Send the link below via email or IMCopy
Present to your audienceStart remote presentation
- Invited audience members will follow you as you navigate and present
- People invited to a presentation do not need a Prezi account
- This link expires 10 minutes after you close the presentation
- A maximum of 30 users can follow your presentation
- Learn more about this feature in the manual
Do you really want to delete this prezi?
Neither you, nor the coeditors you shared it with will be able to recover it again.
Make your likes visible on Facebook?
Connect your Facebook account to Prezi and let your likes appear on your timeline.
You can change this under Settings & Account at any time.
Platonic Solids in Every Dimension
Thomas Elioton 7 April 2011
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.