Math ImageryThe 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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"Wisdom," by Stephen Ren261 viewsThis image is centered on -1.5798, and is magnified 6500x from the original fractal. It is generated with 100 iterations of tetration. See hi-resolution versions of Tetration Fractals. --- Stephen Ren
"Towards Infinite Smallness - 2," by Irene Rousseau ( views47 x 47 x 8 cm, Venetian glass on wood hand cut by artist, 2016

This mosaic wall sculpture consists of tessellated (tightly fitted together) patterns and each sculpture is a 3D disc with 18.5" diameter. The hyperbolic sculpture is composed of planes, each suggests negative curvature and represents the concept of infinite smallness contained within the bounding edge. Hyperbolic geometry can be represented as the points in a circular disc with hyperbolic distance defined. A hyperbolic plane has infinite structures within boundaries.In my hyperbolic sculpture repeating patterns which decrease in size at the bounding edge metaphorically represent infinity. Symmetry is a transformation that preserves the distance as shown on the surface tiling patterns of the sculpture. --- Irene Rousseau
"Linked Tetra Frames," by Carlo Séquin (University of California, Berkeley)256 views48 x 40 x 30 cm, pvc-pipes, painted, 2016

With Walt van Ballegooijen we have studied the number of topologically different ways in which two identical tetrahedral wire-frames could be interlinked. Not counting mirror images separately, we have found eleven different configurations. For each topological configuration there is a maximal clearance between the infinitely thin mathematical wire frames. If the edges are expanded to a diameter identical to this clearance, then a rigid structure results. This particular configuration has been scaled up to accommodate fattened edges of 22mm diameter, the size of the PVC pipes employed. --- Carlo Séquin
"A Frieze Sampler," by Frank A. Farris, Santa Clara University, CA256 viewsDigital print on aluminum, 20" x 24", 2016

Friezes take their name from an architectural feature: a band of decoration typically along the top of a wall. In the theory of plane symmetry, the first famous classification result says that there exist exactly seven types of frieze patterns. Every pattern ever constructed by anyone, as long as it repeats exactly along one direction, can be classified as belonging to one of these seven types. Mathematicians, like many humans, like to collect exactly one artifact of each type. Here is my sampler of seven friezes, made from source photographs of California wildflowers. I especially like the fourth one down, which exemplifies a glide reflection symmetry—a flip-and-slide motion that leaves the pattern unchanged. --- Frank A. Farris
"Borromean racks," by Saul Schleimer (University of Warwick, Coventry, UK) and Henry Segerman (Oklahoma State University, Stillwater, OK)255 views10 x 10 x 10 cm, 3D printed nylon plastic, 2013

This sculpture consists of three identical pieces. Each has two identical rods; each rod is a rectangular prism with racks on three of its four sides. These racks mesh with one or two racks on the other pieces, for a total of 12 meshings. The pieces interlink in the fashion of the Borromean rings; their long axes form a standard orthogonal frame. When we take the directions of the racks into account, the sculpture has a handedness. At its middle position it realizes its maximal symmetry, a dihedral group of order six. Curiously, the Borromean Racks give an example of a triple of gears that mesh pairwise, but are not frozen. Challenge: Does this pattern of racks extend to tile three-space? If so, how many degrees of freedom does it have? --- Saul Schleimer and Henry Segerman

"Autumn Moths from a Hike in the Hills," by Frank A. Farris, Santa Clara University, CA255 viewsDigital print on aluminum, 20" x 20", 2016

The big surprise about wallpaper patterns is that there are exactly 17 types. After making many examples of each type, I still return to the type of this pattern as a favorite; it’s called p31m. Notice the three-fold rotational symmetry, but also the alternation of mirror symmetry with three-way pinwheels. For me, this gives a beautiful balance of sameness and variation. It’s symmetric, yet organic. I took the source photograph on an autumn day, hiking in the golden hills above San Jose, CA. This piece shows that interesting patterns can arise from photographs with rather limited color palettes. --- Frank A. Farris
"Mobius Frame with 2 Holes (View II)," by Gwen Fisher ( viewsMaterials: seed beads, Nymo nylon thread

This Mobius Frame is woven from box stitch (also known as 3D right angle weave), which essentially takes the Cartesian tiling of 3-space with cubes, and places one bead on each edge of some subset of the tiling. With box stitch, rows and columns of cubes (attached face to face) can be woven into any continuous arrangement. Then, I add extra beads at the vertices of each cube to give the object more structure and decoration. This Mobius Frame represents two distinct mathematical objects. First, one can view this object as assembled from cube (or cube-like) shapes. We might be tempted to try to build such an object from wood using three long beams and two short beams. However, like the Impossible Triangle, this Mobius Frame cannot be built in 3D using all straight lines and right angles. In connecting the beams at their ends, the sides of the beams need to twist. The flexibility of the thread connecting the beads allows the beaded frame to twist to accommodate the necessary turns to build this object in 3D. The second way to view this object is to see it as a patch of an infinite surface with no thickness and two holes. Since the surface has no thickness, ignore the layer of purple beads in the middle. The blue and green coloring of the largest faces shows that this surface has two distinct faces. --- Gwen Fisher (
"Coordinate Axis, Highly Unlikely Square and Highly Unlikely Triangle," by Gwen Fisher ( viewsMaterials: seed beads, Nymo nylon thread

These three pieces are woven from box stitch (also known as 3D right angle weave), which essentially takes the Cartesian tiling of 3-space with cubes, and places one bead on each edge of some subset of the tiling. With box stitch, rows and columns of cubes (attached face to face) can be woven into any continuous arrangement. Then, I add extra beads at the vertices of each cube to give the object more structure and decoration. The Coordinate Axis shows the basic structure of box stitch, and is also suitable for a game of children's Jacks. The Highly Unlikely Square and Triangle are beaded versions of the Impossible Triangle of Roger Penrose that was made famous by M.C. Escher. Compared with a regular square frame or triangular frame like you might hang on your wall, these frames have one quarter turn on each side. To see the effect of these twists, imagine painting a regular square frame with four colors to identify four paths: inside, outside, front and back. A similar coloring on the Highly Unlikely Square identifies four paths or faces, one of which is outlined with gold seed beads. Starting at the corner closest to the camera traveling clockwise, the golden face is outside, back, inside, front. In fact, all four faces are congruent. The effect of the quarter turns on the Highly Unlikely Triangle is different; there is only one face that travels around the triangle four times. -- Gwen Fisher (
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American Mathematical Society