. Its rinf is an introverted dipentagon.

125. Blipiketer - (blip PIKE e tur) also called blipike twister, or bloatoplated icosatwister. It is the p{3,5/4} twister. It has 12 rings, 20 bloated trigonal twisters, and 30 bloated dyad twisters. Its rinf is an introverted retrodipentagon.

126. Ipiketer - (ip PIKE e tur) also called ipike twister, or invertiplated icosatwister. It is the p{3/2,5/4} twister. It has 12 rings, 20 bloated retrotrigonal twisters, and 30 bloated dyad twisters. Its rinf is an introverted retrodipentagon.

127. Gopiditer - (go PID it tur) also called gopid twister, or great plated dodecatwister. It is the p{5,5/2} twister. It has 12 rings, 12 bloated pentagonal twisters, and 30 bloated dyad twisters. Its rinf is an introverted distar.

128. Gaquapiditer - (GA qua PID it tur) also called gaquapid twister, or great quasiplated dodecatwister. It is the p{5/4,5/2} twister. It has 12 rings, 12 bloated retropentagonal twisters, and 30 bloated dyad twisters. Its rinf is an introverted distar.

129. Gablapiditer - (GA bla PID it tur) also called gablapid twister, or great bloatoplated dodecatwister. It is the p{5,5/3} twister. It has 12 rings, 12 bloated pentagonal twisters, and 30 bloated dyad twisters. Its rinf is an introverted retrodistar.

130. Gipiditer - (gip PID it tur) also called gipid twister, or great invertiplated dodecatwister. It is the p{5/4,5/3} twister. It has 12 rings, 12 bloated retropentagonal twisters, and 30 bloated dyad twisters. Its rinf is an introverted retrodistar.

131. Sipiditer - (sip PID it tur) also called sipid twister, or stelliplated dodecatwister. It is the p{5/2,5} twister. It has 12 rings, 12 bloated star twisters, and 30 bloated dyad twisters. Its rinf is an introverted dipentagon.

132. Quispiditer - (quis PID it tur) also called quispid twister, or quasistelliplated dodecatwister. It is the p{5/3,5} twister. It has 12 rings, 12 bloated retrostar twisters, and 30 bloated dyad twisters. Its rinf is an introverted dipentagon.

133. Blispiditer - (blis PID it tur) also called blispid twister, or bloatostelliplated dodecatwister. It is the p{5/2,5/4} twister. It has 12 rings, 12 bloated star twisters, and 30 bloated dyad twisters. Its rinf is an introverted retrodipentagon.

134. Ispiditer - (iss PID it tur) also called ispid twister, or invertistelliplated dodecatwister. It is the p{5/3,5/4} twister. It has 12 rings, 12 bloated retrostar twisters, and 30 bloated dyad twisters. Its rinf is an introverted retrodipentagon.

135. Gopiter - (GO pit tur) also called gopi twister, or great plated icosatwister. It is the p{3,5/2} twister. It has 12 rings, 20 bloated trigonal twisters, and 30 bloated dyad twisters. Its rinf is an introverted distar.

136. Gaquipiter - (ga QUIP it tur) also called gaquipi twister, or great quasiplated icosatwister. It is the p{3/2,5/2} twister. It has 12 rings, 20 bloated retrotrigonal twisters, and 30 bloated dyad twisters. Its rinf is an introverted distar.

137. Gablipiter - (ga BLIP it tur) also called gablipi twister, or great bloatoplated icosatwister. It is the p{3,5/3} twister. It has 12 rings, 20 bloated trigonal twisters, and 30 bloated dyad twisters. Its rinf is an introverted retrodistar.

138. Gipiter - (GI pit tur) also called gipi twister, or great invertiplated icosatwister. It is the p{3/2,5/3} twister. It has 12 rings, 20 bloated retrotrigonal twisters, and 30 bloated dyad twisters. Its rinf is an introverted retrodistar.

139. Gispiditer - (gis PID it tur) also called gispid twister, or great stelliplated dodecatwister. It is the p{5/2,3} twister. It has 12 rings, 12 bloated star twisters, and 30 bloated dyad twisters. Its rinf is an introverted ditrigon.

140. Gaqspiditer - (goks PID it tur) also called gaqspid twister, or great quasistelliplated dodecatwister. It is the p{5/3,3} twister. It has 12 rings, 12 bloated retrostar twisters, and 30 bloated dyad twisters. Its rinf is an introverted ditrigon.

141. Gablispiditer - (GA blis PID it tur) also called gablispid twister, or great bloatostelliplated dodecatwister. It is the p{5/2,3/2} twister. It has 12 rings, 12 bloated star twisters, and 30 bloated dyad twisters. Its rinf is an introverted retroditrigon.

142. Gise Piditer - (gise PID it tur) also called gise pid twister, or great invertistelliplated dodecatwister. It is the p{5/3,3/2} twister. It has 12 rings, 12 bloated retrostar twisters, and 30 bloated dyad twisters. Its rinf is an introverted retroditrigon.

Next comes the special cases (the ditrigonaries and similars are here).

143. Tritter - (TRIT tur) also called trit twister, or tetraretrotetratwister. It is the {3/2,3,3} twister. It has 4 rings, 12 strips, 4 trigon twisters, and 4 retrotrigon twisters. Its rinf is a retro ditrigon (and may actually be a triangle). It has tetteric symmetry.

144. Tablitter - (ta BLIT tur) also called tablit twister, or tetrabloated tetratwister. It is the b{3/2,3,3} twister. It has 4 rings, 12 strips, 4 bloated trigon twisters, and 4 bloated retrotrigon twisters. Its rinf is an introverted retro ditrigon. Its rings occur where three bloated retrotwisters (metallic sides in the pic) meet. It has tetteric symmetry.

145. Sacroter - (sa CRO tur) also called sacro twister, or small cubiretro-octatwister. It is the {3/2,4,4} twister. It has 6 rings, 24 strips, 6 square twisters, and 8 retrotrigon twisters. Its rinf is a retro ditetragon (square with triangles dangling off corners). It has cubiteric symmetry.

146. Gadtacoter - (GAD ta CO tur) also called gadtaco twister, or great ditetragonary cubioctatwister. It is the {3/2,4/3,4/3} twister. It has 6 rings, 24 strips, 6 retrosquare twisters, and 8 retrotrigon twisters. Its rinf is a quasitruncated square. It has cubiteric symmetry.

147. Gacroter - (ga CRO tur) also called gacro twister, or great cubiretro-octatwister. It is the {4/3,3,4} twister. It has 6 rings, 24 strips, 6 retrosquare twisters, and 8 trigon twisters. Its rinf is a retro ditetragon. It has cubiteric symmetry.

148. Sadtacoter - (SAD ta CO tur) also called sadtaco twister, or small ditetragonary cubioctatwister. It is the {3,4,4/3} twister. It has 6 rings, 24 strips, 6 square twisters, and 8 trigon twisters. Its rinf is a quasitruncated square. It has cubiteric symmetry.

149. Cricter - (CRIC tur) also called cric twister, or cubiretrocubitwister. It is the {4/3,4,3} twister. It has 8 rings, 24 strips, 6 retrosquare twisters, and 6 square twisters. Its rinf is a retroditrigon. It has cubiteric symmetry.

150. Sacbloter - (sac BLO tur) also called sacblo twister, or small cubibloated octatwister. It is the b{3/2,4,4} twister. It has 6 rings, 24 strips, 6 bloated square twisters, and 8 bloated retrotrigon twisters. Its rinf is an introverted retro ditetragon (square with triangles dangling off corners). It has cubiteric symmetry.

151. Gadtablicoter - (GAD ta blic CO tur) also called gadtablico twister, or great ditetragonary bloated cubioctatwister. It is the b{3/2,4/3,4/3} twister. It has 6 rings, 24 strips, 6 bloated retrosquare twisters, and 8 bloated retrotrigon twisters. Its rinf is an introverted quasitruncated square. It has cubiteric symmetry.

152. Gacbloter - (gac BLO tur) also called gacblo twister, or great cubibloated octatwister. It is the b{4/3,3,4} twister. It has 6 rings, 24 strips, 6 bloated retrosquare twisters, and 8 bloated trigon twisters. Its rinf is an introverted retro ditetragon. It has cubiteric symmetry.

153. Sadtablicoter - (SAD ta blic CO tur) also called sadtablico twister, or small ditetragonary bloated cubioctatwister. It is the b{3,4,4/3} twister. It has 6 rings, 24 strips, 6 bloated square twisters, and 8 bloated trigon twisters. Its rinf is an introverted quasitruncated square. It has cubiteric symmetry.

154. Cablicter - (ca BLIC tur) also called cablic twister, or cubibloated cubitwister. It is the b{4/3,4,3} twister. It has 8 rings, 24 strips, 6 bloated retrosquare twisters, and 6 bloated square twisters. Its rinf is an introverted retroditrigon. It has cubiteric symmetry.

155. Siriditer - (si RID it tur) also called sirid twister, or small icosiretrododecatwister. It is the {3/2,5,5} twister. It has 12 rings, 60 strips, 20 retrotrigon twisters, and 12 pentagonal twisters. Its rinf is a retrodipentagon. This and the remaining polytwisters have doteric symmetry.

156. Driditer - (DRID it tur) also called drid twister, or dodecaretrododecatwister. It is the {5/4,5,5} twister. It has 12 rings, 60 strips, 12 retropentagon twisters, and 12 pentagonal twisters. Its rinf is a retrodipentagon.

157. Giriditer - (ji RID it tur) also called girid twister, or great icosiretrododecatwister. It is the {5/4,3,5} twister. It has 12 rings, 60 strips, 20 trigon twisters, and 12 retropentagonal twisters. Its rinf is a retrodipentagon.

158. Sidraditer - (sid DRAD it tur) also called sidrad twister, or small dodecaretrododecatwister. It is the {5/4,5,3} twister. It has 20 rings, 60 strips, 12 pentagon twisters, and 12 retropentagon twisters. Its rinf is a retroditrigon.

159. Dipiditer - (di PID it tur) also called dipid twister, or dipentagonary icosidodecatwister. It is the {3,5,5/4} twister. It has 12 rings, 60 strips, 20 trigon twisters, and 12 pentagonal twisters. Its rinf is a retrodipentagon.

160. Siiter - (SITE tur) also called sii twister, or small icosicositwister. It is the {3/2,3,5} twister. It has 12 rings, 60 strips, 20 trigon twisters, and 20 retrotrigonal twisters. Its rinf is a retrodipentagon.

161. Midtiditer - (mid TID it tur) also called midtid twister, or medial ditrigonary icosidodecatwister. It is the {3/2,5,3} twister. It has 20 rings, 60 strips, 20 retrotrigon twisters, and 12 pentagonal twisters. Its rinf is a ditrigon.

162. Gidtiditer - (gid TID it tur) also called gidtid twister, or great ditrigonary icosidodecatwister. It is the {3,5,3/2} twister. It has 20 rings, 60 strips, 20 trigon twisters, and 12 pentagonal twisters. Its rinf is a retroditrigon.

163. Sidtiditer - (sid TID it tur) also called sidtid twister, or small ditrigonary icosidodecatwister. It is the {5/2,3,3} twister. It has 20 rings, 60 strips, 20 trigon twisters, and 12 star twisters. Its rinf is a ditrigon.

164. Gidartiditer - (GID ar TID it tur) also called gidartid twister, or great diretrotrigonary icosidodecatwister. It is the {5/3,3/2,3/2} twister. It has 20 rings, 60 strips, 20 retrotrigon twisters, and 12 retrostar twisters. Its rinf is a retroditrigon.

165. Quidtiditer - (quid TID it tur) also called quidtid twister, or quasi ditrigonary icosidodecatwister. It is the {5/3,3/2,3} twister. It has 20 rings, 60 strips, 20 retrotrigon twisters, and 12 retrostar twisters. Its rinf is a ditrigon.

166. Sidartiditer - (SID ar TID it tur) also called sidartid twister, or small diretrotrigonary icosidodecatwister. It is the {5/3,3,3/2} twister. It has 20 rings, 60 strips, 20 trigon twisters, and 12 retrostar twisters. Its rinf is a retroditrigon.

167. Giiter - (GITE tur) also called gii twister, or great icosicositwister. It is the {3/2,3,5/3} twister. It has 12 rings, 60 strips, 20 trigon twisters, and 20 retrotrigonal twisters. Its rinf is a retrodistar.

168. Sidpiditer - (sid PID it tur) also called sidpid twister, or small dipentagonary icosidodecatwister. It is the {5/3,3,5} twister. It has 12 rings, 60 strips, 20 trigon twisters, and 12 retrostar twisters. Its rinf is a retrodipentagon.

169. Ditdiditer - (dit DID it tur) also called ditdid twister, or ditrigonary dodecadodecatwister. It is the {5/3,5,3} twister. It has 20 rings, 60 strips, 12 pentagon twisters, and 12 retrostar twisters. Its rinf is a retroditrigon.

170. Gidpiditer - (gid PID it tur) also called gidpid twister, or great dipentagonary icosidodecatwister. It is the {3,5,5/3} twister. It has 12 rings, 60 strips, 20 trigon twisters, and 12 pentagonal twisters. Its rinf is a retrodistar.

171. Gidraditer - (gid DRAD it tur) also called gidrad twister, or great dodecaretrododecatwister. It is the {5/3,5/2,3} twister. It has 20 rings, 60 strips, 12 retrostar twisters, and 12 star twisters. Its rinf is a ditrigon.

172. Sidpaditer - (SID pa DITE tur) also called sidpadi twister, or small dipentagonary dodekicositwister. It is the {5/3,3,5/2} twister. It has 12 rings, 60 strips, 12 retrostar twisters, and 20 trigonal twisters. Its rinf is a distar.

173. Gidpaditer - (GID pa DITE tur) also called gidpadi twister, or great dipentagonary dodekicositwister. It is the {5/2,3,5/3} twister. It has 12 rings, 60 strips, 12 star twisters, and 20 trigonal twisters. Its rinf is a retrodistar.

174. Dipdiditer - (dip DID it tur) also called dipdid twister, or dipentagonary dodecadodecatwister. It is the {5/2,5/3,5/3} twister. It has 12 rings, 60 strips, 12 star twisters, and 12 retrostar twisters. Its rinf is a retrodistar.

175. Sibriditer - (si BRID it tur) also called sibrid twister, or small icosibloatoretrododecatwister. It is the b{3/2,5,5} twister. It has 12 rings, 60 strips, 20 bloated retrotrigon twisters, and 12 bloated pentagonal twisters. Its rinf is an introverted retrodipentagon.

176. Dabriditer - (da BRID it tur) also called dabrid twister, or dodecabloatoretrododecatwister. It is the b{5/4,5,5} twister. It has 12 rings, 60 strips, 12 bloated retropentagon twisters, and 12 bloated pentagonal twisters. Its rinf is an introverted retrodipentagon.

177. Gibriditer - (ga BRID it tur) also called gibrid twister, or great icosibloatoretrododecatwister. It is the b{5/4,3,5} twister. It has 12 rings, 60 strips, 20 bloated trigon twisters, and 12 bloated retropentagonal twisters. Its rinf is an introverted retrodipentagon.

178. Sidbraditer - (sid BRAD it tur) also called sidbrad twister, or small dodecabloatoretrododecatwister. It is the b{5/4,5,3} twister. It has 20 rings, 60 strips, 12 bloated pentagon twisters, and 12 bloated retropentagon twisters. Its rinf is an introverted retroditrigon.

179. Dipbliditer - (dip BLID it tur) also called dipblid twister, or dipentagonary bloated icosidodecatwister. It is the b{3,5,5/4} twister. It has 12 rings, 60 strips, 20 bloated trigon twisters, and 12 bloated pentagonal twisters. Its rinf is an introverted retrodipentagon.

180. Sabliiter - (sa BLEET tur) also called sibli twister, or small bloated icosicositwister. It is the b{3/2,3,5} twister. It has 12 rings, 60 strips, 20 bloated trigon twisters, and 20 bloated retrotrigonal twisters. Its rinf is an introverted retrodipentagon.

181. Midtabliditer - (MID ta BLID it tur) also called midtablid twister, or medial ditrigonary bloated icosidodecatwister. It is the b{3/2,5,3} twister. It has 20 rings, 60 strips, 20 bloated retrotrigon twisters, and 12 bloated pentagonal twisters. Its rinf is an introverted ditrigon.

182. Gidtabliditer - (GID ta BLID it tur) also called gidtablid twister, or great ditrigonary bloated icosidodecatwister. It is the b{3,5,3/2} twister. It has 20 rings, 60 strips, 20 bloated trigon twisters, and 12 bloated pentagonal twisters. Its rinf is an introverted retroditrigon.

183. Sidtabliditer - (SID ta BLID it tur) also called sidtablid twister, or small ditrigonary bloated icosidodecatwister. It is the b{5/2,3,3} twister. It has 20 rings, 60 strips, 20 bloated trigon twisters, and 12 bloated star twisters. Its rinf is an introverted ditrigon.

184. Gidblatiditer - (GID bla TID it tur) also called gidblatid twister, or great dibloatotrigonary icosidodecatwister. It is the b{5/3,3/2,3/2} twister. It has 20 rings, 60 strips, 20 bloated retrotrigon twisters, and 12 bloated retrostar twisters. Its rinf is an introverted retroditrigon.

185. Quidtabliditer - (QUID ta BLID it tur) also called quidtablid twister, or quasi ditrigonary bloated icosidodecatwister. It is the b{5/3,3/2,3} twister. It has 20 rings, 60 strips, 20 bloated retrotrigon twisters, and 12 bloated retrostar twisters. Its rinf is an introverted ditrigon.

186. Sidblatiditer - (SID bla TID it tur) also called sidblatid twister, or small dibloatotrigonary icosidodecatwister. It is the b{5/3,3,3/2} twister. It has 20 rings, 60 strips, 20 bloated trigon twisters, and 12 bloated retrostar twisters. Its rinf is an introverted retroditrigon.

187. Gabliiter - (ga BLEET tur) also called gibli twister, or great bloated icosicositwister. It is the b{3/2,3,5/3} twister. It has 12 rings, 60 strips, 20 bloated trigon twisters, and 20 bloated retrotrigonal twisters. Its rinf is an introverted retrodistar.

188. Sidpabliditer - (SID pa BLID it tur) also called sidpablid twister, or small dipentagonary bloated icosidodecatwister. It is the b{5/3,3,5} twister. It has 12 rings, 60 strips, 20 bloated trigon twisters, and 12 bloated retrostar twisters. Its rinf is an introverted retrodipentagon.

189. Ditbladiditer - (DIT bla DID it tur) also called ditbladid twister, or ditrigonary bloated dodecadodecatwister. It is the b{5/3,5,3} twister. It has 20 rings, 60 strips, 12 bloated pentagon twisters, and 12 bloated retrostar twisters. Its rinf is an introverted retroditrigon.

190. Gidpabliditer - (GID pa BLID it tur) also called gidpablid twister, or great dipentagonary bloated icosidodecatwister. It is the b{3,5,5/3} twister. It has 12 rings, 60 strips, 20 bloated trigon twisters, and 12 bloated pentagonal twisters. Its rinf is an introverted retrodistar.

191. Gidbraditer - (gid BRAD it tur) also called gidbrad twister, or great dodecabloatoretrododecatwister. It is the b{5/3,5/2,3} twister. It has 20 rings, 60 strips, 12 bloated retrostar twisters, and 12 bloated star twisters. Its rinf is an introverted ditrigon.

192. Sidpabloditer - (SID pa BLO dit tur) also called sidpablodi twister, or small dipentagonary bloated dodekicositwister. It is the b{5/3,3,5/2} twister. It has 12 rings, 60 strips, 12 bloated retrostar twisters, and 20 bloated trigonal twisters. Its rinf is an introverted distar.

193. Gidpabloditer - (GID pa BLO dit tur) also called gidpablodi twister, or great dipentagonary bloated dodekicositwister. It is the b{5/2,3,5/3} twister. It has 12 rings, 60 strips, 12 bloated star twisters, and 20 bloated trigonal twisters. Its rinf is an introverted retrodistar.

194. Dipbladiditer - (DIP bla DID it tur) also called dipbladid twister, or dipentagonary bloated dodecadodecatwister. It is the b{5/2,5/3,5/3} twister. It has 12 rings, 60 strips, 12 bloated star twisters, and 12 bloated retrostar twisters. Its rinf is an introverted retrodistar.

There are also two more infinite groups to add to the regular dysters, they are the redysters (which are the rectified dysters) and the bloredysters (the bloated rectified dysters).

Beyond the 4th dimension

Polytwisters also exist in all even dimensions greater than 4 - but details are a bit sketchy. One known polytwister in 6 dimensions is related to the polypeton "jak" (which is best known as 221, which is the Gosset polytope with 27 vertices), this "jak-twister" has 9 identical rings. I'm not exactly sure what its sides and rinf are yet. At present I'm not sure if there are any regular polytwisters past 4 dimensions, or even uniform - but there are polytwisters with identical rings as well as their duals with identical sides. But one thing looks certain, the 6-D polytwisters don't mimic the polytera in the same way as 4-D polytwisters do the polyhedra, because if they did - there would be no jak-twister.

Duals, Conjugates and Regiments

Below shows how the 36 regular polytwisters pair up as duals and conjugates and group into regiments.

Duals

The following are self duals: tetter and itter.

The following are dual pairs: quitter-blitter, cubiter-octer, quicter-bloter, blicter-quoter, icter-ioter, doter-iketer, quadoter-bliketer, bladoter-quiketer, idoter-iyiketer, gaditer-sissiditer, gaquiditer-soblessiditer, gabliditer-siquissiditer, giaditer-sansiditer, giketer-gissiditer, gaquiter-goblessiditer, gabliter-gaquissiditer, and giyiter-gansiditer.

Conjugates

The following are conjugate pairs: tetter-itter, quitter-blitter, cubiter-icter, quicter-blicter, octer-ioter, quoter-bloter, doter-gansiditer, quadoter-goblessiditer, bladoter-gaquissiditer, idoter-gissiditer, bliditer-goblessiditer, iketer-giyiter, quiketer-gabliter, bliketer-gaquiter, iyiketer-giketer, gaditer-sansiditer, gaquiditer-soblessiditer, gabliditer-siquissiditer, and giaditer-sissiditer.

Regiments

Regiments doesn't even form, simply because of the curved nature of the strips, for instance, in the iketer the strips bow out a certain amount, but in the gaditer they bow out even further - this causes the strips to dis-align, even though the rings align up - if it wasn't for this interesting fact, strange polytwisters like gacroter wouldn't even form.

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