Math 2.0

Geometry Object

24-cell

Builds on Schläfli symbolHurwitz quaternions

Description

The 24-cell is the convex polytope in ℝ4\R^4 whose vertices are the 2424 points obtained from (±1,±1,0,0)(\pm 1, \pm 1, 0, 0) by permuting the coordinates. It is a convex regular polytope with Schläfli symbol {3,4,3}\set{3, 4, 3}. It has 2424 vertices, 9696 edges, 9696 triangular faces and 2424 cells, which are regular octahedra.

Check of the vertex and edge counts: there are (42)=6\binom{4}{2} = 6 ways to place the two nonzero coordinates and 44 choices of signs, so 2424 vertices. All vertices lie at distance 2\sqrt2 from the origin. The nearest neighbours of (1,1,0,0)(1,1,0,0) are the 88 points (1,0,±1,0)(1,0,\pm1,0), (1,0,0,±1)(1,0,0,\pm1), (0,1,±1,0)(0,1,\pm1,0), (0,1,0,±1)(0,1,0,\pm1), at distance 2\sqrt2. So each vertex lies on 88 edges, and there are 24⋅8/2=9624 \cdot 8 / 2 = 96 edges. The Euler characteristic check 24−96+96−24=024 - 96 + 96 - 24 = 0 holds, as for every convex 44-polytope.

Remarks

The vertex figure is a cube, in agreement with the symbol: the last two entries {4,3}\set{4, 3} are the symbol of the cube. The symbol reads the same backwards, and indeed the 24-cell is self-dual. Apart from polygons and simplices, it is the only self-dual convex regular polytope, and it has no analogue in other dimensions. Its symmetry group has order 11521152. Another model: the 2424 units ±1\pm 1, ±i\pm i, ±j\pm j, ±k\pm k and 12(±1±i±j±k)\frac12(\pm 1 \pm i \pm j \pm k) of the Hurwitz quaternions are the vertices of a 24-cell.

Reading path

To understand this page from the ground up, read these 12 pages in order. The foundations come first.

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Euler's four-square identity, statement, no evidence levelEuler's four-square i…statementGroup, concept, no evidence levelGroupconceptOrder (ring theory), concept, no evidence levelOrder (ring theory)conceptQuaternion algebra, concept, no evidence levelQuaternion algebraconceptSchläfli symbol, concept, no evidence levelSchläfli symbolconceptLagrange's four-square theorem, statement, no evidence levelLagrange's four-squar…statementRing, definition, no evidence levelRingdefinitionUnit of a ring, definition, no evidence levelUnit of a ringdefinitionDivision ring, definition, no evidence levelDivision ringdefinitionThe quaternions, example, no evidence levelThe quaternionsexampleHurwitz quaternions, object, no evidence levelHurwitz quaternionsobject24-cell, object, no evidence level24-cellobject
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Euler's four-square identity statement
  2. Group concept
  3. Order (ring theory) concept
  4. Quaternion algebra concept
  5. Schläfli symbol concept
  6. Lagrange's four-square theorem statement
  7. Ring definition
  8. Unit of a ring definition
  9. Division ring definition
  10. The quaternions example
  11. Hurwitz quaternions object
  12. 24-cell object

Further reading

  1. Web https://en.wikipedia.org/wiki/24-cell
About this page
TagSH9NBM8JRR
URIm2://SH9NBM8JRR
Kindobject
WritingW0 stub
MSC51M20, 52B15, 52B11
Instance ofRegular polytope
UsesSchläfli symbol, Hurwitz quaternions
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