The Cat’s Eye Nebula is a sun-like star located 3000 light-years away in the constellation Draco. At its center is this cloud-like, luminous object, which was formed by a series of pulses at 1500-year intervals, during which the star gently ejected eleven or more gaseous rings. The outermost visible ring measures 1.2 light-years in diameter.
NASA, ESA, HEIC, and The Hubble Heritage Team (STScI/AURA)
This contemporary French artist has painted puzzles and geometric objects against the backdrop of two sixteenth-century publications on mathematics, Albrecht Dürer’s Melencholia I (1514), and Wenzel Jamnitzer’s Perspectiva Corporum Regularium (Perspective of regular solids; 1568). Dürer’s engraving has been interpreted as being about Plato’s dialogue on the impossibility of defining Beauty (Hippias Major, ca. 380–367 BC). The seated woman holding a compass (personifying Geometry) has failed to construct the Platonic symbol of a perfect universe—a dodecahedron—producing instead the large irregular polygon (on the left), over which Dürer placed a motto expressing Geometry’s gloomy attitude towards Plato’s (abandoned) search for Beauty: Melencholia I. Like Dürer, Jamnitzer lived in Nuremberg, where he was a goldsmith in the court of the German emperor. In 1568 he published this book of geometric patterns that he based on Plato’s dialogue on cosmology, Timaeus (366–360 BC), and Euclid’s Elements (ca. 300 BC), illustrating his text with 120 geometric forms based on the five Platonic solids.
Courtesy Sylvie Donmoyer
Scientists discover mathematical patterns in nature, such as the paths taken by electrons as they flow over the hills and valleys of tiny “landscapes” that are measured in microns (one micron equals one millionth of a meter). Paths of electrons in this digital print were recorded by Eric J. Heller, who studies rogue waves (freak waves, killer waves) on large and small scales. When a wave of electrons flows through a computer, a freak wave in a semiconductor can suddenly threaten the smooth functioning of the device.
Courtesy of Eric J. Heller
The ancient Babylonians divided the circular band of stars in the ecliptic into twelve constellations—the signs of the zodiac—one for each moon. They also divided the circle into 360 units, presumably approximating the days of the year—12 “months” (from Greek for “moons”) × 30 days. This diagram shows the constellations arranged counter-clockwise as they appear in the night sky (moving, for example, from Gemini in May, to Cancer in June, to Leo in July). Established in ancient times, these dates are somewhat different today because of the earth’s axial precession. (For example, sunrise today enters the constellation Cancer in late July rather than late June.) The pictures of the twelve signs are woodcuts made in 1491 for the first printed edition of the Italian astronomer Guido Bonatti’s manuscript Liber astronomiae (Book of astronomy, 1277).
Courtesy Umbra Studio, New York
The British art critic Roger Fry, who was a friend of the logician Bertrand Russell, believed that the essence of painting—like the essence of mathematics—is its logical form (as opposed to its color): “Mantegna, by negation of all the more brilliant and seductive qualities of paint, by reduction of expression to its simplest terms of flat slightly contrasting tones, bounded by contours of the utmost purity and perfection, has found precisely the method to give to his figures their mysterious spiritual life” (1905).
Courtesy of the Getty's Open Content Program.
In 1905, Albert Einstein discovered the symmetry of mass and energy—mass can be converted into energy, and vice versa (E = mc2). Then in the early decades of the twentieth century, physicists and mathematicians, including Einstein, gathered in Zurich and employed group theory in their exploration of the symmetry of nature. Swiss artists such as Gerstner created patterns that resonate with these mathematical descriptions of nature in terms of symmetry. Like the mathematicians, these artists established basic aesthetic building-blocks—units of color and form—and arranged them using rules that preserve proportion and balance. In 1956 Gerstner devised a modular system—a movable palette with 196 hues in 28 groups—for experimenting with progressions that link form with color. Gerstner’s palette of 196 squares has 28 groups with 7 squares each. Shown here are four of myriad possible arrangements, which the artist describes using the mathematician’s terms: groups, permutations, algorithms, and invariance.
Courtesy Karl Gerstner
Western mathematics proceeds by increasing abstraction and generalization. In the Renaissance the Italian architect Filippo Brunelleschi invented linear perspective, a method to project geometric objects onto a “picture plane” from a given viewpoint. Three centuries later the French mathematician Jean-Victor Poncelet generalized perspective into projective geometry for planes that are tipped or rotated. Then in the early twentieth century the Dutchman L.E.J. Brouwer generalized Poncelet’s projective geometry to projections onto surfaces that are stretched or distorted into any shape—so-called rubber-sheet geometry—provided that the plane remains continuous (with no holes or tears), which is the subject of this photograph. The contemporary artist Jim Sanborn created it by projecting a pattern of concentric circles onto a large rock formation at night from about 1/2 mile away. He then took this photograph with a long exposure at moonrise.
Courtesy Jim Sanborn
Erik Demaine earned a doctorate in computer science with a thesis on origami (traditional Japanese paper folding), and then went on to study folds that are not straight but curved. His research has direct applications to explaining how proteins fold, and he has collaborated extensively with mathematicians and molecular biologists to solve this crucial puzzle. The hope is that once biologists fully understand protein folding, they can quickly and accurately design proteins to target particular disease-causing viruses. In addition to his work on the mathematics of origami, Erik Demaine, together with his father the artist Martin Demaine, create origami sculpture such as this.
Courtesy Erik Demaine
With the development of railroads in the nineteenth century, the topic of finding an optimal route for a journey was of practical interest. The topic entered the mathematics literature in 1930, when the Viennese mathematician Karl Menger described it as the “messenger problem” (das Botenproblem) of finding an optimal delivery route. It was soon dubbed the “travelling salesman’s problem”: given a list of cities and the distances between each pair, find the shortest route that visits each city once and returns to the city of origin. The American mathematician Robert Bosch drew this continuous line based on the solution to a 5000-city instance of the travelling salesman problem. From a distance, the print appears to depict a black cord against a grey background in the form of a Celtic knot. But on close inspection the apparent “grey” is actually a continuous white line moving on top of a black background. The white line never crosses over itself—it is a network rather than a knot—and so the punny answer to the title is “Not.”
Courtesy Robert Bosch
In the early 1880s, the German mathematician Georg Cantor found that he could define a set by giving a recursive procedure to create it. For example, create a set of points by taking a line segment consisting of the infinity of dimensionless points between 0 and 1. Divide the line into thirds and remove the middle third as an open set (all points on the line except the end-points). Repeat the process on the remaining two line segments, and so on ad infinitum. This so-called Cantor (middle-third) set, which has the nickname “Cantor dust,” contains all points between 0 and 1 that are not deleted in this infinite process. After Cantor defined his middle-third set in terms of points on a line in one-dimensional space, twentieth-century mathematicians defined two and three-dimensional versions of Cantor dust, the Sierpiński carpet and the Menger sponge.
Courtesy Sylvie Donmoyer
Mathmatics and Art: A Cultural History by Lynn Gamwell is now available through Princeton University Press.
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