Lotus, 1974 - The Geometric Sculptures of Morton C. BradleyThe sculpture below, Lotus, 1974, is a form based on the icosidodecahedron in which pentapyramids are nested within each other, each one rotated so that its edge meets the midface of the larger piece. It is painted using 50 different colors in systematic gradations. More »
Rawles HallThe Department of Mathematics resides in Rawles Hall, a building built with Indiana limestone, and in this regard very characteristic of Indiana University's Bloomington Campus. More »
manuscriptThis is the first page of Bernhard Riemann's autograph of his article "On the Number of Prime Numbers less than a Given Quantity" ("Über die Anzahl der Primzahlen unter einer gegebenen Grösse."), which appeared in the Monatsberichte der Berliner Akademie, in November 1859. More »
Möbius bandThis Möbius band is twisted three times, each time by 180 degrees, before its short sides are attached to each other. Topologically, it is the same as the usual Möbius band where the band is twisted only once before attaching the short sides to each other; it is just embedded in 3-space in a different way. More »
A Perfect SquareA perfect square is a large square subdivided into smaller squares, where no two of the smaller squares are allowed to be of equal size. It was originally very unclear whether perfect squares existed, but in 1939 R. Sprague found one with 55 smaller squares. More »
The GyroidThe Gyroid was discovered in 1970 by the NASA engineer Alan Schoen. It is a minimal surface ("soap film") that extends periodically in three independent directions in space. More »
Costa's minimal surfaceIn 1984, Celso Costa wrote down a formula for a minimal surface that would awaken the slumbering mathematical discipline of minimal surfaces. Up to then, the known complete, embedded minimal surfaces of finite topology were just the plane, the catenoid and the helicoid, and many mathematicians believed that that was that. More »
Riemann's minimal surfaceIn a posthumous paper from 1867, Bernhard Riemann discusses which minimal surfaces have horizontal cross sections that are all lines or circles. Known examples were the catenoid and the helicoid. Using elliptic functions, Riemann solved the problem completely and wrote down a new 1-parameter family of such surfaces. More »
Charles Darwin's first drawing of an evolutionary treeThe figure is Charles Darwin's first drawing of an evolutionary tree from his First Notebook on Transmutation of Species (1837) on view at the Museum of Natural History in Manhattan. More »
mazeConsider a square grid on a torus. Choose a spanning tree uniformly at random. If we use this spanning tree to form walls, we obtain an interesting maze on the torus. This maze has the homology of a punctured torus, since the walls are contractible to a point. More »
SymmetryThe torus has a symmetry of period 2 given by a 180° rotation about the axis in the picture. This symmetry has 4 fixed points. Less obvious is a symmetry of period 3 with exactly 3 fixed points. More »
square tiling generated by 10x10 electrical networkPlane electrical networks are connected to square tilings of rectangles. In this instance, a 10x10 grid is used as an electrical network between two opposite corners. More »
peanoA uniform spanning tree in a 99x99 square grid is surrounded by a Peano-like curve. The hue denotes the progress along the curve. It has a very interesting scaling limit, i.e., a limiting random curve in a square if we take finer and finer lattice meshes inside the square. More »
free and wired uniform spanning forests for the (2, 3, 7)-triangle tessellationThis shows two random spanning forests, one on a Cayley graph of the (2, 3, 7)-triangle tessellation of the hyperbolic plane, and the other on its plane dual. More »
windChoose a uniform spanning tree T in a 200x200 square grid. Associate to the path from the lower left corner to a vertex x its net number h(x) of turns (also called its winding number), i.e., the number of times it turns left minus the number of times it turns right. More »
pathConsider a 200x200 square grid. A spanning tree is a subset of edges that, for any vertices x and y, allows for exactly one path joining x and y using these edges. More »
color decompositionAssume you plan a roof over a triangular building with sides equal to 11 yards. Each side of the building allows only two peaks, producing a pattern of 4-4-3 yards on each side. You would also require that the lines indicating "hips" and "valleys" (preferably no valleys) should be parallel to the sides of the triangle. More »
the results of numerical simulationsThese figures show the results of numerical simulations performed at the Institute for Scientific Computing and Applied Mathematics at Indiana University. They are related to three-dimensional simulations of a flow in a limited domain, corresponding to simplified models of geophysical flows concerning weather forecast (short term forecast) or climate forecast (long term forecast). More »
Kakeya Needle SetA Besicovitch set, or Kakeya set, is any set of points in Euclidean space which contains a unit line segment in every direction. Quite surprisingly, there exist Besicovitch sets with zero Lebesgue measure. More »
cardsPlayers a, b, and c were dealt cards from a three card deck. The cards are labeled clubs, hearts, and diamonds. The deal is private, and the actual result was that a received clubs, b diamonds, and c hearts. After this, a peeked at b's card, thereby learning c's card, too. More »
A computer program that trades places with all possible inputsIn 1938, my academic grandfather Stephen Kleene published a paper containing a result called the Second Recursion Theorem. This result has many consequences in many areas of mathematical logic. It also implies that in any programming language, there is a computer program self such that running self outputs self itself. The program below, trade, has an even stronger property. More »
Recursion, Fractals, and Category TheoryConsider the equation f(x) = g(f(x),x). This defines f(x) in terms of itself, and it falls under the rubric of recursion. If we take g to be a function symbol with no interpretation, then we can take the solution f(x) to be the tree shown in blue. But we might also start with one or another fixed interpretation of g. For example, let A be the complete metric space of compact subsets of the unit square. More »
Spectra of an unstable detonation waveThe image depicts the spectra, or resonant frequencies, of an unstable detonation wave, plotted in the complex plane, where the vertical line to the left is the imaginary axis. Each point λ marked in the figure corresponds to a time-exponential solution eλtw(x) of the linearized perturbation equations about a steadily propagating detonation front, which is growing or decaying as eRe(&#955); t. More »
Trefoil KnotThis is an image of the simplest nontrivial knot, the trefoil knot. Knot theory, a subject within the general area of Topology, is the mathematical study of knotting. In this study, the tools of mathematics, from algebra to geometry, are applied to answer basic questions about knots. More »
Traveling WaveThe Kuramoto-Sivashinsky equation,a partial differential equation (PDE), is a canonical model for pattern formation, arising in such varied applications as plasma dynamics, propagation of flame fronts, inclined thin film flow, and turbulence. One interesting class of solutions consists of periodic traveling waves $u(x,t)=\bar u(x-ct)$: periodic wave-forms propagating at speed $c$ without changing shape. These satisfy the simpler ordinary differential equation (ODE). More »
A dogboneThis is an example of a picture, called a dogbone, frequently seen in mathematical articles about cut-and-paste problems in analysis on manifolds. More »
bifurcationThe figures show a global bifurcation for the Kuramoto-Sivashinsky equation. An unstable manifold (red) bursts through a pair of stable manifolds upon a slight change of parameter. More »
Rational FunctionThe image below shows preimages of the upper and lower half-planes under several iterates of the rational function. More »
Sierpinsky CarpetThe Julia set of a rational map can be a so-called Sierpinski carpet. The colored regions in the figure below form a countable collection of open disks in the sphere for which (i) each boundary of which is a simple closed curve, (ii) the closures of no two distinct disks intersect, (iii) the diameters of the disks tend to zero. More »
The Geometric Sculptures of Helaman FergusonThe Department of Mathematics is pleased to have and share this sculpture by artist and mathematician Helaman Ferguson. The Figure Eight Knot Complement III, 1992, is direct-carved in white Carrara marble, with a honeycomb texture and esker (raised ridge). More »
Four Dragon Curves Cover the PlaneThe four colored dragon curves begin in the middle and represent paths between integer points on the plane. Here is an explanation of how the red figure works. The first 2 steps are east, north (EN). To get the first 4 steps, we take EN, write it backwards to get NE, and then rotate counterclockwise to get WN. We put WN after the first two steps (EN) to get ENWN, and this is the first 4 steps on the red dragon. More »
Cubic Surfaces and Negative CurvatureThe Clebsch Diagonal Surface partly shown above belongs to one of the 45 families of (possibly singular) cubic surfaces in real projective 3-space. A few of the 27 straight lines that live on this surface are also drawn. More »
The Julia set of a mappingThe Julia set of a mapping is the set that carries the nontrivial (chaotic) dynamics of the map. If the mapping depends on a parameter c the associated Julia set Jc may or may not behave continuously. More »