4-polytopes
A 4-polytope, also known as a polychoron, polycell, or polyhedroid, represents a four-dimensional polytope. This geometric entity is both connected and closed, comprising various lower-dimensional polytopal components, including vertices, edges, faces (polygons), and cells (polyhedra). It is worth noting that each face of a 4-polytope is shared by precisely two cells. The discovery of 4-polytopes dates back to before 1853 by Ludwig Schläfli, a prominent Swiss mathematician.
Regular 4-polytopes
A regular 4-polytope refers to a regular polytope in four dimensions. These remarkable structures are the four-dimensional counterparts of regular polyhedra in three dimensions and regular polygons in two dimensions.
In total, there exist sixteen regular 4-polytopes, comprising six convex polytopes and ten star polytopes. The convex polytopes possess a fully enclosed and convex shape, while the star polytopes exhibit a more intricate and star-like configuration.
Regular convex 4-polytopes
In the mid-19th century, Ludwig Schläfli, a Swiss mathematician, provided the initial description of convex regular 4-polytopes, identifying precisely six such figures.
The convex regular 4-polytopes serve as the four-dimensional counterparts to the Platonic solids in three dimensions and the convex regular polygons in two dimensions.
Among the six regular convex 4-polytopes, five exhibit clear analogies to their corresponding Platonic solids. However, the sixth one, known as the 24-cell, lacks a direct regular analogue in three dimensions. Nonetheless, a pair of irregular solids, namely the cuboctahedron and its dual, the rhombic dodecahedron, offer partial analogues to the 24-cell in complementary ways. Together, they can be seen as a three-dimensional analogue of the 24-cell.
Each convex regular 4-polytope is enclosed by a collection of three-dimensional cells, all of which are Platonic solids sharing the same type and size. These cells are systematically arranged, fitting together face-to-face in a regular manner.
5-cell, pentachoron, pentatope, 4-simplex
In the realm of four-dimensional geometry, the counterpart of the tetrahedron finds its representation in the 5-cell, alternatively known as the pentachoron, pentatope, or the 4-simplex.
The Schläfli symbol for the 5-cell is \(\{3,3,3\}\).
Vertex-first projection:
- 5 balls
- 5 y2
- 6 g2
Orthogonal projection:
- 5 balls
- 2 b2
- 4 y2
- 2 r1
- 2 r2
16-cell, hexadecachoron, 4-orthoplex
The 16-cell, stands as a regular convex 4-polytope and serves as the four-dimensional counterpart to a Platonic solid. It is one of the six regular convex 4-polytopes initially described by Ludwig Schläfli, the Swiss mathematician, during the mid-19th century. Additionally, it is alternatively known as the hexadecachoron or the 4-orthoplex, as named by John Horton Conway.
This polytope belongs to a family of polytopes referred to as cross-polytopes or orthoplexes and draws an analogy to the octahedron in three dimensions.
The Schläfli symbol for the 16-cell is \(\{3,3,4\}\).
Vertex-first projection:
- 7 balls
- 6 b2
- 12 g2
Three different orthogonal projections (same for all three):
- 8 balls
- 6 b2
- 6 r1
- 6 r2
- 6 y2
8-cell, octachoron, tesseract, 4-cube
The 8-cell, or tesseract represents the four-dimensional equivalent of a cube, analogous to how a cube relates to a square. Just as a cube comprises six square faces, a tesseract’s hypersurface consists of eight cubical cells. The dual polytope of a tesseract is known as the 16-cell.
The Schläfli symbol for the 8-cell is \(\{4,3,3\}\).
Cell-first projection:
- 16 balls
- 12 b1
- 12 b2
- 8 y0
Vertex-first projection:
- 15 balls
- 28 y2
24-cell, icositetrachoron, octaplex, polyoctahedron
The boundary of the 24-cell consists of 24 octahedral cells, with six cells converging at each vertex and three cells meeting at each edge. This arrangement yields a total of 96 triangular faces, 96 edges, and 24 vertices. Notably, the vertex figure of the 24-cell takes the shape of a cube. Remarkably, the 24-cell possesses self-duality.
Unlike the other five convex regular 4-polytopes, the 24-cell does not have a direct counterpart among the five regular Platonic solids in three dimensions. It stands as a unique regular polytope that lacks a regular analogue in the adjacent dimension, whether lower or higher. However, it can be understood as the analogue of a pair of irregular solids: the cuboctahedron and its dual, the rhombic dodecahedron.
The Schläfli symbol for the 24-cell is \(\{3,4,3\}\).
Cell-first projection:
- 8 balls
- 16 b2
- 28 g2
Vertex-first projection:
- 15 balls
- 12 b2
- 32 y2
600-cell, hexacosichoron, tetraplex, polytetrahedron
The 600-cell’s boundary consists of 600 tetrahedral cells, with 20 cells converging at each vertex. This arrangement gives rise to a structure comprising 1200 triangular faces, 720 edges, and 120 vertices. In essence, the 600-cell serves as the four-dimensional analogue of the icosahedron, mirroring the icosahedron’s characteristic of five triangles converging at each vertex. The dual polytope of the 600-cell is known as the 120-cell.
The Schläfli symbol for the 600-cell is \(\{3,3,5\}\).
Vertex-first projection:
- 75 balls
- 120 b1
- 120 y1
- 72 r0
- 72 r1
120-cell, hecatonicosachoron, dodecacontachoron, dodecaplex, polydodecahedron
The boundary of the 120-cell consists of 120 dodecahedral cells, with four cells converging at each vertex. This configuration results in a structure comprising 720 pentagonal faces, 1200 edges, and 600 vertices. In terms of analogy, the 120-cell serves as the four-dimensional counterpart to the regular dodecahedron. Just as a dodecahedron features 12 pentagonal facets, with three facets around each vertex, the 120-cell exhibits 120 dodecahedral facets, with three facets around each edge. The dual polytope of the 120-cell is the 600-cell.
The Schläfli symbol for the 120-cell is \(\{5,3,3\}\).
Cell-first projection:
- 330 balls
- 180 b2
- 200 y2
- 120 r1
- 120 r2