
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 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 mathematicsorigami, computergenerated
landscapes, tesselations, fractals, anamorphic art, and
more.
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"Random Walk on Sphere," by Daniel GriesPaths meander on a sphere with each small step taken at a random turning angle. The turn angle at each step is required to stay near the previous turn angle, as if the steering wheel of a car can only be turned slowly from one position to the next as the car drives forward. Computing random walks on a sphere has to be handled carefully with local coordinates so all random directions have an equal probability. Coded in Processing.  Daniel Gries (Hopkins School, New Haven, CT)May 28, 2015


"Parametric Surface," by Daniel GriesA parametric surface like the one shown here is defined by mapping two variables into threedimensional space. This surface was the result of playing with the points on a sphere, perturbing them in different directions by some sinusoidal functions. The surface is rendered so as to appear to be lit from different positions by differently colored point lights. Coded in Processing.  Daniel Gries (Hopkins School, New Haven, CT)May 28, 2015


"Perlin Lines," by Daniel GriesLines emerge from regular grid positions in the canvas, and proceed to grow according to a vector field defined by Perlin noise, a generative texture first created by Ken Perlin for computer animation in the 1982 movie "Tron". As the lines grow in this image, they slowly change color from red to yellow, producing the fiery effect. Coded in Processing.  Daniel Gries (Hopkins School, New Haven, CT)May 28, 2015


"Jellyfish 2," by Daniel GriesThis image is based on a morphing fractal curve method, but shaped into an abstract jellyfish through the use of parametric curves and other mathematical tricks. Additive color blending creates the lighting effect. Coded in JavaScript.  Daniel Gries (Hopkins School, New Haven, CT)May 28, 2015


"Jellyfish 1," by Daniel GriesThis image is based on a morphing fractal curve method, but shaped into an abstract jellyfish through the use of parametric curves and other mathematical tricks. Additive color blending creates the lighting effect. Coded in JavaScript.  Daniel Gries (Hopkins School, New Haven, CT)May 28, 2015


"Chaos Game Fractal 3," by Daniel GriesThis image is created with the "Fractal Flame" algorithm, a modification by Scott Draves of the classic Chaos Game algorithm. In this algorithm, a cloud of points is formed when a particle leaps from one position to the next as it is transformed by a linear transformation, randomly selected from a small fixed collection. The colors are a result of a special averaging of colors associated to each transformation. Coded in JavaScript.  Daniel Gries (Hopkins School, New Haven, CT)May 28, 2015


"Chaos Game Fractal 2," by Daniel GriesThis image is created with the "Fractal Flame" algorithm, a modification by Scott Draves of the classic Chaos Game algorithm. In this algorithm, a cloud of points is formed when a particle leaps from one position to the next as it is transformed by a linear transformation, randomly selected from a small fixed collection. The colors are a result of a special averaging of colors associated to each transformation. Coded in JavaScript.  Daniel Gries (Hopkins School, New Haven, CT)May 28, 2015


"Chaos Game Fractal 1," by Daniel GriesThis image is created with the "Fractal Flame" algorithm, a modification by Scott Draves of the classic Chaos Game algorithm. In this algorithm, a cloud of points is formed when a particle leaps from one position to the next as it is transformed by a linear transformation, randomly selected from a small fixed collection. The colors are a result of a special averaging of colors associated to each transformation. Coded in JavaScript.  Daniel Gries (Hopkins School, New Haven, CT)May 28, 2015


"Stripes," by Daniel GriesA curve which is defined through a fractal subdivision process, and then smoothed out, is drawn from the top of the canvas to the bottom. This curve then sweeps from the left of the canvas to the right, as it morphs into other similarly defined fractal curves, and intermittently changes from red to white. Some mathematical tricks are used to create the most turbulence in the middle of the picture, while maintaining fixed straight edges along the sides of the rectangle.  Daniel Gries (Hopkins School, New Haven, CT)May 28, 2015


"TetraTangle of Four BowTie Links," by Carlo SĂ©quin (University of California, Berkeley)ABS plastic, printed on an FDM machine, 2014
Four sets of three mutually parallel, 3sided prisms, pointing in 4 different tetrahedral directions, form the core of the TETRAXISÂ® puzzle. When two triangular prismendfaces that share a common vertex are closed off with a connecting sweep, a loose "bowtie" is formed. If all twelve pairs of adjoining triangular endfaces are joined in this manner, the result is a link of 4 mutually interlocking, twisted, prismatic bowtie loops. This represents an alternating 12crossing link that has the same connectivity as the â€śTetraTangle,â€ť which I constructed from 4â€ťdiameter cardboard tubes in 1983. The new geometry is has been realized as 4 differently colored sets of 6 tubular snaptogether parts each, fabricated on an FDM machine.  Carlo SĂ©quin (http://www.cs.berkeley.edu/~sequin/)
Apr 06, 2015


"Dance of Stars II," by Reza Sarhangi (Towson University, Towson, MD)Heavy paper, 2014
Dance of Stars II is a decorated Great Stellated Dodecahedron, with SchlĂ¤fli Symbol (5/2, 3), which has been patterned by the sĂ˘zeh module tiles, that are used in the majority of tiling that conforms to local fivefold symmetries. In an article that appeared in Science, the authors proposed the possibility of the use of a set of tiles, girih tiles, by the medieval craftsmen, for the composition of the underlying pattern. I used girih tiles and left the dashed outlines in the final tessellation. I also included rectilinear patterns that appear as additional smallbrick pattern in the decagonal Gunbadi Kabud tomb tower in Maragha, Iran.  Reza Sarhangi
Apr 06, 2015


"15 Irregular Hexahedra," by Aaron Pfitzenmaier (student)Paper, 2014
Honorable Mention, 2015 Mathematical Art Exhibition
This model was made from 180 units of four different types. It consists of 15 irregular hexahedra interlocked together. Each hexahedron has 2fold dihedral symmetry and the positioning of each hexahedron is based on a pair of opposite edges on an icosahedron. This compound has the most complex weaving pattern out of anything I have designed, and is an example of a model where I extensively used the ray tracer POVRay as well as a computer program I wrote to aid in the design and folding/assembly process.  Aaron Pfitzenmaier (http://bit.ly/aaronsorigami)
Apr 06, 2015


"Hyperbolic Catacombs," by Roice Nelson (Austin, TX) and Henry Segerman (Oklahoma State University, Stillwater)Digital Print, 2014
This picture visualizes the regular, selfdual {3,7,3} honeycomb in the upper half space model of hyperbolic 3space. The cells are {3,7} tilings and the vertex figure is a {7,3} tiling. The cells have infinite volume: the vertices are "ultraideal", living beyond the boundary of hyperbolic space. The intersection of each cell with the boundary is an infinite collection of heptagons, together with a disk. The white ceiling and each red "creature" are isometric cells; for all other cells we only show the intersection with the boundary of hyperbolic space, on the floor of the catacombs. Every disk on the floor containing a {7,3} tiling is associated with an ultraideal vertex of the honeycomb.  Roice Nelson (http://google.com/+roicenelson) and
Henry Segerman (http://segerman.org)
Apr 06, 2015


"Longest Crease/Perfect Shuffle1," by Sharol Nau (Northfield, MN)Folded Book, 2014
A classical calculus problem, the socalled Paper Creasing Problem is essential to the design of these sculptures. Pages in a book provide a series of rectangular sheets of paper which are creased by matching one corner; say the lower righthand corner to a point on the opposite edge where the sheets have been bound. Waves are obtained through incremental changes in the length of the crease from page to page. Two sets of points have been used for these new examples. In each case every other page begins its sequence at a different point. The result of the two series interleaved is a socalled perfect shuffle.  Sharol Nau (http://www.sharolnau.com)
Apr 06, 2015


"Penrose Pursuit 2," by Kerry Mitchell (Phoenix, AZ)Digital print onto aluminum panel, 2014
Best photograph, painting, or print
2015 Mathematical Art Exhibition
Underlying this image is a nonperiodic Penrose tiling, using the kite and dart tiles. Each tile is rendered using pursuit curves. To accommodate the concave dart tile, it was split into two triangular halves. Each half was filled with three pursuit curves, while the kite tiles have four.  Kerry Mitchell (http://kerrymitchellart.com)
Apr 06, 2015


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