Mathematical Imagery

From Math to Design

http://www.ams.org/mathimagery/displayimage.php?album=22&pos=0

• ’The Lake' is an object rising from ripples in a lake. The object is formed by placing 5 pointed stars on the transparent faces of a dodecahedron. The sine wave and harmonic ripples in the lake as well as the dodecahedron elements are rendered 3D models. The models are digitally composed with a scanned background. The mountains could also be fractal and algorithmically generated, but in this work the mountains are part of the base background scan which gives a better sense of depth to the artwork

'Totem' represents the frontier, the uncharted, the often surprising and almost mystic nature of mathematical discovery. The totem is composed of ellipsoids ((x2/a2)+(y2/b2)+(z2/c2)) = 1, ray-trace rendered over an algorithmically generated fractal skyscape. Atmospheric effects were calculated as well such as scattering, moisture etc. The totem signifying the last guidepost to the unknown

• Metal sculpture, 4.5" diameter. "The base of this sculpture is rhombic dodecahedron (polyhedron with 12 rhombic faces with cubical symmetry). Each of the 12 faces was transformed into a curved shape with 4 twisted arms, which connects to other shapes at vertices of valence 3 and 4. The boundary of the resulting body forms quite a complex knot. My artistic passions are purely mathematical images and sculptures, which express a certain vision of forms and shapes, my interpretations of distance, transformations and space. In my opinion, mathematics is not simply a profession, but rather a way of thinking, a way of life."

• Metal sculpture, 4.0" diameter. "Stellation of rhombic triacontahedron with 30 identical rhombic faces makes base for this sculpture. All internal intersections of rhombic faces were carefully eliminated by cutting away parts of rhombuses. The resulting 3D body was given organic shape by replacing straight faces with smooth subdivided surface. My artistic passions are purely mathematical images and sculptures, which express a certain vision of forms and shapes, my interpretations of distance, transformations and space. In my opinion, mathematics is not simply a profession, but rather a way of thinking, a way of life." --- Vladimir Bulatov, Independent Artist, Corvallis, OR

• Digital print, 11 x 5 . "Given a pair of start and target configurations, each � �consisting of n pairwise disjoint disks in the plane, what is the minimum number of moves that suffice for transforming the start configuration into the target configuration? In one move a disk slides in the plane without intersecting any other disk, so that its center moves along an arbitrary (open) continuous curve. One can easily show that 2n moves always suffice, while the above construction shows pairs of configurations that require 2n-o(n) moves for this task, for every sufficiently large n. Disks in the start configuration are white, and disks in the target configuration are shaded

• Third Prize, 2009 Mathematical Art Exhibition. Digital print, 19" x 12". Fathauer makes limited-edition prints inspired by tiling, fractals, and knots. He employs mathematics in his art to express his fascination with certain aspects of our world, such as symmetry, complexity, chaos, and infinity.

"The starting point for this iterated knot is a nine-crossing knot that has been carefully arranged to allow seamless iteration. Four regions of this starting knot are replaced with a scaled-down copy of the full starting knot, incorporated in such a way that the iterated knot is still unicursal. These same four regions are then replaced with a scaled-down copy of the iterated knot, resulting in a complex knot possessing self similarity."

• Stoneware, 12" x 8" x 12". This sculpture was made by starting with a cut circular band of clay and then bending and twisting before rejoining the cut ends. Props were used to preserve the shape while drying. The form was then sanded, low fired, sanded, and then high fired.

• Limestone, 9" diameter by 4 depth. "This sculpture was �carved from a circular piece of limestone. The form is based on the shape of the soap film minimal surface on a configuration of a wire trefoil knot. There is a nice interaction of the form and space with light and shadow."

• Mobile--Gauze, papyrus, silver and wood, 10" x 10" x 15". "'Crane' represents the continuous dimensional transition from a point, represented by a silver sphere, to a line, a plane and finally a crane. This transition is depicted in parallel to the evolution of the creative process which starts with an idea, represented by the same silver sphere, and which through refinements and trials culminates in the bird as well. [My] projects consist in artistic explorations that happen in parallel to the teaching/learning of mathematics (measure theory or complex analysis, for example). I search to generate art using mathematics and art inspired in the mathematics that I share with my students in order to motivate them to learn more mathematics, to make some extra connections, and to create some art of their own."

• 28 folded circles, 16 x6 x5 . "Nine inch paper plate circles are folded and reformed � � �into multiple units that have been arranged in one of many possible combinations of joining. Consistently following the development it began to take on a skeleton-like appearance and by decreasing the diameters of the circles it began to form a twisting conical helix, much like a sea slug, thus the name Skelug. Most all of my explorations with the circle start with folding three diameters, developing the equilateral triangular grid, reforming and joining multiples, which often reveals structural forms observable in nature."

• 52 folded 9 paper plate circles, 13 x13 x13 . "Forty circles have been folded, � � � �reformed to an in/out variation of a truncated tetrahedron, then octahedronally joined in pairs, and arranged in an icosahedron pattern. This revealed an interesting form of the icosadodecahedron with open pentagon stars. In this case twelve circles were reformed and added to suggest mouth-like openings found in sea anemones or in opening flower buds. This gives function to the open pentagons. Much of what I explore with folding circles are the structural functions of geometry found in life forms that correlate to the movement forms of the folded circle."

• Nylon (selective laser sintering), 3.5 x 3.5 x 3.5 . "This is a sculptural � � �interpretation, made by selective laser sintering, of two copies of the (10,3)-a lattice. Modern layered fabrication processes allow the construction of two interlocked components which are free to move slightly relative to each other, within the constraints of their being linked. The two copies are congruent, though mirror images. Each interpenetrates the tunnels of the other in a surprisingly complex manner. The 5x5x5 selection from the infinite lattice was made in such a way that the sculpture can stand vertically on a corner. See more works at http://www.georgehart.com."

• Digital print, 8" x 8 . "The artwork 'Spiral Squares' was originally created on a TI-84 Plus graphing �calculator. The design was uniquely created by using two equations only of linear function with restricted domain. These two equations have different parameters of the equation of each line segment, such as slope, y-intercept, domain x-left value, and domain x-right value, organized in a table. Each line segment is drawn by picking up its respected parameter value from the table. Once all the values from the table are exhausted the complete spiral square will appear on the calculator screen. The artwork is simple but truly illustrates the mathematical concepts. A single Spiral Square was then created on a computer using Geometer Sketchpad software. The artwork, shown here, is the simple translation effect of the single spiral square horizontally and vertically thrice."

• One folded square sheet of paper, 5" x 5" x 4". "This is one of the simplest pieces in my ongoing series of bowls formed by pleat tessellations. Every fold is a straight line segment parallel to an edge of the square sheet, and no fold is ever undone. The curved surface is purely a result of the intrinsic tension in the sheet of paper trying to unfold itself. This simple bowl is constructed by using alternating pairs of vertical and horizontal pleats, from the edges toward the center."

• First Prize, 2009 Mathematical Art Exhibition. One folded square sheet of paper, 10" x 10" x 5". "The wave is one of the pleat tessellations that continues to amaze me even years after I first folded it. The peculiar symmetry and the tension caused by locking the edges causes two of its corners to bulge in opposite directions, while the remaining two corners remain fairly flat. As in the simple bowl, the pleat sequences all begin at the edges and proceed towards the center of the sheet, but the difference is that all horizontal pleats are oriented the same way, and similarly all the vertical pleats."

• "The image represents the behavior of mathematical feedback loops, and more particularly the iteration of a complex function. The figure is our rendition of a visually interesting quartic variant of a Ushiki Phoenix Julia set. As with other fractals, the image exhibits a wealth of detail upon successive magnifications. The image Infinite Curl 7 has been made in collaboration � �with Dr. Clifford Pickover, the author of more than thirty books about mathematics, art, and science."

• Digital print, 9.9" x 10.1". Kraskek's interest is focused on the shapes' inner relations, on the relations between the shapes and between them and a regular pentagon. Her artworks also illustrate properties such as golden mean relations, self similarity, ten- and fivefold symmetry, Fibonacci sequence, inward infinity and perceptual ambiguity. She employs contemporary computer technology as well as classical painting techniques.

• Digital print, 12.6" x 11.9". Kraskek's interest is focused on the shapes' inner relations, on the relations between the shapes and between them and a regular pentagon. Her artworks also illustrate properties such as golden mean relations, self similarity, ten- and fivefold symmetry, Fibonacci sequence, inward infinity and perceptual ambiguity. She employs contemporary computer technology as well as classical painting techniques.

"The implicit decagon constituted of five smaller decagons expresses tenfold and fivefold rotational symmetry. The image where golden heart-like shapes are exposed shows self-similarity, the richness of relations between the decagons, pentagonal stars, Penrose rhombs, kites and darts with the golden ratio used several times as a scale factor."

• Computer-generated graphic art; digital print, 14 x 14 . "This complex and intricate design is created very simply � �from just one element the ellipse. The width and height are varied and the sets are rotated, but the result is not at �all what one might expect. In particular, the interior circles are generated entirely by the interaction of the parts. There are no circles drawn at all and the complexity of the design is entirely natural and unpredicted. It is my intention to use the computer as a tool to generate designs that are not only aesthetically pleasing, but that also reveal the order, structure and beauty inherent in mathematical objects. Additionally, if an attractive design can be made from the simplest of elements, then the generating process itself becomes an object of beauty as well. Complexity from a simple beginning via an elemental algorithm is a common, fascinating and universal process

• Digital print, 16" x 16". "This computer graphic represents three superimposed tessellations. The edges of a tessellation (6,6,7) are hidden below two nets consisting of tessellations (7,7,7) and (3,3,3,3,3,3,3), both dual to the original one. My inspiration stems from the rich geometric structures found in tessellations of the hyperbolic plane. Mathematical objects can be manipulated in many ways (superimposing, dualizing, breaking symmetry) to create aesthetically pleasing computer graphics brought to life by the unusual combination of colors."

• Lightjet print, 18" x 24". "Underlying this artwork is a two-dimensional plot of the 'typical behavior' of a chaotic dynamical system, a strange attractor. The base image is 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 the particular form is chosen, it is rendered as a high-resolution 16-bit grayscale image, colorized using gradient mapping and edited to enhance contrast, control composition, and add special effects. I love experimenting in the fuzzy overlap between art, mathematics, and programming. The computer is my canvas, and this is algorithmic artwork--a partnership mediated not by the brush or pencil but by the shared language of software. Seeking to extract and visualize the beauty that I glimpse beneath the surface of equations, I create custom interactive programs and use them to explore algorithms, and ultimately to generate artwork. "

• Lightjet print, 24" x 18". "Underlying this artwork is a two-dimensional plot of the 'typical behavior' of a chaotic dynamical system, a strange attractor. The base image is 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 the particular form is chosen, it is rendered as a high-resolution 16-bit grayscale image, colorized using gradient mapping and edited to enhance contrast, control composition, and add special effects. I love experimenting in the fuzzy overlap between art, mathematics, and programming. The computer is my canvas, and this is algorithmic artwork--a partnership mediated not by the brush or pencil but by the shared language of software. Seeking to extract and visualize the beauty that I glimpse beneath the surface of equations, I create custom interactive programs and use them to explore algorithms, and ultimately to generate artwork."

• Digital photography, 11 x 14 . "This work is a collage of photos taken during the fireworks � �display at Fair St. Louis on July 4, 2008. Each firework is somewhat self-similar and recursive in nature, with a common pattern appearing at both the center and the outer edges, and each piece having almost the same appearance. The shape is complex even on a small scale. The dimension of a firework is difficult to comprehend since its shape is constantly changing over time, but is a three-dimensional display. The change over time can be viewed and even is part of the overall image because of the appearance of the smoke left behind in the same shape as the colored flame. These art pieces are the product of a student research project I was a part of, exploring the relationship between art and math by a study of fractals."

• Second Prize, 2009 Mathematical Art Exhibit. Bronze, 9" tall. "The Figure-8 Knot is the second simplest knot, which can be drawn in the plane with as few as four crossings. When embedded in 3D space it makes a nice constructivist sculpture. This particular realization has been modeled as a B-spline along which a crescent-shaped cross section has been swept. The orientation of the cross section has been chosen to form a continuous surface of negative Gaussian curvature."

• Bronze, 8" tall. "The Chinese Button Knot is a nine-crossing knot, number 9-40 in the knot table. It actually has more symmetries than one would infer from the usual depiction in these tables. This has been brought out in this 3D sculpture, which has one 3-fold and three 2-fold rotational symmetry axes. It has been implemented as an alternating over-under path on the surface of a sphere, realized by a ribbon of continuous negative Gaussian curvature."

• Fortran, photosilkscreens, photolithographs, photographs, etc., 8" x 10". "I explore dynamic factor of line. I find computers to be a perfect tool to explore the regularity of nature. I use the computer on different levels. First I draw abstract geometric designs for executing my computer programs. Then I add photographic content using scanners and digital cameras. The programs that produce two-dimensional artwork serve as a point of departure for photolithographs and photo silkscreened prints on canvas and paper. All of these approaches are combined for image creation with the use of painterly markings."

• Hand-made ceramic tile, 15" diameter. "Islamic star pattern based on a tessellation of 18 and 12 pointed stars in a hexagonal repeat. My primary artistic interest is in designing repeatable patterns--I particularly enjoy creating geometric star and floral designs, which stem from my fascination with Islamic art." --- Nathan Voirol, CAD Drafter / Freelance Artist, Santa Barbara, CA

• Silkscreen print on paper, 20" x 24". "Islamic star pattern based on a tessellation of a 54 pointed star surrounded by 9 and 18 pointed stars in a hexagonal repeat. My primary artistic interest is in designing repeatable patterns--I particularly enjoy creating geometric star and floral designs, which stem from my fascination with Islamic art." --- Nathan Voirol, CAD Drafter / Freelance Artist, Santa Barbara, CA

• Laminated canvas and acrylic paint, 7.5" x 14" x 7.5". "'Flow 1' is created by intersecting two Golden Triangles (base angles of 72 and vertex angle 36 ). The plane of each � �triangle is partially bisected and then curved to create an aesthetically pleasing form. One triangle is smooth; the other has a textured surface. The sculpture changes our perception of a static and planar geometric shape and makes for a dynamic visual experience. The curves move the eye around the form and suggest multiple points of view." --- Elizabeth Whiteley, Studio artist, Washington, DC

• Museum board and acrylic paint, 7.5" x 13.5" x 10.5". "'Flow 4' is created by the close proximity of two Golden Triangles (base angles of 72 and vertex �angle 36 ). The plane of each triangle is curved in opposing directions to �create an aesthetically pleasing form. The sculpture changes our perception of a static and planar geometric shape and makes for a dynamic visual experience. The curves move the eye around the form and suggest multiple points of view." --- Elizabeth Whiteley, Studio artist, Washington, DC