
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 awardwinning 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 mathematicsorigami, computergenerated
landscapes, tesselations, fractals, anamorphic art, and
more.
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"Sierpinski Tetrahedron (View II)" 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.beadinfinitum.com)Apr 07, 2008


"Sierpinski Tetrahedron (View I)" 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.beadinfinitum.com)Apr 07, 2008


"72 Pencils," by George W. Hart (www.georgehart.com)"72 Pencils" is a geometric construction of 72 pencils, assembled into a work of art. The form is an arrangement of four intersecting hexagonal tubes that penetrate each other in a fascinating threedimensional lattice. For some viewers, part of the interest lies in the form of the interior. The four hexagonal tubes are hollow, so the sculpture as a whole is hollow. But, what shape is its cavity? What would someone on the inside see? To the mathematician, the answer is "the rhombic dodecahedron," a geometric solid bounded by twelve rhombuses. See two other views, showing how it looks along various axes of symmetry, at www.georgehart.comwww.georgehart.com.  George W. Hart (www.georgehart.com)
Mar 14, 2008


"Frabjous," by George W. Hart (www.georgehart.com)This is an 11inch diameter sculpture made of lasercut wood (aspen). It is assembled from thirty identical pieces. Each is an elongated Sshaped form, with two openings. The aspen is quite light in color but the lasercut edges are a rich contrasting brown. The openings add nicely to the whirling effect. The appearance is very different as one moves around it. This is an image of how it appears looking straight down one of the vortices. The word "frabjous" comes, of course, from "The Jabberwocky" of Lewis Carroll. "O frabjous day! Callooh! Callay!"  George W. Hart (www.georgehart.com)
Mar 14, 2008


Superimposition of Polar Surfaces1, by Dejenie A. LakewThe superimposition of two polar surfaces:
rho = 2sin4[theta]
rho = 5/3 cos4[theta] (wireframed) with some compositions of tilts and turns.
The two polar surfaces are generated in such a way that one is a derivative surface of the other but with different polar radius.Feb 28, 2008


Superimposition of Polar Surfaces2, by Dejenie A. LakewSuperimpositions of polar surfaces. Equations:
rho = 2cos(3exp sin 3[theta])*exp sin 3[theta]*exp(sin 3(exp sin 3[theta])),
rho = exp(sin 3(exp sin 3[theta]),
rho = 3cos(3exp sin 3[theta])*exp sin 3[theta]*exp(sin 3(exp sin 3[theta]))
with compositions of tilts and turns.
Here the first polar surface is the derivative of the second surface and the third surface is a spatial reflection of the first through the origin with wire frames, but with a larger spatial radius.
Feb 28, 2008


Superimposition of Polar Surfaces3, by Dejenie A. LakewSuperimposition of polar surfaces. Equations:
rho = 2cos(3exp sin 3[theta])*exp sin 3[theta]*exp(sin 3(exp sin 3[theta])),
rho = 3 + 2cos(3exp sin 3[theta])*exp sin 3[theta]*exp(sin 3(exp sin 3[theta]))
with several compositions of tilts and turns.
Feb 28, 2008


Superimposition of Polar Surfaces4, by Dejenie A. LakewSuperimpositions of polar surfaces. Equations:
rho = 10sin8[theta]
rho = 10cos8[theta] with a number of compositions of tilts and turns.Feb 28, 2008


Superimposition of Polar Surfaces5, by Dejenie A. LakewSuperimposition of three polar surfaces. Equations:
rho = 2cos(3exp sin 3[theta])*exp sin 3[theta]*exp(sin 3(exp sin 3[theta])),
rho = 3 + 2cos(3exp sin 3[theta])*exp sin 3[theta]*exp(sin 3(exp sin 3[theta]))
rho = 3cos(3exp sin 3[theta])*exp sin 3[theta]*exp(sin 3(exp sin 3[theta]) followed by many compositions of tilts and turns.
Feb 28, 2008


Superimposition of Polar Surfaces6, by Dejenie A. LakewSuperimposition of three polar surfaces. Equations:
rho = 2cos(3exp sin 3[theta])*exp sin 3[theta]*exp(sin 3(exp sin 3[theta])),
rho = 3 + 2cos(3exp sin 3[theta])*exp sin 3[theta]*exp(sin 3(exp sin 3[theta]))
rho = 6*(the outer sphere wireframed) with many compositions of tilts and turns as rigid transformations.
Feb 28, 2008


"Helios [var. 1198505515]," by Nathan SelikoffThis artwork is based on a rendering of a strange attractor, and is inspired by extreme ultraviolet images of our sun. Helios is part of the "Aesthetic Explorations of Attractor Space" series, more of which can be seen at www.nathanselikoff.com/strangeattractors/.
Underlying each image in this series of work is a twodimensional plot of the "typical behavior" of a chaotic dynamical system. Of course, there is nothing typical about a strange attractor, as it is chaotic and has a fractal structure. The base images are computed with a set of iterated functions, which serve as a numerical approximation to integrating the underlying differential equations. The iterated functions contain four coefficients, which are controlled by sliders in interactive custom software and control the appearance of the attractor. Once a particular form is settled on, it is rendered as a highresolution 16bit grayscale image. Finally, in Photoshop, the render is colorized using gradient mapping and edited to enhance contrast, control composition, and add special effects. The number in the artwork title encodes the moment at which the attractor was "discovered" and archived for rendering.
Feb 26, 2008


"SA_1188475827," by Nathan SelikoffAnother strange attractor, this one existing in three dimensions, comes to life with rich fiery colors that enhance the eastern Asian feel of the swirling lines. See more images at www.nathanselikoff.com/. Feb 26, 2008


"SA_1188415571," by Nathan SelikoffThis threedimensional strange attractor is reminiscent of Hubble images of the Eagle Nebula, though it is a purely mathematical construct. See more images at www.nathanselikoff.com/.Feb 26, 2008


"Death Mask 2," by Nathan SelikoffThis artwork, from the "Faces of Chaos" series, is a twodimensional plot of the Lyapunov exponent of a chaotic dynamical system. The Lyapunov exponent is a measure of how chaotic the system is, and in this case, the system is a strange attractor with a fourdimensional phase space. Two of the dimensions are static, and the other two vary in the x and y directions of the image. A custom program renders four 16bit grayscale images, which represent the different "components" of the spectrum of Lyapunov exponents. These images are combined in Photoshop using a pseudocolor technique to bring out subtle coloration in the final artwork. See more images from this series at www.nathanselikoff.com/facesofchaos/. Feb 26, 2008


"Prime colorings of the sphere, Euclidean plane and the hyperbolic plane," by Chaim GoodmanStrauss, University of Arkansas (http://mathbun.com/main.php)Tilings of the sphere, the Euclidean plane and the hyperbolic plane are shown. In each case, we have triangular faces, but on the sphere, the triangles meet in fives; in the Euclidean plane, the triangles meet in sixes, and in the hyperbolic plane, they meet in sevens. To a great degree, this is forced. It is impossible, for example, to have a tiling of the sphere with triangles meeting only in sevens (try it!).
In each case, a primefold coloring of the pattern is shown. It is helpful to realize that there are more similarities than differences among the three geometries.
The symmetry of the hyperbolic plane pictured above was known to Felix Klein by 1878, and has a tremendous number of interesting topological, geometric and algebraic properties.
This image is from "The Symmetries of Things"� by John H. Conway, Heidi Burgiel and Chaim GoodmanStrauss (AK Peters, 2008).
Jan 31, 2008


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