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

July 28 – 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

July 27 – 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

207

July 26 – 207

Mathematica 10.1 Geometry of a quartic polynomial. Quartic describes an object of the “fourth order” A Quartic function is a polynomial function of degree 4. From a script by Salvador Jesús Gutiérrez Martínez.

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.

205

July 24 – 205

Mathematica 10.1. Geometry of an n cube. An n cube or measure polytope, is an hypercube – a generalization of a 3-cube to n dimensions. It is a regular polytope with mutually perpendicular sides. From a script by Andrew Mueller

204

July 23 – 204

Mathematica 10.1. Multiplication Tables for the Group of Integers Modulo n. Modulo are natural numbers in which all integers having the same remainder are considered equivalent: Euclid set the foundation of modular arithmetic in the third century BC. C. F. Gauss developed the modern approach to modular arithmetic in the early 1800s. Modular arithmetic permeates the fields of number theory, group theory, knot theory, abstract algebra, cryptography, computer science and even the visual and musical arts. From a script by Jaime Rangel-Mondragon.

203

July 22 – 203

Mathematica 10.1. Rotating a cuboid to form 4 Möbius bands. A Möbius band, named after nineteenth century German mathematician A. F.Möbius, is a one-sided nonorientable surface obtained by cutting a closed band into a single strip. Interestingly, in music theory, the space of all two-note chords, known as dyads, takes the shape of a Möbius strip. From a script by Sándor Kabai.