by DK
The genesis of such tools, the elliptic functions, began in England with 17th-century mathematicians John Wallis and Isaac Newton. Working independently, they developed a method for calculating the arc length, or length of a section, of any ellipse. With later contributions, their technique was developed into the elliptic functions and became a way of analyzing many kinds of complex curves and oscillating systems beyond the simple ellipse.
Practical applications
In 1828, Norwegian Neils Abel and German Carl Jacobi, again working independently, showed wider applications for elliptic functions in both mathematics and physics. For example, these functions appear in the 1995 proof of Fermat’s last theorem, and the latest public-key cryptography systems. Since Abel died at 26, just months after making his major discoveries, many of these applications were developed by Jacobi. Jacobi’s elliptic functions are complex, but a more simple form, the p-function, was introduced in 1862 by German mathematician Karl Weierstrass. P-functions are used in classical and quantum mechanics.
Elliptic functions are used to define the trajectories of spacecraft such as the Dawn probe, which explored the dwarf planet Ceres and the asteroid Vesta in the asteroid belt.
CARL GUSTAV JACOB JACOBI
Born in Potsdam, Prussia, in 1804, Carl Gustav Jacob Jacobi was initially tutored by an uncle. Having learned all that school could teach him by the age of 12, he had to wait until he was 16 to be allowed to attend Berlin University, and spent the intervening years teaching himself mathematics. He continued to do so when he found the university courses too basic. He graduated within a year, and in 1832 he became a professor at the University of Königsberg. Falling ill in 1843, Jacobi returned to Berlin, where he was supported by a pension from the king of Prussia. In 1848, he ran unsuccessfully for parliament as a liberal candidate and the offended king temporarily withdrew his support. In 1851, aged just 46, Jacobi contracted smallpox and died.
Key work
1829 Fundamenta nova theoria functionum ellipticarum (The foundations of a new theory of elliptic functions)
See also: Huygens’s tautochrone curve • Calculus • Newton’s laws of motion • Cryptography • Proving Fermat’s last theorem
IN CONTEXT
KEY FIGURE
János Bolyai (1802–60)
FIELD
Geometry
BEFORE
1733 In Italy, mathematician Giovanni Saccheri fails to prove Euclid’s parallel postulate from his other four postulates.
1827 Carl Friedrich Gauss publishes his Disquisitiones generales circa superficies curvas (General Investigations of Curved Surfaces), defining the “intrinsic curvature” of a space, which can be deduced from within the space.
AFTER
1854 Bernhard Riemann describes the kind of surface that has hyperbolic geometry.
1915 Einstein describes gravity as curvature in spacetime in his general theory of relativity.
The parallel postulate (PP) is the fifth of five postulates from which Euclid deduced his theorems of geometry in his Elements. The PP was controversial among the ancient Greeks, since it did not seem as self-evident as Euclid’s other postulates, nor was there an obvious way of verifying it. However, without the PP, many elementary theorems in geometry could not be proved. Over the next 2,000 years, mathematicians would stake their reputations on attempts to resolve the issue. In the 5th century CE, the philosopher Proclus argued that the PP was a theorem that could be derived from the other postulates and should therefore be struck out.
During the Golden Age of Islam (8th–14th century), mathematicians attempted to prove the PP. Persian polymath Nasir al-Din al-Tusi showed that the PP is equivalent to stating that the sum of angles in any triangle is 180°, but the PP nonetheless remained controversial. In the 1600s, new translations of Elements reached Europe, and Giovanni Saccheri showed that if the PP was untrue, then the sum of angles in a triangle was always either less than or greater than 180°.
By the early 1800s, Hungarian János Bolyai and Russian Nicolai Lobachevsky independently proved the validity of a “hyperbolic” non-Euclidean geometry in which the PP did not hold but the other four of Euclid’s postulates did. Bolyai claimed to have “created another world out of nothing,” but the idea was not well received in its time. Gauss acknowledged its validity, but claimed to have discovered it first. Gauss’s idea of the “intrinsic curvature” of a surface or space was an important tool in establishing this new world, but he left little evidence of having developed non-Euclidean geometry himself. He did, however, consider that the Universe might be non-Euclidean. Subsequent advances by Bernhard Riemann, Eugenio Beltrami, Felix Klein, David Hilbert, and others mean that today, non-Euclidean geometries are no longer seen as exotic, and physicists have given serious consideration to whether our Universe is indeed flat (Euclidean) or curved.
Leave the science of parallels alone. I was ready to… remove the flaw from geometry [but] turned back when I saw that no man can reach the bottom of this night.
Wolfgang Bolyai
Father of János Bolyai
Artistic explorations
Hyperbolic geometry also features in art. Models devised by Henri Poincaré inspired many graphic works by M. C. Escher, while some mathematicians, notably Daina Taimina, have used art and craft techniques to make these “new worlds” intuitively graspable.
Crochet models of hyperbolic surfaces created by Daina Taimina are more tactile than paper models. She claims that the crocheting process helps develop geometrical intuition.
DAINA TAIMINA
Born in Latvia in 1954, Daina Taimina began her career in the fields of computer science and the history of mathematics. After teaching for 20 years at the University of Latvia, she moved to Cornell University in the United States in 1996, where a chance encounter opened up a new area of interest. Taimina attended a geometry workshop led by David Henderson in which he demonstrated how to make a paper model of a hyperbolic surface. Henderson himself had learned the technique from pioneering American topologist William Thurston.
Taimina went on to make her own models of hyperbolic surfaces using crochet to assist in her teaching. The models were a success, breaking the stereotype of mathematics as a field unrelated to arts and crafts. Taimina has since embarked on a second career as a mathematician–artist.
Key work
2004 Experiencing Geometry with David W. Henderson
See also: Euclid’s Elements • Projective geometry • Topology • 23 problems for the 20th century • Minkowski space
IN CONTEXT
KEY FIGURE
Évariste Galois (1811–32)
FIELDS
Algebra, number theory
BEFORE
1799 Italian mathematician Paolo Ruffini considers the sets of permutations of roots as an abstract structure.
1815 Augustin-Louis Cauchy, a French mathematician, develops his theory of permutation groups.
AFTER
1846 Galois’ work is published posthumously by fellow Frenchman Joseph Liouville.
1854 British mathematician Arthur Cayley extends the work of Galois to a full theory of abstract groups.
1872 German mathematician Felix Klein defines geometry in terms of group theory.
Group theory is a branch of algebra that pervades modern mathematics. Its genesis was largely due to French mathematician Évariste Galois, who developed it in order to understand why only some polynomial equations could be solved algebraically. In so doing, he not only gave a definitive answer to a historical quest that had begun in ancient Babylon, but also laid the foundations of abstract algebra.
Galois’ approach to this problem was to relate it to a question in another area of mathematics. This can be a powerful strategy when the other area is well understood. In this case, however, Galois first had to develop the theory of the “simpler” area (the theory of groups) in order to tackle the more difficult problem (solubility of equations). The link he made between the two areas is now called Gal
ois theory.
Arithmetic of symmetries
A group is an abstract object—it consists of a set of elements and an operation that combines them, subject to some axioms. When these elements include shapes, groups can be thought of as encoding symmetry. Simple symmetries—such as those of a regular polygon—are intuitively graspable. For example, an equilateral triangle with the vertices A, B, and C can be rotated in three ways (through 120°, 240°, or 360°) around its center, and be reflected in three different lines. Each of these six transformations fits the triangle onto itself—it looks exactly the same, except that the vertices are permuted (rearranged). A clockwise rotation of 120° sends vertex A to where B was, B to C, and C to A, while a reflection in the vertical line through A swaps vertices B and C. The three rotations and the reflections give all possible symmetries of the triangle ABC.
One way to see the symmetries of the triangle is to consider all of the possible permutations of the vertices. A rotation or reflection can send the vertex A to one of three points (including itself). From each of these possibilities, the vertex B has two available destinations. The destination of the third vertex is now determined because the triangle is rigid, so there are 3 × 2 = 6 possibilities. The symmetry groups of polygons can be thought of as permutations of a set of elements. The symmetry group of the equilateral triangle is a member of a small group called D3.
ÉVARISTE GALOIS
Born in 1811, Évariste Galois lived a brief but fiery and brilliant life. He was already familiar as a teenager with the works of Lagrange, Gauss, and Cauchy, but failed (twice) to enter the prestigious École Polytechnique—possibly due to his mathematical and political impetuousness, though no doubt affected by the suicide of his father.
In 1829, Galois enrolled at the École Préparatoire, only to be expelled in 1830 for his politics. A staunch republican, he was arrested in 1831 and imprisoned for eight months. Shortly after his release in 1832, he became involved in a duel—it is unclear whether this was over a love affair or politics. Badly wounded, he died the next day, leaving behind just a handful of mathematical papers which contain the foundations of group theory, finite field theory, and what is now called Galois theory.
Key works
1830 Sur la théorie des nombres (On Number Theory)
1831 Premier Mémoire (First Memoir)
The equilateral triangle has six symmetries. They are rotation (ρ) through 120°, 240°, and 360° and reflection (σ) through a vertical line through A, B, or C. The diagram above shows the results of applying one symmetry after another to e, the identity element (rotation through 0°), and how they are written—ρ2σ (the last equilateral triangle in the diagram) means “rotate through 120 degrees twice and reflect.”
Axioms of group theory
Group theory has four main axioms. The first is the identity axiom; it states that a unique element exists that does not change any element in the group when combined with it. With the ABC triangle, the identity is the rotation of 0°. The second axiom is the inverse axiom. It says that every element has a unique inverse element; combining the two yields the identity element.
The third axiom concerns associativity, which means that the result of operations on elements does not depend on the order in which they are applied. For example, if you combine any set of three elements with a multiplication operator, you can perform the operations in any order. So if the elements 1, 2, and 3 are members of a group, then (1 × 2) × 3 = 2 × 3 = 6, and 1 × (2 × 3) = 1 × 6 = 6, all giving the same result.
The fourth axiom is closure, meaning that a group should have no elements outside the group as a result of performing the operations. One example of a group obeying all four axioms is the set of integers {…, -3, -2, -1, 0, 1, 2, 3, …} with the operation of addition. The unique identity element is 0, and the inverse of any integer n is ˗n as n + ˗n = 0 = ˗n + n. The addition of integers is associative, and the set is also closed, because adding any of the integers together gives another integer.
Groups can also have a further attribute known as commutativity. If a group is commutative, it is known as an Abelian group. This means that its elements can be swapped around without changing the result. Integers added in any order will give the same result (6 + 7 = 13 and 7 + 6 = 13), so the set of integers with the operation of addition is an Abelian group.
The possible rotations of a Rubik’s Cube form a mathematical group with 43,252,003,274,489,856,000 elements, but solving the cube from any position requires no more than 26 turns of 90°.
Wherever groups disclosed themselves, or could be introduced, simplicity crystallized out of comparative chaos.
Eric Temple Bell
Scottish mathematician
Galois groups and fields
Groups are just one kind of abstract algebraic structure among many. Closely related structures include rings and fields, which are also defined in terms of a set with operations and axioms. A field contains two operations; complex numbers (with the operations of addition and multiplication) are a field. The field of complex numbers is the territory in which solutions to polynomial equations are found.
Galois theory relates the solvability of a polynomial equation (whose roots are elements of a field) to a group—specifically, to the permutation group that encodes possible rearrangements of its roots. Galois showed that this group, now called a Galois group, must have one kind of structure if the equation is algebraically solvable, and a different kind of structure if it is not. Galois groups of quartic equations and simpler polynomials are solvable, but those of higher degree polynomials are not. Modern algebra is an abstract study of groups, rings, fields, and other algebraic structures.
Group theory continues to develop in its own right and has many applications. Group theory is used to study symmetries in chemistry and physics, for example, and can be used in public key cryptography, which secures much of today’s digital communication.
We need a super- mathematics in which the operations are as unknown as the quantities they operate on… such a super-mathematics is the Theory of Groups.
Arthur Eddington
British astrophysicist
Group theory in physics
The ATLAS detector at the CERN accelerator is designed to study subatomic particles, including those predicted by group theory.
The Universe, as we understand it through physics, is full of symmetries, and group theory is proving a powerful tool for both understanding and prediction. Physicists use the Lie groups, named after the 19th-century Norwegian mathematician Sophus Lie. Lie groups are continuous, not discrete—for example, they model the infinite number of rotational symmetries, such as those associated with a circle, rather than the finite number of transformations of a polygon.
In 1915, German algebraist Emmy Noether demonstrated how Lie groups related to conservation laws (such as the conservation of energy). By the 1960s, physicists began to use group theory to classify subatomic particles. But the mathematical groups they used included a combination of symmetries that no known particles had. Scientists tried looking for a particle with that combination of symmetries, and found the Omega minus particle. More recently, the Higgs boson has filled another such gap.
See also: The algebraic resolution of equations • Emmy Noether and abstract algebra • Finite simple groups
IN CONTEXT
KEY FIGURE
William Rowan Hamilton (1805–65)
FIELD
Number systems
BEFORE
1572 Italy’s Rafael Bombelli creates complex numbers by combining real numbers, based on the unit 1, with imaginary numbers, based on the unit i.
1806 Jean-Robert Argand creates a geometrical interpretation of complex numbers by plotting them as coordinates to create the complex plane.
AFTER
1888 Charles Hinton invents the tesseract, which is an extension of the cube into four spatial dimensions. A tesseract has four cubes, six squares, and four edges meeting at every corner.
An extension of complex numbers, quaternions are used to model, control, and describe motion in three dimensions, which is essential in, for example, creating the graphics of a video game, planning a space probe’s trajectory, and calculating the direction in which a smartphone is pointing. Quaternions were the brainchild of William Rowan Hamilton, an Irish mathematician who was interested in how to model movement mathematically in three-dimensional space. In 1843, in a flash of inspiration, he realized that the “third dimension problem” could not be solved with a three-dimensional number, but needed a four-dimensional one (a quaternion).
Movements and rotations
Complex numbers are two-dimensional: they are made up of a real and an imaginary part, for example, 1 + 2i. As a result, the two parts of any complex number can act as coordinates, and the number can be plotted on a surface or plane. The two-dimensional complex plane extends the one-dimensional number line by combining real numbers with imaginary units. The plotting of complex numbers then enables the calculation of motion and rotation in two dimensions. Any linear motion from point A to B can be expressed as the addition of two complex numbers. Adding more numbers creates a sequence of movements across the plane. To describe rotation, complex numbers are multiplied together. Every multiplication by i, the imaginary unit, results in a 90° rotation, and a rotation of any other angle is due to some factor or fraction of i.
Once complex numbers were understood, the next challenge for mathematicians was to create a number that worked the same way in a three-dimensional space. The logical answer was to add a third number line, j, which ran at 90 degrees to both the real and imaginary number lines, but no one could figure out how such a number added, multiplied, and so on.