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Platonic Solids in Every Dimension
Transcript of Platonic Solids in Every Dimension
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.