page_title: 24-cell
section_title: Relationships among interior polytopes
hierarchical_section_title: 24-cell / Geometry / Constructions / Relationships among interior polytopes
caption_attribution_description: English: Johannes Kepler discovered that the tetrahedron can be inscribed in the cube, occupying four of the cube's eight corners, with its edges as the diagonals of the six square faces. His drawing of the relationship between the two solids appears on page 181 of book V of his 'Harmonics of the World', published in 1619.
context_page_description: In four-dimensional geometry, the 24-cell is the convex regular 4-polytope with Schläfli symbol {3,4,3}. It is also called C₂₄, or the icositetrachoron, octaplex, icosatetrahedroid, octacube, hyper-diamond or polyoctahedron, being constructed of octahedral cells.
  The boundary of the 24-cell is composed of 24 octahedral cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The vertex figure is a cube. The 24-cell is self-dual. It and the tesseract are the only convex regular 4-polytopes in which the edge length equals the radius.
  The 24-cell does not have a regular analogue in 3 dimensions. It is the only one of the six convex regular 4-polytopes which is not the four-dimensional analogue of one of the five regular Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the cuboctahedron and its dual the rhombic dodecahedron.
context_section_description: The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core. The tesseracts are inscribed in the 24-cell such that their vertices and edges lie on the surface of the 24-cell (they are elements of the 24-cell), but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell such that only their vertices lie on the surface: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior 16-cell edges have length √2.
  The 16-cells are also inscribed in the tesseracts: their √2 edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells. This is reminiscent of the way, in 3 dimensions, two tetrahedra can be inscribed in a cube, as discovered by Kepler. In fact it is the exact dimensional analogy (the demihypercubes), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.
  The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable 4-dimensional interstices between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are 4-pyramids, alluded to above.
caption: Kepler's drawing of tetrahedra inscribed in the cube.[15]
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