by DK
Langlands began studying the relationship between integers and periodic functions as part of research into patterns in prime numbers. He was awarded the Abel Prize in 2018 for his “visionary” Program.
Key works
1967 Euler Products
1967 Letter to André Weil
1976 On the Functional Equations Satisfied by Eisenstein Series
2004 Beyond Endoscopy
See also: Fourier analysis • Elliptic functions • Group theory • The prime number theorem • Emmy Noether and abstract algebra • Proving Fermat’s last theorem
IN CONTEXT
KEY FIGURE
Paul Erdős (1913–96)
FIELD
Number theory
BEFORE
1929 Hungarian author Frigyes Karinthy postulates the concept of six degrees of separation in his short story, Láncszemek (Chains).
1967 American social psychologist Stanley Milgram conducts experiments on the interconnectedness of social networks.
AFTER
1996 The Bacon number is introduced on an American TV show. It indicates the number of degrees of separation an actor has from American actor Kevin Bacon.
2008 Microsoft conducts the first experimental study into the effects of social media on connectedness.
Hungarian mathematician Paul Erdős wrote and cowrote around 1,500 academic papers in his lifetime. He worked with more than 500 others in the global mathematical community across different branches of mathematics, including number theory (the study of integers) and combinatorics— a field of mathematics concerned with the number of permutations that are possible in a collection of objects. His motto, “Another roof, another proof,” referred to his habit of staying at the homes of fellow mathematicians in order to “collaborate” for a while.
The Erdős number, first used in 1971, indicates how far a mathematician is removed from Erdős in their published work. To qualify for an Erdős number, a person has to have written a mathematical paper—someone who coauthored a paper with Erdős would have an Erdős number of 1. Someone who worked with a coauthor (but not with Erdős directly) would have an Erdős number of 2, and so on. Albert Einstein has an Erdős number of 2; Paul Erdős’s number is 0.
Oakland University runs the Erdős Number Project, which analyzes collaboration among research mathematicians. The average Erdős number is around 5. The rarity of an Erdős number higher than 10 indicates the degree of collaboration within the mathematical community.
Erdős has an amazing ability to match problems with people. Which is why so many mathematicians benefit from his presence.
Béla Bollobás
Hungarian–British mathematician
See also: Diophantine equations • Euler’s number • Six degrees of separation • Proving Fermat’s last theorem
IN CONTEXT
KEY FIGURE
Roger Penrose (1931–)
FIELD
Applied geometry
BEFORE
4000 BCE Sumerian buildings incorporate tessellations into wall decorations.
1619 Johannes Kepler conducts the first documented study of tessellations.
1891 Russian crystallographer Evgraf Fyodorov proves there are only 17 possible groups that form periodic tilings of the plane.
AFTER
1981 Dutch mathematician Nicolaas Govert de Bruijn explains how to construct Penrose tilings from five families of parallel lines.
1982 Israeli engineer Dan Shechtman discovers quasi-crystals whose structure is similar to Penrose tilings.
Tile patterns have been a feature of art and construction for millennia, especially in the Islamic world. The need to fill two-dimensional space as efficiently as possible led to the study of tessellations—the fitting together of polygons with no gaps or overlap. Some natural structures, such as a honeycomb, tessellate.
There are three regular shapes that tessellate on their own, without the need for another shape: the square, equilateral triangle, and regular hexagon. However, many irregular shapes also tessellate, and semiregular tessellations involve more than one regular shape. The pattern of such tessellations usually repeats. This is known as a “periodic tessellation.”
Nonperiodic tessellations, in which the pattern does not repeat, are harder to find, although some regular shapes can be combined to create nonperiodic tessellations. British mathematician Roger Penrose investigated whether any polygons could only lead to nonperiodic tessellations. In 1974, he created tiles using kite and dart shapes. The kite and dart must be exactly the same shape as the ones shown (above); the area of the kite to that of the dart is expressed by the golden ratio. Although no part of the tiling matches another part exactly, the pattern does repeat on a larger scale in a similar way to a fractal.
Penrose tiling consists of kites and darts, producing a nonperiodic tessellation. Shapes with five-fold symmetry, such as pentagons and stars, can also be identified.
See also: The golden ratio • The problem of maxima • Fractals
IN CONTEXT
KEY FIGURE
Benoit Mandelbrot (1924–2010)
FIELDS
Geometry, topology
BEFORE
c. 4th century BCE Euclid sets out the foundations of geometry in Elements.
AFTER
1999 The study of “allometric scaling” applies fractal growth to metabolic processes within biological systems, leading to valuable medical applications.
2012 In Australia, the largest 3-D map of the sky suggests that the Universe is fractal up to a point, with clusters of matter within larger clusters, but ultimately matter is distributed evenly.
2015 Fractal analysis is applied to electrical power networks, leading to the modeling of the frequency of power failure.
A geometry able to include mountains and clouds now exists… Like everything in science this new geometry has very, very deep and long roots.
Benoit Mandelbrot
After Euclid, scholars and mathematicians modeled the world in terms of simple geometry: curves and straight lines; the circle, ellipse, and polygons; and the five Platonic solids—the cube, the tetrahedron, the octahedron, the dodecahedron, and the icosahedron. For much of the past 2,000 years, the prevailing assumption has been that most natural objects—mountains, trees, and so on—can be deconstructed into combinations of these shapes to ascertain their size. However, in 1975, Polish-born mathematician Benoit Mandelbrot drew attention to fractals—nonuniform shapes that echo larger and smaller shapes in a structure such as a jagged mountaintop. Fractals, a word derived from the Latin fractus, meaning “broken,” would eventually lead to the topic of fractal geometry.
A computer graphic shows a fractal pattern derived from the Mandelbrot set. Mesmerizingly beautiful, such images produced with fractal-generating software make popular screen savers.
A new geometry
Although it was Mandelbrot who brought fractals to the attention of the world, he was building on the findings of earlier mathematicians. In 1872, German mathematician Karl Weierstrass had formalized the mathematical concept of “continuous function,” meaning that changes in the input result in roughly equal changes in the output. Composed entirely of corners, the Weierstrass function has no smoothness anywhere, however much it is magnified. This was seen at the time as a mathematical abnormality that, unlike the sensible Euclidean shapes, had no real-world relevance.
In 1883, another German mathematician, Georg Cantor, built on work by British mathematician Henry Smith to demonstrate how to create a line that is nowhere continuous and has zero length. He did so by drawing a straight line, removing the middle third (leaving two lines and a gap), and then repeating the process ad infinitum. The result is a line composed entirely of disconnected points. Like the Weierstrass function, this “Cantor set” was considered unsettling by the mathematical establishment, who branded these new shapes “pathological”—meaning “lacking usual properties.”
In 1904, Swedish mathematician Helge von
Koch constructed a shape known as the Koch curve or “Koch snowflake,” which repeated a triangular motif at an ever smaller size. This was followed in 1916 by the Sierpinski triangle, or Sierpinski gasket, composed entirely of triangular holes.
All these shapes possess self-similarity, which is a key property of fractal geometry. This means that enlargement of a portion of the shape reveals smaller replicas with equal detail. Mathematicians realized that this was a fundamental property of natural growth—a repetition of a pattern on many scales, from the macro to the micro.
In 1918, German mathematician Felix Hausdorff proposed the existence of fractional dimensions. Whereas the simple line, plane, and solid occupy one, two, and three dimensions respectively, these new shapes could be given non-whole-number dimensions. For example, the British coastline could, in theory, be measured with a one-dimensional rope, but inlets would require string, and crevices require thread. This implies that the coastline cannot be measured in one dimension. The British coastline has a Hausdorff dimension of 1.26, like the Koch curve.
BENOIT MANDELBROT
Born into a Jewish family in Warsaw in 1924, Benoit Mandelbrot left Poland in 1936 to escape the Nazis. His family went first to Paris and then to the south of France. After World War II, Mandelbrot gained scholarships to study in France and then the US, before returning to Paris, where he was awarded a doctorate in mathematical sciences from the city’s university in 1952.
In 1958, Mandelbrot joined IBM in New York, where his role as a researcher gave him the space and facilities to develop new ideas. In 1975, he coined the term “fractal,” and in 1980 he unveiled the Mandelbrot set, a structure that became synonymous with the new science of fractal geometry. The topic gained popular appeal in 1982 with the publication of his book The Fractal Geometry of Nature. Mandelbrot received many honors and prizes for his work, including France’s Légion d’honneur in 1989. He died in 2010.
Key work
1982 The Fractal Geometry of Nature
Dynamic self-similarities
French mathematician Henri Poincaré found that dynamical systems (systems that change over time) also had fractal properties of self-similarity. By their nature, dynamical states are “nondeterministic”: two systems that are nearly identical can lead to very different behaviors even when the initial conditions are also almost identical. This phenomenon is popularly known as the “butterfly effect,” after the frequently cited example of the massive effect a single butterfly can theoretically have on a weather system when it causes a small disturbance by flapping its wings. The differential equations devised by Poincaré to prove his theory implied the existence of dynamical states that possess self-similarity much like fractal structures. Large-scale weather systems, such as major cyclonic flows, for instance, repeat themselves on much smaller scales, right down to gusts of wind.
In 1918, French mathematician Gaston Julia, a former student of Poincaré, explored the concept of self-similarity when he began to map the complex plane (the coordinate system based on complex numbers) under a process called iteration—entering a value into a function, obtaining an output, and then plugging that back into the function. Along with George Fatou, who undertook similar research independently, Julia found that by taking a complex number, squaring it, adding a constant (a fixed number or a letter standing for a fixed number) to it, and then repeating the process, some initial values would diverge to infinity while others would converge to a finite value. Julia and Fatou mapped these different values on a complex plane, noting which ones converged and which ones diverged. The boundaries between these regions were self-replicating, or fractal. With the limited computational power available at the time, Julia and Fatou were unable to see the true significance of their discovery, but they had found what would become known as the Julia set.
The Mandelbrot set
In the late 1970s, Benoit Mandelbrot used the term “fractal” for the first time. Mandelbrot had become interested in the work of Julia and Fatou while working at the IT company IBM. With the computer facilities available at IBM, he was able to analyze the Julia set in great detail, noting that some values of the constant (c) gave “connected” sets, in which each of the points is joined to another, and others were disconnected. Mandelbrot mapped each value of c on the complex plane, coloring the connected sets and the disconnected sets in different colors. This led, in 1980, to the creation of the Mandelbrot set.
Beautifully complex, the Mandelbrot set displays self-similarity at all scales: magnification reveals smaller replicas of the Mandelbrot set itself. In 1991, Japanese mathematician Mitsuhiro Shishikura proved that the boundary of the Mandelbrot set has a Hausdorff dimension of 2.
Infinite complexity is suggested by the self-similarities of a Romanesco cauliflower. The natural world is full of fractals, from ferns and sunflowers to ammonites and seashells.
Application of fractals
Fractal geometry has allowed mathematicians to describe the irregularity of the real world. Many natural objects exhibit self-similarity, including mountains, rivers, coastlines, clouds, weather systems, blood circulatory systems, and even cauliflowers. Being able to model these diverse phenomena using fractal geometry enables us to better understand their behavior and evolution, even if that behavior is not entirely deterministic.
Fractals have applications in medical research, such as understanding the behavior of viruses and the development of tumors. They are also used in engineering, particularly in the development of polymer and ceramic materials. The structure and evolution of the Universe can also be modeled on fractals, as can the fluctuations of economic markets. As the range of applications grows, along with ever-increasing computational capacity, fractals are becoming integral to our understanding of the seemingly chaotic world in which we live.
Fractals and the arts
Under the Wave off Kanagawa by Japanese artist Katsushika Hokusai (1760–1849) employs the concept of self-similarity to dramatic effect.
Self-similarity on infinite scales is explored in philosophy and the arts, often to produce a meditative effect. It is a key tenet of Buddhist meditation and mandalas (symbols used in rituals to represent the Universe), and is also used to suggest the infinite nature of God in Islamic decoration, such as tilework. Self-similarity is even suggested in the poem “Auguries of Innocence” by the 19th-century British poet William Blake, which begins with the line “To see a world in a grain of sand.”
The work of the Japanese artist Katsushika Hokusai, with its swirling repeated motifs, is often cited as an example of fractal use in art, as is the architecture of Catalan artist Antoni Gaudí.
The musical “rave” scene in the US and UK in the late 1980s and early ’90s was linked to a surge of interest in fractal art. Nowadays there are many fractal-generating computer programs, making it possible for the general public to create fractals.
See also: The Platonic solids • Euclid’s Elements • The complex plane • Non-Euclidean geometries • Topology
IN CONTEXT
KEY FIGURES
Kenneth Appel (1932–2013), Wolfgang Haken (1928–)
FIELD
Topology
BEFORE
1852 South African law student Francis Guthrie asserts that four colors are needed to color a map so that adjacent areas are not the same color.
1890 British mathematician Percy Heawood proves that five colors are sufficient to color any map.
AFTER
1997 In the US, Neil Robertson, Daniel P. Sanders, Robin Thomas, and Paul Seymour provide a simpler proof of the four-color theorem.
2005 Microsoft researcher Georges Gonthier proves the four-color theorem with general purpose theorem-proving software.
Cartographers have long known that any map, however complicated, can be colored in with just four colors, so that no two nations or regions sharing a border are the same color. Although five colors can seem to be necessary, there is always a way of recoloring the map using only four colors. Mathematici
ans searched for a proof for this deceptively simple theorem for more than 120 years, making it one of the most enduring unsolved theorems in mathematics.
The first person to formulate the four-color theorem is thought to have been Francis Guthrie, a South African law student. He had colored a map of the English counties using just four colors and believed that the same could be done with any map, however complex. In 1852, he asked his brother Frederick, who was studying under mathematician Augustus De Morgan in London, if his theory could be proved. Admitting that he could not prove the theorem, De Morgan shared it with Irish mathematician William Hamilton. Hamilton went on to attempt to prove the theorem himself, but did not succeed.
False start
In 1879, British mathematician Alfred Kempe claimed a proof for the four-color theorem in the scientific journal Nature. Kempe received plaudits for this work, and two years later became a Fellow of the Royal Society partly on the strength of his proof. However, in 1890, fellow British mathematician Percy Heawood found a hole in Kempe’s proof, and Kempe himself acknowledged that he had made a mistake that he could not rectify. Heawood did prove correctly that no more than five colors were needed to color any map.
Mathematicians continued to work on the problem, and gradual progress was made. In 1922, Philip Franklin proved that any map with 25 regions or fewer was four-colorable. The figure of 25 was slowly increased; Norwegian mathematician Øystein Ore and American mathematician Joel Stemple together achieved 39 in 1970, and Frenchman Jean Mayer lifted the figure to 95 in 1976.