# Isogonal polytope

An **isogonal polytope** or **vertex-transitive polytope** is a polytope (or polytope-like object) whose vertices are identical under its symmetry group. In other words, given any two vertices, there is a symmetry of the polytope that transforms one into the other.

Possibly, the earliest mention of the concept of an isogonal figure is found in the Rigveda (ca. 1500-1000 BCE), an ancient text written in Vedic Sanskrit, where it states *tásmint sākáṃ triśatā́ ná śaṅkávo 'rpitā́ḥ ṣaṣṭír ná calācalā́saḥ* (On it are placed together three hundred and sixty like pegs. They shake not in the least.), referring to the transitivity of the so-called "pegs" (*śaṅkú*) which connect the vertices of the circle to its center.^{[citation needed]}

## Properties and terminology[edit | edit source]

An isogonal polytope has a singular vertex figure. (However, polytopes with a single vertex figure are not necessarily isogonal, such as the pseudo-uniform polyhedra.) The dual of an isogonal polytope is an isotopic polytope, which are made out of one facet type.

Restricting discussion to finite planar polytopes, all vertices of an isogonal n -polytope lie on an (n - 1)-hypersphere, and every proper element of rank r is inscribed in an (r - 1)-hypersphere. The convex hull of an isogonal polytope is also isogonal.

An isogonal polytope does not necessarily have all its elements isogonal. An example is the triangular gyroprism, which in general form has scalene triangles. They do not even need to be abstractly isogonal, such as in the scaliform polychora.

Subsets of isogonal polytopes include:

- regular polytopes, which are flag-transitive
- weakly regular polytopes, which are transitive for every individual rank
- quasiregular polytopes, which are isogonal and have only regular facets and quasi-regular vertex figures
- noble polytopes, which are isogonal and isotopic
- uniform polytopes, which are isogonal, have a single edge length, and have only uniform elements, plus other constraints
- semiregular polytopes, which are isogonal and only have regular facets
- semi-uniform polytopes, which are isogonal and have only isogonal elements, but may have multiple edge lengths
- scaliform polytopes, which are isogonal and have only one edge length

Isogonality applies directly to abstract polytopes, compounds, tilings, skew polytopes, exotic polytopoids, and certain types of incidence geometries where the notion of a vertex is specified (such as complex polytopes).

## Classification[edit | edit source]

Isogonal polytopes are rarely studied as a group on their own, as they become very broad in high dimensions.

### Isogonal polygons[edit | edit source]

The isogonal polygons are easily characterized, comprising the regular polygons and the semi-uniform polygons. The latter are formed by taking a regular polygon with an even number of sides and altering every other side length. There are uncountably many semi-uniform polygons due to the continuum of side length ratios.

### Isogonal polyhedra[edit | edit source]

The (finite and planar) isogonal polyhedra are a very broad set that has not been fully characterized by any reasonable standard.^{[1]} As almost all such polyhedra can have their edge lengths continuously varied, one question would be to classify all abstract polyhedra that have isogonal realizations. Unfortunately, even the sub-problem of classifying all abstractly distinct noble polyhedra is open.

The convex isogonal polyhedra are fully characterized, and are all formed by varying the edge lengths of the convex uniform polyhedra.

In 1997, Grunbaum investigated the 3D isogonal prismatoids, which include the gyroprisms and crown polyhedra.

### Isogonal polychora[edit | edit source]

Isogonality is an even weaker condition for 4D and above. There are convex isogonal polychora that cannot be formed by edge length modification of the convex uniform polychora. Most notably, many but not all convex isogonal swirlchora fit this description.

## Types of convex isogonal polytopes[edit | edit source]

### Polygons[edit | edit source]

- Regular polygons (infinite, half symmetry variants exist for even-sided polygons with two alternating edge lengths)

### Polyhedra[edit | edit source]

- Platonic solids (5 total)
- Tetrahedron (lower symmetry variants are the tetragonal disphenoid and the rhombic disphenoid)
- Cube (lower symmetry variants are the square prism, the cuboid, and the rectangular trapezoprism)
- Octahedron (a lower symmetry variant is the triangular antiprism)
- Dodecahedron (no lower symmetry variants)
- Icosahedron (lower symmetry variants are the pyritohedral icosahedron and the snub tetrahedron)

- Archimedean solids (13 total)
- Truncated tetrahedron
- Cuboctahedron (a lower symmetry variant is the small rhombitetratetrahedron)
- Truncated cube
- Truncated octahedron (a lower symmetry variant is the great rhombitetratetrahedron)
- Small rhombicuboctahedron (a lower symmetry variant is the pyritosnub cube)
- Great rhombicuboctahedron
- Snub cube
- Icosidodecahedron
- Truncated dodecahedron
- Truncated icosahedron
- Small rhombicosidodecahedron
- Great rhombicosidodecahedron
- Snub dodecahedron

- Polygonal prisms (infinite, half symmetry variants exist for even-sided polygons with bases alternating two edge lengths, and can either be parallel or gyrated with respect to each other)
- Polygonal antiprisms (a lower symmetry variant are the gyroprisms) (infinite, half symmetry variants also exist with the two bases rotated so that the base-first projection envelope is not a regular polygon)

### Polychora[edit | edit source]

- Regular polychora (6 total)
- Pentachoron (a lower symmetry variant is the 5-2 step prism/5-2 gyrochoron)
- Tesseract (lower symmetry variants are the cubic prism, square-square duoprism, rectangular-square duoprism, rectangular duoprism and the rectangular-rectangular duoprism)
- Hexadecachoron (lower symmetry variants are the tetrahedral antiprism, rectangular duotegum, digonal-digonal duoantiprism, rhombic disphenoidal antiprism and the 8-3 step prism)
- Icositetrachoron
- Hecatonicosachoron
- Hexacosichoron

- Non-regular uniform polychora (41 total)
- Rectified pentachoron (a lower symmetry variant is the 5-2 double step prism)
- Truncated pentachoron
- Decachoron (a lower symmetry variant is the pentapentachoron)
- Small rhombated pentachoron
- Great rhombated pentachoron
- Small prismatodecachoron (a lower symmetry variant is the small disprismatopentapentachoron and the triangular-prismatic 10-3 double gyrostep prism)
- Prismatorhombated pentachoron
- Great prismatodecachoron (a lower symmetry variant is the great disprismatopentapentachoron)
- Rectified tesseract (a lower symmetry variant is the runcic tesseract)
- Truncated tesseract
- Truncated hexadecachoron (a lower symmetry variant is the cantic tesseract)
- Tesseractihexadecachoron (a lower symmetry variant is the runcicantic tesseract)
- Small rhombated tesseract
- Great rhombated tesseract
- Small disprismatotesseractihexadecachoron (a lower symmetry variant is the square duoexpandoprism)
- Prismatorhombated tesseract
- Prismatorhombated hexadecachoron
- Great disprismatotesseractihexadecachoron
- Rectified icositetrachoron (lower symmetry variants are the cantellated hexadecachoron, prismatorhombated demitesseract and the transitional hexafold ditetraswirlchoron)
- Truncated icositetrachoron (lower symmetry variants are the cantitruncated hexadecachoron and the omnitruncated demitesseract)
- Tetracontoctachoron (a lower symmetry variant is the icositetraicositetrachoron)
- Small rhombated icositetrachoron (a lower symmetry variant is the cantic snub icositetrachoron)
- Great rhombated icositetrachoron
- Small prismatotetracontoctachoron (a lower symmetry variant is the small disprismatoicositetraicositetrachoron)
- Prismatorhombated icositetrachoron (a lower symmetry variant is the runcicantic snub icositetrachoron)
- Great prismatotetracontoctachoron (a lower symmetry variant is the great disprismatoicositetraicositetrachoron)
- Snub disicositetrachoron (lower symmetry variants are the snub rhombatohexadecachoron and the snub demitesseract)
- Rectified hecatonicosachoron
- Rectified hexacosichoron (a lower symmetry variant is the snub tetrahedral hecatonicosachoron)
- Truncated hecatonicosachoron
- Truncated hexacosichoron
- Hexacosihecatonicosachoron
- Small rhombated hecatonicosachoron
- Small rhombated hexacosichoron
- Great rhombated hecatonicosachoron
- Great rhombated hexacosichoron
- Small disprismatohexacosihecatonicosachoron
- Prismatorhombated hecatonicosachoron
- Prismatorhombated hexacosichoron
- Great disprismatohexacosihecatonicosachoron
- Grand antiprism

- Polyhedral prisms (17 total, variations are the same as the polyhedral bases)
- Tetrahedral prism
- Octahedral prism
- Dodecahedral prism
- Icosahedral prism
- Truncated tetrahedral prism
- Cuboctahedral prism
- Truncated cubic prism
- Truncated octahedral prism
- Small rhombicuboctahedral prism
- Great rhombicuboctahedral prism
- Snub cubic prism
- Icosidodecahedral prism
- Truncated dodecahedral prism
- Truncated icosahedral prism
- Small rhombicosidodecahedral prism
- Great rhombicosidodecahedral prism
- Snub dodecahedral prism

- Scaliform polychora (4 total)
- Rectified isotoxal decachoric and tetracontoctachoric polychora (4 total)
- Truncated isotoxal decachoric and tetracontoctachoric polychora (4 total)
- Decachoric and tetracontoctachoric doublings (40 total)
- Bidecachoron/Bitetracontoctachoron (the former has a lower symmetry variant as a 10-3 step prism)
- Biambodecachoron/Biambotetracontoctachoron (the former has a lower symmetry variant as a 10-3 double step prism)
- Bitruncatodecachoron/Bitruncatotetracontoctachoron
- Bimesotruncatodecachoron/Bimesotruncatotetracontoctachoron
- Small birhombatodecachoron/Small birhombatotetracontoctachoron
- Medial birhombatodecachoron/Medial birhombatotetracontoctachoron
- Great birhombatodecachoron/Great birhombatotetracontoctachoron
- Small bicantitruncatodecachoron/Small bicantitruncatotetracontoctachoron
- Medial bicantitruncatodecachoron/Medial bicantitruncatotetracontoctachoron
- Great bicantitruncatodecachoron/Great bicantitruncatotetracontoctachoron
- Transitional bicantitruncatodecachoron/Transitional bicantitruncatotetracontoctachoron
- Biprismatodecachoron/Biprismatotetracontoctachoron
- Small biprismatorhombatodecachoron/Small biprismatorhombatotetracontoctachoron
- Medial biprismatorhombatodecachoron/Medial biprismatorhombatotetracontoctachoron
- Great biprismatorhombatodecachoron/Great biprismatorhombatotetracontoctachoron
- Transitional biprismatorhombatodecachoron/Transitional biprismatorhombatotetracontoctachoron
- Small biomnitruncatodecachoron/Small biomnitruncatotetracontoctachoron
- Great biomnitruncatodecachoron/Great biomnitruncatotetracontoctachoron
- Prismatic transitional biomnitruncatodecachoron/Prismatic transitional biomnitruncatotetracontoctachoron
- Tetrahedral transitional biomnitruncatodecachoron/Cubic transitional biomnitruncatotetracontoctachoron

- Chiral decachoric and tetracontoctachoric doublings (16 total)
- Small biomnisnub decachoron/Small biomnisnub tetracontoctachoron
- Medial biomnisnub decachoron/Medial biomnisnub tetracontoctachoron
- Great biomnisnub decachoron/Great biomnisnub tetracontoctachoron
- Small gyrowedged biomnisnub decachoron/Small gyrowedged biomnisnub tetracontoctachoron
- Great gyrowedged biomnisnub decachoron/Great gyrowedged biomnisnub tetracontoctachoron
- Antipodial biomnisnub decachoron/Antipodial biomnisnub tetracontoctachoron
- Prismatic transitional biomnisnub decachoron/Prismatic transitional biomnisnub tetracontoctachoron
- Tetrahedral transitional biomnisnub decachoron/Cubic transitional biomnisnub tetracontoctachoron

- Alternated polychora based on regular and prismatic symmetries (15 total)
- Snub decachoron
- Small omnisnub bidecachoron
- Great omnisnub bidecachoron
- Prismatic transitional omnisnub bidecachoron
- Tetrahedral transitional omnisnub bidecachoron
- Snub tesseract
- Snub tetracontoctachoron
- Small omnisnub bitetracontoctachoron
- Great omnisnub bitetracontoctachoron
- Prismatic transitional omnisnub bitetracontoctachoron
- Cubic transitional omnisnub bitetracontoctachoron
- Snub hexacosihecatonicosachoron
- Pyritohedral icosahedral antiprism (a lower symmetry variant is the snub tetrahedral antiprism)
- Snub cubic antiprism
- Snub dodecahedral antiprism

- Chiral pyritohedral prismatic polychora (1 total)
- Oxitic symmetric polychora (12 total)
- Bitetrahedral tetracontoctachoron
- Snub bitetrahedral tetracontoctachoron
- Bitruncatotetrahedral tetracontoctachoron
- Small omnisnub bitetrahedral tetracontoctachoron
- Great omnisnub bitetrahedral tetracontoctachoron
- Small disphenoidal omnisnub bitetrahedral tetracontoctachoron
- Medial disphenoidal omnisnub bitetrahedral tetracontoctachoron
- Great disphenoidal omnisnub bitetrahedral tetracontoctachoron
- Antipodial omnisnub bitetrahedral tetracontoctachoron
- Antiwedged omnisnub bitetrahedral tetracontoctachoron
- Gyrowedged omnisnub bitetrahedral tetracontoctachoron
- Triangular transitional omnisnub bitetrahedral tetracontoctachoron

- Ixitic symmetric polychora (13 total)
- Tetrahedral-snub tetrahedral hecatonicosachoron
- Hecatonicosadiminished hecatonicosachoron
- Ambotruncatotetrahedral hecatonicosachoron
- Small snub truncatotetrahedral hecatonicosachoron
- Medial snub truncatotetrahedral hecatonicosachoron
- Great snub truncatotetrahedral hecatonicosachoron
- Truncatosnub tetrahedral hecatonicosachoron
- Small snub rhombatotetrahedral hecatonicosachoron
- Medial snub rhombatotetrahedral hecatonicosachoron
- Great snub rhombatotetrahedral hecatonicosachoron
- Small snub tetrahedral-transitional hecatonicosachoron
- Great snub tetrahedral-transitional hecatonicosachoron
- Snub truncatotetrahedral-transitional hecatonicosachoron

- Ixoic symmetric polychora (at least 15 total)
- Bitetrahedral diacositetracontachoron
- Small bitetratetrahedral diacositetracontachoron
- Medial bitetratetrahedral diacositetracontachoron
- Snub bitetrahedral diacositetracontachoron
- Great bitetratetrahedral diacositetracontachoron
- Antiwedged bitetratetrahedral diacositetracontachoron
- Small biambotetrahedral diacositetracontachoron
- Great biambotetrahedral diacositetracontachoron
- Transitional biambotetrahedral diacositetracontachoron
- Bitruncatotetrahedral diacositetracontachoron
- Biambotruncatotetrahedral diacositetracontachoron
- Omnisnub tetrahedral hecatonicosachoron
- Antipodial omnisnub bitetrahedral diacositetracontachoron
- Disphenoidal omnisnub bitetrahedral diacositetracontachoron
- Tetrahedral transitional omnisnub bitetrahedral diacositetracontachoron

- Edge-snubs (2 total)
- Antiprismatic prisms (infinite, variations are the same as the antiprismatic bases)
- Duoprisms (infinite, variations are the same as the polygonal bases)
- Duotegums (infinite, the bases must have identical isogonal polygons with additional step prism variations)
- Rectified duoprisms (infinite)
- Truncated duoprisms (infinite)
- Duoantiprisms (infinite, lower symmetry variants have one or both antiprism bases twisted)
- Prismantiprismoids (infinite, contains a ring of alternating prisms and antiprisms, lower symmetry variants have twisted antiprisms)
- Truncatoprismantiprismoids (infinite, contains two rings of alternating prisms and trapezoprisms)
- Snub prismantiprismoids (infinite, alternations of the truncatoprismantiprismoids)
- Ditetragoltriates (infinite, contains two orthogonal rings of identical prisms)
- Antiditetragoltriates (infinite, contains two antialigned orthogonal rings of two types of prisms each)
- Duoexpandoprisms (infinite, contains two orthogonal rings of two types of prisms each)
- Duotruncatoprisms (infinite, contains two orthogonal rings of identical prisms whose bases are truncated polygons)
- Duotruncatoalterprisms (infinite, contains two orthogonal rings of alternating prisms and cupolae)
- Duotruncatoalterantiprisms (infinite, contains two orthogonal rings of alternating antiprisms and cupolae)
- Duotransitionalterprisms (infinite, contains two orthogonal rings of alternating prisms and trapezorhombihedra)
- Duotransitionalterantiprisms (infinite, contains two orthogonal rings of alternating antiprisms and bicupolae)
- Double antiprismoids (infinite, contains two orthogonal rings of identical antiprisms)
- Double gyroantiprismoids (infinite, contains two antialigned orthogonal rings of identical antiprisms)
- Double chiroantiprismoids (infinite, contains two antialigned gyrated orthogonal rings of identical antiprisms)
- Double prismantiprismoids (infinite, contains two orthogonal rings of alternating prisms and antiprisms)
- Double gyroprismantiprismoids (infinite, contains two antialigned orthogonal rings of alternating prisms and antiprisms)
- Double snub prismantiprismoids (infinite, convex hulls of two orthogonal snub prismantiprismoids)
- Duoprismatic swirlprisms (infinite)
- Double duoprismatoswirlprisms (infinite)
- Alterantiprismatic swirlprisms (infinite)
- Double alterantiprismatic swirlprisms (infinite)
- Prismantiprismatic swirlprisms (infinite)
- Double prismantiprismatoswirlprisms (infinite)
- Truncatoprismantiprismatic swirlprisms (infinite)
- Double truncatoprismantiprismatoswirlprisms (infinite)
- Step prisms (infinite, variations are the same as the pentachoron)
- Step prism compound hulls (infinite)
- Swirlchoron (infinite)
- Swirlchoron compound hulls (infinite)

### Polytera[edit | edit source]

- Regular polytera (3 total)
- Non-regular uniform polytera (55 total)
- Rectified hexateron
- Dodecateron
- Truncated hexateron
- Bitruncated hexateron
- Small rhombated hexateron
- Small birhombated dodecateron
- Great rhombated hexateron
- Great birhombated dodecateron
- Small prismated hexateron
- Prismatotruncated hexateron
- Prismatorhombated hexateron
- Great prismated hexateron
- Small cellated dodecateron
- Cellirhombated dodecateron
- Celliprismated hexateron
- Celligreatorhombated hexateron
- Celliprismatotruncated dodecateron
- Great cellated dodecateron
- Demipenteract
- Rectified penteract
- Rectified triacontaditeron
- Penteractitriacontaditeron
- Truncated penteract
- Truncated triacontaditeron
- Bitruncated penteract
- Bitruncated triacontaditeron
- Small rhombated penteract
- Small rhombated triacontaditeron
- Small birhombated penteractitriacontaditeron
- Great rhombated penteract
- Great rhombated triacontaditeron
- Great birhombated penteractitriacontaditeron
- Small prismated penteract
- Small prismated hexateron
- Prismatotruncated penteract
- Prismatotruncated triacontaditeron
- Prismatorhombated penteract
- Prismatorhombated triacontaditeron
- Great prismated penteract
- Great prismated triacontaditeron
- Small cellated penteractitriacontaditeron
- Cellirhombated penteractitriacontaditeron
- Celliprismated penteract
- Celliprismated triacontaditeron
- Celligreatorhombated penteract
- Celligreatorhombated triacontaditeron
- Celliprismatotruncated penteractitriacontaditeron
- Great cellated penteractitriacontaditeron
- Truncated demipenteract
- Small rhombated demipenteract
- Great rhombated demipenteract
- Small prismated demipenteract
- Prismatotruncated demipenteract
- Prismatorhombated demipenteract
- Great prismated demipenteract

- Uniform polychoric prisms (46 total)
- Pentachoric prism
- Hexadecachoric prism
- Icositetrachoric prism
- Hecatonicosachoric prism
- Hexacosichoric prism
- Rectified pentachoric prism
- Truncated pentachoric prism
- Decachoric prism
- Small rhombated pentachoric prism
- Great rhombated pentachoric prism
- Small prismatodecachoric prism
- Prismatorhombated pentachoric prism
- Great prismatodecachoric prism
- Rectified tesseractic prism
- Truncated tesseractic prism
- Truncated hexadecachoric prism
- Tesseractihexadecachoric prism
- Small rhombated tesseractic prism
- Great rhombated tesseractic prism
- Small disprismatotesseractihexadecachoric prism
- Prismatorhombated tesseractic prism
- Prismatorhombated hexadecachoric prism
- Great disprismatotesseractihexadecachoric prism
- Rectified icositetrachoric prism
- Truncated icositetrachoric prism
- Tetracontoctachoric prism
- Small rhombated icositetrachoric prism
- Great rhombated icositetrachoric prism
- Small prismatotetracontoctachoric prism
- Prismatorhombated icositetrachoric prism
- Great prismatotetracontoctachoric prism
- Snub disicositetrachoric prism
- Rectified hecatonicosachoric prism
- Rectified hexacosichoric prism
- Truncated hecatonicosachoric prism
- Truncated hexacosichoric prism
- Hexacosihecatonicosachoric prism
- Small rhombated hecatonicosachoric prism
- Small rhombated hexacosichoric prism
- Great rhombated hecatonicosachoric prism
- Great rhombated hexacosichoric prism
- Small disprismatohexacosihecatonicosachoric prism
- Prismatorhombated hecatonicosachoric prism
- Prismatorhombated hexacosichoric prism
- Great disprismatohexacosihecatonicosachoric prism
- Grand antiprismatic prism

- Scaliform polytera (14 total, includes some alterprisms based on decachoric, demitesseractic and tetracontoctachoric symmetries)
- Tridiminished rectified hexateron
- Partially-biexpanded hexateron
- Partially-expanded demipenteract
- Small rhombated pentachoric alterprism
- Great rhombated pentachoric alterprism
- Prismatorhombated pentachoric alterprism
- Rectified tesseractic alterprism
- Truncated hexadecachoric alterprism
- Tesseractihexadecachoric alterprism
- Icositetrachoric antiprism
- Rectified icositetrachoric alterprism
- Small rhombated icositetrachoric alterprism
- Great rhombated icositetrachoric alterprism
- Prismatorhombated icositetrachoric alterprism

- Scaliform polychoric prisms (4 total)
- Edge-snubs (2 total)
- Alternated polytera based on regular and prismatic symmetries (at least 17 total)
- Snub dodecateron
- Omnisnub bidodecateron
- Snub prismatotriacontaditeron
- Snub penteract
- Snub decachoric antiprism
- Small omnisnub bidecachoric antiprism
- Great omnisnub bidecachoric antiprism
- Prismatic transitional omnisnub bidecachoric antiprism
- Tetrahedral transitional omnisnub bidecachoric antiprism
- Snub tesseractic antiprism
- Snub disicositetrachoric antiprism
- Snub tetracontoctachoric antiprism
- Small omnisnub bitetracontoctachoric antiprism
- Great omnisnub bitetracontoctachoric antiprism
- Prismatic transitional omnisnub bitetracontoctachoric antiprism
- Cubic transitional omnisnub bitetracontoctachoric antiprism
- Snub hexacosihecatonicosachoric antiprism

- Rectified isotoxal dodecateric polytera (3 total)
- Truncated isotoxal dodecateric polytera (3 total)
- Dodecateric doublings (at least 6 total)
- Decachoric, demitesseractic and tetracontoctachoric alterprisms (2 total)
- Chiral decachoric and tetracontoctachoric alterprisms (16 total)
- Small biomnisnub decachoric alterprism/Small biomnisnub tetracontoctachoric alterprism
- Medial biomnisnub decachoric alterprism/Medial biomnisnub tetracontoctachoric alterprism
- Great biomnisnub decachoric alterprism/Great biomnisnub tetracontoctachoric alterprism
- Small gyrowedged biomnisnub decachoric alterprism/Small gyrowedged biomnisnub tetracontoctachoric alterprism
- Great gyrowedged biomnisnub decachoric alterprism/Great gyrowedged biomnisnub tetracontoctachoric alterprism
- Antipodial biomnisnub decachoric alterprism/Antipodial biomnisnub tetracontoctachoric alterprism
- Prismatic transitional biomnisnub decachoric alterprism/Prismatic transitional biomnisnub tetracontoctachoric alterprism
- Tetrahedral transitional biomnisnub decachoric alterprism/Cubic transitional biomnisnub tetracontoctachoric alterprism

- Pyritoicositetrachoric alterprisms (3 total)
- Toxitic polychoric alterprisms (3 total)
- Ixitic polychoric alterprisms (16 total)
- Tetrahedral hecatonicosachoric alterprism
- Snub tetrahedral hecatonicosachoric alterprism
- Truncated tetrahedral hecatonicosachoric alterprism
- Tetrahedral-snub tetrahedral hecatonicosachoric alterprism
- Hecatonicosadiminished hecatonicosachoric alterprism
- Ambotruncatotetrahedral hecatonicosachoric alterprism
- Small snub truncatotetrahedral hecatonicosachoric alterprism
- Medial snub truncatotetrahedral hecatonicosachoric alterprism
- Great snub truncatotetrahedral hecatonicosachoric alterprism
- Truncatosnub tetrahedral hecatonicosachoric alterprism
- Small snub rhombatotetrahedral hecatonicosachoric alterprism
- Medial snub rhombatotetrahedral hecatonicosachoric alterprism
- Great snub rhombatotetrahedral hecatonicosachoric alterprism
- Small snub tetrahedral-transitional hecatonicosachoric alterprism
- Great snub tetrahedral-transitional hecatonicosachoric alterprism
- Snub truncatotetrahedral-transitional hecatonicosachoric alterprism

- Octahedral-square prismatic quotient prisms (5 total)
- Disphenoids (infinite, consists of two identical orthogonal regular polygons with a height between them)
- Prisms (infinite, consists of all the prisms of the 4D isogonal categories not mentioned above and are not duoprisms themselves)
- Polygonal-polyhedral duoprisms (infinite, includes polygon-antiprism duoprisms)
- Duoprismatic prisms (infinite)
- Polygonal-polyhedral duoantiprisms (infinite)
- Duoantiprismatic antiprisms (infinite)
- Polygonal-polyhedral prismantiprismoids (infinite)
- Double antiprismoidal antiprisms (infinite)
- Truncated tetrahedral duoalterprisms (infinite, consists of a ring of truncated tetrahedral alterprisms)
- Truncated tetrahedral prismalterprismoids (infinite, similar to the truncated tetrahedral duoalterprisms but with alternating truncated tetrahedral prisms and truncated tetrahedral alterprisms)
- Duoantiwedges (infinite, all members are scaliform except for the digonal duoantiwedge, which is the square disphenoid)
- Duotegmatic alterprisms (infinite, related to the duotegums)
- Duoprismatic antialterprisms (infinite, related to the antiditetragoltriates)
- Duoprismatic cupoliprisms (infinite, related to the duoexpandoprisms)
- Duoprismatic truncatocupoliprisms (infinite, related to the duotruncatoprisms)
- Duoprismatic cupolialterprisms (infinite, related to the duotruncatoalterprisms)
- Duoantiprismatic antialterprisms (infinite, related to the double gyroantiprismoids)
- Snub duoantiprismatic alterprisms (infinite, related to the double snub trapezoprismoids)
- Prismantiprismoidal alterprisms (infinite, related to the double prismantiprismoids)
- Prismantiprismoidal antialterprisms (infinite, related to the double gyroprismantiprismoids)
- Truncatoprismantiprismoidal alterprisms (infinite, related to the double truncatoprismantiprismoids)
- Snub prismantiprismoidal alterprisms (infinite, related to the double snub prismantiprismoids)
- Duoprismatic swirlprismatic alterprisms (infinite, related to the double duoprismatoswirlprisms)
- Alterantiprismatic swirlprismatic alterprisms (infinite, related to the double alterantiprismatoswirlprisms)
- Prismantiprismatic swirlprismatic alterprisms (infinite, related to the double prismantiprismatoswirlprisms)
- Truncatoprismantiprismatic swirlprismatic alterprisms (infinite, related to the double truncatoprismantiprismatoswirlprisms)
- Step prism alterprisms (infinite)
- Swirlchoron alterprisms (infinite)

## Types of convex isogonal elementary tessellations[edit | edit source]

### Tilings[edit | edit source]

- Regular tilings (3 total)
- Square tiling (lower symmetric variants such as the cantellated square tiling x4o4y, or the rectangular tiling that is the honeycomb product of two ifferent sized apeirogons, also exist)
- Triangular tiling (lower symmetric variants can be formed by alternating the variants of the hexagonal tiling)
- Hexagonal tiling (lower symmetric variants are the truncated triangular tiling o6x3y and the omnitruncated cyclotriangular tiling x3y3z3*a)

- Uniform tilings (8 total)
- Truncated square tiling (a subsymmetric form is the omnitruncated form x4y4z)
- Snub square tiling (a lower symmetric variant is formed by alternating the omnitruncate)
- Trihexagonal tiling (a subsymmetric variant is the cyclotruncated triangular tiling x3y3o3*a)
- Truncated hexagonal tiling
- Small rhombitrihexagonal tiling (a subsymmetric variant is the pyritosnub(?) hexagonal tiling x6s3s)
- Great rhombitrihexagonal tiling
- Snub trihexagonal tiling
- Elongated triangular tiling

### Honeycombs[edit | edit source]

- Regular honeycombs (1 total)
- Cubic honeycomb (subsymmetric forms include various honeycomb products, such as the square prismatic honeycomb or the product of three apeirogons of different sizes)

- Uniform honeycombs (27 total)
- Rectified cubic honeycomb (plus subsymmetric variants with demicubic and quarter-cubic symmetry)
- Truncated cubic honeycomb
- Bitruncated cubic honeycomb (plus subsymmetric variants with single cubic, demicubic, and quarter-cubic symmetry)
- Small rhombated cubic honeycomb (plus variant iwth demicubic symmetry)
- Great rhombated cubic honeycomb (plus variants with demicubic symmetry)
- Prismatorhombated cubic honeycomb
- Great prismated cubic honeycomb
- Tetrahedral-octahedral honeycomb
- Truncated tetrahedral-octahedral honeycomb (plus variants with quarter-cubic symmetry)
- Cyclotruncated tetrahedral-octahedral honeycomb (plus variants with quarter-cubic symmetry)
- Great rhombated tetrahedral-octahedral honeycomb
- Small rhombated tetrahedral-octahedral honeycomb
- Gyrated tetrahedral-octahedral honeycomb
- Elongated tetrahedral-octahedral honeycomb
- Gyroelongated tetrahedral-octahedral honeycomb
- Gyrated triangular prismatic honeycomb
- Gyroelongated triangular prismatic honeycomb
- Truncated square prismatic honeycomb
- Snub square prismatic honeycomb
- Triangular prismatic honeycomb
- Hexagonal prismatic honeycomb
- Trihexagonal prismatic honeycomb
- Truncated hexagonal prismatic honeycomb
- Small rhombitrihexagonal prismatic honeycomb
- Great rhombitrihexagonal prismatic honeycomb
- Snub trihexagonal prismatic honeycomb
- Elongated triangular prismatic honeycomb

- Scaliform honeycombs (23 total)
- Gyrated rectified cubic honeycomb
- Prismatorhombisnub bicubic honeycomb
- Dissected rectified cubic honeycomb (6Q3-2S3-gyro)
- Dissected gyrated rectified cubic honeycomb (6Q3-2S3-ortho)
- Dissected elongated rectified cubic honeycomb (3Q3-S3-2P6-2P3-gyro)
- Dissected elongated gyrated rectified cubic honeycomb (3Q3-S3-2P6-2P3-ortho)
- Dissected small rhombated tetrahedral-octahedral honeycomb (3Q4-T-2P8-P4)
- Truncated square alterprismatic honeycomb (6Q4-2T)
- Bisected tetrahedral-octahedral honeycomb (10Y4-8T-0)
- Bisected gyrated tetrahedral-octahedral honeycomb (10Y4-8T-3)
- Alternate-bisected tetrahedral-octahedral honeycomb (10Y4-8T-2-alt)
- Helical-bisected tetrahedral-octahedral honeycomb (10Y4-8T-2-hel)
- Alternate-bisected gyrated tetrahedral-octahedral honeycomb (10Y4-8T-1-alt)
- Helical-bisected gyrated tetrahedral-octahedral honeycomb (10Y4-8T-1-hel)
- Elongated bisected tetrahedral-octahedral honeycomb (5Y4-4T-4P4)
- Paraelongated bisected tetrahedral-octahedral honeycomb (5Y4-4T-6P3-sq-para)
- Skew-elongated bisected tetrahedral-octahedral honeycomb (5Y4-4T-6P3-sq-skew)
- Bisected elongated tetrahedral-octahedral honeycomb (5Y4-4T-6P3-tri-0)
- Bisected elongated gyrated tetrahedral-octahedral honeycomb (5Y4-4T-6P3-tri-3)
- Alternate-bisected elongated tetrahedral-octahedral honeycomb (5Y4-4T-6P3-tri-2-alt)
- Helical-bisected elongated tetrahedral-octahedral honeycomb (5Y4-4T-6P3-tri-2-hel)
- Alternate-bisected elongated gyrated tetrahedral-octahedral honeycomb (5Y4-4T-6P3-tri-1-alt)
- Helical-bisected elongated gyrated tetrahedral-octahedral honeycomb (5Y4-4T-6P3-tri-1-hel)

- Rectified isotoxal honeycombs (2 total)
- Truncated isotoxal honeycombs (2 total)
- Cubic honeycomb doublings (20 total)
- Bicubic honeycomb
- Biambocubic honeycomb
- Bitruncatocubic honeycomb
- Bimesotruncatocubic honeycomb
- Small birhombatocubic honeycomb
- Medial birhombatocubic honeycomb
- Great birhombatocubic honeycomb
- Small bicantitruncatocubic honeycomb
- Medial bicantitruncatocubic honeycomb
- Great bicantitruncatocubic honeycomb
- Transitional bicantitruncatocubic honeycomb
- Biprismatocubic honeycomb
- Small biprismatorhombatocubic honeycomb
- Medial biprismatorhombatocubic honeycomb
- Great biprismatorhombatocubic honeycomb
- Transitional biprismatorhombatocubic honeycomb
- Small biomnitruncatocubic honeycomb
- Great biomnitruncatocubic honeycomb
- Prismatic transitional biomnitruncatocubic honeycomb
- Cubic transitional biomnitruncatocubic honeycomb

- Chiral cubic honeycomb doublings (8 total)
- Small biomnisnub cubic honeycomb
- Medial biomnisnub cubic honeycomb
- Great biomnisnub cubic honeycomb
- Small gyrowedged biomnisnub cubic honeycomb
- Great gyrowedged biomnisnub cubic honeycomb
- Antipodial biomnisnub cubic honeycomb
- Prismatic transitional biomnisnub cubic honeycomb
- Cubic transitional biomnisnub cubic honeycomb

- Bisnub cubic honeycomb doublings (5 total)
- Tetrahedral-octahedral honeycomb doublings (12 total)
- Bitetrahedral honeycomb
- Biambotetrahedral honeycomb
- Small bitruncatotetrahedral honeycomb
- Medial bitruncatotetrahedral honeycomb
- Great bitruncatotetrahedral honeycomb
- Transitional bitruncatotetrahedral honeycomb
- Bimesotruncatotetrahedral honeycomb
- Biprismatosnub cubic honeycomb
- Birhombatotetrahedral honeycomb
- Transitional birhombatotetrahedral honeycomb
- Prismatic transitional biomnitruncatotetrahedral honeycomb
- Tetrahedral transitional biomnitruncatotetrahedral honeycomb

- Alternated honeycombs based on regular and prismatic symmetries (13 total)
- Bisnub cubic honeycomb
- Snub rectified cubic honeycomb
- Snub bitetrahedral honeycomb
- Snub bicubic honeycomb
- Snub bimesocubic honeycomb
- Bicantisnub cubic honeycomb (topologically equivalent to an alternated transitional bicantitruncatocubic honeycomb)
- Snub biprismatocubic honeycomb
- Small omnisnub bicubic honeycomb
- Great omnisnub bicubic honeycomb
- Prismatic transitional omnisnub bicubic honeycomb
- Cubic transitional omnisnub bicubic honeycomb
- Snub square antiprismatic honeycomb
- Snub trihexagonal antiprismatic honeycomb

- Edge-snub honeycombs (8 total)

## Optimization[edit | edit source]

Many isogonal polytopes can be continuously deformed until they have a single edge length and therefore become scaliform. However, not all isogonal polytopes have this property, such as the snub decachoron.

Most isogonal polytopes can be **optimized** in the sense of minimizing the edge length variations. However, some isogonal polytopes cannot be optimized in any meaningful way, such as the antiditetragoltriates, because their optimized forms are not topologically identical to these shapes. If an isogonal polytope has no variations in its highest symmetry, such as the triangular duoantiprism, it is considered to be optimized.

There are two methods of optimization: the **absolute-value method** and the **ratio method**, which may yield different solutions. Both methods rely on a set variable, defined by a constant. All regular, uniform and scaliform polytopes are optimized polytopes in both of the two methods.

### Absolute-value method[edit | edit source]

This method uses all possible sums of absolute values of differences between two edge length types, and then computing the minimum value of the resulting function. The number of sums required for an isogonal polytope with *n* edge types is *n*(*n*-1)/2. The feasibility of this method depends on the number of variables and the number of edge lengths.

As an example, a truncated octahedron in B_{3} symmetry can be defined by all permutations and sign changes of (0, *a*, *b*), where *a* and *b* are nonzero and distinct from each other. The variable *a* can be set to a constant value such as 1, yielding (0, 1, *b*). The two edge lengths are given by:

*d1*= √2 (distance between (0, 1,*b*) and (1, 0,*b*))*d2*= √2|b-1| (distance between (0, 1,*b*) and (0,*b*, 1))

Since there are only two edge lengths, only one absolute value sum is needed: f(x) = |√2-√2|b-1||. The minimum of this function is attained when *b* is equal to 2, which evaluates to 0. Therefore, any optimized truncated octahedron in this sense is equilateral and hence uniform.

### Ratio method[edit | edit source]

This method is dependent on the polytope's largest and smallest edge lengths. It involves dividing the largest edge length by the smallest edge length, giving a value equal to or greater than 1, and then finding the lowest possible value. Due to the unpredictability of edge lengths per given variables, a convenient method is to enumerate all divisions between two edge types and their reciprocals, and then finding the lowest possible value greater than (or equal to) 1 that is not within the "area" of the functions. For an isogonal polytope with *n* edge lengths, *n*(*n*-1) divisions are required. For topologically similar isogonal polytopes, the largest and smallest edge lengths may vary between edge length types depending on the variable used.

We can use the truncated octahedron example from earlier. For *b* > 2, √2|b-1| is larger than √2, and for 1 < *b* < 2, √2|b-1| is smaller than √2. Dividing the appropriate edge lengths results in a minimum ratio of 1:1 at *b* = 2, implying that any optimized truncated octahedron in this sense is equilateral and hence uniform.

It is more useful in most cases than the absolute-value method, especially when an isogonal polytope has no uniform realization. The optimized form in this case is the closest to what a "uniform" variant would look like, whereas the result obtained through the absolute-value method has a bigger ratio in some cases. Notable examples are the step prisms, which are more reliably optimized generally than the absolute-value method.