page_title: 24-cell
section_title: Hypercubic chords
hierarchical_section_title: 24-cell / Geometry / Coordinates / Hypercubic chords
caption_attribution_description: English: The 24 vertices of the 24-cell are distributed at four different chord lengths from each other: √1, √2, √3, or √4. The √1 edges occur in 16 hexagonal great circles. The √2 chords occur in 18 square great circles. The √3 chords occur in 32 triangular great circles. The √4 chords occur as 12 vertex-to-vertex diameters, the 24 radii around the 25th central vertex.
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 vertices of the 24-cell are distributed at four different chord lengths from each other: √1, √2,  √3 and √4.
  Each vertex is joined to 8 others by an edge of length 1, spanning 60° =  of arc. Next nearest are 6 vertices located 90° =  away, along an interior chord of length √2. Another 8 vertices lie 120° =  away, along an interior chord of length √3. The opposite vertex is 180° = π away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center can be treated as a 25th canonical apex vertex, which is 1 edge length away from all the others.
  To visualize how the interior polytopes of the 24-cell fit together (as described below), keep in mind that the four chord lengths (√1, √2, √3, √4) are the long diameters of the hypercubes of dimensions 1 through 4: the long diameter of the square is √2; the long diameter of the cube is √3; and the long diameter of the tesseract is √4. Moreover, the long diameter of the octahedron is √2 like the square; and the long diameter of the 24-cell itself is √4 like the tesseract.
caption: Vertex geometry of the radially equilateral[b] 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.
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