Regular 4-polytopes

In geometry, a tesseract or 4-cube is a four-dimensional hypercube, analogous to a two-dimensional square and a three-dimensional cube. Just as the perimeter of the square consists of four edges and the surface of the cube consists of six square faces, the hypersurface of the tesseract consists of eight cubical cells, meeting at right angles. The tesseract is one of the six convex regular 4-polytopes.

In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol {3,3,3}. It is a 5-vertex four-dimensional object bounded by five tetrahedral cells. It is also known as a C5, hypertetrahedron, pentachoron, pentatope, pentahedroid, tetrahedral pyramid, or 4-simplex (Coxeter's \alpha_4 polytope), the simplest possible convex 4-polytope, and is analogous to the tetrahedron in three dimensions and the triangle in two dimensions. The 5-cell is a 4-dimensional pyramid with a tetrahedral base and four tetrahedral sides.

thumb|right|Net (polyhedron)|Net In geometry, the 120-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {5,3,3}. It is also called a C120, dodecaplex (short for "dodecahedral complex"), hyperdodecahedron, polydodecahedron, hecatonicosachoron, dodecacontachoron and hecatonicosahedroid.

In geometry, the 16-cell is the regular convex 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,3,4}. It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century. It is also called C16, hexadecachoron, or hexdecahedroid.

thumb|right|Net (polyhedron)|Net In geometry, the 600-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,3,5}. It is also known as the C600, hexacosichoron, hexacosihedroid and hypericosahedron. It is also called a tetraplex (abbreviated from "tetrahedral complex") and a polytetrahedron, being bounded by tetrahedral cells.

In four-dimensional geometry, the 24-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,4,3}. It is also called C24, or the icositetrachoron, octaplex (short for "octahedral complex"), icosatetrahedroid, octacube, hyper-diamond or polyoctahedron, being constructed of octahedral cells.

{| class="wikitable" align="right" style="margin-left:10px" width="250" !bgcolor=#e7dcc3 colspan=2|11-cell |- |bgcolor=#ffffff align=center colspan=2|240pxThe 11 hemi-icosahedra with vertices labeled by indices 0..9,t. Faces are colored by the cell it connects to, defined by the small colored boxes. |- |bgcolor=#e7dcc3|Type||Abstract regular 4-polytope |- |bgcolor=#e7dcc3|Cells||11 hemi-icosahedron150px |- |bgcolor=#e7dcc3|Faces||55 {3} |- |bgcolor=#e7dcc3|Edges||55 |- |bgcolor=#e7dcc3|Vertices||11 |- |bgcolor=#e7dcc3|Vertex figure||hemi-dodecahedron |- |bgcolor=#e7dcc3|Schläfli symbol||\{\{3,

only regular space-filling tessellation of the cube

{| class="wikitable" align="right" style="margin-left:10px" width="250" !bgcolor=#e7dcc3 colspan=2|57-cell |- |bgcolor=#e7dcc3|Type||Abstract regular 4-polytope |- |bgcolor=#e7dcc3|Cells||57 hemi-dodecahedra150px |- |bgcolor=#e7dcc3|Faces||171 {5} |- |bgcolor=#e7dcc3|Edges||171 |- |bgcolor=#e7dcc3|Vertices||57 |- |bgcolor=#e7dcc3|Vertex figure||hemi-icosahedron |- |bgcolor=#e7dcc3|Schläfli type||{5,3,5} |- |bgcolor=#e7dcc3|Symmetry group||order 3420Abstract L2(19) |- |bgcolor=#e7dcc3|Dual||self-dual |- |bgcolor=#e7dcc3|Properties||Regular |} In mathematics, the 57-cell (pentacontaheptachoron)

regular Schläfli-Hess 4-polytope with 600 vertices