24-Cell Perspective Projection

This is a wireframe model of a perspective projection into 3 dimensional space of the vertices of a regular 4 dimensional regular solid, or 4-polytope, called the 24-cell or octaplex. The 24-cell (also called the icositetrachoron, polyoctahedron, or hyperdiamond) is a4 dimensional  regular 3polytope, or polychoron. It has a Schläfli symbol {3,4,3}, which means that the polygon faces of the cells have 3 equal sides, meeting 4 at a vertex, to form an octahedral cell, 3 of which meet around each edge in 4 dimensions. The boundary of the figure is 24 octahedral cells. All together the, octaplex has 96 triangular faces, 96 edges, and 24 vertices. The vertex figure (the shape exposed when a 4D corner is sliced off) is a cube, with 6 octahedra meeting at each vertex. The 24-cell is self-dual, and is the only self-dual polytope that is not a simplex (the multi-dimensional analog of the tetrahedron). It is one the the 6 regular convex 4-polytopes, the others being the 5-cell (simplex), the 8-cell (the 4D analog of the octahedron), the 16-cell, or tesseract, the 4D analog of the cube, the 120-cell, the analog of the dodecahedron, and the 600 cell, the analog of the icosahedron. The straight edges of the 24-cell are preserved in this perspective projection into 3-space, but the lengths of the edges and the angles between them are not preserved. This perspective projection is termed a Schlegel diagram of the polychoron.