The Math Book
Page 29
In addition to his work on primes, Riemann helped to formulate the rules for applying calculus to complex functions (functions using complex numbers). His revolutionary understanding of space was used by Einstein in developing relativity theory. Despite his success, Riemann struggled financially. He could finally afford to marry when he was awarded a full professorship by Göttingen in 1862. Just a month later, he fell ill and his health deteriorated until he died of tuberculosis in 1866.
Key work
1868 Über die Hypothesen, welche der Geometrie zu Grunde liegen (On the Hypotheses Which Lie at the Foundation of Geometry)
See also: • Mersenne primes • Imaginary and complex numbers • The complex plane • The prime number theorem
IN CONTEXT
KEY FIGURE
Georg Cantor (1845–1918)
FIELD
Number theory
BEFORE
450 BCE Zeno of Elea uses a series of paradoxes to explore the nature of infinity.
1844 French mathematician Joseph Liouville proves that a number can be transcendental—have an infinite number of digits arranged with no repeating pattern and without an algebraic root.
AFTER
1901 Bertrand Russell’s barber paradox exposes the weakness of set theory’s ability to define numbers.
1913 The infinite monkey theorem explains that given infinite time, random input will eventually produce all possible outcomes.
Infinity was a concept that mathematicians had long instinctively mistrusted. It was only in the late 1800s that Georg Cantor was able to explain it with mathematical rigor. He found there was more than one kind of infinity—an infinite variety, in fact—and that some were larger than others. In order to describe these differing infinities, he introduced “transfinite” numbers.
While he was studying set theory, Cantor aimed to create definitions for every number to infinity. This need arose from the discovery of transcendental numbers, such as π and e, which are irrational, infinitely long, and are not themselves an algebraic root. Between every algebraic number—including the integers, fractions, and certain irrational numbers (such as )—there is an infinite number of transcendentals.
Counting infinities
To help identify where a number is located, Cantor drew a distinction between two kinds of numbers: cardinals, which are the counting numbers 1, 2, 3… that denote the size of a set; and ordinals, such as 1st, 2nd, or 3rd, which list order.
Cantor created a new transfinite cardinal number—aleph (ℵ), the first letter of the Hebrew alphabet—to denote a set containing an infinite number of elements. The set of integers that includes the natural numbers, negative integers, and zero, was given the cardinality of ℵ0, the smallest transfinite cardinal, as these are theoretically countable numbers but are actually impossible to count completely. A set with a cardinality of ℵ0 starts with a first item, and ends with a ω (omega) item, a transfinite ordinal number. The number of items in a set with a cardinality of ℵ0 is ω.
Adding to that set makes a new set of ω + 1. A set of all countable ordinals, such as ω + 1, ω + 1 + 2, ω + 1 + 2 + 3…, will contain ω1 items. This set cannot be counted, making this infinity larger than countable ones, so it is said to have a cardinality of ℵ1.
The set of all ℵ1 sets contains ω2 items, with a cardinality of ℵ2. In this way, Cantor’s set theory creates infinities nestled inside each, expanding forever.
These concentric rings show the different types of numbers, which correspond to different types of infinities. Each ring describes a set of numbers. For example, the set of natural numbers is a small subset of rational numbers, which in turn combine with the set of irrational numbers to make the full set of real numbers.
GEORG CANTOR
Born in St. Petersburg, Russia, in 1845, Georg Cantor moved with his family to Germany in 1856. An outstanding scholar (and violinist), he studied in Berlin and Göttingen. He was later made a professor of mathematics at the University of Halle.
Although much admired by today’s mathematicians, Cantor was something of a pariah among his contemporaries. His theory of transfinite numbers clashed with traditional mathematical beliefs and the criticisms of leading mathematicians damaged his career. His work was also criticized by the clergy, but Cantor, who was deeply religious, saw his research as a glorification of God.
Overwhelmed by depression, Cantor was institutionalized for much of his later life. He began to receive plaudits in the early 1900s, but lived out his old age in poverty. He died of a heart attack in 1918.
Key work
1915 Contributions to the founding of the theory of transfinite numbers
See also: Irrational numbers • Zeno’s paradoxes of motion • Negative numbers • Imaginary and complex numbers • Calculus • The logic of mathematics • The infinite monkey theorem
IN CONTEXT
KEY FIGURE
John Venn (1834–1923)
FIELD
Statistics
BEFORE
c. 1290 Catalan mystic Ramon Llull devises classification systems using devices such as trees, ladders, and wheels.
c. 1690 Gottfried Leibniz creates classification circles.
1762 Leonhard Euler describes the use of logic circles, now known as “Euler circles.”
AFTER
1963 American mathematician David W. Henderson outlines the connection between symmetrical Venn diagrams and prime numbers.
2003 In the US, Jerrold Griggs, Charles Killian, and Carla Savage show that symmetrical Venn diagrams exist for all primes.
In 1880, British mathematician John Venn introduced the idea of the Venn diagram in his paper “On the Diagrammatic and Mechanical Representation of Propositions and Reasonings.” The Venn diagram is a way of grouping things in overlapping circles (or other curved shapes) to show the relationship between them.
Overlapping circles
The Venn diagram considers two or three different sets or groups of things with something in common, such as all living things, or all planets of the solar system. Each set is given its own circle and the circles are overlapped. Objects in each set are then arranged in the circles so that objects that belong in more than one set are placed where the circles overlap.
Two-circle Venn diagrams can represent categorical propositions, such as “All A are B,” “No A are B,” “Some A are B,” and “Some A are not B.” Three-circle diagrams can also represent syllogisms, in which there are two categorical premises and a categorical conclusion. For example: “All French people are European. Some French people eat cheese. Therefore, some Europeans eat cheese.”
As well as being a widely used tool for sorting data in everyday life, in contexts ranging from school classrooms to boardrooms, Venn diagrams are an integral part of set theory, due to their distinctive ability to express relationships.
Great ideas are the ones that lie in the intersection of the Venn diagram of ‘is a good idea’ and ‘looks like a bad idea.’
Sam Altman
American entrepreneur
See also: Syllogistic logic • Probability • Calculus • Euler’s number • The logic of mathematics
IN CONTEXT
KEY FIGURE
Édouard Lucas (1842–91)
FIELD
Number theory
BEFORE
1876 Édouard Lucas proves that the Mersenne number 2127 - 1 is prime. This is still the largest prime ever found without using a computer.
AFTER
1894 Lucas’s work on recreational mathematics is posthumously published in four volumes.
1959 American writer Erik Frank Russell publishes “Now Inhale,” a short story about an alien allowed to play a version of the Tower of Hanoi before his execution.
1966 In an episode of the BBC’s Doctor Who, the villain, The Celestial Toymaker, forces the Doctor to play a ten-disk version of the game.
French mathematician Édouard Lucas is believed to have invented his Tower of Hanoi game in 1883. T
he aim of the puzzle is simple. The challenger is presented with three poles, one of which holds three disks in order of size, with the largest disk on the bottom. The three disks must be moved one disk at a time so as to recreate the starting arrangement on a different pole using the smallest possible number of moves, with the restriction that players can only place a disk on top of a larger disk or on to an empty pole.
Solving the puzzle
With just three disks, the Tower of Hanoi can be solved in just seven moves. With any number of disks, the formula 2n ˗ 1 will give the minimum number of moves (where n is equal to the number of disks). One solution to the challenge employs binary numbers (0 and 1). Each disk is represented by a binary digit, or bit. A value of 0 indicates that a disk is on the starting pole; 1 shows that it is on the final pole. The sequence of bits changes at each move.
According to legend, if monks at a certain temple in either India or Vietnam (depending on the version of the tale) succeed in moving 64 disks from one pole to another in line with the rules, the world will end. However, even using the best strategy and moving one disk per second, they would take 585 billion years to complete the game.
A form of the Tower of Hanoi is a popular toy for small children. Versions with eight disks are often used to test developmental skills of older children.
See also: Wheat on a chessboard • Mersenne primes • Binary numbers
IN CONTEXT
KEY FIGURE
Henri Poincaré (1854–1912)
FIELD
Geometry
BEFORE
1736 Leonhard Euler solves the historical topological problem of “The Seven Bridges of Königsberg.”
1847 Johann Listing coins the term “topology” as a mathematical subject.
AFTER
1925 Russian mathematician Pavel Aleksandrov establishes the basis for studying the essential properties of topological spaces.
2006 Grigori Perelman’s proof of the Poincaré conjecture is confirmed.
Topology is, in simple terms, the study of shapes without measurements. In classical geometry, if a pair of shapes has equal corresponding lengths and angles, and you can slide, reflect, or rotate one of the shapes into the other, they are congruent— a mathematical way of saying they are identical. To a topologist, however, two shapes are identical—or invariant, in topological terminology—if they can be molded one into the other by continuous stretching, twisting, or bending, but with no cutting, piercing, or sticking together. This has led to topology being called “rubber-sheet geometry.”
For more than 2,000 years, from the time of Euclid, c. 300 BCE, geometry was concerned with classifying shapes by their lengths and angles. In the 18th and early 19th centuries, some mathematicians began to look at geometric objects differently, considering the global properties of shapes beyond the confines of lines and angles. Out of this grew the mathematical field of topology, which by the early 1900s had moved far from the notion of “shape” to embrace abstract algebraic structures. The most ambitious and influential exponent of this was French mathematician Henri Poincaré, who used complex topology to throw new light on the “shape” of the Universe itself.
Birth of a new geometry
In 1750, Leonhard Euler revealed that he had been working on a formula for polyhedra—three-dimensional figures with four or more planes, such as a cube or pyramid—that involved their vertices, edges, and faces rather than lines and angles. What he postulated became known as Euler’s polyhedral formula: V + F - E = 2, where V is the number of vertices, F the number of faces, and E the number of edges. The formula suggested that all polyhedra shared basic characteristics.
However, in 1813, another Swiss mathematician, Simone L’Huilier, noted that Euler’s formula was not true for all polyhedra; it was false for polyhedra with holes and for nonconvex polyhedra—shapes with some diagonals (linked by vertices) not contained within or on the surface. L’Huilier devised a system whereby every shape had its own “Euler characteristic”— (V - E + F)—and shapes with the same Euler characteristic were related regardless of how much they might be manipulated.
The term “topology”—derived from the Greek topos, meaning “a place”—was introduced to the mathematical world by German mathematician Johann Listing in 1847 in his treatise Vorstudien zur Topologie (Introductory Studies in Topology), although he had used the word in correspondence at least 10 years earlier. In particular, Listing was interested in shapes that did not satisfy Euler’s formula or defied the conventions of having distinct “outside” and “inside” surfaces. He even devised a version of the Möbius strip—a surface that has only one side and one edge—a few months before August Möbius.
Around the same period, another German mathematician, Bernhard Riemann, devised new geometrical coordinate systems that extended beyond the limits of the 2-D and 3-D systems devised by René Descartes. Riemann’s new framework enabled mathematicians to explore shapes in four dimensions or higher, including seemingly “impossible” shapes.
One such shape was the “Klein bottle,” devised in 1882 by German mathematician Felix Klein. He imagined joining two Möbius strips together to create a shape that has only one surface, is nonorientable (has no “left” or “right”), and, unlike a Möbius strip, has no edge or boundary curve. As it has no intersections, the shape can only truly exist in four-dimensional space. If the shape is represented in 3-D, it has to intersect itself, which is where it starts to look like a bottle. Topologists applied the term “2-manifold” to shapes such as the Möbius strip and Klein bottle to describe their surfaces, which are two-dimensional surfaces embedded within a space of higher dimension (the Möbius strip can exist inside three dimensional space, but the Klein bottle can only exist properly in four).
Euler’s formula, V + F - E = 2, works for most polyhedra, including a cube. Its values of V = 8, F = 6, and E = 12, when fed into the formula, produce the calculation 8 + 6 - 12 which equals 2.
Algebraic topology allows us to read qualitative forms and their transformations.
Stephanie Strickland
American poet
To a topologist, a coffee mug is identical in shape to a doughnut, because by pulling, stretching, and bending one, you could mold one into the shape of the other.
A universal conjecture
The shape of the Universe has long been a source of speculation. We appear to inhabit a 3-D world, but to make any sense of its shape we need to take ourselves outside this, into four dimensions. In the same way, to gain a sense of the shape of a 2-D surface, we need to look down on it in three dimensions. A starting point would be to imagine that we inhabit a Universe that is a 3-D surface embedded within four dimensions. Taking this one step further, you could consider that this 3-D surface is actually a sphere embedded in a 4-D space, also known as a “3-sphere.” A “2-sphere” is equivalent to a “normal” sphere (such as a ball) in a 3-D space.
In 1904, Henri Poincaré went even further, producing a theory that would help to lay a topological basis for understanding the shape of the Universe. He proposed what became known as the Poincaré conjecture: “every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.” A “3-manifold” is a shape that appears 3-D when its surface is enlarged but exists within higher dimensions, and “simply connected” means that it has no holes—like an orange but not a doughnut. A “closed” shape is finite, with no boundaries—like a sphere. Finally, “homeomorphic” describes shapes that can be molded into each other, such as a mug and a doughnut. A doughnut and an orange, however, are not homeomorphic because of the hole in the doughnut.
According to Poincaré, if it could be could shown that the Universe did not contain holes, then you could model it as a “3-sphere.” To establish whether it contained holes, you could, in theory, conduct an experiment with string. Imagine you are an explorer traveling around the Universe from a set point, and unraveling a ball of string as you go. When you get back to your starting point, you see the end of the string tha
t you started with. You take both ends, and start to gather in the string, pulling both ends. If the Universe is “simply connected,” then you would be able to gather in the whole string, like a loop following the smooth contours of a sphere; if you had passed through holes or gaps, then the string could get “snagged.” For example, if the Universe were shaped like a doughnut, and, in your travels, you wrapped your string around its girth, the string would get caught. You would not be able to gather in the string without pulling it beyond the Universe.
The BlackDog™ robot is designed to carry loads over rough terrain. The robot’s moves are computed using algebraic topology that can predict and model the surrounding “space.”
Shaping the future
Topology developments still continued during the 1900s. In 1905, French mathematician Maurice Fréchet devised the idea of a metric space—a set of points along with a “metric” that defines the distance between them.
Also at the turn of the 20th century, German mathematician David Hilbert invented the idea of a space that took the Euclidean spaces of two and three dimensions and generalized them to infinite dimensions. Mathematics could then be done in any dimension in much the same way as in a 3-D coordinate system. This area of topological mathematics has become known as “infinite-dimensional topology.”
The field of topology is now vast, embracing abstract algebraic structures far removed from a simple notion of “shape.” It has wide-ranging applications in areas such as genetics and molecular biology, such as helping to unravel the “knots” created around DNA by certain enzymes.