Back in year 2000, I was looking at the small swirlprism that George Olshevsky discovered earlier. I then noticed that it was possible to circumscribe 12 equatorial rings around the vertices in such a way that the rings were parallel to the 12 girdles of the small swirlprism. Every point in this set of 12 rings are congruent. I checked to see what the convex hull of these rings would look like (icosatwister), as well as using the rings to carve out cylindrical curves (dodecatwister). Thus began my search of the "polytwisters".

Polytwisters are curvaceous four dimensional polytope-like objects with an interesting twist, instead of having vertices as their simplest 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 bowed out or in extreme cases bloated big time) which is then twisted 360 degrees and then curved back in on itself into a cylindrical ring. Polytwisters are actually the "lovechild" of polyhedra and Hopf Fibration. There is a one-to-one correspondence between the points of a sphere and a set of equatorial rings on a glome (4-D sphere). These rings fill the entire surface of the glome in a swirling manner and they never intersect - this is called Hopf Fibration. In a sense, a fibrated glome is a sphere of swirling equators and could also be called a "spheretwister". So what would happen if we started with a dodecahedron instead of a sphere and took the "Hopf function" of it? - you would get something we could call a "Hopf dodecahedron". This object looks a lot like the dodecatwister, except that it's facets bow inwards and it is concave, while the dodecatwister remains convex and has cylindrical curves. 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.

Above are two projections of the dodecatwister, the first is a cutaway version, the other shows two orthogonal twisters. Below is a dodecatwister unfolded. These are actually approximations, the actual dodecatwister will have smooth curves. These were done with Stella4D.

**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 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 a 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 and their rings are completely buried and unseen from the outside). 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.

**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}. Above are the dysters with up to five sides, along with a few 13 sided ones.

**37. Tatetter** - (tuh TET tur) also called tatet twister, or tetratetratwister. Its symbol is (3, 3/2) 2. It has 6 rings, 12 strips, 4 trigonal twisters and 4 retrotrigonal twisters. It's ring figure (rinf) is a very skinny bowed rectangle. *This thing has a strange feminine beauty to it.* It has tetteric symmetry. The (3, 3) 2 and (3/2, 3/2) 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. Its symbol is (3, 3/2) 2/3. 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 (3, 3) 2/3 and (3/2, 3/2) 2/3 are bloter and ioter respectively and isn't counted here.

**39. Coter** - (CO tur) also called co twister, or cuboctatwister. Its symbol is (4, 3) 2. 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. Its symbol is (4/3, 3) 2. 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 (4, 3/2) 2. 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. Its symbol is (4/3, 3/2) 2. 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. Its symbol is (4, 3) 2/3. 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. Its symbol is (4/3, 3) 2/3. 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. Its symbol is (4, 3/2) 2/3. 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. Its symbol is (4/3, 3/2) 2/3. 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. Its symbol is (5, 3) 2. 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. Its symbol is (5/4, 3) 2. 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. Its symbol is (5, 3/2) 2. 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. Its symbol is (5/4, 3/2) 2. 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. Its symbol is (5, 3) 2/3. 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. Its symbol is (5/4, 3) 2/3. 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. Its symbol is (5, 3/2) 2/3. 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. Its symbol is (5/4, 3/2) 2/3. 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. Its symbol is (5, 5/2) 2. 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. Its symbol is (5/4, 5/2) 2. 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. Its symbol is (5, 5/3) 2. 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. Its symbol is (5/4, 5/3) 2. 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. Its symbol is (5, 5/2) 2/3. 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. Its symbol is (5/4, 5/2) 2/3. 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. Its symbol is (5, 5/3) 2/3. 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. Its symbol is (5/4, 5/3) 2/3. 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. Its symbol is (5/2, 3) 2. 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. Its symbol is (5/3, 3) 2. 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. Its symbol is (5/2, 3/2) 2. 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. Its symbol is (5/3, 3/2) 2. 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. Its symbol is (5/2, 3) 2/3. 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. Its symbol is (5/3, 3) 2/3. 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. Its symbol is (5/2, 3/2) 2/3. 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. Its symbol is (5/3, 3/2) 2/3. It has 12 retrostarpod twisters and 20 retrotripodic twisters. It's rinf is a bloated X.

**71. Vitter** - (VIT tur) also called vit twister, or sheaved tetratwister. Its symbol is (2, 3) 3 (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. Its symbol is (2, 3/2) 3. 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. Its symbol is (2, 3) 3/2. 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. Its symbol is (2, 3/2) 3/2. 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. Its symbol is (2, 4) 3. 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. Its symbol is (2, 4/3) 3. 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. Its symbol is (2, 4) 3/2. 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. Its symbol is (2, 4/3) 3/2. 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. Its symbol is (2, 3) 4. 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. Its symbol is (2, 3/2) 4. 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. Its symbol is (2, 3) 4/3. 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. Its symbol is (2, 3/2) 4/3. 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. Its symbol is (2, 5) 3. 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. Its symbol is (2, 5/4) 3. It has 20 rings, 12 retropentagonal twisters, and 30 dyad twisters. Its rinf is a ditrigon. This thing looks like it would hurt - REAL bad!

**85. Blividiter** - (bliv VID it tur) also called blivid twister, or bloatosheaved dodecatwister. Its symbol is (2, 5) 3/2. 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. Its symbol is (2, 5/4) 3/2. 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. Its symbol is (2, 3) 5. 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. Its symbol is (2, 3/2) 5. 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. Its symbol is (2, 3) 5/4. 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. Its symbol is (2, 3/2) 5/4. 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. Its symbol is (2, 5) 5/2. 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. Its symbol is (2, 5/4) 5/2. 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. Its symbol is (2, 5) 5/3. 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. Its symbol is (2, 5/4) 5/3. 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. Its symbol is (2, 5/2) 5. 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. Its symbol is (2, 5/3) 5. 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. Its symbol is (2, 5/2) 5/4. 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. Its symbol is (2, 5/3) 5/4. 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. Its symbol is (2, 3) 5/2. 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. Its symbol is (2, 3/2) 5/2. 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. Its symbol is (2, 3) 5/3. 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. Its symbol is (2, 3/2) 5/3. 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. Its symbol is (2, 5/2) 3. 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. Its symbol is (2, 5/3) 3. 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. Its symbol is (2, 5/2) 3/2. 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. Its symbol is (2, 5/3) 3/2. It has 12 rings, 12 retrostar twisters, and 30 dyad twisters. Its rinf is a retroditrigon.

**107. Potter** - (POT tur) also called pot twister, or plated tetratwister. Its symbol is (2, 3) 3/4. 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. Its symbol is (2, 3/2) 3/4. 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. Its symbol is (2, 3) 3/5. 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. This one reminds me of the Ivory Tower from the Neverending Story.

**110. Ipiter** - (IPE it tur) also called ipit twister, or invertiplated tetratwister. Its symbol is (2, 3/2) 3/5. 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. Its symbol is (2, 4) 3/4. 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. Its symbol is (2, 4/3) 3/4. 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. Its symbol is (2, 4) 3/5. 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. Its symbol is (2, 4/3) 3/5. 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. Its symbol is (2, 3) 4/5. 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. Its symbol is (2, 3/2) 4/5. 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. Its symbol is (2, 3) 4/7. 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. Its symbol is (2, 3/2) 4/7. 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. Its symbol is (2, 5) 3/4. 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. Its symbol is (2, 5/4) 3/4. 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. Its symbol is (2, 5) 3/5. 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. Its symbol is (2, 5/4) 3/5. 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. Its symbol is (2, 3) 5/6. 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. Its symbol is (2, 3/2) 5/6. 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. Its symbol is (2, 3) 5/9. 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. Its symbol is (2, 3/2) 5/9. 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. Its symbol is (2, 5) 5/7. It has 12 rings, 12 bloated pentagonal twisters, and 30 bloated dyad twisters. Its rinf is an introverted distar. This one looks like a bouquet of flowers.

**128. Gaquapiditer** - (GA qua PID it tur) also called gaquapid twister, or great quasiplated dodecatwister. Its symbol is (2, 5/4) 5/7. 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. Its symbol is (2, 5) 5/8. 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. Its symbol is (2, 5/4) 5/8. 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. Its symbol is (2, 5/2) 5/6. 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. Its symbol is (2, 5/3) 5/6. 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. Its symbol is (2, 5/2) 5/9. 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. Its symbol is (2, 5/3) 5/9. 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. Its symbol is (2, 3) 5/7. 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. Its symbol is (2, 3/2) 5/7. 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. Its symbol is (2, 3) 5/8. 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. Its symbol is (2, 3/2) 5/8. 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. Its symbol is (2, 5/2) 3/4. 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. Its symbol is (2, 5/3) 3/4. 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. Its symbol is (2, 5/2) 3/5. 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. Its symbol is (2, 5/3) 3/5. It has 12 rings, 12 bloated retrostar twisters, and 30 bloated dyad twisters. Its rinf is an introverted retroditrigon.

**143. Tritter** - (TRIT tur) also called trit twister, or tetraretrotetratwister. Its symbol is (3/2, 3) 3. 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. Its symbol is (3/2, 3) 3/5. 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. Its symbol is (3/2, 4) 4. 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. Sadtacoter** - (SAD ta CO tur) also called sadtaco twister, or small ditetragonary cubioctatwister. Its symbol is (3, 4) 4/3. It has 6 rings, 24 strips, 6 square twisters, and 8 trigon twisters. Its rinf is a quasitruncated square. It has cubiteric symmetry.

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

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

**149. Cricter** - (CRIC tur) also called cric twister, or cubiretrocubitwister. Its symbol is (4, 4/3) 3. 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. Its symbol is (3/2, 4) 4/7. 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. Sadtablicoter** - (SAD ta blic CO tur) also called sadtablico twister, or small ditetragonary bloated cubioctatwister. Its symbol is (3, 4) 4/5. 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.

**152. Gacbloter** - (gac BLO tur) also called gacblo twister, or great cubibloated octatwister. Its symbol is (3, 4/3) 4/7. 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. Gadtablicoter** - (GAD ta blic CO tur) also called gadtablico twister, or great ditetragonary bloated cubioctatwister. Its symbol is (3/2, 4/3) 4/5. 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.

**154. Cablicter** - (ca BLIC tur) also called cablic twister, or cubibloated cubitwister. Its symbol is (4, 4/3) 3/5. 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. Sidtiditer** - (sid TID it tur) also called sidtid twister, or small ditrigonary icosidodecatwister. Its symbol is (5/2, 3) 3. It has 20 rings, 60 strips, 20 trigon twisters, and 12 star twisters. Its rinf is a ditrigon. This and the remaining polytwisters have doteric symmetry.

**156. Gidditditer** - (GID dit DI tur) also called gidditdi twister, or great ditrigonary dodekicosatwister (previously called gidartiditer). Its symbol is (5/2, 3/2) 3/2. It has 20 rings, 60 strips, 20 retrotrigon twisters, and 12 retrostar twisters. Its rinf is a retroditrigon.

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

**158. Gadtiditer** - (gad TID it tur) also called gadtid twister, or grand ditrigonary icosidodecatwister (previously called sidartiditer). Its symbol is (5/3, 3) 3/2. It has 20 rings, 60 strips, 20 trigon twisters, and 12 retrostar twisters. Its rinf is a retroditrigon.

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

**160. Gadtabliditer** - (GAD ta BLID it tur) also called gadtablid twister, or grand ditrigonary bloated icosidodecatwister (previously called quidtabliditer which now goes to another polytwister). Its symbol is (5/3, 3/2) 3/5. It has 20 rings, 60 strips, 20 bloated retrotrigon twisters, and 12 bloated retrostar twisters. Its rinf is an introverted ditrigon.

**161. Sidditbladiter** - (SID dit bla DI tur) also called sidditbladi twister, or small ditrigonary bloatododekicosatwister (previously called sidblatiditer). Its symbol is (5/3, 3) 3/4. It has 20 rings, 60 strips, 20 bloated trigon twisters, and 12 bloated retrostar twisters. Its rinf is an introverted retroditrigon.

**162. Gaquidtabliditer** - (ga QUID ta BLID it tur) also called gaquidtablid twister, or great quasiditrigonary bloatoicosidodecatwister (previously called gidblatiditer). Its symbol is (5/2, 3) 3/5. It has 20 rings, 60 strips, 20 bloated trigon twisters, and 12 bloated star twisters. Its rinf is an introverted ditrigon.

**163. Sidtabliditer** - (SID ta BLID it tur) also called sidtablid twister, or small ditrigonary bloated icosidodecatwister. Its symbol is (5/2, 3/2) 3/4. It has 20 rings, 60 strips, 20 bloated retrotrigon twisters, and 12 bloated star twisters. Its rinf is an introverted retroditrigon.

**164. Sabliiter** - (sa BLEET tur) also called sibli twister, or small bloated icosicositwister. Its symbol is (3, 3/2) 5/7 twister. It has 12 rings, 60 strips, 20 bloated trigon twisters, and 20 bloated retrotrigonal twisters. Its rinf is an introverted retrodistar.

**165. Gaquidtiditer** - (GAH quid TID it tur) also called gaquidtid twister, or great quasiditrigonary icosidodecatwister. Its symbol is (5/4, 3/2) 3/2. It has 20 rings, 60 strips, 12 retropentagonal twisters, and 20 retrotrigonal twisters. Its rinf is a retroditrigon. This is one of the new ones.

**166. Sidditditer** - (SID dit DI tur) also called sidditdi twister, or small ditrigonary dodekicosatwister. Its symbol is (5/4, 3) 3. It has 20 rings, 60 strips, 12 retrodipentagonal twisters, and 20 trigonal twisters. Its rinf is a retroditrigon. This is a new one.

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

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

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

**170. Quidtabliditer** - (QUID ta BLID it tur) also called quidtablid twister, or quasi ditrigonary bloated icosidodecatwister. Its symbol is (5, 3) 3/4. It has 20 rings, 60 strips, 20 bloated trigon twisters, and 12 bloated pentagonal twisters. Its rinf is an introverted retroditrigon. This one is new.

**171. Gidditbladiter** - (GID it bla DI tur) also called gidditbladi twister, or great ditrigonary bloated dodekicosatwister. Its symbol is (5, 3/2) 3/5. It has 20 rings, 60 strips, 20 bloated retrotrigon twisters, and 12 bloated pentagonal twisters. Its rinf is an introverted ditrigon. This one is new.

**172. Midtabliditer** - (MID ta BLID it tur) also called midtablid twister, or medial ditrigonary bloated icosidodecatwister. Its symbol is (5/4, 3/2) 3/4. It has 20 rings, 60 strips, 20 bloated retrotrigon twisters, and 12 bloated retropentagonal twisters. Its rinf is an introverted retroditrigon.

**173. Gidtabliditer** - (GID ta BLID it tur) also called gidtablid twister, or great ditrigonary bloated icosidodecatwister. Its symbol is (5/4, 3) 3/5. It has 20 rings, 60 strips, 20 bloated trigon twisters, and 12 bloated retropentagonal twisters. Its rinf is an introverted ditrigon.

**174. Gabliiter** - (ga BLEET tur) also called gibli twister, or great bloated icosicositwister. Its symbol is (3, 3/2) 5/9. It has 12 rings, 60 strips, 20 bloated trigon twisters, and 20 bloated retrotrigonal twisters. Its rinf is an introverted dipentagon.

**175. Sidriter** - (sid DRI tur) also called sidri twister, or small dodecaretroicosatwister (previously called siriditer which now goes to another polytwister). Its symbol is (3/2, 5) 5. It has 12 rings, 60 strips, 20 retrotrigon twisters, and 12 pentagonal twisters. Its rinf is a retrodipentagon.

**176. Gorditer** - (GOR dit tur) also called gordi twister, or great retrododekicosatwister. Its symbol is (3/2, 5/4) 5/4. It has 12 rings, 60 strips, 20 retrotrigon twisters, and 12 retropentagonal twisters. Its rinf is a retrodipentagon. This is one of the new ones (in symbol only), but I did suspect one of the polytwisters to look like this one due to gadtacoter.

**177. Coditer** - (KO dit tur) also called codi twister, or compact dodeckicosatwister (previously called dipiditer). Its symbol is (3, 5) 5/4. It has 12 rings, 60 strips, 20 trigon twisters, and 12 pentagonal twisters. Its rinf is a retrodipentagon.

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

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

**180. Gibriditer** - (gib BRID it tur) also called gibrid twister, or great icosibloatoretrododecatwister. Its symbol is (3, 5/4) 5/9. It has 12 rings, 60 strips, 20 bloated trigon twisters, and 12 bloated retropentagonal twisters. Its rinf is an introverted retrodipentagon.

**181. Boditer** - (BO dit tur) also called bodi twister, or bloated dodekicosatwister (previously called dipbliditer). Its symbol is (3, 5) 5/6. It has 12 rings, 60 strips, 20 bloated trigon twisters, and 12 bloated pentagonal twisters. Its rinf is an introverted retrodistar.

**182. Sabroditer** - (SAB ro DITE tur) also called sabrodi twister, or small bloatoretrododekicosatwister. Its symbol is (3/2, 5/4) 5/6. It has 12 rings, 60 strips, 20 bloated retrotrigon twisters, and 12 bloated retropentagonal twisters. Its rinf is an introverted retrodistar. This is one of the new ones.

**183. Sidbriter** - (sid BRI tur) also called sidbri twister, or small dodecabloatoretroicosatwister (previously called sibriditer which goes to another polytwister). Its symbol is (3/2, 5) 5/9. It has 12 rings, 60 strips, 20 bloated retrotrigon twisters, and 12 bloated pentagonal twisters. Its rinf is an introverted retrodipentagon.

**184. Gidbraditer** - (gid BRAD it tur) also called gidbrad twister, or great dodecabloatoretrododecatwister. Its symbol is (5/4, 5) 3/5. It has 20 rings, 60 strips, 12 bloated retropentagon twisters, and 12 bloated pentagon twisters. Its rinf is an introverted ditrigon.

**185. Gidriter** - (gid RI tur) also called gidri twister, or great dodecaretroicositwister. Its symbol is (3/2, 5/2) 5/2. It has 12 rings, 60 strips, 12 star twisters, and 20 retrotrigonal twisters. Its rinf is a distar. This one is new.

**186. Sorditer** - (SOR di tur) also called sordi twister, or small retrododekicositwister. Its symbol is (3/2, 5/3) 5/3. It has 12 rings, 60 strips, 12 retrostar twisters, and 20 retrotrigonal twisters. Its rinf is a retrodistar. This one is new.

**187. Scoditer** - (sko DI tur) also called scodi twister, or small compact dodekicositwister (previously called gidpaditer). Its symbol is (3, 5/2) 5/3. It has 12 rings, 60 strips, 12 star twisters, and 20 trigonal twisters. Its rinf is a retrodistar.

**188. Siriditer** - (si RID it tur) also called sirid twister, or small icosiretrododecatwister (previously called sidpaditer). Its symbol is (3, 5/3) 5/2. It has 12 rings, 60 strips, 12 retrostar twisters, and 20 trigonal twisters. Its rinf is a distar.

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

**190. Sibriditer** - (sib BRID it tur) also called sibrid twister, or small icosibloatoretrododecatwister (previously called sidpabloditer which now goes to another polytwister). Its symbol is (3, 5/3) 5/8. It has 12 rings, 60 strips, 12 bloated retrostar twisters, and 20 bloated trigonal twisters. Its rinf is an introverted distar.

**191. Sibditer** - (sib DI tur) also called sibdi twister, or small bloated dodekicosatwister (previously called gidpabloditer which now goes to another polytwister). Its symbol is (3, 5/2) 5/7. It has 12 rings, 60 strips, 12 bloated star twisters, and 20 bloated trigonal twisters. Its rinf is an introverted retrodistar.

**192. Gabroditer** - (GAB ro DI tur) also called gabrodi twister, or great bloatoretrododekicosatwister. Its symbol is (3/2, 5/3) 5/7. It has 12 rings, 60 strips, 12 bloated retrostar twisters, and 20 bloated retrotrigonal twisters. Its rinf is an introverted retrodistar. This one is new.

**193. Gidbriter** - (gid BRI tur) also called gidbri twister, or great dodecabloatoretroicosatwister. Its symbol is (3/2, 5/2) 5/8. It has 12 rings, 60 strips, 12 bloated star twisters, and 20 bloated retrotrigonal twisters. Its rinf is an introverted distar. This one is new.

**194. Sidbraditer** - (sid BRAD it tur) also called sidbrad twister, or small dodecabloatoretrododecatwister. Its symbol is (5/2, 5/3) 3/5. It has 20 rings, 60 strips, 12 bloated star twisters, and 12 bloated retrostar twisters. Its rinf is an introverted ditrigon.

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

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

**197. Dabriditer** - (da BRID it tur) also called dabrid twister, or dodecabloatoretrododecatwister. Its symbol is (5/4, 5) 5/9. It has 12 rings, 60 strips, 12 bloated retropentagon twisters, and 12 bloated pentagonal twisters. Its rinf is an introverted dipentagon.

**198. Dipbladiditer** - (DIP bla DID it tur) also called dipbladid twister, or dipentagonary bloated dodecadodecatwister. Its symbol is (5/2, 5/3) 5/7. It has 12 rings, 60 strips, 12 bloated star twisters, and 12 bloated retrostar twisters. Its rinf is an introverted retrodistar.

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

**200. Quiditdiditer** - (QUID it DID it tur) also called quiditdid twister, or quasiditrigonary dodecadodecatwister. Its symbol is (5/2, 5/4) 3. It has 20 rings, 60 strips, 12 retropentagon twisters, and 12 star twisters. Its rinf is a retroditrigon. This one is new.

**201. Gadtadoditer** - (GAD ta DOE dit tur) also called gadtadod twister, or great ditrigonary dodecadodecatwister. Its symbol is (5/3, 5/4) 3/2. It has 20 rings, 60 strips, 12 retropentagon twisters, and 12 retrostar twisters. Its rinf is a retroditrigon. This one is new.

**202. Sadtadoditer** - (SAD ta DOE dit tur) also called sadtadod twister, or small ditrigonary dodecadodecatwister. Its symbol is (5/2, 5) 3/2. It has 20 rings, 60 strips, 12 pentagon twisters, and 12 star twisters. Its rinf is a retroditrigon. This one is new.

**203. Quiditbladiditer** - (QUID it bla DID it tur) also called quiditbladid twister, or quasiditrigonary bloated dodecadodecatwister. Its symbol is (5/2, 5/4) 3/5. It has 20 rings, 60 strips, 12 bloated retropentagon twisters, and 12 bloated star twisters. Its rinf is an introverted ditrigon. This one is new.

**204. Ditbladiditer** - (DIT bla DID it tur) also called ditbladid twister, or ditrigonary bloated dodecadodecatwister. Its symbol is (5/3, 5) 3/5. It has 20 rings, 60 strips, 12 bloated pentagon twisters, and 12 bloated retrostar twisters. Its rinf is an introverted ditrigon.

**205. Sidtablidoditer** - (sid TAH ble DO dit tur) also called sidtablidod twister, or small ditrigonary bloated dodecadodecatwister. Its symbol is (5/2, 5) 3/4. It has 20 rings, 60 strips, 12 bloated pentagon twisters, and 12 bloated star twisters. Its rinf is an introverted retroditrigon. This one is new.

**206. Gidtablidoditer** - (gid TAH ble DO dit tur) also called gidtablidod twister, or great ditrigonary bloated dodecadodecatwister. Its symbol is (5/3, 5/4) 3/4. It has 20 rings, 60 strips, 12 bloated retropentagon twisters, and 12 bloated retrostar twisters. Its rinf is an introverted retroditrigon. This one is new.

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

**208. Sidpoditer** - (SID po DI tur) also called sidpodi twister, or small dipentagonary dodekicosatwister (previously called gidpiditer which now belongs to polytwister 210). Its symbol is (3, 5) 5/3. It has 12 rings, 60 strips, 20 trigon twisters, and 12 pentagon twisters. Its rinf is a retrodistar.

**209. Gidpoditer** - (GID po DI tur) also called gidpodi twister, or great dipentagonary dodekicosatwister. Its symbol is (3, 5/4) 5/2. It has 12 rings, 60 strips, 20 trigon twisters, and 12 retrostar twisters. Its rinf is a retrodistar. This one is new in symbol only, I did suspect one to look like this due to gacroter.

**210. Gidpiditer** - (gid PID it tur) also called gidpid twister, or great dipentagonary icosidodecatwister. Its symbol is (3, 5/2) 5/4. It has 12 rings, 60 strips, 20 trigon twisters, and 12 star twisters. Its rinf is a retrodipentagon. This one is new.

**211. Gadpablidoter** - (gad PA ble DO tur) also called gadpablid twister, or grand dipentagonary bloated icosidodecatwister. Its symbol is (5/2, 3/2) 5/9. It has 12 rings, 60 strips, 12 bloated retrotrigon twisters, and 12 bloated star twisters. Its rinf is an introverted dipentagon. This one is new.

**212. Gadpabloditer** - (gad PA blo DI tur) also called gadpablodi twister, or grand dipentagonary bloated dodekicosatwister. Its symbol is (3/2, 5/4) 5/7. It has 12 rings, 60 strips, 12 bloated retrotrigon twisters, and 12 bloated retropentagon twisters. Its rinf is an introverted retrodistar. This one is new.

**213. Midpabloditer** - (mid PA blo DI tur) also called midpablodi twister, or medial dipentagonary bloated dodekicosatwister. Its symbol is (3/2, 5) 5/8. It has 12 rings, 60 strips, 12 bloated retrotrigon twisters, and 12 bloated pentagon twisters. Its rinf is an introverted distar. This one is new.

**214. Midpablidoter** - (mid PA ble DO tur) also called midpablid twister, or medial dipentagonary bloated icosidodecatwister. Its symbol is (3/2, 5/3) 5/6. It has 12 rings, 60 strips, 12 bloated retrotrigon twisters, and 12 bloated retrostar twisters. Its rinf is an introverted retrodipentagon. This one is new.

**215. Gadpoditer** - (GAD po DI tur) also called gadpodi twister, or grand dipentagonary dodekicosatwister. Its symbol is (3/2, 5/4) 5/3. It has 12 rings, 60 strips, 20 retrotrigon twisters, and 12 retropentagon twisters. Its rinf is a retrodistar. This one is new.

**216. Gadpiditer** - (gad PID it tur) also called gadpid twister, or grand dipentagonary icosidodecatwister. Its symbol is (3/2, 5/3) 5/4. It has 12 rings, 60 strips, 20 retrotrigon twisters, and 12 retrostar twisters. Its rinf is a retrodipentagon. This one is new.

**217. Midpoditer** - (MID po DI tur) also called midpodi twister, or medial dipentagonary dodekicosatwister. Its symbol is (3/2, 5) 5/2. It has 12 rings, 60 strips, 20 retrotrigon twisters, and 12 pentagon twisters. Its rinf is a distar. This one is new.

**218. Midpiditer** - (mid PID it tur) also called midpid twister, or medial dipentagonary icosidodecatwister. Its symbol is (3/2, 5/2) 5. It has 12 rings, 60 strips, 20 retrotrigon twisters, and 12 star twisters. Its rinf is a dipentagon. This one is new.

**219. Sadpabloditer** - (sad PA blo DI tur) also called sadpablodi twister, or small dipentagonary bloated dodekicosatwister (previously called gidpabliditer). Its symbol is (3, 5) 5/7. It has 12 rings, 60 strips, 20 bloated trigon twisters, and 12 bloated pentagonal twisters. Its rinf is an introverted retrodistar.

**220. Sadpablidoter** - (sad PA ble DO tur) also called sadpablid twister, or small dipentagonary bloated icosidodecatwister. Its symbol is (3, 5/2) 5/6. It has 12 rings, 60 strips, 20 bloated trigon twisters, and 12 bloated star twisters. Its rinf is an introverted retrodipentagon. This one is new.

**221. Gidpabloditer** - (gid PA blo DI tur) also called gidpablodi twister, or great dipentagonary bloated dodekicosatwister. Its symbol is (3, 5/4) 5/8. It has 12 rings, 60 strips, 20 bloated trigon twisters, and 12 bloated retropentagonal twisters. Its rinf is an introverted distar. This one is new.

**222. Gidpablidoter** - (gid PA ble DO tur) also called gidpablid twister, or great dipentagonary bloated icosidodecatwister (previously called sidpabliditer). Its symbol is (3, 5/3) 5/9. It has 12 rings, 60 strips, 20 bloated trigon twisters, and 12 bloated retrostar twisters. Its rinf is an introverted dipentagon.

Below are sample slices of the 222 polytwisters on five pages for easy printing.

**Polyduotwisters** - six dimensional objects with cylindrically curved sides which twist in two orthogonal directions as you follow the curve. Smallest elements are equatorial circles. Examples include the "jak-twister" which is related to the polypeton "jak" (which is best known as 2_{21}, this 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. Its dual is the enneaduotwister. There are polyduotwisters with identical rings as well as their duals with identical sides, which would form dice in the convex cases. I'm not sure if there are regular ones yet. The 6-D polytwisters don't mimic the polytera in the same way as 4-D polytwisters do the polyhedra.

**Polytriotwisters** - eight dimensional objects with cylindrically curved sides which twist in three orthogonal directions as you follow the curve. Smallest elements are equatorial circles. There should be some related to fy (better known as the E8 polytope).

**Polyswirlers** - eight dimensional objects whose sides curve like a glome (4-D sphere) and can roll like a 4-D ball on all of their sides. Their smallest element are equatorial fibrated glomes. These objects will mimic the symetries of the polytera in the same way that polytwisters mimic polyhedra. There are many regular ones including the hexaswirler (hix swirler), penteractswirler (or decaswirler), and the triacontidiswirler (tac swirler). Each regular polyteron has 16 polyswirler analogues. The number of uniform polyswirlers will likely number into the thousands. These objects are based on quarternion fibration.

**Polyteratwisters** - ten dimensional objects with cylindrically curved sides which twist in four orthogonal directions as you follow the curve. Smallest elements are equatorial circles.

**Polypetatwisters** - twelve dimensional objects with cylindrically curved sides which twist in five orthogonal directions as you follow the curve. Smallest elements are equatorial circles.

**Polyduoswirlers** - twelve dimensional objects with glomically curved sides which twist in two orthogonal sets of directions as you follow the curves. Smallest elements are equatorial fibrated glomes.

**Polyectatwisters** - 14 dimensional objects with cylindrically curved sides which twist in six orthogonal directions as you follow the curve. Smallest elements are equatorial circles.

**Polyzettatwisters** - 16 dimensional objects with cylindrically curved sides which twist in seven orthogonal directions as you follow the curve. Smallest elements are equatorial circles.

**Polytrioswirlers** - 16 dimensional objects with glomically curved sides which twist in three orthogonal sets of directions as you follow the curves. Smallest elements are equatorial fibrated glomes.

**Polymaelstroms** - 16 dimensional objects with 8-D spherically curved sides. Smallest elements are equatorial eight dimensional spheres with the S_{7} fibration. These objects are based on octonion fibration. I'm not sure how well these work yet. If they do, they should mimic nine dimensional polytope symmetries and each regular polyyotton should lead to 256 versions of polymaelstroms.

**Poly-n-twisters** - 2(n+1) dimensional objects with cylindrically curved sides which twist in n orthogonal directions as you follow the curve. Smallest elements are equatorial circles.

**Poly-n-swirlers** - 4(n+1) dimensional objects with glomically curved sides which twist in n orthogonal sets of directions as you follow the curves. Smallest elements are equatorial fibrated glomes.

There shouldn't be any poly-n-maelstroms where n>1.

**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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