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  "body": "\n## Description\n\nThe *600-cell* is the convex [[regular polytope|F60PD912D5]] in $\\R^4$ with [[Schläfli symbol|MAE6BF4DFK]] $\\set{3, 3, 5}$.\nIts cells are $600$ regular tetrahedra, five around each edge, and its vertex figure is a regular icosahedron.\nIt has $120$ vertices, $720$ edges, $1200$ triangular faces and $600$ cells.\n\nA model: let $\\varphi = (1 + \\sqrt5)/2$ be the [[golden ratio|0SC8T7548D]].\nTake the $8$ points obtained from $(\\pm1, 0, 0, 0)$ by permuting coordinates, and the $16$ points $\\frac12(\\pm1, \\pm1, \\pm1, \\pm1)$.\nAdd the $96$ points obtained from $\\frac12(\\pm\\varphi, \\pm1, \\pm\\varphi^{-1}, 0)$ by even permutations of the coordinates.\nThese $8 + 16 + 96 = 120$ points lie on the unit sphere, and their convex hull is a 600-cell with edge length $\\varphi^{-1}$.\n\nCheck of the counts: the icosahedral vertex figure has $12$ vertices, so each vertex lies on $12$ edges and there are $120 \\cdot 12 / 2 = 720$ edges.\nEach tetrahedron has $4$ triangles, each lying in $2$ cells, so $600 \\cdot 4 / 2 = 1200$ faces.\nThen $120 - 720 + 1200 - 600 = 0$.\n\n## Remarks\n\nThe 600-cell is the dual of the [[120-cell|3RQEK50CP0]], and its symmetry group is the reflection group $H_4$ of order $14400$.\nRead as unit quaternions, the $120$ vertices of the model above form a group under multiplication, the binary icosahedral group.\nIt is the four-dimensional analogue of the icosahedron $\\set{3, 5}$.\n\n## Explanations\n\n### L1 human\n\nThe 600-cell is a four-dimensional solid made of $600$ regular tetrahedra, with $120$ corners.\nAround each corner the tetrahedra are arranged like the triangles of an icosahedron.\nIt is the four-dimensional cousin of the icosahedron, and its corners have coordinates built from the golden ratio.\n",
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