Infinite Families
These families can extend to all dimensions.
Simplex
- 2D: Triangle
- 3D: Tetrahedron: 4-face body
- 4D: Pentachoron: 5-cell body
- 5D: Hexateron: 6-tera body
To get the simplex of 1 dimension higher,
add a point in equal distances from all existing points.
Hypercube
- 2D: Square
- 3D: Cube: 6-face body
- 4D: Tesseract: 8-cell body
- 5D: Pentaract:10-tera body
To get the hypercube of 1 dimension higher,
translate it along the new axis of distance 1.
Orthoplex
- 2D: Square
- 3D: Octahedron: 8-face body
- 4D: hexadecachoron: 16-cell body
- 5D: triacontaditeron: 32-tera body
To get the orthoplex of 1 dimension higher,
add two points on the new axis where the distances from the origin is 1.
Demihypercube
Deletion of half of vertices of hypercubes.
- 2D: line
- 3D: Tetrahedron; 3D simplex
- 4D: Hexadecachoron; 4D orthoplex
- 5D: a shape made by 16 pentachoron & 10 hexadecachoron
Double simplex
A convex hull of the compound of two simplex in dual positions.
- 2D: Hexagon.
- 3D: Cube.
- 4D: Bidecachoron; a shape made by 30 irregular tetrahedra.
- Dual of 4D: bitruncated pentachoron.
- 5D: a shape made by 20 triangular duotegums.
- Dual of 5D: birectified hexateron.
Rectified simplex
A shape created by the midpoints of the edges of a simplex.
- 2D: Triangle; 2D simplex.
- 3D: Octahedron; 3D orthoplex.
- 4D: Rectified pentachoron; a shape made by 5 Tet's and 5 Oct's.
- 4D dual: a shape made by 10 triangular bipiramid.
- 5D: a shape made by 6 pentachora & 6 rectified pentachora.
- 5D dual: a shape made by 15 tetrahedral tegums.
Demihypercubic honeycomb
Deletion of half of vertices of hypercubic honeycomb.
- 2D: Square tiling.
- 3D: Octet truss; honeycomb with Tet's and Oct's.
- 4D: Hexadecachoron honeycomb.
- 5D: a honeycomb made by 5-demicube and 5-orthoplex.
Basic space-filling shape
A shape which can fill the space. The fewest components join at each vertex.
- 2D: Hexagon. 3 of them meet at each corner.
- 3D: Truncated Octahedron. 4 of them meet at each corner.
- 4D: Omnitruncated Pentachoron. 5 of them meet at each corner.
- 5D: Omnitruncated Hexateron. 6 of them meet at each corner.
