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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"Twelve Pi," by Sarah Berube (Diametric Arts, Shutesbury, MA)5 x 5 x 5 cm, 3D printed steel, 2016

My work currently revolves around the polyhedral symmetry groups. I pursue this form of art because I am strongly drawn to symmetry and the satisfying sense of beauty and perfection it evokes. My design process is an exploration of the properties of polyhedra. The design for this sculpture is comprised of twelve pi symbols arranged with tetrahedral symmetry. At four different locations, the right legs of three pi symbols spiral into each other, forming vortices in lieu of vertices. --- Sarah Berube
May 09, 2017
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"Invertible Infinity," by Ellie Baker (Lexington, MA)45 x 45 cm, silk crepe de chine (custom printed via Spoonflower.com), 2016

This reversible infinity scarf is a specially constructed cloth torus such that its shape is invariant under inversion AND it folds flat into a six-layer equilateral triangle. Since the meridians and longitudes of a torus swap places under inversion, one might think the invariance property dictates construction from a square piece of fabric (with opposite edges sewn together). However, although inversion invariance can be achieved with a square construction, the equilateral triangle folding cannot. Can you figure out a possible shape for the flat fabric layout used? The fabric designs, both P6M wallpaper group patterns that I created with Richter-Gebert’s app iOrnament, are a clue, and permitted sewing the pattern to match at the seams.The mathematical ideas incorporated into the design of this scarf were developed in collaboration with Charles Wampler. Thank you to Carol Maglitta for modeling. More information. -- Ellie Baker
May 09, 2017
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"A Fine Mesh We're In," by Dan Bach (Northern California, USA)35 x 45 cm, inkjet print on canvas, 2016

A central curve (not shown) has tangent, normal, and binormal directions at each point, making a local {T, N, B} frame. Using a trigonometric combination of the N and B vectors, we describe a toroidal mesh of curves with hues of green, yellow, and orange. Spheres of varying colors and sizes are placed along an equatorial helix and some try to escape their bonds. More information. --- Dan Bach
May 09, 2017
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"Toroweave," by David Bachman (Pitzer College, Claremont, CA)10 x 25 x 25 cm, 3D printed "sandstone" (gypsum powder + binder), 2016

These pieces were each created from two copies of a diamond tiling of a torus. The interior of each face of the tilings was removed, and the resulting webs were alternately offset in opposite directions to form a weave. Complementary colors are used to distinguish the two interlinked forms, which are completely disjoint. -- David Bachman
May 09, 2017
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"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
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"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
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"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
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"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
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"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
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"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
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"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
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"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
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"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
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