# Symmetries of Four Dimensions

This page will list the symmetries of objects in four dimensions. It will deal with polytopes (as well as round objects) but not include the tilings and honeycomb symmetries. These symmetries are often called point groups due to being centered on one point, the origin. I will introduce some 'short names' for the symmetries along with a Coxeter notation for it. I'll extend Coxeter's notation to include round symmetries also. This page will link to other pages which detail various symmetries and the isogonal (vertex-transitive) and isotopal (facet-transitive aka dice) shapes there.

# Four Dimensional Symmetries

Symmetries in 4-space can be grouped into these types: Fully 4-D (includes several families: glomic, pennic, tessic, icoic, and hyic families), 3-D derivatives which includes prismatics (which are products of the 3-D symmetries with dyadic symmetry and their derivatives) and pyramid versions (which are the fully 3-D ones all over again), swirl symmetries (includes 11 infinite families that lead to polytwister symmetries as their round version), duoprismic (these get fairly involved and tedious), dysteric (the swirlsymmetries based on prism and pyramid symmetries), and gyrochoric.

## Glome Symmetries

There are four symmetries that are based on a glome, two are fully 4-D, the other two are more like polytwister versions.

Glomic Symmetry - [O,O,O] order 2*OOO. It is the symmetry of a glome.

Kiglomic Symmetry - [O,O,O]⁺ order OOO. Chiral glome symmetry can best be visuallized by having a strange blue sphere that looks red in a mirror.

Hopfic Symmetry - [O,O:O] order 4*OO*O. This is the symmetry of Hopf fibration.

Kihopfic Symmetry - [O,O:O]⁺ order 2*OO*O. Although Hopf fibration is chiral, the underlying symmetry that generates kihopfic is chiral also.

## Pennic Symmetry Family

Pennic Symmetry - [3,3,3] order 120. Also called pentachoric symmetry. There are two sets of five points with tettic symmetry and two sets of ten points with trippic symmetry. There are ten rings with dyadic symmetry that generate 20 trigonic areas and 15 dyadic rings which generate 30 rectic areas. There are ten spheres with trippyric symmetry that generate 60 dyadic areas. Elsewhere (which we'll call the 'white space') generates 120 no sym areas.

Kipennic Symmetry - [3,3,3]⁺ order 60. Also called chiropennic or chiro-pentachoric symmetry. There are two sets of five points with kitettic symmetry and two sets of ten points with kitrippic symmetry. There are ten rings with dyadic symmetry that generate 20 kitriggic areas and 15 rings with dyadic symmetry that generate 30 essic areas. White space creates 60 no sym areas.

Decaic Symmetry - [[3,3,3]] order 240. It is the symmetry of deca and spid. There are ten points with tettic symmetry, 20 points with trippic symmetry, 20 points with trappic symmetry, and 30 points with disphic symmetry. There are 10 rings with rectic symmetry which generate 40 trigonic areas, 15 rings with rectic symmetry which generate 60 rectic symmetry, and 15 rings with square symmetry that generate 120 essic areas. There are ten spheres with trappic symmetry that generate 120 dyadic areas. White space generates 240 no sym areas.

Kidecaic Symmetry - [[3,3,3]]⁺ order 120. Also called chirodecaic symmetry. There are ten points with kitettic symmetry, two sets of 20 with kitrippic symmetry, and 30 with kibrick symmetry. There are 10 rectic rings which generates 40 kitriggic areas and two sets of 15 rectic rings which creates 60 regions with essic symmetry. White space creates 120 no sym areas.

Iodecaic Symmetry - [[3,3,3]⁺] order 120. Also called ionic decaic symmetry. There are ten points with kitettic symmetry, 20 with kitrippic symmetry, 20 with protrappic symmetry, and 30 with prodappic symmetry. There are 10 rectic rings which generates 40 kitriggic areas and 15 rectic rings which creates 60 essic areas. White space creates 120 no sym areas.

## Tessic Symmetry Family

Tessic Symmetry - [4,3,3] order 384. Also called tesseractic symmetry. There are eight points of cubic symmetry, 16 of tettic symmetry, 24 of squippic symmetry, and 32 of trippic symmetry. There are six rings of square symmetry that generates 48 square symmetric areas, 16 rectic rings that create 64 trigonic regions, 24 rectic rings that create 96 rectic areas, and 12 square rings that creates 96 rectic areas. There are four spheres with cubic symmetry which generate 192 dyadic regions and 12 spheres with squippic symmetry which generate 192 dyadic areas. White space creates 384 no sym areas.

Kitessic Symmetry - [4,3,3]⁺ order 192. This is chiral tesseractic symmetry. It has eight points of kicubic symmetry, 16 of kitettic, 24 of kisquippic, and 32 of kitrippic. There are six rings of square symmetry that generates 48 kisquare areas, 16 rectic rings that creates 64 kitriggic symmetry, 24 rectic rings that create 96 essic areas, and 12 square rings that create 96 essic areas. White space creates 192 no sym areas.

Demitessic Symmetry - [31,1,1] order 192. It has three sets of eight points with tettic symmetry and 24 points with brick symmetry. There are 16 rings with essic symmetry that creates 64 trigonic areas and three sets of six rings with square symmetry that generate 48 rectic symmetric regions. There are 12 spheres with brick symmetry that create 96 dyadic regions. White space creates 192 no sym areas.

Kidemitessic Symmetry - [31,1,1]⁺ order 96. This is chiral demitessic symmetry. It has three sets of eight points with kitettic symmetry and 24 points with kibrick symmetry. There are 16 rings with essic symmetry which creates 64 kitriggic areas and three sets of six rings with square symmetry which creates 48 essic areas. White space creates 96 no sym areas.

Tessipyritic Symmetry - [4,[3,3]⁺] order 192. There are eight points with pyritic symmetry, 16 with kitettic symmetry, 24 with dappic symmetry, and 32 with protrippic symmetry. There are six rings of square symmetry that creates 48 rectic areas, 16 rectic rings which creates 64 kitriggic areas, and 12 square rings which creates 96 essic areas. There are four spheres with pyritic symmetry that generates 96 dyadic areas. White space creates 192 no sym areas.

## Icoic Symmetry Family

Icoic Symmetry - [3,4,3] order 1152. It has two sets of 24 points with cubic symmetry and two sets of 96 points with trippic synmmetry. It has 18 rings with square symmetry which generate 144 square symmetric regions, two sets of 16 hexagonal rings which generate 192 trigonic regions each, and 72 rectic rings which generate 288 rectic regions. It has two sets of 12 spheres with cubic symmetry which generate 576 dyadic regions. White space creates 1152 no sym areas.

Kiicoic Symmetry - [3,4,3]⁺ order 576. It has two sets of 24 points with kicubic symmetry and two sets of 96 points with kitrippic symmetry. It has 18 rings with square symmetry which generate 144 kisquare regions, two sets of 16 hexagonal rings which generate 192 kitrigic regions each, and 72 rectic rings which generate 288 essic regions. White space creates 576 no sym areas.

Contic Symmetry - [[3,4,3]] order 2304. It has 48 points with cubic symmetry, 144 points with squappic symmetry, 192 points with trippic symmetry, and 288 points with dappic symmetry. It has 18 rings with octagon symmetry which generate 288 square symmetric regions, 32 hexagonal rings which generate 384 trigonic regions, 72 square rings which generate 576 rectic regions, and 72 octic rings which generate 1152 essic regions. There are 24 spheres with cubic symmetry which generate 1152 dyadic regions. White space creates 2304 no sym areas.

Kicontic Symmetry - [[3,4,3]]⁺ order 1152. It has 48 kicubic points, 144 kisquappic points, 192 kitrippic points, and 288 kibrick points. It has 18 octagonal rings which generate 288 kisquare regions, 32 hexagonal rings which generate 384 kitriggic regions, 72 square rings which generate 576 essic regions, and two sets of 36 octic rings which generate 576 essic regions each. White space creates 1152 no sym areas.

Iocontic Symmetry - [[3,4,3]⁺] order 1152. It has 48 kicubic points, 144 prosquappic points, 192 kitrippic points, and 288 prodappic points. It has 18 octagonal rings which generate 288 kisquare regions, 32 hexagonal rings which generate 384 kitriggic regions, and 72 square rings which generate 576 essic regions. White space creates 1152 no sym areas.

Icopyritic Symmetry - [3,4,3⁺] order 576. It has 24 pyritic points, 24 tettic points, and 96 protrippic points. It has 18 square symmetric rings which generate 144 rectic regions, 16 hexagonal symmetric rings which generate 192 kitriggic regions, and 16 kihexagonic rings which generate 96 triggic regions. It also has 12 spheres with pyritic symmetry which generate 288 dyadic regions. White space creates 576 no sym areas.

Toxitic Symmetry - [3⁺,4,3⁺] order 288. It has two sets of 24 kitettic points. It has 18 square symmetric rings which generate 144 essic areas and two sets of 16 kihexagonal rings which generate 96 kitrigonal regions each. White space creates 288 no sym areas.

Oxitic Symmetry - [[3⁺,4,3⁺]] order 576. It has 48 kitettic points and 144 kibrick points. It has 18 octagonal rings which generate 288 essic areas, 32 kihexagonal rings which generate 192 kitrigonal regions, and 36 square rings which generate 288 essic regions. White space creates 576 no sym areas.

## Hyic Symmetry Family

Hyic Symmetry - [5,3,3] order 14400. This is the symmetry of the 120-cell and is quite common in the uniform polychora. This symmetry has 120 points of doic symmetry, 600 points of tettic symmetry, 720 points of pippic symmetry, and 1200 points of trippic symmetry. It has 72 rings with decagonal symmetry that generate 1440 spots with pentagon symmetry, 200 rings with hexagonal symmetry that generate 2400 trigonic regions, and 450 rings with square symmetry that create 3600 rectic areas. There are also 60 spheres with doic symmetry that create 7200 dyadic regions. White space creates 14400 points with no symmetry.

Kihyic Symmetry - [5,3,3]⁺ order 7200. This is the chiral 120-cell symmetry. It has 120 points of kidoic symmetry, 600 points of kitettic symmetry, 720 points of kipippic symmetry, and 1200 points of kitrippic symmetry. It has 72 rings with decagonal symmetry that create 1440 kipeggic regions, 200 hexagonal rings that generate 2400 kitriggic regions, and 450 square rings that create 3600 essic regions. White space creates 7200 spots with no symmetry.

Ixitic Symmetry - 5[3⁺,4,3⁺] order 1440. It has two sets of 120 points with kitettic symmetry. It has 80 rings with kihiggic symmetry which generate 480 kitriggic areas and 90 rings with square symmetry that generate 720 essic areas. White space generates 1440 no sym areas.

Ixoic Symmetry - 5[[3⁺,4,3⁺]] order 2880. It has 240 points with kitettic symmetry, two sets of 480 points with kitrippic symmetry, and 720 points with kibrick symmetry. It has 80 hexagonal rings which generate 960 kitriggic areas and 90 octagonal rings that generate 1440 essic areas. White space creates 2880 no sym areas.

# Prismattic 4-D Symmetries

These symmetries include prisms and pyramids of the main 3-D symmetries.

## Spherinder Symmetry Family

Sphindic Symmetry - [O,O,2] order 4*OO. This is the symmetry of the spherinder.

Prosphindic Symmetry - [[O,O]⁺,2] order 2*OO. This is the product of kispheric symmetry with dyadic.

Kisphindic Symmetry - [O,O,2]⁺ order 2*OO. This is the chiral version of sphindic symmetry.

Sphonic Symmetry - [O,O,1] order 2*OO. This is the symmetry of the sphone (spherical cone), it is a copy of the 3-D spheric symmetry.

Kisphonic Symmetry - [O,O,1]⁺ order OO. This is a 4-D copy of the kispheric symmetry.

## Tepic Symmetry Family

Tepic Symmetry - [3,3,2] order 48. This is the tetrahedron prism symmetry. It has two tettic points, two sets of four trippic points, and six brick symmetric points. It has four dyadic rings that generate eight trigonic regions, three rectic rings which create 12 rectic regions, and six horizontal dyadic rings that create 12 rectic regions. It has six rectpyric spheres that generate 24 dyadic regions and one tettic sphere that create 24 dyadic regions. White space creates 48 no sym areas.

Protepic Symmetry - [[3,3]⁺,2] order 24. This is the prism of chiral tet symmetry. It has two kitettic points, two sets of four protrippic points, and six espic points. It has four dyadic rings that create eight kitriggic regions and three rectic rings that create 12 essic regions. White space creates 24 no sym areas.

Kitepic Symmetry - [3,3,2]⁺ order 24. This is chiral tepe symmetry. It has two kitettic points, two sets of four kitrippic points, and six kibrick points. It has four dyadic rings that create eight kitriggic regions, three rectic rings that create 12 essic regions, and six dyadic rings that create 12 essic points. There's one ghost sphere with tettic symmetry which is part of white space. White space creates 24 no sym areas.

Tetapic Symmetry - [3,3⩫2] order 48. This is the tet antiprism symmetry. It has two tettic spots and six dappic spots. It has four essic rings which generate 8 trigonic regions, three rectic rings which create 12 rectic areas, and three square symmetric rings that create 24 essic areas. It has one ghost sphere with tet-skewed cubic symmetry and six rectic spheres which generate 24 dyadic areas. White space creates 48 no sym areas.

Protetapic Symmetry - [[3,3]⩫2] order 24. It has two kitettic spots and six prodappic spots. It has four essic rings which generate 8 kitriggic spots and three rectic rings that generate 12 essic spots. It has one kitet-skewed kicubic ghost sphere. White space creates 24 no sym areas.

Kitetapic Symmetry - [3,3⩫2]⁺ order 24. It has two kitettic spots and six kibrick spots. It has four essic rings which generate 8 kitriggic spots, three rectic rings that generate 12 essic spots, and three rectic rings that generate 12 essic regions. It has one kitet-skewed pyritic ghost sphere. White space creates 24 no sym areas.

Tetpyric Symmetry - [3,3,1] order 24. This is the tet pyramid symmetry and is the 4-D version of tettic symmetry. It has two distinct tettic points. It has four no sym rings which generate four trigonic regions and three dyadic rings that create six rectic regions. It has six dyadic spheres that create 12 dyadic regions and white space creates 24 no sym areas.

Kitetpyric Symmetry - [3,3,1]⁺ order 12. This is the 4-D version of kitettic symmetry. It has two distinct kitettic points. It has four no sym rings which generate four kitriggic areas and three dyadic rings which create six essic areas. White space creates 12 no sym areas.

## Cupic Symmetry Family

Cupic Symmetry - [4,3,2] order 96. This is the cube prism symmetry. It has two points with cubic symmetry, six with squippic symmetry, 8 with trippic symmetry, and 12 points with brick symmetry. It has three rectic rings that generate 12 square symmetric areas, four rectic rings which generate 16 triggic regions, six rectic vertical rings which create 24 rectic regions, three square symmetric rings which create 24 rectic regions, and six rectic horizontal rings which generate 24 rectic regions. It has one sphere with cubic symmetry which generate 48 dyadic regions, three spheres with squippic symmetry which generate 48 dyadic regions, and six spheres with brick symmetry which generate 48 dyadic regions. White space creates 96 no sym areas.

Procupic Symmetry - [[4,3]⁺,2] order 48. This is the symmetry of the snic prism. It has two kicubic points, six prosquippic points, 8 protrippic points, and 12 espic points. It has three rectic rings that generate 12 kisquare regions, four rectic rings which generate 16 kitriggic regions, and six rectic rings which create 24 essic regions. It has one sphere with kicubic symmetry which creates 24 dyadic regions. White space creates 48 no sym areas.

Kicupic Symmetry - [4,3,2]⁺ order 48. This is the symmetry of the snic alterprism. It has two kicubic points, six kisquippic points, 8 kitrippic points, and 12 kibrick points. It has three rectic rings that generate 12 kisquare regions, four rectic rings which generate 16 kitriggic regions, and six rectic rings which create 24 essic regions. It has one ghost sphere with cubic symmetry. White space creates 48 no sym areas.

Pyripic Symmetry - [4,3⁺,2] order 48. It has two pyritic points, six brick symmetric points, and 8 protrippic points. It has three rectic rings which generate 12 rectic areas (vertical), four rectic rings which generate 16 kitriggic areas, and three rectic rings which generate 12 rectic areas (horizontal). It has one pyritic sphere which generates 24 dyadic areas and three brick symmetric spheres which generate 24 dyadic areas. White space creates 48 no sym areas.

Pyrapic Symmetry - [4,[3,2]⁺] order 48. It has two pyritic points, six dappic points, and 8 kitrippic points. It has three rectic rings which generate 12 rectic areas, four rectic rings which generate 16 kitriggic areas, and six rectic rings that create 24 essic regions. It has three dappic spheres which generate 24 dyadic areas and a ghost sphere with cubic symmetry. White space creates 48 no sym areas.

Cupyric Symmetry - [4,3,1] order 48. This is the symmetry of the cube pyramid and is a 4-D copy of cubic symmetry. It has two distinct cubic points. It has three dyadic rings which generate six square symmetric areas, four dyadic rings which create 8 triggic areas, and six dyadic rings which create 12 rectic areas. It has three spheres with squippyric symmetry which generate 24 dyadic regions and six spheres with rectpyric symmetry which generate 24 dyadic regions. White space creates 48 no sym areas.

Kicupyric Symmetry - [4,3,1]⁺ order 24. This is the 4-D copy of kicubic symmetry. It has two distinct kicubic points. It has three dyadic rings which generate six kisquare areas, four dyadic rings which create 8 kitriggic areas, and six dyadic rings which create 12 essic areas. White space creates 24 no sym areas.

Pyritopyric Symmetry - [4,3⁺,1] order 24. This is the 4-D copy of pyritic symmetry. It has two distinct pyritic points. It has three dyadic rings which create six rectic areas and four dyadic rings which create 8 kitriggic areas. It has three spheres with rectpyric symmetry which creates 12 dyadic regions. White space creates 24 no sym areas.

## Dopic Symmetry Family

Dopic Symmetry - [5,3,2] order 240. This is the symmetry of the dodecahedron prism. It has 2 doic points, 12 pippic points, 20 trippic points, and 30 brick points. It has 6 rectic rings that generate 24 pentagonal areas, 10 rectic rings that generate 40 triggic areas, and two sets of 15 rectic rings which generate 60 rectic areas (one goes through the two doic points, the other is horizontal). There are 15 spheres with brick symmetry which generate 120 dyadic regions and one equatorial sphere with doic symmetry that creates 120 dyadic regions. White space creates 240 no sym regions.

Prodopic Symmetry - [[5,3]⁺,2] order 120. This is the symmetry of the snid prism. It has 2 kidoic points which are lined up, 12 propippic points, 20 protrippic points, and 30 espic points. It has 6 rectic rings that generate 24 kipeggic areas, 10 rectic rings which create 40 kitriggic areas, and 15 rectic rings which create 60 essic areas. There is one sphere with kidoic symmetry which generates 60 dyadic regions. White space creates 120 no sym regions.

Kidopic Symmetry - [5,3,2]⁺ order 120. It has 2 kidoic points, 12 kipippic points, 20 kitrippic points, and 30 kibrick points. It has 6 rectic rings that generate 24 kipeggic regions, 10 rectic rings that create 40 kitriggic regions, and 15 rectic rings which creates 60 essic regions. There's one ghost sphere with doic symmetry which is part of white space. White space creates 120 no sym regions.

Dopyric Symmetry - [5,3,1] order 120. This is the symmetry of the dodecahedron pyramid and is a 4-D copy of doic symmetry. It has two sets of one doic points. It has six dyadic rings which creates 12 pentagonal areas, ten dyadic rings that create 20 triggic areas, and 15 dyadic rings that create 30 rectic areas. There are 15 spheres with rectic symmetry which creates 60 dyadic regions. White space creates 120 no sym areas.

Kidopyric Symmetry - [5,3,1]⁺ order 60. This is the symmetry of the snid pyramid and is a 4-D copy of kidoic symmetry. It has two sets of one kidoic points. It has six dyadic rings which creates 12 kipeggic areas, ten dyadic rings that create 29 kitriggic areas, and 15 dyadic rings that creates 30 essic areas. White space creates 60 no sym areas.

# Polytwister and Swirl Symmetries

## Polytwister Symmetries

Tetteric Symmetry - [3,3:O] order 48*O. This is the symmetry of the tetratwister and is the continuous form of the stettic and astettic symmetries.

Kitetteric Symmetry - [3,3:O]⁺ order 24*O. This is the symmetry of the snit twister and is the continuous form of the skitettic and askitettic symmetries.

Cubiteric Symmetry - [4,3:O] order 96*O. This is the symmetry of the cube twister and is the continuous form of the scubic and ascubic symmetries.

Kicubiteric Symmetry - [4,3:O]⁺ order 48*O. This is the symmetry of the snic twister and is the continuuous form of the skicubic and askicubic symmetries.

Pyriteric Symmetry - [4,3⁺:O] order 48*O. This is the continuous form of the spyritic symmetries.

Doteric Symmetry - [5,3:O] order 240*O. This is the symmetry of the dodecatwister and is the continuous form of the swirldoic symmetries.

Kidoteric Symmetry - [5,3:O]⁺ order 120*O. This is the symmetry of the snid twister and is the continuous form of the skidoic symmetries.

## Swirltettic Symmetries

These are the symmetries that Conway refered to as 1/2[O x D2n]. As n approaches infinity, these symmetries approach tetteric symmetry. I coded these using the quarternion sets ico1 x cn1 and ico2 x cn2. When N is even, then there are six swirl rings with 2N symmetry which contains two sets of 12N points with kibrick symmetry and the rings generate 24N essic areas. When N is odd, the six swirl rings are ghosted and form girdles in white space instead and the points vanish, these girdles have skewed 4N symmetry. When N is divisible by 3, then there are four sets of 8N points with kitrippic symmetry and two sets of four swirl rings with 2N symmetry, these form 16N kitrigic areas for each ring set (4N spots per ring). When N is not divisible by 3, then the points here vanish and the rings are ghosted into white space, but do form girdles with 6N/3 gyrogon symmetry. White space creates 48N no sym areas. There are also two sets of 6N cross rings with rectic symmetry when N is even and one set of 12N cross rings with essic symmetry when odd.

1-Stettic Symmetry - [3,3:1] order 48. White space generates 48 no sym areas. There are two sets of four ghost ring girdles with 6/3-gyrogonal symmetry and a set of six ghost ring girdles with skewsquare symmetry. There are 12 cross rings with essic symmetry generating 24 essic locations.

2-Stettic Symmetry - [3,3:2] order 96. It has two sets of 24 points with kibrick symmetry. It has two sets of four girdles (ghost rings) with 12/3-gyrogonal symmetry. It has six swirling rings with square symmetry generating 48 essic spots which goes through the two types of points. It also has two sets of 12 cross rings with rectangle symmetry which generate 48 essic spots. Both types of cross rings goes through both types of points generating a triaxial cross where they meet the swirling rings. White space generates 96 no sym spots.

3-Stettic Symmetry - [3,3:3] order 144. It has four sets of 24 points which have kitrippic symmetry. It has 6 girdles (ghost ring which is part of white space) with skew-dodecagonic symmetry. It has two sets of four rings that swirl about each other with hexagonal symmetry which generate 48 kitrigonal regions (12 per ring). It has 36 cross rings with essic symmetry which generate 72 essic regions. Each of the swirl rings have three cross rings meeting in triplicate at two opposite points, there are six triples of rings (18 in all) that meet at each swirl ring. White space generates 144 no sym spots.

4-Stettic Symmetry - [3,3:4] order 192. It has two sets of 48 points which have kibrick symmetry. There are two sets of four girdles which doesn't form rings, these have 24/3-gyrogon symmetry. It has six swirling rings with octagonal symmetry which generate 96 spots with essic symmetry, these go through both types of points. It also has two sets of 24 crossing rings with rectic symmetry which generate 96 essic locations each one goes through one of the point types each. The white space generates 192 no symmetric spots.

5-Stettic Symmetry - [3,3:5] order 240. It has two sets of four swirling girdles with 30/3-gyrogonal symmetry. There are also six swirling girdles with skew icosagonal symmetry. These girdles doesn't form rings they are part of the 'white space' where no symmetry areas reside. There are however 60 cross rings with essic symmetry generating 120 spots all together with essic symmetric. All other spots (the 'white space') form 240 areas with no symmetry.

6-Stettic Symmetry - [3,3:6] order 288. It has four sets of 48 points with kitrippic symmetry. Also two sets of 72 points with kibrick symmetry. There are two sets of four swirling rings which have dodecagonal symmetry and generate 96 kitrigic areas each. There are also six swirling rings with dodecagonal symmetry which generate 144 essic spots. There's also two sets of 36 crossing rings which have rectic symmetry generating 144 spots each with essic symmetry. White space form 288 areas with no symmetry.

7-Stettic Symmetry - [3,3:7] order 336. There are 84 cross rings with essic symmetry generating 168 essic areas. White space generates 336 no sym areas. There are two sets of four ghost girdles with 42/3-gyrogonal symmetry and a set of six ghost girdles with 28-skewgon symmetry.

8-Stettic Symmetry - [3,3:8] order 384. It has two sets of 96 points which have kibrick symmetry. There are two sets of four girdles which doesn't form rings, these have 48/3-gyrogon symmetry. It has six swirling rings with 16-gonal symmetry which generate 192 spots with essic symmetry, these goes through both types of points. It also has two sets of 48 crossing rings with rectic symmetry which generate 192 essic locations, each one goes through one of the point types each. The white space generates 384 no sym spots.

9-Stettic Symmetry - [3,3:9] order 432. It has four sets of 72 points which have kitrippic symmetry. It has 6 girdles (ghost ring which is part of white space) with skew-36-gonic symmetry. It has two sets of four rings that swirl about each other with 18-gonal symmetry which generate 144 kitrigonal regions (36 per ring). It has 108 cross rings with essic symmetry which generate 216 essic regions. Each of the swirl rings have three cross rings meeting in triplicate at two opposite points, there are 18 triples of rings (54 in all) that meet at each swirl ring. White space generates 432 no sym spots.

10-Stettic Symmetry - [3,3:10] order 480. It has two sets of 120 points which have kibrick symmetry. There are two sets of four girdles which doesn't form rings, these have 60/3-gyrogon symmetry. It has six swirling rings with 20-gonal symmetry which generate 240 spots with essic symmetry, these goes through both types of points. It also has two sets of 60 crossing rings with rectic symmetry which generate 240 essic locations, each one goes through one of the point types each. The white space generates 480 no sym spots.

12-Stettic Symmetry - [3,3:12] order 576. It has four sets of 96 points with kitrippic symmetry. Also two sets of 144 points with kibrick symmetry. There are two sets of four swirling rings which have 24-gonal symmetry and generate 192 kitrigic areas each. There are also six swirling rings with 24-gonal symmetry which generate 288 essic spots. There's also two sets of 72 crossing rings which have rectic symmetry generating 288 spots each with essic symmetry. White space form 576 areas with no symmetry.

15-Stettic Symmetry - [3,3:15] order 720. It has four sets of 120 points which have kitrippic symmetry. It has 6 girdles (ghost ring which is part of white space) with skew-60-gonic symmetry. It has two sets of four rings that swirl about each other with 30-gonal symmetry which generate 240 kitrigonal regions (60 per ring). It has 180 cross rings with essic symmetry which generate 360 essic regions. Each of the swirl rings have three cross rings meeting in triplicate at two opposite points, there are 30 triples of rings (90 in all) that meet at each swirl ring. White space generates 720 no sym spots.

18-Stettic Symmetry - [3,3:18] order 864. It has four sets of 144 points with kitrippic symmetry. Also two sets of 216 points with kibrick symmetry. There are two sets of four swirling rings which have 36-gonal symmetry and generate 288 kitrigic areas each. There are also six swirling rings with 36-gonal symmetry which generate 432 essic spots. There's also two sets of 108 crossing rings which have rectic symmetry generating 432 spots each with essic symmetry. White space form 864 areas with no symmetry.

## Swirlkitettic Symmetries

These are the symmetries that Conway refered to as [T x C2n]. As n approaches infinity, these symmetries approach kitetteric symmetry. I coded these using the quarternion set ico1 x cn1. When N is even, then there are six swirl rings with ki-2N symmetry which generate 12N essic areas. When N is odd, the six swirl rings are ghosted and form girdles in white space instead which have skewed ki4N symmetry. When N is divisible by 3, then there are two sets of four swirl rings with ki-2N symmetry, these form 8N kitrigic areas for each ring set (2N spots per ring). When N is not divisible by 3, then the rings are ghosted into white space, but do form girdles with ki-6N/3 gyrogon symmetry. White space creates 24N no sym areas. There are no cross rings.

1-Skitettic Symmetry - [3,3:1]⁺ order 24. All spots generate 24 areas with no symmetry. There are two sets of four ghost girdles with ki-6/3 gyrogon symmetry and a set of six ghost girdles with skewed kisquare symmetry.

2-Skitettic Symmetry - [3,3:2]⁺ order 48. There are six swirl rings with kisquare symmetry which generate 24 essic areas. There are two sets of four ghost girdles with ki-12/3 gyrogon symmetry. White space generates 48 no sym areas.

3-Skitettic Symmetry - [3,3:3]⁺ order 72. There are two sets of four rings which has kihigic symmetry and a set of six ghost girdles with skewed ki12-gon symmetry. White space generates 72 no sym areas.

4-Skitettic Symmetry - [3,3:4]⁺ order 96. There are six rings with kioctagon symmetry which generates 48 essic areas and two sets of four ghost girdles with ki-24/3 gyrogon symmetry. White space generates 96 no sym areas.

5-Skitettic Symmetry - [3,3:5]⁺ order 120. All spots generate 120 areas with no symmetry. Girdles form, but with no distinction of what forms at them. There are two sets of four girdles with ki 30/3-gyrogonic symmetry and a set of six girdles with skewed 20-kigonic symmetry swirling about each other.

6-Skitettic Symmetry - [3,3:6]⁺ order 144. There are two sets of four swirling rings with kidodecagon symmetry which generate 48 essic spots each, and six swirl rings with kidodecagonal symmetry which generate 72 essic spots. Any point on the rings will generate 12 spots for each similar ring. All other spots generate 144 spots with no symmetry.

7-Skitettic Symmetry - [3,3:7]⁺ order 168. Everywhere is white space which generates 168 no sym areas. There are ghost girdles, six of one type with skewed ki-28-gon symmetry and two sets of four with ki-42/3-gyrogon symmetry.

8-Skitettic Symmetry - [3,3:8]⁺ order 192. There are six rings with ki-16-gon symmetry which generates 96 essic areas and two sets of four ghost girdles with ki-48/3 gyrogon symmetry. White space generates 192 no sym areas.

9-Skitettic Symmetry - [3,3:9]⁺ order 216. There are two sets of four rings which has ki-18-gonic symmetry and a set of six ghost girdles with skewed ki36-gon symmetry. White space generates 216 no sym areas.

10-Skitettic Symmetry - [3,3:10]⁺ order 240. There are six rings with ki-20-gon symmetry which generates 120 essic areas and two sets of four ghost girdles with ki-60/3 gyrogon symmetry. White space generates 240 no sym areas.

12-Skitettic Symmetry - [3,3:12]⁺ order 288. There are two sets of four swirling rings with ki24-gon symmetry which generate 96 essic spots each, and six swirl rings with ki24-gonal symmetry which generate 144 essic spots. Any point on the rings will generate 24 spots for each similar ring. All other spots generate 288 spots with no symmetry.

15-Skitettic Symmetry - [3,3:15]⁺ order 360. There are two sets of four rings which has ki30-gonic symmetry and a set of six ghost girdles with skewed ki60-gon symmetry. White space generates 360 no sym areas.

18-Skitettic Symmetry - [3,3:18]⁺ order 432. There are two sets of four swirling rings with ki36-gon symmetry which generate 144 essic spots each, and six swirl rings with ki36-gonal symmetry which generate 216 essic spots. Any point on the rings will generate 36 spots for each similar ring. All other spots generate 432 spots with no symmetry.

## Altswirltettic Symmetries

These are the symmetries that Conway refered to as 1/6[O x D6n]. As n approaches infinity, these symmetries approach tetteric symmetry. I coded these using the quarternion sets hex4 x cn1, hex5 x cn1+1/3, hex6 x cn1+2/3, hex1 x cn2, hex2 x cn2+1/3, and hex3 x cn2+2/3. When N is even, then there are six swirl rings with 2N symmetry which contains two sets of 12N points with kibrick symmetry and the rings generate 24N essic areas. When N is odd, the six swirl rings are ghosted and form girdles in white space instead and the points vanish, these girdles have skewed 4N symmetry. When N is not divisible by 3, then there are two sets of 8N points with kitrippic symmetry and two sets of four swirl rings with 2N symmetry, these form 16N kitrigic areas (4N spots per ring), there's also a set of four girdles with 6N/3 gyro symmetry which are ghosted into white space. When N is divisible by 3, then the points here vanish and the rings are ghosted into white space, but do form girdles with 6N/3 gyrogon symmetry. White space creates 48N no sym areas. There are also cross rings which generate 24N essic areas for each cross ring set. The cross rings have rectic symmetry when N is even and essic symmetry when N is odd. When N is odd there's one set of 12N cross rings. When N is even then there are two sets of 6N cross rings.

1-Astettic Symmetry - [3;3:1] order 48. There are two sets of eight points with kitrippic symmetry. Four swirl rings going through the points have rectic symmetry generating 16 kitrig areas. There are also four ghost girdles with 6/3 gyrogon symmetry and six ghost girdles with skew square symmetry. There are twelve cross rings with essic symmetry generating 24 essic spots. White space generates 48 areas with no sym.

2-Astettic Symmetry - [3;3:2] order 96. There are two sets of 16 points with kitrippic symmetry and two sets of 24 points with kibrick symmetry. There are four swirling rings going through the kitrippic points with square symmetry generating 32 kitrig symmetric spots. There are six swirling rings going through the kibrick points with square symmetry generating 48 essic areas. Four ghost girdles also exist with 12/3 gyrogon symmetry. There are also two sets of twelve rectic cross rings that intersect each other, they go through both sets of kibrick points and one set of the kitrippic points each. These generate 48 essic areas each. White space generates 96 no sym spots.

3-Astettic Symmetry - [3;3:3] order 144. There are two sets of four ghost girdles with 18/3 gyrogon symmetry and one set of six ghost girdles with skew dodecagon symmetry which are part of white space. There are 36 cross rings with essic symmetry which generate 72 essic areas. White space creates 144 no sym spots.

4-Astettic Symmetry - [3;3:4] order 192. There are two sets of 32 points with kitrippic symmetry and two sets of 48 points with kibrick symmetry. There are four swirling rings with octagon symmetry which generate 64 kitrig areas and six swirling rings with octagon symmetry which generate 96 essic areas. There are also four 24/3 gyrogonal ghost girdles. There are two sets of 24 cross rings with rectic symmetry which generate 96 essic areas, one type for each of the kibrick point sets and meets these points four times, two times in two orientations. White space creates 192 no sym areas.

5-Astettic Symmetry - [3;3:5] order 240. There are two sets of 40 points with kitrippic symmetry. There are four swirl rings with decagonal symmetry that generate 80 kitrigic areas. There are also four 30/3 gyrogonal and six skew icosagonal ghost girdles. There are 60 cross rings with essic symmetry generating 120 essic areas. White space generates 240 no sym areas.

6-Astettic Symmetry - [3;3:6] order 288. There are two sets of 72 points with kibrick symmetry. There are two sets of four ghost girdles with 36/3-gyrogonal symmetry and a set of six rings with dodecagonal symmetry which generate 144 essic areas. There are two sets of 36 cross rings with rectic symmetry that generate 144 essic areas each. White space generates 288 no sym areas.

7-Astettic Symmetry - [3;3:7] order 336. There are two sets of 56 kitrippic points. There are four swirl rings with 14-gon symmetry which generate 112 kitrigic areas which go through the kitrippic points. There are also four ghost girdles with 42/3 gyrogonal symmetry and six ghost girdles with skew 28-gon symmetry which are part of white space. It has 84 cross rings with essic symmetry which generate 168 essic areas. White space generates 336 no sym areas.

8-Astettic Symmetry - [3;3:8] order 384. There are two sets of 64 points with kitrippic symmetry and two sets of 96 points with kibrick symmetry. There are four swirling rings with 16-gon symmetry which generate 128 kitrig areas and six swirling rings with 16-gon symmetry which generate 192 essic areas. There are also four 48/3 gyrogonal ghost girdles. There are two sets of 48 cross rings with rectic symmetry which generate 192 essic areas, one type for each of the kibrick point sets and meets these points four times, two times in two orientations. White space creates 384 no sym areas.

9-Astettic Symmetry - [3;3:9] order 432. There are two sets of four ghost girdles with 54/3 gyrogon symmetry and one set of six ghost girdles with skew 36-gon symmetry which are part of white space. There are 108 cross rings with essic symmetry which generate 216 essic areas. White space creates 432 no sym spots.

10-Astettic Symmetry - [3;3:10] order 480. There are two sets of 80 points with kitrippic symmetry and two sets of 120 points with kibrick symmetry. There are four swirling rings with 20-gon symmetry which generate 160 kitrig areas and six swirling rings with 20-gon symmetry which generate 240 essic areas. There are also four 60/3 gyrogonal ghost girdles. There are two sets of 60 cross rings with rectic symmetry which generate 240 essic areas, one type for each of the kibrick point sets and meets these points four times, two times in two orientations. White space creates 480 no sym areas.

12-Astettic Symmetry - [3;3:12] order 576. There are two sets of 144 points with kibrick symmetry. There are two sets of four ghost girdles with 72/3-gyrogonal symmetry and a set of six rings with 24-gonal symmetry which generate 288 essic areas. There are two sets of 72 cross rings with rectic symmetry that generate 288 essic areas each. White space generates 576 no sym areas.

15-Astettic Symmetry - [3;3:15] order 720. There are two sets of four ghost girdles with 90/3 gyrogon symmetry and one set of six ghost girdles with skew 60-gon symmetry which are part of white space. There are 180 cross rings with essic symmetry which generate 360 essic areas. White space creates 720 no sym spots.

18-Astettic Symmetry - [3;3:18] order 864. There are two sets of 216 points with kibrick symmetry. There are two sets of four ghost girdles with 108/3-gyrogonal symmetry and a set of six rings with 36-gonal symmetry which generate 432 essic areas. There are two sets of 108 cross rings with rectic symmetry that generate 432 essic areas each. White space generates 864 no sym areas.

## Altswirlkitettic Symmetries

These are the symmetries that Conway refered to as 1/3[T x C3n]. As n approaches infinity, these symmetries approach kitetteric symmetry. I coded these using the quarternion sets hex4 x cn1, hex5 x cn+1/3, and hex6 x cn+2/3. When N is even, then there are six swirl rings with ki-2N symmetry which generate 12N essic areas. When N is odd, the six swirl rings are ghosted and form girdles in white space instead which have skew ki-4N gon symmetry. When N is not divisible by 3, then there are four swirl rings with ki-2N symmetry, these form 8N kitrigic areas (2N spots per ring). There are also four girdles which are ghosted into white space with ki 6N/3 gyrogon symmetry. When N is divisible by 3, then the rings are ghosted into white space, but do form girdles with ki 6N/3 gyrogon symmetry. White space creates 24N no sym areas. There are no cross rings.

1-Askitettic Symmetry - [3;3:1]⁺ order 24. There are four swirling rings with essic symmetry generating eight spots with kitrigic symmetry. There's also four ghost ring girdles with ki 6/3-gyrogon symmetry and six ghost ring girdles with skew kisquare sym which are part of white space. The white space generates 24 no sym areas.

2-Askitettic Symmetry - [3;3:2]⁺ order 48. There are four swirling rings with kisquare symmetry generating 16 spots with kitrig symmetry. There are also six rings with kisquare symmetry generating 24 spots with essic symmetry. There are also four ghost ring girdles with ki 12/3 gyrogon symmetry. White space generates 48 no sym areas.

3-Askitettic Symmetry - [3;3:3]⁺ order 72. Everywhere is white space generating 72 no symmetric areas. There are two sets of 4 girdles with ki 18/3 gyrogon symmetry and a set of 6 girdles with skew kidodecagon symmetry, all ghosted.

4-Askitettic Symmetry - [3;3:4]⁺ order 96. There are four swirling rings with kioctagonal symmetry generating 32 kitrig areas. There are four ghosted girdles also with ki 24/3 gyrogon sym. There are six swirling rings with kioctagon symmetry generating 48 essic areas. White space generates 96 no sym spots.

5-Askitettic Symmetry - [3;3:5]⁺ order 120. There are four swirling rings with kidecagonal symmetry generating 40 kitrig areas. There are also four ghost girdles with ki 30/3 gyrogon symmetry and six more ghost girdles with skew kiicosagon symmetry. White space generates 120 no sym areas.

6-Askitettic Symmetry - [3;3:6]⁺ order 144. There are six swirling rings with kidodecagonal symmetry generating 72 essic areas and two sets of four ghost girdles with ki 36/3 gyrogon symmetry, the ghost girdles are part of white space. White space generates 144 no sym areas.

7-Askitettic Symmetry - [3;3:7]⁺ order 168. There are four rings with ki14gon symmetry generating 56 kitrig areas. The four ghost girdles have ki 42/3 gyrogon sym and the six ghost girdles hav skew ki28gon symmetry. White space generates 168 no sym areas.

8-Askitettic Symmetry - [3;3:8]⁺ order 192. There are four swirling rings with ki16-gonal symmetry generating 64 kitrig areas. There are four ghosted girdles also with ki 48/3 gyrogon sym. There are six swirling rings with ki16-gon symmetry generating 96 essic areas. White space generates 192 no sym spots.

9-Askitettic Symmetry - [3;3:9]⁺ order 216. Everywhere is white space generating 216 no symmetric areas. There are two sets of 4 girdles with ki 54/3 gyrogon symmetry and a set of 6 girdles with skew ki36-gon symmetry, all ghosted.

10-Askitettic Symmetry - [3;3:10]⁺ order 240. There are four swirling rings with ki20-gonal symmetry generating 80 kitrig areas. There are four ghosted girdles also with ki 60/3 gyrogon sym. There are six swirling rings with ki20-gon symmetry generating 120 essic areas. White space generates 240 no sym spots.

12-Askitettic Symmetry - [3;3:12]⁺ order 288. There are six swirling rings with ki24-gonal symmetry generating 144 essic areas and two sets of four ghost girdles with ki 72/3 gyrogon symmetry, the ghost girdles are part of white space. White space generates 288 no sym areas.

15-Askitettic Symmetry - [3;3:15]⁺ order 360. Everywhere is white space generating 360 no symmetric areas. There are two sets of 4 girdles with ki 90/3 gyrogon symmetry and a set of 6 girdles with skew ki60-gon symmetry, all ghosted.

18-Askitettic Symmetry - [3;3:18]⁺ order 432. There are six swirling rings with ki36-gonal symmetry generating 216 essic areas and two sets of four ghost girdles with ki 108/3 gyrogon symmetry, the ghost girdles are part of white space. White space generates 432 no sym areas.

## Swirlcubic Symmetries

1-Scubic Symmetry - [4,3:1] order 96.

2-Scubic Symmetry - [4,3:2] order 192.

3-Scubic Symmetry - [4,3:3] order 288.

4-Scubic Symmetry - [4,3:4] order 384.

5-Scubic Symmetry - [4,3:5] order 480.

6-Scubic Symmetry - [4,3:6] order 576.

7-Scubic Symmetry - [4,3:7] order 672.

8-Scubic Symmetry - [4,3:8] order 768.

9-Scubic Symmetry - [4,3:9] order 864.

10-Scubic Symmetry - [4,3:10] order 960.

12-Scubic Symmetry - [4,3:12] order 1152.

15-Scubic Symmetry - [4,3:15] order 1440.

16-Scubic Symmetry - [4,3:16] order 1536.

18-Scubic Symmetry - [4,3:18] order 1728.

20-Scubic Symmetry - [4,3:20] order 1920.

24-Scubic Symmetry - [4,3:24] order 2304.

## Swirlkicubic Symmetries

1-Skicubic Symmetry - [4,3:1]⁺ order 48.

2-Skicubic Symmetry - [4,3:2]⁺ order 96.

3-Skicubic Symmetry - [4,3:3]⁺ order 144.

4-Skicubic Symmetry - [4,3:4]⁺ order 192.

5-Skicubic Symmetry - [4,3:5]⁺ order 240.

6-Skicubic Symmetry - [4,3:6]⁺ order 288.

7-Skicubic Symmetry - [4,3:7]⁺ order 336.

8-Skicubic Symmetry - [4,3:8]⁺ order 384.

9-Skicubic Symmetry - [4,3:9]⁺ order 432.

10-Skicubic Symmetry - [4,3:10]⁺ order 480.

12-Skicubic Symmetry - [4,3:12]⁺ order 576.

15-Skicubic Symmetry - [4,3:15]⁺ order 720.

16-Skicubic Symmetry - [4,3:16]⁺ order 768.

18-Skicubic Symmetry - [4,3:18]⁺ order 864.

20-Skicubic Symmetry - [4,3:20]⁺ order 960.

24-Skicubic Symmetry - [4,3:24]⁺ order 1152.

## Altswirlcubic Symmetries

1-Ascubic Symmetry - [4;3:1] order 96.

2-Ascubic Symmetry - [4;3:2] order 192.

3-Ascubic Symmetry - [4;3:3] order 288.

4-Ascubic Symmetry - [4;3:4] order 384.

5-Ascubic Symmetry - [4;3:5] order 480.

6-Ascubic Symmetry - [4;3:6] order 576.

7-Ascubic Symmetry - [4;3:7] order 672.

8-Ascubic Symmetry - [4;3:8] order 768.

9-Ascubic Symmetry - [4;3:9] order 864.

10-Ascubic Symmetry - [4;3:10] order 960.

12-Ascubic Symmetry - [4;3:12] order 1152.

15-Ascubic Symmetry - [4;3:15] order 1440.

16-Ascubic Symmetry - [4;3:16] order 1536.

18-Ascubic Symmetry - [4;3:18] order 1728.

20-Ascubic Symmetry - [4;3:20] order 1920.

24-Ascubic Symmetry - [4;3:24] order 2304.

## Altswirlkicubic Symmetries

1-Askicubic Symmetry - [4;3:1]⁺ order 48.

2-Askicubic Symmetry - [4;3:2]⁺ order 96.

3-Askicubic Symmetry - [4;3:3]⁺ order 144.

4-Askicubic Symmetry - [4;3:4]⁺ order 192.

5-Askicubic Symmetry - [4;3:5]⁺ order 240.

6-Askicubic Symmetry - [4;3:6]⁺ order 288.

7-Askicubic Symmetry - [4;3:7]⁺ order 336.

8-Askicubic Symmetry - [4;3:8]⁺ order 384.

9-Askicubic Symmetry - [4;3:9]⁺ order 432.

10-Askicubic Symmetry - [4;3:10]⁺ order 480.

12-Askicubic Symmetry - [4;3:12]⁺ order 576.

15-Askicubic Symmetry - [4;3:15]⁺ order 720.

16-Askicubic Symmetry - [4;3:16]⁺ order 768.

18-Askicubic Symmetry - [4;3:18]⁺ order 864.

20-Askicubic Symmetry - [4;3:20]⁺ order 960.

24-Askicubic Symmetry - [4;3:24]⁺ order 1152.

## Swirlpyritic Symmetries

1-Spyritic Symmetry - [4,3⁺:1] order 48.

2-Spyritic Symmetry - [4,3⁺:2] order 96.

3-Spyritic Symmetry - [4,3⁺:3] order 144.

4-Spyritic Symmetry - [4,3⁺:4] order 192.

5-Spyritic Symmetry - [4,3⁺:5] order 240.

6-Spyritic Symmetry - [4,3⁺:6] order 288.

7-Spyritic Symmetry - [4,3⁺:7] order 336.

8-Spyritic Symmetry - [4,3⁺:8] order 384.

9-Spyritic Symmetry - [4,3⁺:9] order 432.

10-Spyritic Symmetry - [4,3⁺:10] order 480.

12-Spyritic Symmetry - [4,3⁺:12] order 576.

15-Spyritic Symmetry - [4,3⁺:15] order 720.

16-Spyritic Symmetry - [4,3⁺:16] order 768.

18-Spyritic Symmetry - [4,3⁺:18] order 864.

20-Spyritic Symmetry - [4,3⁺:20] order 960.

24-Spyritic Symmetry - [4,3⁺:24] order 1152.

## Swirldoic Symmetries

These are the symmetries that Conway refered to as [I x D2n]. As n approaches infinity, these symmetries approach doteric symmetry. I coded these using the quarternion sets ex x cn1 and ex x cn2. When N is even, then there are 30 swirl rings with 2N symmetry which contains two sets of 60N points with kibrick symmetry and the rings generate 120N essic areas. When N is odd, the 30 swirl rings are ghosted and form girdles in white space instead and the points vanish, these girdles have skewed 4N symmetry. When N is divisible by 5, then there are two sets of 24N points with kipippic symmetry and a set of twelve swirl rings with 2N symmetry, these form 48N kipegic areas (4N spots per ring). When N is not divisible by 5, then the points here vanish and the rings are ghosted into white space, but do form girdles with 10mN/5 gyrogon symmetry where m is the modulus of N mod 5, 1 and 4 are considered equivalent as well as 2 and 3, but they gyrate in different directions. When N is divisible by 3, then there are two sets of 40N points with kitrippic symmetry and a set of 20 swirl rings with 2N symmetry, these form 4N kitrigic areas for each ring (80N all together). When N is not divisible by 3, then the points vanish and the rings are ghosted into white space, but do form girdles with 6N/3 gyrogon symmetry. White space creates 240N no sym areas. There are also two sets of 15N cross rings with square symmetry when N is even and one set of 30N cross rings with kisquare symmetry when odd.

1-Swirldoic Symmetry - [5,3:1] order 240. There are 12 ghost girdles with 10/5 gyrogonal symmetry, 20 ghost girdles with 6/3 gyrogonic symmetry, and 30 ghost girdles with skew tetragonal symmetry all in white space which generate 240 no sym areas. There are also 30 cross rings with kisquare symmetry which create 120 essic areas.

2-Swirldoic Symmetry - [5,3:2] order 480. This is the symmetry of spiddit. There are two sets of 120 points with kibrick symmetry which are in 30 rings where four spots from each set alternate. The rings have square symmetry and create 240 essic areas. There are 12 ghost girdles with 20/(5/2) gyrogonal symmetry and 20 ghost girdles with 12/3 gyrogonal symmetry. The ghost girdles are in white space. There are two sets of 30 cross rings with square symmetry which creates 240 essic regions each. White space creates 480 no sym areas.

3-Swirldoic Symmetry - [5,3:3] order 720. This is the symmetry of sispatit. There are two sets of 120 points with kitrippic symmetry which are in 20 rings where 6 spots of each set alternate. The rings have hexagon symmetry and create 240 kitrigic regions. There are 12 ghost girdles with 30/(5/2) gyrogonal symmetry and 30 ghost girdles with skew 12-gon symmetry, both in white space. There are 90 cross rings with kisquare symmetry which generate 360 essic regions. White space creates 720 no-sym regions.

4-Swirldoic Symmetry - [5,3:4] order 960. There are two sets of 240 points with kibrick symmetry which are in 30 rings where eight spots from each set alternate. The rings have octagon symmetry and create 480 essic areas. There are 12 ghost girdles with 40/5 gyrogonal symmetry and 20 ghost girdles with 24/3 gyrogonal symmetry. The ghost girdles are in white space. There are two sets of 60 cross rings with square symmetry which creates 480 essic regions each. White space creates 960 no sym areas.

5-Swirldoic Symmetry - [5,3:5] order 1200. Also called sispic symmetry, for it is the symmetry of sisp. It has two sets of 120 kipippic spots which are in 12 rings, where ten spots in both sets alternate. The rings have decagon symmetry and generate 20 kipegic regions each, totalling 240 spots. There are 20 ghost girdles with 30/3-gyrogonic symmetry and 30 ghost girdles with skew 20-gonal symmetry, these are in white space. There are 150 cross rings with kisquare symmetry which form 600 essic areas, five cross rings meet at each kipippic spot. White space creates 1200 no sym areas.

6-Swirldoic Symmetry - [5,3:6] order 1440. It has two sets of 240 kitrippic spots which are in 20 rings, where 12 spots in each alternate. The rings have dodecagonal symmetry and generate 24 kitrigic regions each, totaling 480 areas. There are two sets of 360 spots with kibrick symmetry which show up in thirty rings, where 12 spots in each set alternate. These rings also have 12-gon symmetry and generate 24 essic regions each, totaling 720 areas. There are also 12 ghost girdles with 60/5-gyrogonal symmetry which are in white space. There are two sets of 90 cross rings with square symmetry which form 720 essic areas, three meet at each kitrippic spot. White space creates 1440 no sym areas.

7-Swirldoic Symmetry - [5,3:7] order 1680. There are 12 ghost girdles with 70/(5/2)-gyrogonal symmetry, 20 ghost girdles with 42/3-gyrogonal symmetry, and 30 ghost girdles with skew 28-gonal symmetry all in white space which generate 1680 no sym areas. There are 210 cross rings with kisquare symmetry which generate 840 essic areas.

8-Swirldoic Symmetry - [5,3:8] order 1920. There are two sets of 480 points with kibrick symmetry which are in 30 rings where 16 spots from each set alternate. The rings have 16-gon symmetry and create 960 essic areas. There are 12 ghost girdles with 80/(5/2) gyrogonal symmetry and 20 ghost girdles with 48/3 gyrogonal symmetry. The ghost girdles are in white space. There are two sets of 120 cross rings with square symmetry which creates 960 essic regions each. White space creates 1920 no sym areas.

9-Swirldoic Symmetry - [5,3:9] order 2160. There are two sets of 360 points with kitrippic symmetry which are in 20 rings where 18 spots of each set alternate. The rings have 18-gon symmetry and create 720 kitrigic regions. There are 12 ghost girdles with 90/5 gyrogonal symmetry and 30 ghost girdles with skew 36-gon symmetry, both in white space. There are 270 cross rings with kisquare symmetry which generate 1080 essic regions. White space creates 2160 no-sym regions.

10-Swirldoic Symmetry - [5,3:10] order 2400. It has two sets of 240 kipippic spots which are in 12 rings, where 20 spots in both sets alternate. The rings have 20-gon symmetry and generate 40 kipegic regions each, totalling 480 spots. There are two sets of 600 points with kibrick symmetry which are in 30 rings where 20 spots from each set alternate. The rings have 20-gon symmetry and create 1200 essic areas. There are 20 ghost girdles with 60/3-gyrogonic symmetry which are in white space. There are two sets of 150 cross rings with square symmetry which form 1200 essic areas. White space creates 2400 no sym areas.

12-Swirldoic Symmetry - [5,3:12] order 2880. It has two sets of 480 kitrippic spots which are in 20 rings, where 24 spots in each alternate. The rings have 24-gonal symmetry and generate 48 kitrigic regions each, totaling 960 areas. There are two sets of 720 spots with kibrick symmetry which show up in thirty rings, where 24 spots in each set alternate. These rings also have 24-gon symmetry and generate 48 essic regions each, totaling 1440 areas. There are also 12 ghost girdles with 120/(5/2)-gyrogonal symmetry which are in white space. There are two sets of 180 cross rings with square symmetry which form 1440 essic areas, three meet at each kitrippic spot. White space creates 2880 no sym areas.

15-Swirldoic Symmetry - [5,3:15] order 3600. It has two sets of 360 kipippic spots which are in 12 rings, where 30 spots in both sets alternate. The rings have 30-gon symmetry and generate 60 kipegic regions each, totalling 720 spots. There are two sets of 600 points with kitrippic symmetry which are in 20 rings where 30 spots of each set alternate. The rings have 30-gon symmetry and create 1200 kitrigic regions. There are 30 ghost girdles with skew 60-gonal symmetry, these are in white space. There are 450 cross rings with kisquare symmetry which form 1800 essic areas, five cross rings meet at each kipippic spot. White space creates 3600 no sym areas.

18-Swirldoic Symmetry - [5,3:18] order 4320. It has two sets of 720 kitrippic spots which are in 20 rings, where 36 spots in each alternate. The rings have 36-gonal symmetry and generate 72 kitrigic regions each, totaling 1440 areas. There are two sets of 1080 spots with kibrick symmetry which show up in thirty rings, where 36 spots in each set alternate. These rings also have 36-gon symmetry and generate 72 essic regions each, totaling 2160 areas. There are also 12 ghost girdles with 180/(5/2)-gyrogonal symmetry which are in white space. There are two sets of 270 cross rings with square symmetry which form 2160 essic areas each, three meet at each kitrippic spot. White space creates 4320 no sym areas.

20-Swirldoic Symmetry - [5,3:20] order 4800. It has two sets of 480 kipippic spots which are in 12 rings, where 40 spots in both sets alternate. The rings have 40-gon symmetry and generate 80 kipegic regions each, totalling 960 spots. There are two sets of 1200 points with kibrick symmetry which are in 30 rings where 40 spots from each set alternate. The rings have 40-gon symmetry and create 2400 essic areas. There are 20 ghost girdles with 120/3-gyrogonic symmetry which are in white space. There are two sets of 300 cross rings with square symmetry which form 2400 essic areas each. White space creates 4800 no sym areas.

25-Swirldoic Symmetry - [5,3:25] order 6000. It has two sets of 600 kipippic spots which are in 12 rings, where 50 spots in both sets alternate. The rings have 50-gon symmetry and generate 100 kipegic regions each, totalling 1200 spots. There are 20 ghost girdles with 150/3-gyrogonic symmetry and 30 ghost girdles with skew 100-gonal symmetry, these are in white space. There are 750 cross rings with kisquare symmetry which form 3000 essic areas, five cross rings meet at each kipippic spot. White space creates 6000 no sym areas.

30-Swirldoic Symmetry - [5,3:30] order 7200. It has two sets of 720 kipippic spots which are in 12 rings, where 60 spots in both sets alternate. The rings have 60-gon symmetry and generate 120 kipegic regions each, totalling 1440 spots. There are two sets of 1200 points with kitrippic symmetry which are in 20 rings where 60 spots of each set alternate. The rings have 60-gon symmetry and create 2400 kitrigic regions. There are two sets of 1800 points with kibrick symmetry which are in 30 rings where 60 spots from each set alternate. The rings have 60-gon symmetry and create 3600 essic areas. There are two sets of 450 cross rings with square symmetry which form 1800 essic areas each, five cross rings of one or the other set meet at each kipippic spot. White space creates 7200 no sym areas.

## Swirlkidoic Symmetries

These are the symmetries that Conway refered to as [I x Cn]. As n approaches infinity, these symmetries approach kidoteric symmetry. I coded these using the quarternion sets ex x cn1. When N is even, then there are 30 swirl rings with ki-2N-gon symmetry which generate 60N essic areas. When N is odd, the 30 swirl rings are ghosted and form girdles in white space instead, these girdles have skewed ki-4N-gon symmetry. When N is divisible by 5, then there is a set of twelve swirl rings with ki-2N-gon symmetry, these form 24N kipegic areas (2N spots per ring). When N is not divisible by 5, then the rings are ghosted into white space, but do form girdles with 10mN/5 kigyrogon symmetry where m is the modulus of N mod 5, 1 and 4 are considered equivalent as well as 2 and 3, but they gyrate in different directions. When N is divisible by 3, then there is a set of 20 swirl rings with ki-2N-gon symmetry, these form 2N kitrigic areas for each ring (40N all together). When N is not divisible by 3, then the rings are ghosted into white space, but do form girdles with 6N/3 kigyrogon symmetry. There are no point locations nor cross rings. White space creates 120N no sym areas.

1-Skidoic Symmetry - [5,3:1]⁺ order 120. There are 12 ghost girdles with 10/5-kigyrogonal symmetry, 20 ghost girdles with 6/3 kigyrogonal symmetry and 30 ghost girdles with skewed kisquare symmetry, all of which are in white space. White space generates 120 no sym areas. All areas are white space here.

2-Skidoic Symmetry - [5,3:2]⁺ order 240. There are 30 swirl rings with kisquare symmetry which create 120 essic spots. There are 12 ghost girdles with 20/(5/2)-kigyrogonal symmetry and 20 ghost girdles with 12/3 kigyrogonal symmetry, which are in white space. White space generates 240 no sym areas.

3-Skidoic Symmetry - [5,3:3]⁺ order 360. There are 20 swirl rings with kihexagon symmetry which create 120 kitrigic areas. There are 12 ghost girdles with 30/(5/2)-kigyrogonal symmetry and 30 ghost girdles with skewed ki-12-gonal symmetry, which are in white space. White space generates 360 no sym areas.

4-Skidoic Symmetry - [5,3:4]⁺ order 480. There are 30 swirl rings with kioctagonal symmetry which create 240 essic spots. There are 12 ghost girdles with 40/5-kigyrogonal symmetry and 20 ghost girdles with 24/3 kigyrogonal symmetry, which are in white space. White space generates 480 no sym areas.

5-Skidoic Symmetry - [5,3:5]⁺ order 600. Also called kisispic symmetry. There are 12 swirl rings with kidecagonal symmetry which generate 120 kipegic regions. There are 20 ghost girdles with 30/3 kigyrogonal symmetry and 30 ghost girdles with skewed ki-20-gonal symmetry, all of which are in white space. White space generates 600 no sym areas.

6-Skidoic Symmetry - [5,3:6]⁺ order 720. There are 20 swirl rings with ki-12-gon symmetry which create 240 kitrigic areas. There are 30 swirl rings with ki-12-gon symmetry which create 360 essic areas. There are 12 ghost girdles with 60/5-kigyrogonal symmetry which are in white space. White space generates 720 no sym areas.

7-Skidoic Symmetry - [5,3:7]⁺ order 840. There are 12 ghost girdles with 70/5-kigyrogonal symmetry, 20 ghost girdles with 42/3 kigyrogonal symmetry and 30 ghost girdles with skewed ki-28-gonal symmetry, all of which are in white space. White space generates 840 no sym areas. All areas are white space here.

8-Skidoic Symmetry - [5,3:8]⁺ order 960. There are 30 swirl rings with ki-16-gonal symmetry which create 480 essic spots. There are 12 ghost girdles with 80/(5/2)-kigyrogonal symmetry and 20 ghost girdles with 48/3 kigyrogonal symmetry, which are in white space. White space generates 960 no sym areas.

9-Skidoic Symmetry - [5,3:9]⁺ order 1080. There are 20 swirl rings with ki-18-gon symmetry which create 360 kitrigic areas. There are 12 ghost girdles with 90/5-kigyrogonal symmetry and 30 ghost girdles with skewed ki-36-gonal symmetry, which are in white space. White space generates 1080 no sym areas.

10-Skidoic Symmetry - [5,3:10]⁺ order 1200. There are 12 swirl rings with ki-20-gonal symmetry which generate 240 kipegic regions. There are 30 swirl rings with ki-20-gon symmetry which create 600 essic areas. There are 20 ghost girdles with 60/3 kigyrogonal symmetry which are in white space. White space generates 1200 no sym areas.

12-Skidoic Symmetry - [5,3:12]⁺ order 1440. There are 20 swirl rings with ki-24-gon symmetry which create 480 kitrigic areas. There are 30 swirl rings with ki-24-gon symmetry which create 720 essic areas. There are 12 ghost girdles with 120/(5/2) -kigyrogonal symmetry which are in white space. White space generates 1440 no sym areas.

15-Skidoic Symmetry - [5,3:15]⁺ order 1800. There are 12 swirl rings with ki-30-gonal symmetry which generate 360 kipegic regions. There are 20 swirl rings with ki-30-gonal symmetry which creates 600 kitrigic regions. There are 30 ghost girdles with skewed ki-60-gonal symmetry which are in white space. White space generates 1800 no sym areas.

18-Skidoic Symmetry - [5,3:18]⁺ order 2160. There are 20 swirl rings with ki-36-gon symmetry which create 720 kitrigic areas. There are 30 swirl rings with ki-36-gon symmetry which create 1080 essic areas. There are 12 ghost girdles with 180/(5/2) -kigyrogonal symmetry which are in white space. White space generates 2160 no sym areas.

20-Skidoic Symmetry - [5,3:20]⁺ order 2400. There are 12 swirl rings with ki-40-gonal symmetry which generate 480 kipegic regions. There are 30 swirl rings with ki-40-gon symmetry which create 1200 essic areas. There are 20 ghost girdles with 120/3 kigyrogonal symmetry which are in white space. White space generates 2400 no sym areas.

25-Skidoic Symmetry - [5,3:25]⁺ order 3000. There are 12 swirl rings with ki-50-gonal symmetry which generate 600 kipegic regions. There are 20 ghost girdles with 150/3 kigyrogonal symmetry and 30 ghost girdles with skewed ki-100-gonal symmetry, all of which are in white space. White space generates 3000 no sym areas.

30-Skidoic Symmetry - [5,3:30]⁺ order 3600. There are 12 swirl rings with ki-60-gonal symmetry which generate 720 kipegic regions. There are 20 swirl rings with ki-60-gonal symmetry which creates 1200 kitrigic regions. There are 30 swirl rings with ki-60-gonal symmetry which create 1800 essic areas. White space generates 3600 no sym areas.

# Duoprismic Symmetries

More to come later

# Dysteric and Swirlprismic Symmetries

## Dysteric Symmetries

N Dysteric Symmetry - [N,2:O] order 8N*O.

N Prodysteric Symmetry - [N⁺,2:O] order 4N*O.

N Kidysteric Symmetry - [N,2:O]⁺ order 4N*O.

N Apiteric Symmetry - [2N,2⁺:O] order 8N*O.

N Proapiteric Symmetry - [2N⁺,2⁺:O] order 4N*O.

N Pyristeric Symmetry - [N,1:O] order 4N*O.

N Kipyristeric Symmetry - [N,1:O]⁺ order 2N*O.

## Swirlprismic Symmetries

More to come later

# Gyrochoric Symmetries

More to come later

```Page created by Jonathan Bowers, 2021
e-mail = hedrondude at suddenlink dot net ```