212 July 31 – 212

Mathematica 10.1. Non-Periodic Tiling. From a script by Stephen Wolfram. This particular set of 12 tiles is the smallest set that can be arranged to cover the plane—but only in a non-periodic pattern. Non-periodic tilings tile the plane in a non-repeating manner and can also tile it in a regular periodic manner. Ending My “Mathematica-series” with a script from S.Wolfram is also an homage to the creator of this wonderful program and the many writers who helped me convert mathematics problems into artistic visualization all month long.

211 July 30 – 211

Mathematica 10.1. Four-dimensional immersion of the Klein bottle. Although the 4D immersion has no self-intersections, projecting into 3D space makes it appear as if there were. The Klein bottle is a closed non-orientable surface that has no inside or outside and was originally described by Felix Klein in 1882. From a script by Richard Hennigan.

210 July 29 – 210

Mathematica 10.1 The “last geometric statement of Jacobi” says that the conjugate locus of a non-umbilic point on the triaxial ellipsoid has four cusps – which was later proved by Itoh and Kiyohara. Here is an artistic interpretation of a locus of conjugate points along a “spray” of unit speed geodesics emanating from a “base point” on the ellipsoid. From an original script by Thomas Waters.

209

Mathematica 10.1. Scherk’s surface are periodic minimal surfaces – a singly and a doubly periodic surface. The two surfaces are conjugates of each other. They are named after mathematician H. Scherk who first described them in 1835. From a script by Enrique Zeleny.

208

Mathematica 10.1 Closed periodic paths: a ray is reflected multiple times inside an ellipse. The black dots are the two foci of the ellipse. From a script by Michael Trott

206 July 25 – 206

Mathematica 10.1 The volume of the regular octahedron is four times that of the regular tetrahedron through decomposition. The large octahedron has a side that is twice the length of any of the small octahedra. So the volume of the large octahedron is eight times as much as a small one. But the large octahedron is made of six small octahedra and eight tetrahedra. So the eight tetrahedra must have a volume equal to two small octahedra, and the ratio is 4 to 1. From an original script by Dan Suttin.