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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Orndorff2-caption.jpg
"OSU Triptych No. 2," (view 2) by Robert OrndorffPaper and acrylic, 20 x 46 cm, 2015
Honorable Mention, 2016 Mathematical Art Exhibition

My paper folding art is math plus paper. Folded paper is simply tangible math. This is a permanent manifestation of an ephemeral artwork, namely, one solution for a specific one-straight-cut problem. Such problems are usually stated as follows: How must one fold a paper rectangle into a flat figure such that one straight cut through all of the layers will produce a given planar straight-line graph? Here the problem has been solved with paper and then represented in acrylic. To a significant degree the work relies on transmitted and reflected light, and so it never looks the same twice. The figure (the letters "OSU") has been divided into three frames. The crease patterns for the left and right letters are pedestrian but that for the central letter is sublime. --- Robert Orndorff
Jun 23, 2016
A_Bird_in_Flight.jpg
"A Bird in Flight (2016)," by Hamid Naderi YeganehThis image shows all circles of the form:
(x-A(k))^2+(y-B(k))^2=(R(k))^2,

for k=-10000, -9999, ... , 9999, 10000, where

A(k)=(3k/20000)+sin((π/2)(k/10000)^7)((cos(41πk/10000))^6)+(1/4)((cos(41πk/10000))^16)((cos(πk/20000))^12)sin(6πk/10000),

B(k)=-cos((π/2)(k/10000)^7)(1+(3/2)(cos(πk/20000)cos(3πk/20000))^6)((cos(41πk/10000))^6)+(1/2)(cos(3πk/100000)cos(9πk/100000)cos(18πk/100000))^10,

R(k)=(1/50)+(1/10)((sin(41πk/10000)sin(9πk/100000))^2)+(1/20)((cos(41πk/10000))^2)((cos(πk/20000))^10).
--- Hamid Naderi Yeganeh
Mar 23, 2016
Ring.jpg
"Ring," by Hamid Naderi YeganehThis image shows 5,600 ellipses. For each k=1,2,3,...,5600 the foci of the k-th ellipse are:
A(k)+iB(k)+C(k)e^(44πik/5600)
and
A(k)+iB(k)-C(k)e^(44πik/5600)
and the eccentricity of the k-th ellipse is D(k), where
A(k)=(cos(28πk/5600))^3,

B(k)=sin(28πk/5600)+(1/4)(cos((14πk/5600)-(7π/4)))^18,

C(k)=(1/70)+(1/6)+(1/6)sin(28πk/5600),

D(k)=(399/400)-(1/6)(sin(28πk/5600))^8.
--- Hamid Naderi Yeganeh
Mar 23, 2016
8000_Ellipses_1~0.jpg
"8,000 Ellipses," by Hamid Naderi YeganehThis image shows 8,000 ellipses. For each k=1,2,3,...,8000 the foci of the k-th ellipse are:
A(k)+iB(k)+C(k)e^(300πik/8000)
and
A(k)+iB(k)-C(k)e^(300πik/8000)
and the eccentricity of the k-th ellipse is D(k), where
A(k)=(3/4)sin(2πk/8000)cos(6πk/8000)+(1/4)sin(28πk/8000),

B(k)=(3/4)cos(2πk/8000)cos(8πk/8000)+(1/4)cos(28πk/8000),

C(k)=(1/18)+(1/20)cos(24πk/8000),

D(k)=(49/50)-(1/7)(sin(10πk/8000))^4.
--- Hamid Naderi Yeganeh
Mar 23, 2016
Heart.jpg
"Heart," by Hamid Naderi YeganehThis image shows 2,500 ellipses. For each k=1,2,3,...,2500 the foci of the k-th ellipse are:
A(k)+iB(k)+C(k)e^(68πik/2500)
and
A(k)+iB(k)-C(k)e^(68πik/2500)
and the eccentricity of the k-th ellipse is D(k), where
A(k)=(-3/2)((sin(2πk/2500))^3)+(3/10)((sin(2πk/2500))^7),

B(k)=sin((2πk/1875)+(π/6))+(1/4)(sin((2πk/1875)+(π/6)))^3,

C(k)=(2/15)-(1/8)cos(πk/625),

D(k)=(49/50)-(1/7)(sin(4πk/2500))^4.
--- Hamid Naderi Yeganeh
Mar 23, 2016
Butterfly_3.jpg
"Butterfly (3)," by Hamid Naderi YeganehThis image shows 40,000 circles. For k=1,2,3,...,40000 the center of the k-th circle is (X(k), Y(k)) and the radius of the k-th circle is R(k), where

X(k)=(3/2)((cos(141πk/40000))^9)(1-(1/2)sin(πk/40000))(1-(1/4)((cos(2πk/40000))^30)(1+(cos(32πk/40000))^20))(1-(1/2)((sin(2πk/40000))^30)((sin(6πk/40000))^10)((1/2)+(1/2)(sin(18πk/40000))^20)),

Y(k)=cos(2πk/40000)((cos(141πk/40000))^2)(1+(1/4)((cos(πk/40000))^24)((cos(3πk/40000))^24)(cos(21πk/40000))^24),

R(k)=(1/100)+(1/40)(((cos(141πk/40000))^14)+(sin(141πk/40000))^6)(1-((cos(πk/40000))^16)((cos(3πk/40000))^16)(cos(12πk/40000))^16).
--- Hamid Naderi Yeganeh
Mar 23, 2016
Butterfly_1.jpg
"Butterfly (1)," by Hamid Naderi YeganehThis image shows 40,000 circles. For k=1,2,3,...,40000 the center of the k-th circle is (X(k), Y(k)) and the radius of the k-th circle is R(k), where

X(k)=(6/5)((cos(141πk/40000))^9)(1-(1/2)(sin(πk/40000))^3)(1-(1/4)((cos(2πk/40000))^30)(1+(2/3)(cos(30πk/40000))^20)-((sin(2πk/40000))^10)((sin(6πk/40000))^10)((1/5)+(4/5)(cos(24πk/40000))^20)),

Y(k)=cos(2πk/40000)((cos(141πk/40000))^2)(1+(1/4)((cos(πk/40000))^24)((cos(3πk/40000))^24)(cos(19πk/40000))^24),

R(k)=(1/100)+(1/40)(((cos(2820πk/40000))^6)+(sin(141πk/40000))^2)(1-((cos(πk/40000))^16)((cos(3πk/40000))^16)(cos(12πk/40000))^16).
--- Hamid Naderi Yeganeh
Mar 23, 2016
Olive_Branch.jpg
"Olive Branch," by Hamid Naderi YeganehThis image shows 4,000 circles. For k=1,2,3,...,4000 the center of the k-th circle is (X(k), Y(k)) and the radius of the k-th circle is R(k), where

X(k)=(2k/4000)+(1/28)sin(42πk/4000)+(1/9)((sin(21πk/4000))^8)+(1/4)((sin(21πk/4000))^6)sin((2π/5)(k/4000)^12),

Y(k)=(1/4)(k/4000)^2+(1/4)(((sin(21πk/4000))^5)+(1/28)sin(42πk/4000))(cos((π/2)(k/4000)^12)),

R(k)=(1/170)+(1/67)((sin(42πk/4000))^2)(1-((cos(21πk/4000))^4)).
--- Hamid Naderi Yeganeh
Mar 23, 2016
jmm16-sequin.jpg
"Snap-together Super-Bottle of Genus 4/σ," by Carlo Séquin (University of California, Berkeley)ABS plastic, printed on an FDM machine, 16 x 20 x 14 cm, 2015

Stimulated by the LEGO-Knot project, I aimed to design a set of modular parts that permits to compose not only various handle-bodies, but also single-sided surfaces of higher genus. The modular parts employed in my sculptures are tubular 3-way junctions, where one of the tubular stubs exposes the opposite side of the surface shown by the other two stubs. Depending on how the parts are connected, the resulting compositions remains orientable or becomes single-sided; in the latter case they correspond to sums of multiple Klein bottles; which I call "Super-Bottles." The two identical parts of which the sculpture is composed can be put together in three different ways. In two cases the resulting surface is single-sided (σ = 1) and in the third case it is double-sided (σ = 2). The genus of the resulting surface is 4/σ. The configuration shown is a non-orientable surface of genus 4, corresponding to the connected sum of two Klein Bottles, with two punctures. The insets show the two individual parts, and an assembly of them resulting in a 2-hole torus of genus 2 (with two punctures). --- Carlo Séquin
Mar 09, 2016
jmm16-segerman.jpg
"Stereographic projection (grid)," by Henry Segerman (Oklahoma State University, Stillwater)3D printed nylon plastic, lamp, 10 x 9 x 9 cm, 2014

The light rays from the lamp are partly blocked by the shrinking design on the sphere; the resulting shadow is a regular tiling of the plane by squares. This illustrates how stereographic projection transforms the sphere, minus the north pole, into the plane. Note how shapes are slightly distorted near the south pole, and dramatically distorted near the north pole. --- Henry Segerman
Mar 09, 2016
jmm16-sarhangi.jpg
"Kokabi Stars," by Reza Sarhangi (Towson University, Towson, MD)Tile, 50 x 50 cm, 2015

I am interested in Persian geometric art and its historical methods of construction. Kokabi Star (the great pentagram) can be constructed using the lines of the 10/3 star polygon. Patterning this star can be achieved using different approaches. Some of the presented stars in this artwork have been made based on the actual tiling on existing buildings. Some others have been constructed based on old treatises and scrolls. Some of the patterns have been created using the traditional compass-straightedge process. Modularity is another approach in this regard. Moreover, the two decorated quasicrystal patterns of Star and Sun (the only two quasicrystal patterns with global five-fold rotational symmetry) and their striking relationships with Kokabi Star have been presented. Is this relationship a theorem? --- Reza Sarhangi
Mar 09, 2016
jmm16-riddle.jpg
"Levy Dragon Outside Tapestry," by Larry Riddle (Agnes Scott College, Decatur, GA)Back stitch embroidery on 18 point canvas, 31 x 31 cm, 2013

I have been working with needle crafts since graduate school. I have also been interested in fractals and fractal geometry for more than 20 years. The Levy Dragon Outside Tapestry consists of four copies of the Levy Dragon built from the edges of a square. The iteration steps are repeated on each of the four sides of the square with the initial triangle motif placed outside the square. This back stitch design shows the twelfth iteration for this outside construction.The outside tapestry was done in two shades of blue to better show each of the four copies of the Levy dragon. --- Larry Riddle
Mar 09, 2016
jmm16-reimann.jpg
"Walnut Star," by David Reimann (Albion College, Albion, MI)Walnut veneer and brass fasteners, 38 x 38 x 38 cm, 2015

I am interested in creating patterns that convey messages at multiple levels and scales using a wide variety of mathematical elements and media. This form is based on the small rhombicosidodecahedron, an Archimedean solid with 120 edges. The underlying polyhedral edges have been replaced by 4.75 cm squares made from laser-cut paper-backed walnut veneer and connected at their corners with brass split-pin fasteners. The 62 faces (squares, hexagons, and pentagons) and 60 vertices of the underlying polyhedron are transformed into open negative space. The expansion of linear edges into squares results in a sphere-like shape with 20 knobs. --- David Reimann

Mar 09, 2016
jmm16-orndorff.jpg
"OSU Triptych No. 2," by Robert OrndorffPaper and acrylic, 20 x 46 cm, 2015
Honorable Mention, 2016 Mathematical Art Exhibition

My paper folding art is math plus paper. Folded paper is simply tangible math. This is a permanent manifestation of an ephemeral artwork, namely, one solution for a specific one-straight-cut problem. Such problems are usually stated as follows: How must one fold a paper rectangle into a flat figure such that one straight cut through all of the layers will produce a given planar straight-line graph? Here the problem has been solved with paper and then represented in acrylic. To a significant degree the work relies on transmitted and reflected light, and so it never looks the same twice. The figure (the letters "OSU") has been divided into three frames. The crease patterns for the left and right letters are pedestrian but that for the central letter is sublime. --- Robert Orndorff
Mar 09, 2016
jmm16-mitchell.jpg
"Shield 1," by Kerry Mitchell (Phoenix, AZ)Digital print on aluminum panel, 40 x 40 cm, 2014

My work is composed primarily of computer generated, mathematically-inspired, abstract images. I draw from the areas of geometry, fractals and numerical analysis, and combine them with image processing technology. This image was created using a dynamic version of the Chaos Game algorithm. The Chaos Game is a simple example illustrating chaotic motion and strange attractors. It is typically implemented using three anchor points, which become the vertices of a Sierpinski triangle. The Dynamic Chaos Game allows the anchor points to move each iteration. Here, the image comprises seven panels. In each, the three anchor points slid along line segments, with different speeds. The pixels were then colored according how frequently that point was visited during the iteration. --- Kerry Mitchell
Mar 09, 2016
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