page_title: 24-cell
section_title: Visualization
hierarchical_section_title: 24-cell / Visualization
caption_attribution_description: English: 24-cell face-stacked ring of 6 octahedra
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 is bounded by 24 octahedral cells.  For visualization purposes, it is convenient that the octahedron has opposing parallel faces (a trait it shares with the cells of the tesseract and the 120-cell).  One can stack octahedrons face to face in a straight line bent in the 4th direction into a great circle with a circumference of 6 cells. The cell locations lend themselves to a hyperspherical description. Pick an arbitrary cell and label it the "North Pole". Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd "South Pole" cell. This skeleton accounts for 18 of the 24 cells (2 + 8×2). See the table below.
  There is another related great circle in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the hexagonal geodesics described above. One can easily follow this path in a rendering of the equatorial cuboctahedron cross-section.
  Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere. The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial "equatorial" cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices.  This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a tesseract (8-cell), although they touch at their vertices instead of their faces.
  The 24-cell can be partitioned into disjoint sets of four of these 6-cell great circle rings, forming a discrete Hopf fibration of four interlocking rings. One ring is "vertical", encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.
  Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees.  This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously.
  One can also follow a great circle route, through the octahedrons' opposing vertices, that is four cells long.  These are the square geodesics along four √2 chords described above. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section.  The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell.  This great circle is self dual.  This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells.  The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract.  Each of these subsets can be further equipartitioned into two interlocking great circle chains, four cells long.  Collectively these three subsets now produce another, six ring, discrete Hopf fibration.
caption: An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator
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