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.

"Quilt," by Galina Bulatova (Artist, Corvallis, OR)Glass, 7.5” x 7.5”, 2009. This tessellation is the traditional coloring and layout of the Churn Dash pattern. The Churn Dash quilt block is an interesting pattern that is composed of right triangles, rectangles, and a single square. This symmetrical design is based on a square and can be recolored in this manner to produce a tessellating motif. Geometric design and especially geometric patterns always intrigued me. I was excited by seeing how patterns changed as various symmetrical arrangements were applied to them. My pieces are created by layering and melting together various shapes and colours of glass in a fusing kiln, which heats the glass to 1500 degrees Fahrenheit. --- Galina Bulatova (Artist, Corvallis, OR) http://bulatov.org/

May 10, 2010

"Overlapping Circles #25," by Anne Burns (Long Island University, Brookville, NY)Digital print, 19" X 13", 2009. The subgroup of Möbius Transformations that maps the unit circle onto itself is composed with Möbius Transformations that map the unit circle into itself to produce overlapping circles. The program was written in Actionscript. I am fascinated by art, mathematics and nature and try to combine all three in my work. --- Anne Burns (Long Island University, Brookville, NY) http://myweb.cwpost.liu.edu/aburns/ May 10, 2010

"Ready to Fly High," by Mingjang Chen (National Chiao Tung University, Hsinchu, Taiwan)Digital print by PowerPoint, 17" x 22", 2008. Complete r-partite graph is the graph with vertices set consisting of r disjoint sets such that any two vertices in different sets are connected by an edge and not for vertices in the same set. The work is a complete bipartite graph, following by a rotation on each line segments. One part of vertices is positioned on two adjacent line segments with equal distance; another part of vertices is positioned on an oval. There are 27 vertices on one part and 24 vertices on the oval. Hence, there are 27*24 line segments in this work. The transparency of these line segments is high up 95%. Structural Cloning Method (SCM) implemented on PowerPoint is a Human Computer Interface which handles geometry transformations on a huge number of drawing elements; it can be used to arrange complicate elements. Based on SCM, we can explore symmetry patterns and fractal patterns in different ways, and so math art becomes interesting. Complete r-partite graph is a common used graph, the number of edges in a complete r-partite graph is very large, and its edges always cover the space among vertices when visualizing. Once we tune up the transparency of these line segments, the transparency of overlapped areas become various degrees, this effect makes overlapped areas look like light-spots or rays if the background is in dark color, however the line segments are in light color. Parts of vertices can be arranged in various structures such that the overlapped transparent line segments can generate various and amazing patterns. --- Mingjang Chen (National Chiao Tung University, Hsinchu, Taiwan) May 10, 2010

"Sculpture System No. 5 (2009)" by Richard Grimes (www.richardgrimes.net) and Edmund Harriss (University Of Leicester)Deltahedra are polyhedra where all the faces are regular triangles. Sculpture system 5 is a system to build any deltahedron using a triangular shapes that hinge together. Using just twenty of these shapes a huge variety of polyhedra can be made. The actual sculpture was built by a group of volunteers who also able to design the final shape that was actually built. --- Edmund Harriss Feb 09, 2010

"Ammann Scaling (2006)," by Edmund Harriss (University of Leicester)Printed on Canvas 24" x 24". The Ammann-Beenker tiling is the eight-fold sibling of the more famous, five-fold Penrose rhomb tiling. It was discovered independently by R. Ammann and F. Beenker. Like the Penrose tiling, the Ammann-Beenker can be constructed by two particular methods. The first method is uses the substitution rule, and the second method is to construct the tiling as a planar slice of a four dimensional lattice (in much the same way that a computer draws a line using the pixels of its screen) and then project this to the plane. See more information at www.mathematicians.org.uk/eoh/Art/Ammann_Text.pdf. This was a commission for the School of Mathematical Sciences at Queen Mary. It is one of a pair with Ammann Squares, exploring aspects of the Ammann-Beenker Tiling. It appeared in the June 2007 issue of Notices of the AMS. --- Edmund Harriss Feb 04, 2010

"CurveTile (2004)," by Edmund Harriss (University of Leicester)A pattern built of just four images (two up to rotation). You can make some some patterns yourself at www.mathematicians.org.uk/eoh/Curvetile/Curvetile.html. --- Edmund HarrissFeb 04, 2010

Sculpture System No. 5 (2009)" by Richard Grimes (www.richardgrimes.net) and Edmund Harriss (University of Leicester)Deltahedra are polyhedra where all the faces are regular triangles. Sculpture system 5 is a system to build any deltahedron using triangular shapes that hinge together. A huge variety of polyhedra can be made using just twenty of these shapes. The actual sculpture was built by a group of volunteers who also designed the final shape that was actually built. --- Edmund Harris Feb 04, 2010

"Octagonal Gasket (2006)," by Edmund Harriss (University of Leicester)Printed on Canvas 36" x 36". This image is based on a version of the Sierpinski Gasket with octagons rather than triangles. --- Edmund Harris Feb 04, 2010

"Two Squares (2006)," by Edmund Harriss (University of Leicester)Printed on Canvas 36" x 36". This is based on the Ammann-Beenker Tiling. Along with Ammann Squares this work explores the extension of the work of Raymond Brownell (www.raymondbrownell.com) to more complicated geometry. The Ammann-Beenker tiling is the eight-fold sibling of the more famous, five-fold Penrose rhomb tiling. It was discovered independently by R. Ammann and F. Beenker. Like the Penrose tiling, the Ammann-Beenker can be constructed by two particular methods. The first method is uses the substitution rule, and the second method is to construct the tiling as a planar slice of a four dimensional lattice (in much the same way that a computer draws a line using the pixels of its screen) and then project this to the plane.
See more information at www.mathematicians.org.uk/eoh/Art/Ammann_Text.pdf.

--- Edmund Harriss Feb 04, 2010

"Fiddler Crab, opus 446" crease pattern, by Robert J. Lang. Copyright Robert J. Lang (www.langorigami.com).Crease patterns (CPs) provide a one-step connection from the unfolded square to the folded form, compressing hundreds of creases, and sometimes hours of folding, into a single diagram! A CP can sometimes be more illuminating than a detailed folding sequence, conveying not just "how to fold," but also how the figure was originally designed. Mathematical and geometric CPs usually show all the creases, but representational origami rarely shows every crease in the finished form, as it would make the crease pattern impossibly busy. Instead, the crease pattern gives the creases needed to fold the "base," that is, a geometric shape that has the right number and arrangements of flaps. It is still left up to the folder to add thinning and shaping folds. See the final "Fiddler Crab" origami work in this album.Aug 19, 2009

"Night Hunter, opus 469" crease pattern, by Robert J. Lang. Copyright Robert J. Lang (www.langorigami.com).Crease patterns (CPs) provide a one-step connection from the unfolded square to the folded form, compressing hundreds of creases, and sometimes hours of folding, into a single diagram! A CP can sometimes be more illuminating than a detailed folding sequence, conveying not just "how to fold," but also how the figure was originally designed. Mathematical and geometric CPs usually show all the creases, but representational origami rarely shows every crease in the finished form, as it would make the crease pattern impossibly busy. Instead, the crease pattern gives the creases needed to fold the "base," that is, a geometric shape that has the right number and arrangements of flaps. It is still left up to the folder to add thinning and shaping folds. See the final "Night Hunter" origami work in this album.Aug 19, 2009

"DNAQuilt," by Mike Field (University of Houston)"DNAQuilt" is a repeating pattern of type pgg. As is the case of the other repeating patterns that have a pgg component, this type of symmetry is particularly dynamic as there are no lines of symmetry in the pattern--only glide-reflection symmetries. Although lines of reflection can be artistically interesting in two-color repeating patterns (for example, in "RedCenter" and "UncertainEnd"), too many lines of symmetry--as in patterns with p4m (square) symmetry--can tend to lead to 'pretty' but ultimately rather dull and static results (at least in patterns without two-color symmetry). Mathematically speaking. the pattern is a visual representation of the invariant measure of a deterministic dynamical system defined on the two-dimensional torus. The pattern is lifted to the plane to obtain a repeating pattern. --- Mike FieldMay 21, 2009

"InHotPursuit," by Mike Field (University of Houston)"InHotPursuit" is a section of a planar repeating pattern of type cm and the pattern was generated using an iterated function system defined on the two-dimensional torus. The resulting pattern on the torus was lifted to the plane to obtain a repeating pattern. The coloring reflects an invariant measure on the attractor of the iterated function system. This image is a bit surprising for an iterated function system as the textures and detail are more suggestive of a deterministic system (the torus maps used to generate the iterated function system are quite discontinuous). The original image was created in 2003. --- Mike FieldMay 21, 2009

"Thorns," by Mike Field (University of Houston)"Thorns" is a bounded symmetric pattern in the plane with symmetry Z_5. It is a visual representation of the invariant measure on the attractor of a rational Z_5-equivariant planar map. The original image was created in 1996 and was perhaps my first serious attempt to investigate ways one could use methods based on symmetry, dynamics and chaos to achieve artistic ends. --- Mike FieldMay 21, 2009

"UncertainEnd," by Mike Field (University of Houston)"UncertainEnd" is a section of a planar repeating pattern of type p'_{c}gg (or, in Coxeter notation, cmm/pgg). Ignoring the colors, the underlying pattern is of type cmm and is the superposition of two colored patterns, each of type pgg. The pattern was generated using an iterated function system defined on the two-dimensional torus. The resulting pattern on the torus was lifted to the plane to obtain a repeating pattern. The coloring reflects invariant measures on each of the underlying patterns of type pgg and takes account of overlap, as well as symmetry, using algorithms designed for revealing detail hidden in the dynamics. The original image was created in 2001. --- Mike FieldMay 21, 2009