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 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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lipson-figeight.jpg
"Figure eight knot," copyright Andrew Lipson. Made of Lego ®I think this is the most difficult single construction I have ever made out of Lego®. Those long sweeping curves, hanging unsupported in space... It's only when you get about 2/3 of the way up that you start to discover exactly which bits 1/3 of the way up aren't strong enough. And there are never enough 1x3 bricks... But I didn't cheat anywhere. The figure-eight knot has a nice tetrahedral skew-symmetry which the model illustrates quite well. On my website you can find more pictures and an LDRAW .DAT file generated by my program for this sculpture. Beware--the .DAT file builds it out of 1x1 bricks. Actually constructing this out of larger bricks so that it holds together is a (non-trivial) exercise! Lego® is a trademark of The Lego Group. --- Andrew Lipson (http://www.andrewlipson.com/mathlego.htm)Feb 06, 2012
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"Escher's 'Belvedere'," copyright Andrew Lipson. Made of Lego ®Daniel Shiu and I worked on this as a joint project. We discovered a few nasty surprises that Escher had hidden in the picture (other than the obvious one). And we had to get the camera position just right for the picture to come out OK. The domes on top, and the slightly protruding cell wall at the near end of the bottom level, were both interesting exercises in half-brick spacing, and many of those useful 1x2 plate offset bricks with the single stud on top were used. We took a small liberty with the guy in the red hat at the bottom of the picture. In Escher's original, he's holding an "impossible cube", but in our version he's holding an impossible Lego® square. Well, OK, not quite impossible if you've got a decent pair of pliers (ouch). See photos of the construction in progress . Lego® is a trademark of The Lego Group. On my website I post images of M.C. Escher's original works (C) Cordon Art, Baarn, the Netherlands on his website, used with permission, so that you may compare with the Lego® creations. All rights reserved. --- Andrew Lipson (http://www.andrewlipson.com/mathlego.htm)Feb 06, 2012
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"Escher's 'Ascending and Descending'," copyright Andrew Lipson. Made of Lego ®Daniel Shiu and I worked on this as a joint project. There are no camera tricks, but the picture has to be taken from exactly the right place, and the final photograph was slightly distorted to emphasize the perspective effect. I'm especially pleased with the way the roof in the top left of the picture came out. See photos of the construction in progress. Lego® is a trademark of The Lego Group. On my website I post images of M.C. Escher's original works (C) Cordon Art, Baarn, the Netherlands on his website, used with permission, so that you may compare with the Lego® creations. All rights reserved. --- Andrew Lipson (http://www.andrewlipson.com/mathlego.htm)Feb 06, 2012
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"RedCenter," (detail) by Mike Field (University of Houston)"RedCenter" is a section of a planar repeating "two-color" pattern of type pmm' (or pmm/pm in Coxeter notation). The underlying repeating pattern has reflection symmetries and two-fold rotational symmetries as well as translation symmetries and, less obviously, glide reflection symmetries. Roughly speaking, half the symmetries preserve colors and half interchange colors. (The 46 two-color repeating patterns of the plane were originally classified by H. J. Woods of the Textile Physics Laboratory, University of Leeds, in 1935-36.) The pattern was generated using a determinsitic torus map and the coloring reflects the density of two invariant measures on the torus. The name "RedCenter" is suggested by Uluru (Ayers Rock) in Central Australia. --- Mike FieldAug 26, 2011
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"Sierpinski Tetrahedron (View V)" in glass bugle beads, size 11/0 and 8/0 seed beads, Fireline thread, by Gwen L. Fisher, California Polytechnic State University, San Luis Obispo and beAd Infinitum There are several ways to build a polyhedron with beads. One technique that will always work is to align the hole of a bead along each edge of the polyhedron. Then, the thread connects the beads at the vertices of the polyhedron. The most stable polyhedron is the tetrahedron because it is made of all triangles. In a beaded tetrahedron, there are three sets of beads in each loop, like the three sides of a triangle. Any regular tetrahedral beaded bead will naturally require six identical sets of beads, one set for each of the six edges of the tetrahedron. In this case, a set is three beads: a short, a long and a short. Rather than give an example of the simplest tetrahedron, I have used a more complex design based on the structure resulting from the third iteration in the construction of the "Sierpinski Tetrahedron" with its 64 little tetrahedrons. Adding a bead at each interior vertex is necessary to stabilize the structure and make it more rigid. --- Gwen L. Fisher (www.beadfinitum.com)Aug 26, 2011
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"Sierpinski Tetrahedron (View IV)" in glass bugle beads, size 11/0 and 8/0 seed beads, Fireline thread, by Gwen L. Fisher, California Polytechnic State University, San Luis Obispo and beAd Infinitum There are several ways to build a polyhedron with beads. One technique that will always work is to align the hole of a bead along each edge of the polyhedron. Then, the thread connects the beads at the vertices of the polyhedron. The most stable polyhedron is the tetrahedron because it is made of all triangles. In a beaded tetrahedron, there are three sets of beads in each loop, like the three sides of a triangle. Any regular tetrahedral beaded bead will naturally require six identical sets of beads, one set for each of the six edges of the tetrahedron. In this case, a set is three beads: a short, a long and a short. Rather than give an example of the simplest tetrahedron, I have used a more complex design based on the structure resulting from the third iteration in the construction of the "Sierpinski Tetrahedron" with its 64 little tetrahedrons. Adding a bead at each interior vertex is necessary to stabilize the structure and make it more rigid. --- Gwen L. Fisher (www.beadfinitum.com)Aug 26, 2011
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"Sierpinski Tetrahedron (View III)" in glass bugle beads, size 11/0 and 8/0 seed beads, Fireline thread, by Gwen L. Fisher, California Polytechnic State University, San Luis Obispo and beAd Infinitum There are several ways to build a polyhedron with beads. One technique that will always work is to align the hole of a bead along each edge of the polyhedron. Then, the thread connects the beads at the vertices of the polyhedron. The most stable polyhedron is the tetrahedron because it is made of all triangles. In a beaded tetrahedron, there are three sets of beads in each loop, like the three sides of a triangle. Any regular tetrahedral beaded bead will naturally require six identical sets of beads, one set for each of the six edges of the tetrahedron. In this case, a set is three beads: a short, a long and a short. Rather than give an example of the simplest tetrahedron, I have used a more complex design based on the structure resulting from the third iteration in the construction of the "Sierpinski Tetrahedron" with its 64 little tetrahedrons. Adding a bead at each interior vertex is necessary to stabilize the structure and make it more rigid. --- Gwen L. Fisher (www.beadfinitum.com)Aug 26, 2011
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"Floating Pentangle Construction," by Bente Simonsen (Landeryd, Sweden)Digital print, 20" x 24", 2010

Impossible pentangle construction, 2D and 3D mix-illusion. --- Bente Simonsen (http://geometric-impossibilities.blogspot.se)
Mar 10, 2011
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"Thorn Dice Set," by Chuck Stover (Lansing, MI)Printed stainless steel and bronze, 6" x 8" x 1", 2010

A set of polyhedral dice with edges defined by interlocking vines of steel. --- Chuck Stover

Mar 10, 2011
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"Martin Gardner - Master Puzzler," by Bruce Torrence (Randolph-Macon College, Ashland, VA)Archival inkjet print, 20" x 20", 2010

This portrait of Martin Gardner (1914-2010) was made by coloring the individual tiles on a kite and dart Penrose tiling. This particular tiling exhibits fivefold rotational symmetry (can you find the center?), and was created by "deflating" a wheel of five kites eight times. Gardner's oft-cited January 1977 Scientific American column introduced the public to Penrose's aperiodic tiles. --- Bruce Torrence (http://faculty.rmc.edu/btorrenc)
Mar 10, 2011
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"Visualizing Abstract Quantity," by Anna Ursyn (University of Northern Colorado, Greeley)Archival print, 8" x 10", 2010

Unspoken fears. --- Anna Ursyn (http://www. Ursyn.com)
Mar 10, 2011
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"Möbius strip patterned by 48 different striped squares," by Anna Virágvölgyi (Budapest, Hungary)Folded paper, 100 x 100 x 100 mm, 2010

Diagonally striped tiles of this arrangement create concentrically striped squares. The number of squares is the number of all possible triplets of three symbols (no symbols are paired): 3*2*2 = 12. The surface of the Möbius strip is diced with this different 12 squares. The edge of the strip is diced with another whole set of such triplets. This arrangement would be realized on tori as well. --- Anna Virágvölgyi
Mar 10, 2011
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"Infinity," by Mary Wahr (Manistee Area Public Schools, MI)Pen and ink, 18" x 21", 2009

This is an abstract pen and ink rendering of a broccoflower. It is the first piece of art completed for my thesis and was the starting point of two years of research and art. Since my fractal ideas are accomplished without arithmetic, I needed to incorporate the components that define a fractal. This piece of art reflects the concepts of infinity, iteration, self-similarity and scaling. --- Mary Wahr
Mar 10, 2011
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"iteration," by Trygve Wastvedt (St. Olaf College, Northfield, MN)Bronze, plaster, wax, concrete, 18" x 18" x 8", 2010

"iteration" is a series of identical humanoid figures cast in bronze, plaster, wax, and concrete. The form is a near honeycomb so that the individual pieces stack together to fill space. Though geometric, the form still evokes human emotions, which allows the sculpture to ask social and relational questions. --- Trygve Wastvedt (http://www.trygvewastvedt.com)
Mar 10, 2011
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"Ten Triangular Prisms," by Magnus Wenninger (Saint John's Abbey, Collegeville, MN)Paper, 9" x 9" x 9", 2010

Robert Webb's Stella program is now the computer program I use for the construction of all the 
polyhedron models I have recently been making. It is the program par excellence I now use for 
the discovery of any new polyhedra, especially any I have never made before. The photo shows a model of Ten Triangular Prisms, recently made by me. I found the Stella version on
 a web page called '75 Uniform Polyhedra' done by Roger Kaufman. It is #32 on this web page. The Stella
 version gives me a 3D computer view in 10 colors and allows me to choose the size of the model and
 thus also the size and shape of the net to be used for the construction of the model. However, I wanted
 my model to be done using only 5 colors. This is where the artwork comes into play. The model now
 shows each prism with its faces in one color of the five. Thus it becomes uniquely artistic in appearance. --- Magnus Wenninger (http://www.saintjohnsabbey.org/wenninger/)
Mar 10, 2011
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