Just a few geometry-related things I like.

    • Henry Segerman has various math-related creations, one of which is this fascinating visualization of the stereographic projection with a 3D-printed object.

      This is a candidate in NSF’s visualization challenge; you can see some other candidates competing in the same section here. Segerman’s website has a lot of other interesting things.
    • Wikipedia says that the term stereographic projection was coined by François d’Aguilon. A search for his Six Books of Optics gives an page where you can flip through the book, and find the following illustration, which is another visualization of the stereographic projection.
      Stereographic_ProjectionHere is a direct link to the page. This was apparently drawn for the book by the famous painter Rubens. How cool is that?
    • Another famous projection of the sphere is onto a cylinder, instead of the plane, as in the image below.
      Archimedes realized that a region on the sphere and its projected version on the cylinder have the same area, which allows one to calculate the area of the sphere using the area of the cylinder (which is easier to figure out). He liked his discovery so much that he requested to have a picture of a sphere inscribed in a cylinder on his tomb. (Amusingly, in his book on PDEs, V.I. Arnold calls this projection the Archimedes symplectomorphism.)

      The Roman philosopher/politician Cicero writes about finding Archimedes’ tomb in Sicily, 137 years after Archimedes’ death. Search for “obscure mathematician” (ha!) here to read a translation. Here is a painting by Pierre-Henri de Valenciennes, depicting Cicero’s (claimed) discovery:

    • Finally, another one of Archimedes’ findings, demonstrated beautifully in an animation by Zachary Abel, who summarizes it as “hemisphere + cone = cylinder”.
      Taken from Abel’s blog.

More on Archimedes some other time, hopefully.

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