# Symmetries Up To Three Dimensions

This page will list the symmetries of objects up to three 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. Also included will be a complete list of isogonal (vertex-transitive) and isotopal (facet-transitive aka dice) polyhedra as well as round versions.

## Zero Dimensional Symmetries

Pointic Symmetry - ⁺ order 1, this is the symmetry of a point. It basicly has no symmetry.

There is one isogonal / isotopal shape here, actually there is ONLY one shape here, BEHOLD . the amazing shape called 'point' - it has no vertices nor facets so it is both isogonal and isotopal by default.

## One Dimensional Symmetries

In one dimension, reflections become possible.

Dyadic Symmetry - [] order 2, this is the symmetry of a line segment. It is commonly called bilateral symmetry or reflective symmetry.

Monic Symmetry - []⁺ order 1, this is the symmetry of a ray. It basicly has no symmetry and is a copy of the 0-D point symmetry.

- The dyad is the only shape that is isogonal or isotopal, it has two vertices which are also its facets.

# Two Dimensional Symmetries

(Clint Eastwood voice) You got to ask yourself two questions: 'What number do you want (0 and up)?' and 'Do you want fries with that?' (assuming fries are reflections) - 'well do ya? PUNK' (exit Clint Eastwood voice). Rotations become possible in two dimensions.

Above are the symmetry graphs of the 2-D symmetries, all of the chiral symmetries have similar looking graphs. Below are examples of the polygonal symmetry groups.

Cyclic Symmetry - [O] order 2*O, this is the symmetry of the circle. Think of O as a continuum instead of zero. It is the limit of the [n] symmetries. Every point on its perimeter leads to a continuum of points, i.e. the entire perimeter in this symmetry, where they have dyadic symmetry at each point.

Kicyclic Symmetry - [O]⁺ order O, this is the rotational symmetries of the circle, it has no reflection symmetries (no fries). To picture an object with this symmetry, imagine a circle that if you feel of the rim clockwise, it feels smooth, but counter-clockwise feels rough. It is the limit of the [n]⁺ symmetries. Kicyclic is short for chirocyclic. Every point on it's perimeter can generate all other points on the perimeter and they have no symmetry but point in the same direction.

Dyadic Symmetry - [1] order 2. This is the same symmetry as the one in one dimension, but has been extended to two. Also called bilateral symmetry or reflective symmetry. It is the symmetry of the heart shape or the letter A. The red and yellow points generate one spot each with dyadic symmetry, while all other points on the perimeter generate two spots with no symmetry.

Monic Symmetry - [1]⁺ order 1. This is the same symmetry as the one in one dimension, but has been extended to two. Also called no symmetry. It is the symmetry of a totally irregular polygon, or even the letter F. Every point on the perimeter is distinct from each other and has no symmetry.

Rectic Symmetry - [2] order 4. Also called rectangular or digonal symmetry. It is the symmetry of a rectangle as well as the letter H. The two red points and two yellow points generate two spots each with dyadic symmetry, all other points on the perimenter lead to four spots with no symmetry.

Essic Symmetry - [2]⁺ order 2. Also called chiro digonal symmetry. It is the symmetry of the letter S as well as Z and N. All spots on the perimeter generate two spots with no symmetry.

Trigonic Symmetry - [3] order 6. Also called triangular or trigic symmetry. It is the symmetry of a triangle or a ditrigon. Red and yellow spots (3 each) generate 3 spots with dyad symmetry, all other spots on perimenter generate six areas with no symmetry.

Kitrigonic Symmetry - [3]⁺ order 3. Also called chiral triangular or kitric symmetry. This is the symmetry of that recycling symbol. All spots on perimeter generate three areas with no symmetry, but facing same direction (as in clockwise or counter clockwise).

Square Symmetry - [4] order 8. Could also be called squaric symmetry. It is the symmetry of a square. All red and yellow spots (4 each) generate four spots with dyadic symmetry, all other spots on perimeter generate 8 places with no symmetry.

Kisquare Symmetry - [4]⁺ order 4. Also called chiral square symmetry. This is the symmetry of those toy pinwheels if you remove the handle. It's also the symmetry of a very infamous symbol found in WWII. All spots on perimeter generate four spots with no symmetry.

Pentagonal Symmetry - [5] order 10. It is the symmetry of a pentagon or a five pointed star, can be called peggic symmetry for short. All red and yellow spots (5 each) form five areas with dyad symmetry, all other perimeter locations generate ten spots with no symmetry.

Kipentagonal Symmetry - [5]⁺ order 5. Also called chiral or rotational pentagonal symmetry. We could even call it kipic for short. This is the symmetry of a star, if you curve the points clockwise (or counterclockwise). All perimeter places generate five spots with no symmetry.

Hexagonal Symmetry - [6] order 12. It is the symmetry of a hexagon as well as snowflakes, can be shortened to higgic symmetry. The red and yellow spots (6 each) generate six spots each with dyad symmetry. Other spots form 12 places with no symmetry in an alternating method, six clockwise, six counter.

Kihexagonal Symmetry - [6]⁺ order 6. Better known as chiral hexagonal symmetry, short form called kihiggic. All perimeter places generate six areas with no symmetry pointing the same way around the circle.

Heptagonal Symmetry - [7] order 14. It is the symmetry of a heptagon, short form heggic.

Kiheptagonal Symmetry - [7]⁺ order 7. Short name - kiheggic.

Octagonal Symmetry - [8] order 16. It is the symmetry of an octagon or a stop sign, short name - occic.

Kioctagonal Symmetry - [8]⁺ order 8. Short name kioccic.

Enneagonal Symmetry - [9] order 18. It is the symmetry of an enneagon, short name ennic symmetry.

Kienneagonal Symmetry - [9]⁺ order 9. Short name - kiennic.

Decagonal Symmetry - [10] order 20. It is the symmetry of a decagon, short name deggic.

Kidecagonal Symmetry - [10]⁺ order 10. Short name kideggic.

N Gonal Symmetry - [N] order 2N. It is the symmetry of a regular polygon with N sides. All red and yellow spots (N each) form N spots with dyad symmetry, all other places on perimeter generate 2N spots with no symmetry that alternate between clockwise and counter pointing.

N Kigonal Symmetry - [N]⁺ order N. This is the chiral N-gonal symmetry. All spots on perimeter generate N areas with no symmetry all pointing either clockwise or counter, all pointing the same way.

All convex regular polygons are both isogonal and isotopal. They have three sides or more. The circle is also both isogonal and isotopal. These have N-gon symmetry for the N sided cases and cyclic symmetry for the circle. These are also isogonal and isotopal under N-kigonal / kicyclic symmetries respectively. The even polygons have variants. Isogonal variants alternate the edge lengths but have the same angles, these include the rectangle, ditrigon, ditetragon etc and they have N/2-gonal symmetry. Their duals which are isotopal have alternating angles, but all edges are the same length, these include the rhombus, triambus, tetrambus, etc and they also have N/2-gonal symmetry. These can be seen above. Round isogonals can be thought of as 'points on minimum element' transitive and their isotopal duals are like the round dice which is contact region transitive. All convex isogonals and isotopals in two dimensions are described above, basically the circle and regular polygons with variants on the even ones.

# Three Dimensional Symmetries

Symmetries in 3-space can be grouped into two types: Fully 3-D (4 pairs of reflections and no reflections and a special one) and 2-D derivatives which includes prismatics (which are products of the 2-D symmetries with dyadic symmetry and their derivatives, five types each) and pyramid versions (which are the 2-D ones all over again). Both types have round symmetries which we'll split off first. The round ones are the spheric (two of the fully 3-D symmetries, cylindrical (three types), and cone (two types) symmetries. Next we'll look at the polyhedral symmetries which will be the remaining seven of the fully 3-D symmetries. Then comes the rest of the symmetries - prismattic and pyramid (pyric). They'll be divided into general, specific (trigonic to hexagonic), and then 2-fold and finally the 1-fold ones. Roto-reflections are possible in 3 dimensions.

## Round Symmetries

There are seven symmetries here, two are spheric, three are cylic, and two are conic. The two conic symmetries doesn't produce isogonals or isotopals.

Spheric Symmetry - [O,O] order 2*OO. It is the symmetry of a sphere.

Kispheric Symmetry - [O,O]⁺ order OO. Chiral sphere symmetry can best be visuallized by having a strange blue sphere that looks red in a mirror. Or if you grasp it and twist your hand clockwise, it feels smooth. Counter clockwise feels rough.

Cylic Symmetry - [O,2] order 4*O. Also called cylindric or cylindrical. It is the symmetry of a cylinder and is a product of cyclic and dyadic symmetries. It is the supreme of [N,2] symmetries.

Kicylic Symmetry - [O,2]⁺ order 2*O. Also called chiro cylindrical. It is only the rotational symmetries of cylic symmetry, no reflections. It is the supremum of [N,2]⁺ and [2N⁺,2⁺] symmetries.

Procylic Symmetry - [O⁺,2] order 2*O. Also called procylindrical. It is the cross product of kicyclic and dyadic symmetries and is the supremum of [N⁺,2] symmetries.

Conic Symmetry - [O,1] order 2*O. It is a 3-D copy of cyclic symmetry. It is the symmetry of a cone and is a copy of the 2-D version.

Kiconic Symmetry - [O⁺,1] order O. It is a 3-D copy of kicyclic symmetry. It is the rotational symmetries of a cone and is a copy of the 2-D version.

## Round Isogonals and Isotopals

There are only three round isoes (isogonals / isotopals).

Sphere - The sphere is both isogonal and isotopal under spheric and kispheric symmetry.

Cylinder - The cylinder is isogonal under cylic, procylic, and kicylic symmetries. It has variants with the same symmetry where it can be pulled into a pole or crushed into a coin.

Spindle - The spindle, which is also called the bicone is isotopal under cylic, procylic, and kicylic symmetries. It has variants with the same symmetry where it can be pulled into a two-spiked thing or crushed into a circular blade like thing.

## Polyhedral Symmetries

Above are the symmetry graphs of the 7 polyhedral symmetries and below are examples of the symmetries.

Doic Symmetry - [5,3] order 120. Also called icosahedral or dodecahedral symmetry. Most uniform polyhedra has this symmetry. On the graph, the white areas of the sphere come in 120s with no symmetry. The red points come in 12s with full pentagonal symmetry, the yellow points come in 20s with full trigon symmetry. The green points come in 30 with rectic symmetry. The blue circles have three types of sections (red-green, green-yellow, yellow-red), the circles have rectic symmetry, but generate dyadic symmetry in 60 locations.

Kidoic Symmetry - [5,3]⁺ order 60. Also called chirodoic or chiro-icosahedral symmetry. It is the symmetry of snid and many dodecahedral snubs. The white areas generate 60 symmetry points with no symmetry for any point in the areas. The red (12), yellow (20), and green (30) points have kipentagonic, kitrigonic, and essic symmetry respectively. This symmetry might also be called snid symmetry.

Cubic Symmetry - [4,3] order 48. It is the symmetry of the cube and polyhedra such as gocco and cotco. The white areas generate 48 identical spots per point and have no symmetry at those points. The red (6), yellow (8), and green (12) points generate square, trigon, and rectic symmetries. The blue circles have square symmetry but generate dyadic symmetric areas in 24 locations. The cyan circles have rectic symmetry and have two types of sections (red-yellow and yellow-green) which also generate dyadic symmetry in 24 spots.

Kicubic Symmetry - [4,3]⁺ order 24. Also called chirocubic symmetry or snic symmetry. It is the symmetry of the snub cube. The white areas generate 24 identical spots with no symmetry (like the snub triangles in snic). The red (6), yellow (8), and green (12) form kisquare, kitrigonic, and essic symmetries.

Pyritic Symmetry - [4,3⁺] order 24. It is the symmetry of a pyrite crystal. It turns the trigonic symmetric regions of cubic symmetry and turns them into kitrigonic symmetry and changes the square symmetric regions to rectic. This is the special one mentioned above. The white areas generate 24 symmetric spots with no symmetry. The red (6) and yellow (8) points generate rectic and kitrigonic symmetric areas, the latter coming in both right and left handed forms. The blue circles have square symmetry but generate dyadic symmetric spots in 12 locations. If a point is close to a red spot to the right and left on the circle, then it is closer to the other red spots in the up-down directions.

Tettic Symmetry - [3,3] order 24. It is the symmetry of tet as well as tut. The white areas generate 24 spots with no symmetry, the red and yellow spots generate trigonic symmetries in four areas each. The green points generate rectic symmetries in six spots. The circles have dyadic symmetry with three types of arcs (yellow-green, green-red, red-yellow) and generate dyadic symmetric locations in 12 spots for every point selected in it.

Kitettic Symmetry - [3,3]⁺ order 12. It is the rotational symmetries of tut without the reflections. It is the symmetry of snit, an ike variant with 4 large triangles, 4 small triangles, and 12 scalenes. The white areas generate 12 spots with the symmetry and have no symmetry at those areas. The red and yellow points generate kitrigonic symmetry at four spots each. The green spots generate six essic symmetric areas.

## Polyhedral Symmetric Isoes

The five regular polyhedra are isogonal and isotopal

Tet - The tetrahedron is both isogonal and isotopal under tettic and kitettic symmetry. It also has iso variants under prismattic symmetries which will be mentioned in a later section. Tet has four equalateral triangles (equit for short).

Cube - The cube is isogonal and isotopal under cubic, kicubic, and pyritic symmetries. It has six squares for sides.. It is also isotopal under tettic and kitettic symmetries which doesn't form variants in these symmetries. This version has six rhombuses, where the rhombus has perfect square shape. Variants do exist in prism symmetries that will be mentioned later.

Oct - The octahedron is isogonal and isotopal under cubic, kicubic, and pyritic symmetries. Oct has eight equits. It is also isogonal under tettic and kitettic symmetries which doesn't form variants in these symmetries. This version has two sets of four equits. It could be called a tetratetrahedron (tit for short). Variants do exist in prism symmetries that will be mentioned later.

Doe - The dodecahedron is isogonal and isotopal under doic and kidoic symmetries, it has 12 pentagons. Isotopal variants also exist under pyritic and kitettic symmetries. The pyritic form has twelve isosceles pentagons. The kitettic form has twelve 'abaac' pentagons.

Ike - The icosahedron is isogonal and isotopal under doic and kidoic symmetries, it has 20 equits. Isogonal variants also exist under pyritic and kitettic symmetries in which it could be called pyritic ike and snub tetrahedron (snit) respectively. Pyritic ike has eight equits and twelve isots (isosceles triangles). Snit has two sets of four equits and twelve scalenes.

The Archimedean solids are isogonal but not isotopal. Their duals, the Catalans are isotopal but not isogonal. Catalans are fair dice.

Tut - The truncated tetrahedron is isogonal under both tettic and kitettic symmetries and can be variated by how truncated it is. It has four ditrigons and four equits. Its dual had 12 isots (isosceles triangles).

Tic - The truncated cube is isogonal under both cubic and kicubic symmetries and can also be variated by how truncated it is. It has six ditetragons and eight equits. Its dual has 24 isots.

Toe - The truncated oct is isogonal under both cubic and kicubic symmetries and like above can be variated by how truncated it is, it has six squares and eight ditrigons, its dual has 24 isots. There's also variations with tettic symmetry which can be called the great rhombitetratetrahedron or 'gratet' for short. Gratet has two degrees of variations, its faces are six rectangles and two sets of four ditrigons, its dual has 24 scalenes.

Tid - The truncated doe is isogonal under doic and kidoic symmetries and can be variated by level of truncation, its faces are 12 dipentagons and 20 equits. Its dual has 60 isots.

Ti - The truncated ike is isogonal under doic and kidoic symmetries and can be variated by truncation level. It has twelve pentagons and 20 ditrigons. Its dual is the Buckyball shape and has 60 isots.

Co - The cuboctahedron is isogonal under cubic and kicubic symmetries, it has eight equits and six squares, its dual is rad the rhombic dodecahedron which has 12 rhombuses. It also has variants that are isogonal under tettic and kitettic symmetries, these variants can be called rhombitetratetrahedron or 'ratet'. They variate by the shape of the rectangles in it, it has six rectangles and two sets of four equits. Its dual has 12 kites for faces.

Id - The icosidodecahedron is isogonal under doic and kidoic symmetries. It has no variations. It has 12 pentagons and 20 equits. Its dual is the rhombic triacontahedron which has 30 rhombuses.

Sirco - Sirco is isogonal under cubic and kicubic symmetries which can be variated by turning the 12 rhombi squares into rectangles of different edge lengths, it has six squares, eight equits, and twelve rectangles. Its dual has 24 kites. There are also variations that are isogonal under pyritic symmetry where the rectangles can turn into trapezoids, it has six rectangles, eight equits, and twelve trapezoids. Its dual has 24 tetragons of the type 'abcd'.

Srid - Srid is isogonal under doic and kidoic symmetries which can turn the squares into rectangles. It has 12 pentagons, 20 equits, and 30 rectangles. Its dual has 60 kites.

Girco - Girco is only isogonal under cubic symmetry with two degrees of variations. It has six ditetragons, eight ditrigons, and twelve rectangles. Its dual has 48 scalenes.

Grid - Grid is only isogonal under doic symmetry with two degrees of variations. It has 12 dipentagons, 20 ditrigons, and 30 rectangles. Its dual has 120 scalenes.

Snic - The snub cube is only isogonal under kicubic symmetry. It has two degrees of variations. It has six rotated squares, eight rotated equits, and 24 scalenes. Its dual has 24 pentagons of the 'abaac' sort.

Snid - The snub dodecahedron is only isogonal under kidoic symmetry. It has two degrees of variations. It has 12 rotated pentagons, 20 rotated equits, and 60 scalenes. Its dual has 60 pentagons of the 'abaac' sort.

## General Prismattic Symmetries

Above are the symmetry graphs of the 7 types of prism and pyramid symmetries, using the decagonal cases, an even number. If we're dealing with an odd number, the pink circles aren't there, they are all blue. Under it are examples of these symmetries - yes I know, the red one looks like it has a face on it as well as its friends.

N Prismic Symmetry - [N,2] order 4N. It is the symmetry of the N-gon prism. This is the cross product of N-gonic and dyadic symmetry. The names of the 7-10 versions are heppic, opic, epic, and dippic symmetries. On the graph, the white areas generate 4N copies of no symmetric areas. The red (2) points generate two areas with N-gon symmetry, the yellow (N) and green (N) points generate N areas with rectic symmetry. The cyan ring (1) has N-gon symmetry and generates 2N areas with dyadic symmetry. If N is even, the blue (N/2) and pink (N/2) rings have rectic symmetry and generate 2N dyadic symmetric areas. If N is odd, the blue (N) rings have dyadic symmetry and produce 2N areas with dyadic symmetry.

N Proprismic Symmetry - [N⁺,2] order 2N. This is the cross product of N kigonic and dyadic symmetry. The 7-10 versions are proheppic, proopic, proepic, prodippic. The white areas generate 2N areas with no symmetry. The two red points generate two spots with N kigonal symmetry. The blue ring has N kigonal symmetry and generate N areas with dyadic symmetry (in a vertical orientation). If you made a prism of a 2-D chiral shape, it will have a proprismic symmetry.

N Kiprismic Symmetry - [N,2]⁺ order 2N. This is the symmetry of the N-gon prism with all reflections removed. It is the same as the N kiapic symmetry (N antiprism with no reflections). The 7-10 versions are kiheppic, kiopic, kiepic, and kidippic. In the graph above, the white areas produce 2N areas with no symmetry. The two red points generate two areas with N kigonal symmetry which seem to twist relative to each other. The N green and N yellow points generate N areas with essic symmetry. Gyroprisms (twisted antiprisms) have this symmetry.

N Apic Symmetry - [2N,2⁺] order 4N. It is the symmetry of the N-gon antiprism. The 7-10 versions are heappic, oapic, eapic, and dappic symmetries. On the graph above, the white areas generate 4N areas with no symmetry. The two red points generate two areas with N gonal symmetry. The 2N yellow points generate 2N areas with essic symmetry which flip between right and left handed forms. The N blue rings have essic symmetry if odd and dyadic symmetry if even and generate 2N areas with dyadic symmetry.

N Proapic Symmetry - [2N⁺,2⁺] order 2N. Although it looks chiral, it does have reflections, glide reflections that is. Think of a chiral polygon above a similar one which is twisted halfway. The 7-10 versions are proheapic, prooapic, proeapic, and prodappic. The white areas generate 2N areas with no symmetry. The two red points generate two N kigonal areas. This symmetry also has central inversion symmetry as well as the glide reflections mentioned earlier. Antiprisms have this symmetry when the whole object is rotated a bit.

N Pyric Symmetry - [N,1] order 2N. This is the same as N-gonal in 2-D. The 7-10 versions are hepyric, opyric, epyric, and depyric. The white areas generate 2N spots with no symmetry. The one red and one yellow points prudce one spot with N-gonal symmetry each. If N is even then the blue (N/2) and pink (N/2) rings have dyadic symmetry and generate N spots with dyadic symmetry. If N is odd, then the blue (N) circles have no symmetry and generate N areas with dyadic symmetry. This is the symmetry of an N-gon pyramid. This symmetry produces no isogonals nor isotopals.

N Kipyric Symmetry - [N⁺,1] order N. This is the same as N-kigonal in 2-D. The 7-10 versions are kihepyric, kiopyric, kiepyric, and kidepyric. The white areas generate N areas with no symmetry. The one red and one yellow points generate one spot each with N-kigonal symmetry. This symmetry produces pyramids of chiral 2-D shapes. No isogonals nor isotopals can form here.

## General Prismattic Symmetric Isoes

Odd N-Gon Prism - These are isogonal under N prismic, N proprismic, and N kiprismic symmetries. They can be variated by stretching or crushing only. Their faces are two N-gons and N rectangles. Their duals are N-gon tegums with 2N isots for faces.

Even N-Gon Prism - These are isogonal under N prismic, N proprismic, N kiprismic symmetries. They can be variated by stretching or crushing. Their faces are two N-gons and N rectangles. Their duals are N-gon tegums with 2N isots for faces. There are also N/2 prismic variants with two di-N/2-gons and two sets of N/2 rectangles, these are the di-N/2-gon prisms. Their duals are N/2-ambus tegums with 2N scalenes for faces. Finally there are N/2-apic variants with two di-N/2-gons and N trapezoids, these are the di-N/2-gon alterprisms. Their duals have 2N scalenes for faces with a zigzaging equator.

N-Gon Antiprism - These are isogonal under N apic, N kiprismic, and N proapic symmetries. They can be variated by stretching or crushing. Their faces are two N-gons and 2N isots. Their duals are N-gon antitegums with 2N kites for faces. There are more N kiprismic variants that can be made by twisting it a bit, these are the N-gon gyroprisms. These versions have two N-gons and 2N scalenes for faces. Their duals faces are 2N 'aabc' tetragons.

## Specific Prismattic Symmetries 3-6 Gonal

Hippic Symmetry - [6,2] order 24. It is the symmetry of the hexagon prism. This is the cross product of higgic and dyadic symmetries. The white areas in the symmetry graph above generate 24 areas with no symmetry. The two red spots produce two spots with higgic symmetry. The yellow and green spots (six each) produce six spots with rectic symmetry. The cyan circle has higgic symmetry and produce 12 spots with dyadic symmetry. The three blue circles and three pink circles have rectic symmetry and produce 12 spots with dyadic symmetry.

Prohippic Symmetry - [6⁺,2] order 12. This is the cross product of kihiggic and dyadic symmetries. The white areas in the graph produce 12 areas with no symmetry. The two red spots generate two areas with kihiggic symmetry. The blue circle has kihiggic symmetry and generate six areas with dyadic symmetry.

Kihippic Symmetry - [6,2]⁺ order 12. This is the symmetry of the hexagon prism with all reflections removed. The white areas generate 12 spots with no symmetry, the two red points generate two spots with kihiggic symmetry that twist relative to each other. The six yellow and six green spots produce six spots each with essic symmetry. Hexagon gyroprisms have this symmetry.

Happic Symmetry - [12,2⁺] order 24. It is the symmetry of the hexagon antiprism. The white areas generate 24 spots with no symmetry. The two red spots produce two spots with higgic symmetry. The 12 yellow spots make 12 areas with essic symmetry which flip back and forth between right and left handed forms. The six blue circles have dyadic symmetry and generate 12 spots with dyadic symmetry.

Prohappic Symmetry - [12⁺,2⁺] order 12. Although it looks chiral, it does have glide reflections. The white areas generate 12 spots with no symmetry and the two red points generate two spots with kihiggic symmetry. Hexagon antiprisms can take on this symmetry even when rotated.

Higpyric Symmetry - [6,1] order 12. This is the same as the 2-D hexagonal symmetry. The white areas generate 12 spots with no symmetry. The red and yellow points (one each) make one spot with higgic symmetry. The blue and pink circles (three each) have dyadic symmetry and generate six spots with dyadic symmetry. If you cut out a snowflake and made a pyramid of it, it will have this symmetry.

Kihigpyric Symmetry - [6⁺,1] order 6. This is the same as the 2-D kihexagonal symmetry. The white areas make six spots with no symmetry. The red and yellow points generate one spot each with kihiggic symmetry.

Pippic Symmetry - [5,2] order 20. It is the symmetry of the pentagon prism. This is the cross product of pegic and dyadic symmetry. The white areas generate 20 spots with no symmetry, the two red points produce two areas with peggic symmetry. The five green and five yellow points produce five areas each with rectic symmetry. The only cyan circle which have peggic symmetry creates ten areas with dyadic symmetry and the five blue circles which have dyadic symmetry with two types of sections (red-green, red-yellow) creates ten areas with dyadic symmetry.

Propippic Symmetry - [5⁺,2] order 10. This is the cross product of kipic and dyadic symmetry. The white areas in the graph above creates ten areas with no symmetry, the two red points creates two areas with kipic symmetry, and the blue circle which has kipic symmetry creates five areas with dyadic symmetry. This symmetry can form pips that are free to rotate.

Kipippic Symmetry - [5,2]⁺ order 10. This is the symmetry of the pentagon prism with all reflections removed. The white areas generate ten areas with no symmetry. The two red points generate two kipic areas. The five yellow and five green points create five areas each with essic symmetry.

Pappic Symmetry - [10,2⁺] order 20. It is the symmetry of the pentagon antiprism. The white areas create 20 spots with no symmetry. The two red spots generate two peggic symmetric areas. The ten yellow point create ten essic symmetric areas. The five blue circles have essic symmetry and creates ten spots with dyadic symmetry.

Propappic Symmetry - [10⁺,2⁺] order 10. Although it looks chiral, it does have glide reflections. The white areas create ten spots with no symmetry and the two red points make two spots with kipic symmetry. This symmetry can create paps that is allowed the freedom to rotate.

Peppyric Symmetry - [5,1] order 10. This is the same as the 2-D pentagonal symmetry. The white areas generate ten spots with no symmetry. The red and yellow points create one spot each with peggic symmetry. The five blue circles have monic symmetry and create five spots with dyadic symmetry.

Kipeppyric Symmetry - [5⁺,1] order 5. This is the same as the 2-D kipentagonal symmetry. The white areas create five spots with no symmetry and the red and yellow points create one area each with kipic symmetry.

Squippic Symmetry - [4,2] order 16. It is the symmetry of the square prism. This is the cross product of square and dyadic symmetry. The white areas on the graph generate 16 places with no symmetry. The two red points create two areas with square symmetry. The four green and four yellow points make four spots each with rectic symmetry. The cyan circle has square symmetry and creates eight spots with dyadic symmetry. The two pink and two blue circles have rectic symmetry and creates eight spots with dyadic symmetry.

Prosquippic Symmetry - [4⁺,2] order 8. This is the cross product of kisquare and dyadic symmetry. The white areas generate eight spots with no symmetry and the two red spots create two areas with kisquare symmetry. The blue circle has kisquare symmetry and creates four areas with dyadic symmetry.

Kisquippic Symmetry - [4,2]⁺ order 8. This is the symmetry of the square prism with all reflections removed. Example is a square gyroprism. The white areas create eight areas with no symmetry and the two red points create two areas with kisquare symmetry. The four green and four yellow points create four spots each with essic symmetry.

Squappic Symmetry - [8,2⁺] order 16. It is the symmetry of the square antiprism. The white areas generate 16 areas with no symmetry. The two red points create two areas with square symmetry that are twisted halfway relative to each other. The eight yellow points generate eight areas with essic symmetry that alternate. The four blue circles have dyadic symmetry and generate eight spots with dyadic symmetry.

Prosquappic Symmetry - [8⁺,2⁺] order 8. Although it looks chiral, it does have glide reflections. The white areas generate eight areas with no symmetry and the two red points create two areas with kisquare symmetry.

Squippyric Symmetry - [4,1] order 8. This is the same as the 2-D square symmetry. The white areas generate eight spots with no symmetry. The red and yellow points create one spot each with square symmetry. The two blue and two pink circles have dyadic symmetry and generate four spots with dyadic symmetry.

Kisquippyric Symmetry - [4⁺,1] order 4. This is the same as the 2-D kisquare symmetry. The white areas generate four spots with no symmetry. The red and yellow points create one spot each with kisquare symmetry.

Trippic Symmetry - [3,2] order 12. It is the symmetry of the triangle prism. This is the cross product of trigonic and dyadic symmetries. The white areas in the graph generates twelve areas with no symmetry. The two red points create two areas with trigonal symmetry. The three yellow and three green points create three rectic areas each. The cyan circle has trigonal symmetry and generates six areas with dyadic symmetry. The three blue circles have dyadic symmetry and creates six dyadic spots.

Protrippic Symmetry - [3⁺,2] order 6. This is the cross product of kitric and dyadic symmetries. The white areas creates six areas with no symmetry. The two red points make two kitric areas and the blue circle has kitric sym and creates three spots with dyadic symmetry.

Kitrippic Symmetry - [3,2]⁺ order 6. This is the symmetry of the triangle prism with all reflections removed. The white areas create six spots with no sym. The two red points create two kitric spots. The three green and three yellow points make three essic spots each.

Trappic Symmetry - [6,2⁺] order 12. It is the symmetry of the triangle antiprism. The white areas create twelve no sym spots. The two red points make two trigonic areas and the six yellow points create six essic spots that alternate. The three blue circles have essic sym and create six spots with dyadic sym.

Protrappic Symmetry - [6⁺,2⁺] order 6. Although it looks chiral, it does have glide reflections. The white areas generate six no sym spots and the two red points make two kitric spots.

Trippyric Symmetry - [3,1] order 6. This is the same as the 2-D trigonal symmetry. The white areas make six areas with no sym. The red and yellow points make one trigonic spot each. The three blue circles have no sym but creates three spots with dyadic sym.

Kitrippyric Symmetry - [3⁺,1] order 3. This is the same as the 2-D kitrigonal symmetry. The white areas create three spots with no sym. The red and yellow points create one area each which is kitric.

## Specific Prismattic Symmetric Isoes

Hip - The hexagon prism is isogonal under hippic, prohippic, and kihippic symmetries. They can be variated by stretching or crushing. Their faces are two hexagons and six rectangles. Their duals are hexagon tegums with 12 isots for faces. There are also trippic variants with two ditrigons and two sets of three rectangles, this is the ditrigon prism. Their duals are triambus tegums with 12 scalenes for faces. Finally there are trappic variants with two ditrigons and six trapezoids, these are the ditrigon alterprisms, or ditra for short. Their duals have 2N scalenes for faces with a zigzaging equator.

Pip - The pentagon prism is isogonal under pippic, propippic, and N kipippic symmetries. They can be variated by stretching or crushing only. Their faces are two pentagons and five rectangles. Their duals are pentagon tegums with ten isots for faces.

Squip - The square prism is a variant of the cube and is isogonal under squippic, prosquippic, and kisquippic symmetries. They can be variated by stretching or crushing. Their faces are two squares and four rectangles. Their duals are square tegums (an oct variant) with 8 isots for faces. There are also lower symmetry variants which will be mentioned in the last section.

Trip - The triangular prism is isogonal under trippic, protrippic, and N kitrippic symmetries. They can be variated by stretching or crushing only. Their faces are two equits and three rectangles. Their duals are triangular tegums with six isots for faces.

Hap - The hexagon antiprism is isogonal under happic, N kihippic, and N prohappic symmetries. They can be variated by stretching or crushing. Their faces are two hexagons and 12 isots. Their duals are hexagon antitegums with 12 kites for faces. There are more kihippic variants that can be made by twisting it a bit, these are the hexagon gyroprisms. These versions have two hexagons and 12 scalenes for faces. Their duals faces are 12 'aabc' tetragons.

Pap - The pentagon antiprism is isogonal under pappic, N kipippic, and N propappic symmetries. They can be variated by stretching or crushing. Their faces are two pentagons and ten isots. Their duals are pentagon antitegums with ten kites for faces. There are more kipippic variants that can be made by twisting it a bit, these are the pentagon gyroprisms. These versions have two pentagons and ten scalenes for faces. Their duals faces are ten 'aabc' tetragons.

Squap - The square antiprism is isogonal under squappic, N kisquippic, and N prosquappic symmetries. They can be variated by stretching or crushing. Their faces are two squares and 8 isots. Their duals are square antitegums with 8 kites for faces. There are more kisquippic variants that can be made by twisting it a bit, these are the square gyroprisms. These versions have two squares and 8 scalenes for faces. Their duals faces are eight 'aabc' tetragons.

Trap - The triangular antiprism is an oct variant and is isogonal under trappic, N kitrippic, and N protrappic symmetries. They can be variated by stretching or crushing. Their faces are two equits and six isots. Their duals are triangular antitegums with six kites for faces which are cube variants. There are more kitrippic variants that can be made by twisting it a bit, these are the triangular gyroprisms, also oct variants. These versions have two equits and six scalenes for faces. Their duals faces are six 'aabc' tetragons and are cube variants.

## Lower Prismattic Symmetries 1-2 Gonal

Brick Symmetry - [2,2] order 8. It is the symmetry of a rectangular box. This is the cross product of rectic and dyadic symmetries. The white areas create eight spots with no sym. The two red, two green, and two yellow points create two spots each with rectic symmetry. The blue, cyan, and pink circles have rectic symmetry and generate four spots each with dyadic sym.

Espic Symmetry - [2⁺,2] order 4. This is the cross product of essic and dyadic symmetries. The name 'espic' is short for S prismic. It is the symmetry of those toy S letter blocks. The white areas create four spots with no sym. The two red points make two spots with essic sym. The blue circle has essic sym and creates two spots with dyadic sym.

Kibrick Symmetry - [2,2]⁺ order 4. This is the symmetry of a brick with all reflections removed. It is also the symmetry of twisted (gyrated) disphenoids. The white areas make four spots with no sym. The two red, two yellow, and two green points create two essic areas each.

Dappic Symmetry - [4,2⁺] order 8. It is the symmetry of the tetragonal disphenoid as well as afdec's verf. The white areas generate eight no sym spots. The two red points create two rectic areas that are rotated 90 degrees from each other. The four yellow points create four essic spots. The two circles have dyadic symmetry and create four spots with dyadic sym.

Prodappic Symmetry - [4⁺,2⁺] order 4. Although it looks chiral, it does have glide reflections. Take two S's and rotate them 90 degrees, then separate into the 3rd dimension - it will have this symmetry. The white areas create four no sym spots. The two red spots make two areas with essic symmetry which are rotated 90 degrees from each other.

Rectpyric Symmetry - [2,1] order 4. This is the same as the 2-D symmetry. The white areas generate four no sym spots. The red and yellow points create one rectic spot each. The blue and pink circles have dyadic sym and creates two spots each with dyadic sym.

Espyric Symmetry - [2⁺,1] order 2. This is the same as the 2-D version. The white areas generate two no sym areas. The red and yellow points create one spot with essic symmetry each.

Inversic Symmetry - [2⁺,2⁺] order 2. It is the 3-D central inversion symmetry. Take an S, cut it in half and pull the two halfs into the third dimension. This shape has this symmetry. Each point on the graph creates two points on opposite sides of the sphere with no symmetry.

Dyadic Symmetry - [1,1] order 2. Also called bilateral. This is the same as the 1-D symmetry and is the symmetry of the human body. The white spots will create two spots with no symmetry, one is right handed the other left handed. The blue circle has no symmetry and generates one spot with dyadic symmetry.

Monic Symmetry - [1⁺,1] order 1. This is the same as the 1-D version. All spots on the graph generates only itself and no other spots and these have no symmetry.

## Lower Prismattic Symmetric Isoes

Brick - Also called block or box, it is isogonal under brick symmetry and is a variant of the cube. They can be variated by changing any of the dimensions. Their faces are three pairs of rectangles. Their duals are rhombus tegums and are oct variants, they have 8 scalenes as faces.

Disph - The (rhombic) disphenoid is isogonal and isotopal under dappic, kibrick, and prodappic symmetries. It is a tet variant. It has four isots for faces. Their duals are also disphenoids. They can be variated by stretching or squashing them. If we twist them we will get more kibrick variants, they have four scalenes and their duals are also gyro disphenoids.

[5⩫ 2] [5⁺⩫ 2] [5~2] [5'~2]

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