Platonic Solids

Platonic Solids are 3D shapes that:

• have faces identical in shape and size
• have the same number of faces meeting at each vertex,
• and are convex.

How many such solids do you think meet these criteria? Surprisingly, there are precisely 5 and you can easily make them at home. With a hands-on approach through construction of your own solids, we will see that this is indeed true! Along the way, we’ll tackle intriguing questions such as whether the sum of angles in a triangle on a Platonic solid always equals 180 degrees, how this compares to a sphere, and what our reality would resemble if lived on such a solid.

Videos:

• A basic overview of what a platonic solid is and what the five solids are:
  • https://www.youtube.com/watch?v=-Ssf__QXKWEA
• An overview of the platonic solids that also include a geometric and a more advanced topological proof of why there is only 5 such solids:
  • https://www.youtube.com/watch?v=04eVLnPrpEYA

Popular works:

• An overview of the platonic solids with good visuals and a proof to why there is only 5:
  • https://www.mathsisfun.com/geometry/platonic-solids-why-five.html
• Another overview with a focus on construction and other topics such as: Symmetries, Schlafli Symbols, and Kepler-Poinsot polyhedra:
  • https://brilliant.org/wiki/regular-polyhedra/#platonic-solids-additional-properties

Technical works:

• An introduction to spherical trigonometry:
  • https://www.math.ucla.edu/~robjohn/math/spheretrig.pdf
• An overview of Angular Defect and Gauss-Bonnet Theorem:
  • https://math.berkeley.edu/~giventh/difgem.pdf