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 ands, 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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"Autumn Moths from a Hike in the Hills," by Frank A. Farris, Santa Clara University, CA91 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

"Heart of the Fractal," by Stephen Ren88 viewsThis image is centered on 0.170938545, and is magnified 6000x from the original fractal. This figure can be found repeated in many areas along the real axis, though it is often deeply blurred in a larger figure. This image is generated with 400 iterations of tetration. See hi-resolution versions of Tetration Fractals. --- Stephen Ren

"Coordinate Axis, Highly Unlikely Square and Highly Unlikely Triangle," by Gwen Fisher (www.beadinfinitum.com)87 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 (www.beadinfinitum.com)

"Negative Flower," by Stephen Ren85 viewsThis image is centered on -2.5, and is focused on the somewhat hexagonal figure found primarily to the left of the imaginary axis. This is magnified 3x from the overall image. This image was generated with 100 iterations of tetration. See hi-resolution versions of Tetration Fractals. --- Stephen Ren

"Mobius Frame with 2 Holes (View II)," by Gwen Fisher (www.beadinfinitum.com)83 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 (www.beadinfinitum.com)

"Rolling Flames," by Stephen Ren75 viewsThis image is centered on 0.9717 + 2.21i, and depicts an endless sequence of “compound-star” figures. This image was produced with 100 iterations of tetration. See hi-resolution versions of Tetration Fractals. --- Stephen Ren

"Center0-0-6," by Stephen Ren62 viewsThis is the main fractal, and all the other images in this gallery are zoomed in from here. This image is centered on the origin, with the real axis ranging from -6 to 6 and the imaginary ranging from -3.375i to 3.375i . This image was generated with 100 iterations of tetration. See hi-resolution versions of Tetration Fractals . --- Stephen Ren

"Wisdom," by Stephen Ren60 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