The Platonic Solids Tetrahedron Octahedron Icosahedron Cube Dodecahedron 3 2 Triangle 4 Tesseract 3 Cubes 90 degrees 116.57 degrees 90 degrees 3 Tetrahedrons 75.5 degrees Hyperpyramid Hypercube 16 Cell 4 Tetrahedrons 120 degrees Hexacosichoron 600 Cell 5 Tetrahedra The Biggest of All 120 Cell 3 Dodecahedra (also way too big) 3 Octahedra 120 Degrees 5 And Higher Simplex Cube Cross-Polytope Simplex 90 Degrees Pentagon Thanks go out to Sergei Tabachnikov and Anatole Katok of MASS at PSU Adrian Ocneanu, for inspiration in visualization 108 Degrees H.S.M. Coxeter, Regular Polytopes. Macmillan Mathematics Paperbacks, 1948 24 Cell Thomas Eliot teliot@willamette.edu Presentation made in Prezi Images made in Stella and Gimp, models built out of Zome and pipecleaners By 70.53 degrees 109.47 degrees 138.19 degrees 60 Degrees Square And infinitely many more Pentachoron Hexadecachoron Octacube Hecatonicosachoron Paul Kunkel and www.whistleralley.com Cross Polytope, or CoCube Universal Regular Polytopes 2d 3d 4d Euclid tjeliot.wordpress.com But most importantly: The internal angle of the polygon How many sides the polygon has a and b are the interior angles of the faces meeting at the edge

c is the interior angle of the third face meeting them at a vertex

In a Regular polytope, a=b=c

is the Dihedral Angle The Convex Regular Polytopes Polytope: bounded on all sides by hyperplanes Convex: select any two points in the shape and the line segment joining them is also in the shape Regular: select any 2d face, and then a 1d edge on that face, and then a 0d vertex on that edge. This is a flag, and can be sent to any other flag can be an isometry. Isometry: any distance preserving transformation of the whole shape. In 3d, these are rotations, reflections, translations, glide reflections, and screw translations. Waaait a tic-that's not 3 faces meeting at a vertex! True-but here's a way around that. Cut the top half off and examine a vertex-3 faces meet there, the 2 triangles and the new square. The angle between the two triangles is the angle we're looking for; now use the Dihedral Formula. Simplex: the convex hull of n+1 equidistant points Cube: the Minkowski sum of the standard basis vectors Cross Polytope: the convex hull of the positive and negative standard basis vectors Convex Hull: the smallest convex set containing them Minkowski Sum: take two vectors and place one at every point on the other. The shape thus traced out is their Minkowski Sum. To add a third vector, trace this shape along that vector, and so on.

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# Platonic Solids in Every Dimension

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