The connection between mathematics and
art goes back thousands of years. Mathematics has been used in the design of Gothic cathedrals, Rose windows, oriental rugs, mosaics and tilings. Geometric forms were fundamental to the cubists and many abstract expressionists, and award-winning sculptors have used topology as the basis for their pieces. Dutch artist M.C. Escher represented infinity, Möbius bands, tessellations, deformations, reflections, Platonic solids, spirals, symmetry, and the hyperbolic plane in his works.

Mathematicians and artists continue to create stunning works in all media and to explore the visualization of mathematics--origami, computer-generated landscapes, tesselations, fractals, anamorphic art, and more.

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"Octahedral Globe from a Window," by Frank A. Farris, Santa Clara University, CAInk jet on paper, 2015.

This coloring of the sphere, based on a photograph of a stained-glass window by Hans Schepker, is invariant under the action of the octahedral group. I created it by mapping the sphere (and the group action) to the plane via stereographic projection and using known techniques for creating complex-valued functions invariant under groups that act on the plane. This image is part of a larger work, "Imaginary Planets." --- Frank Farris Sep 02, 2015

"Alternating Wood Bugs," by Frank A. Farris, Santa Clara University, CAInk jet on paper, 2015.

Had I used a source photograph whose colors reverse exactly when you turn it upside down, the image computed with these wallpaper waves would have exact color-reversing symmetry of type p4g/cmm. (See my book Creating Symmetry for explanation.) However, when you rotate my picture of a freshly cut pine stump, the colors only more-or-less reverse. This causes what I call "approximate color-reversing symmetry." The blond bugs marching northwest have the same outlines as the dark bugs marching northeast, but the details of the insides are quite different. --- Frank Farris Sep 02, 2015

"Random Walk on Sphere," by Daniel GriesPaths meander on a sphere with each small step taken at a random turning angle. The turn angle at each step is required to stay near the previous turn angle, as if the steering wheel of a car can only be turned slowly from one position to the next as the car drives forward. Computing random walks on a sphere has to be handled carefully with local coordinates so all random directions have an equal probability. Coded in Processing. --- Daniel Gries (Hopkins School, New Haven, CT)May 28, 2015

"Parametric Surface," by Daniel GriesA parametric surface like the one shown here is defined by mapping two variables into three-dimensional space. This surface was the result of playing with the points on a sphere, perturbing them in different directions by some sinusoidal functions. The surface is rendered so as to appear to be lit from different positions by differently colored point lights. Coded in Processing. --- Daniel Gries (Hopkins School, New Haven, CT)May 28, 2015

"Perlin Lines," by Daniel GriesLines emerge from regular grid positions in the canvas, and proceed to grow according to a vector field defined by Perlin noise, a generative texture first created by Ken Perlin for computer animation in the 1982 movie "Tron". As the lines grow in this image, they slowly change color from red to yellow, producing the fiery effect. Coded in Processing. --- Daniel Gries (Hopkins School, New Haven, CT)May 28, 2015

"Jellyfish 2," by Daniel GriesThis image is based on a morphing fractal curve method, but shaped into an abstract jellyfish through the use of parametric curves and other mathematical tricks. Additive color blending creates the lighting effect. Coded in JavaScript. --- Daniel Gries (Hopkins School, New Haven, CT)May 28, 2015

"Jellyfish 1," by Daniel GriesThis image is based on a morphing fractal curve method, but shaped into an abstract jellyfish through the use of parametric curves and other mathematical tricks. Additive color blending creates the lighting effect. Coded in JavaScript. --- Daniel Gries (Hopkins School, New Haven, CT)May 28, 2015

"Chaos Game Fractal 3," by Daniel GriesThis image is created with the "Fractal Flame" algorithm, a modification by Scott Draves of the classic Chaos Game algorithm. In this algorithm, a cloud of points is formed when a particle leaps from one position to the next as it is transformed by a linear transformation, randomly selected from a small fixed collection. The colors are a result of a special averaging of colors associated to each transformation. Coded in JavaScript. --- Daniel Gries (Hopkins School, New Haven, CT)May 28, 2015

"Chaos Game Fractal 2," by Daniel GriesThis image is created with the "Fractal Flame" algorithm, a modification by Scott Draves of the classic Chaos Game algorithm. In this algorithm, a cloud of points is formed when a particle leaps from one position to the next as it is transformed by a linear transformation, randomly selected from a small fixed collection. The colors are a result of a special averaging of colors associated to each transformation. Coded in JavaScript. --- Daniel Gries (Hopkins School, New Haven, CT)May 28, 2015

"Chaos Game Fractal 1," by Daniel GriesThis image is created with the "Fractal Flame" algorithm, a modification by Scott Draves of the classic Chaos Game algorithm. In this algorithm, a cloud of points is formed when a particle leaps from one position to the next as it is transformed by a linear transformation, randomly selected from a small fixed collection. The colors are a result of a special averaging of colors associated to each transformation. Coded in JavaScript. --- Daniel Gries (Hopkins School, New Haven, CT)May 28, 2015

"Stripes," by Daniel GriesA curve which is defined through a fractal subdivision process, and then smoothed out, is drawn from the top of the canvas to the bottom. This curve then sweeps from the left of the canvas to the right, as it morphs into other similarly defined fractal curves, and intermittently changes from red to white. Some mathematical tricks are used to create the most turbulence in the middle of the picture, while maintaining fixed straight edges along the sides of the rectangle. --- Daniel Gries (Hopkins School, New Haven, CT)May 28, 2015

"Tetra-Tangle of Four Bow-Tie Links," by Carlo Séquin (University of California, Berkeley)ABS plastic, printed on an FDM machine, 2014
Four sets of three mutually parallel, 3-sided prisms, pointing in 4 different tetrahedral directions, form the core of the TETRAXIS® puzzle. When two triangular prism-end-faces that share a common vertex are closed off with a connecting sweep, a loose "bow-tie" is formed. If all twelve pairs of adjoining triangular end-faces are joined in this manner, the result is a link of 4 mutually interlocking, twisted, prismatic bow-tie loops. This represents an alternating 12-crossing link that has the same connectivity as the “Tetra-Tangle,” which I constructed from 4”-diameter card-board tubes in 1983. The new geometry is has been realized as 4 differently colored sets of 6 tubular snap-together parts each, fabricated on an FDM machine. --- Carlo Séquin (http://www.cs.berkeley.edu/~sequin/) Apr 06, 2015

"Dance of Stars II," by Reza Sarhangi (Towson University, Towson, MD)Heavy paper, 2014
Dance of Stars II is a decorated Great Stellated Dodecahedron, with Schläfli Symbol (5/2, 3), which has been patterned by the sâzeh module tiles, that are used in the majority of tiling that conforms to local fivefold symmetries. In an article that appeared in Science, the authors proposed the possibility of the use of a set of tiles, girih tiles, by the medieval craftsmen, for the composition of the underlying pattern. I used girih tiles and left the dashed outlines in the final tessellation. I also included rectilinear patterns that appear as additional small-brick pattern in the decagonal Gunbad-i Kabud tomb tower in Maragha, Iran. --- Reza Sarhangi Apr 06, 2015

"15 Irregular Hexahedra," by Aaron Pfitzenmaier (student)Paper, 2014
Honorable Mention, 2015 Mathematical Art Exhibition
This model was made from 180 units of four different types. It consists of 15 irregular hexahedra interlocked together. Each hexahedron has 2-fold dihedral symmetry and the positioning of each hexahedron is based on a pair of opposite edges on an icosahedron. This compound has the most complex weaving pattern out of anything I have designed, and is an example of a model where I extensively used the ray tracer POV-Ray as well as a computer program I wrote to aid in the design and folding/assembly process. --- Aaron Pfitzenmaier (http://bit.ly/aaronsorigami)

Apr 06, 2015

"Hyperbolic Catacombs," by Roice Nelson (Austin, TX) and Henry Segerman (Oklahoma State University, Stillwater)Digital Print, 2014
This picture visualizes the regular, self-dual {3,7,3} honeycomb in the upper half space model of hyperbolic 3-space. The cells are {3,7} tilings and the vertex figure is a {7,3} tiling. The cells have infinite volume: the vertices are "ultra-ideal", living beyond the boundary of hyperbolic space. The intersection of each cell with the boundary is an infinite collection of heptagons, together with a disk. The white ceiling and each red "creature" are isometric cells; for all other cells we only show the intersection with the boundary of hyperbolic space, on the floor of the catacombs. Every disk on the floor containing a {7,3} tiling is associated with an ultra-ideal vertex of the honeycomb. --- Roice Nelson (http://google.com/+roicenelson) and
Henry Segerman (http://segerman.org) Apr 06, 2015