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## George Friedrich Bernhard Riemann / REE-mawn /

###### Mathematics Ranking 38th of 46

George Friedrich Bernhard Riemann founded Riemannian geometry, which is of fundamental importance to both mathematics and physics.

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Riemann studied under Carl Gauss at Göttingen, a university in Germany and became professor there. Riemann created Riemannian geometry, which is a form of differential non-Euclidean geometry where space is everywhere positively curved. In Riemann’s geometry, parallel lines do not exist, the angles of a triangle do not add up to 180°, and perpendiculars to the same line converge. It provided Einstein with a basis for his general theory of relativity.

The Riemann hypothesis, about the complex numbers that are roots of a certain transcendental equation, remains one of the unsolved problems of mathematics(1) and is one of the seven millennium problems which has a \$1 million prize for its solution. In the opinion of many mathematicians, the Riemann hypothesis, and its extension to general classes of L-functions, is probably the most important open problem in pure mathematics today.(2) (see end of conclusion for this section for more information about each of the millennium problems).

Non-Euclidean Geometry

“The creation of non-Euclidean geometry was the most consequential and revolutionary step in mathematics since Greek times . . . (because) the axioms of geometry are our basic facts about physical space, and vast branches of mathematics and physical science use the properties of Euclidean geometry.” pgs. 879-880 in Morris Kline’s Mathematical Thought from Ancient to Modern Times.

As presented in the biographies, four people are usually recognized for the discovery of non-Euclidean geometries – Gauss (1777-1855), Lobachevsky (1792-1856), Bolyai (1802-1860), and Riemann (1826-1866).

Gauss invented the term “Non-Euclidean Geometry” but never published anything on the subject.

Gauss’s work in geometry led to the discovery of the two different kinds of non-Euclidean geometry: hyperbolic geometry, which is associated with the Russian mathematician Lobachevsky and the Hungarian mathematician Bolyai, and elliptical geometry, which was discovered by Riemann, a student of Gauss. Although Gauss himself did not develop the complete mathematics of non-Euclidean geometry, he constructed the mathematical framework around which the main body of non-Euclidean geometry had to be built.(3)

To understand the need for non-Euclidean geometry, you must look at the weakest link in the chain of postulates on which Euclidean geometry is built—the famous parallel postulate (the fifth postulate), which states the following:

Given a straight line and a point outside this line, one and only one straight line (in the plane determined by the given line and point) can be drawn through the point parallel to the given line.

As far back as the fifteenth century, mathematicians were uncomfortable with this postulate and so they tried to prove it using the other postulates without success. Gauss was thus attracted to it as a challenge, but he soon became convinced that the parallel postulate is, indeed, a postulate and that it, therefore, cannot be proved; it must be accepted as a truth if one works with Euclidean geometry.(4)

Gauss, like Newton, abhorred controversy and refused to publicize mathematical discoveries that might lead to a conflict of ideas between him and other mathematicians. His letters and his diaries show that he had made many mathematical discoveries that he did not publish.(5) In 1800, for example, he discovered certain important functions known as “elliptic functions” and by 1816 he had extended his early geometry to encompass most of the features of the non-Euclidean geometries discovered later by Lobachevsky, Bolyai, and Riemann.(6) Gauss’s diaries show that he did not accept the Kantian thesis that the Euclidean geometry of space must be accepted as an absolute, a priori truth.(7) He insisted that the nature of the correct geometry of space must be determined by observation as an experimental fact. In line with this approach, Gauss proposed an experiment which could not be performed with the crude technology of that era. He suggested that the three angles of a triangle, formed by the three lines connecting three different stars, be measured to see if their sum is 180 degrees as demanded by Euclidean geometry. This experiment was never performed, but it underscores Gauss’s rejection of an a priori concept about the correct geometry of space.(8)

This discussion leads to the other great mid-nineteenth-century geometers Lobachevsky, Bolyai, and Riemann, who continued Gauss’s work on non-Euclidean geometry and brought it to its modern form. The works of Lobachevsky and Bolyai did not stem directly from Gauss’s work but rather from the rejection of Euclid’s fifth postulate (the parallel postulate), which had troubled mathematicians from the time of its pronouncement by Euclid to the discovery of non-Euclidean geometry in the nineteenth century. For almost two thousand years, outstanding mathematicians had tried to prove it as a theorem rather than accepting it as an independent postulate so that the notions of a non-Euclidean geometry were implicit in the failure of these proofs.(9) In his A Concise History of Mathematics, Dirk Struick notes that attempts to derive Euclid’s fifth postulate go back to Ptolemy, such mathematicians as Nasir-al-din in the Middle Ages and Lambert and Legendre in the eighteenth and nineteenth centuries, respectively, devoted much time to it.(10)

These attempts were not abandoned until after Gauss had developed his surface geometry which convinced Gauss that non-Euclidean geometries are valid but can be constructed only if the fifth postulate of Euclid is replaced by another parallel postulate. Why Gauss stopped at this point, going no further than introducing the label “non-Euclidean geometry” is not clear; he certainly had the mathematical ability to do so.

If Gauss was deterred from developing a complete theory of non-Euclidean geometry because he doubted its reality or its practical usefulness, Lobachevsky, Bolyai, and Riemann were quite the opposite. Being unsatisfied with Euclid’s parallel postulate, each, in his own way, decided to see how far he could go without Euclid’s fifth postulate. Simple geometrical considerations convinced each one that leaving the postulate out altogether would not do because it would be impossible to prove certain basic geometrical theorems.(11) The parallel postulate makes it possible for one to prove, for example, that the sum of the three angles of a triangle is 180 degrees and that the circumference of a circle equals 2 times pi ( about 3.14) times its radius r. But the restrictions imposed on Euclidean geometry by its basic elements, the line and the point, make it an ideal intellectual concept rather than a real one because neither a point nor a line had any reality.(12) Thus Euclidean geometry ultimately would have to give way to a real geometry which could only be produced by replacing Euclid’s fifth postulate by another “parallel” postulate. Bolyai and Lobachevsky, independently, carried this out in a way which led to what we now call hyperbolic geometry. Riemann, by contrast, replaced Euclid’s fifth postulate in quite a different way which led to what we now call elliptical geometry.

Both Bolyai and Lobachevsky replaced Euclid’s parallel postulate by the following statement:

Given a straight line and a point outside this line, an infinite number of straight lines can pass through the given point parallel to the given straight line.

This leads to a self-consistent non-Euclidean geometry called Lobachevskian or hyperbolic geometry.

This parallel postulate cannot apply to a flat surface nor to what we ordinarily call straight lines on a plane. The concept of a straight line must be replaced by the concept of the shortest distance between two points (called a geodesic). In Lobachevskian geometry, straight lines are curves. Triangles consist of three such curves with their concave sides out. In a triangle of this sort, the angles are smaller than they would be if the sides were straight in the Euclidean sense. Thus, in Lobachevskian geometry, the sum of the three angels of a triangle is smaller than 180 degrees and the circumference of a circle is larger than 2 times pi (about 3.14) times its radius. A saddle surface, which has two concave sides (no convex or flat side), is an excellent example of a hyperbolic surface. (see illustration below)

Riemann’s contributions to the development of non-Euclidean geometries stemmed, in part, from Gauss’s tutelage at the University of Göttingen, where Riemann earned his Ph.D. degree in mathematics in 1851. Riemann, like Lobachevsky, abandoned Euclid’s parallel postulate, replacing it by the postulate that no parallel lines exist. Indeed, all straight lines intersect in Riemannian (elliptical) geometry. The geometry on the surface of a sphere is an example of elliptical geometry; “straight lines” (shortest distances) on a sphere are arcs of great circles (circles whose radii equal the radius of the sphere) and they all intersect each other. Because the three sides of a triangle on a sphere are the arcs of three great circles, the sum of the three angles of such a triangle is greater than 180 degrees. One can also prove that the circumference of a circle on a sphere is always smaller than 2 times its radius. The circles of longitude (the meridian circles) on the earth’s surface are the equivalent of straight lines and all intersect each other at the north and south poles.

Whereas Euclidean geometry imposes exact arithmetic relations among the geometrical entities that define figures such as triangles (the sum of the angles must equal exactly 180 degrees) and circles (circumferences must equal exactly 2 pi (about 3.14) times their radii), Lobachevskian and Riemannian geometries allow a range of values for the sum of angles of a triangle and for the relationship between a circle’s circumference and its radius.(13) Thus, Euclidean geometry is like a line on a plane that divides the points on the upper half from those on the lower half. Euclidean geometry can be pictured as the limiting case of both Lobachevskian and Riemannian geometries; as one passes from one Riemannian geometry to the next, in infinitesimal steps, in a direction such that in each succeeding example of Riemannian geometry the sum of the three angles of a typical triangle decreases, we instantaneously pass through Euclidean geometry when the sum equals 180 degrees.(14) You then enter Lobachevskian geometry for which the three angle sum is less than 180 degrees and decreases as we move along.

1

How the total degrees in a triangle compare in elliptical, Euclidean, and hyperbolic space.

Possibly the most important contribution made by Riemann to non-Euclidean geometry was his development of n-dimensional non-Euclidean geometries.(15) He introduced the concept of the Riemannian curvature of n-dimensional geometry and wrote down the mathematics that one has to use to calculate this curvature at any point. His innovation ultimately led to tensor calculus, which is an extension of what is called vector analysis. Riemann’s development of the geometry of n-dimensional space became the basis of Einstein’s general theory of relativity and tensor analysis became its mathematics. The general theory of relativity is a four-dimensional non-Euclidean theory of gravity.

From the work of Lobachevsky, Gauss, Bolyai, and Riemann two new non-Euclidean geometries were discovered and each was found to be just as valid and consistent as Euclidean geometry. Also, it soon became clear that it is impossible to tell which, if any, of the three geometries is the most accurate as a mathematical representation of the real world. Thus, mathematicians were forced to abandon the cherished concept of a single correct geometry and to replace it with the concept of equally consistent and valid alternative geometries.(16) They were also forced to realize that mathematical systems are not merely natural phenomena waiting to be discovered; instead, mathematicians create such systems by selecting consistent axioms and postulates and studying the theorems that can be derived from them.(17)

Riemann Hypothesis

This is the only problem that remains unsolved from David Hilbert’s famous list in 1900 (see list under Hilbert). Mathematicians across the world agree that this obscure-looking question about the possible solutions to a particular equation is the most significant unsolved problem in mathematics.

The Riemann Hypothesis: The Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part ½. It asserts that all the 'non-obvious' zeros of the zeta function are complex numbers with real part 1/2.

Riemann created his hypothesis in 1859 as part of an attempt to answer one of the oldest questions in mathematics: What, if any, is the pattern of the prime numbers among all counting numbers? Around 350 B.C.E., the famous Greek mathematician Euclid proved that the primes continue forever; that is, there are infinitely many of them. Moreover, by inspection, the primes seem to “thin out” and become less common the higher up you go through the counting numbers. But can you conclude any more than that? The answer is yes.

Euclid also proved that every number bigger than 1 (i.e., every positive counting number bigger than 1) is either itself a prime or else can be written as the product of prime numbers in a way that is unique apart from the order in which the primes are written. For example,

21 = 3 x 7,

260 = 2 x 2 x 5 x 13.

The expressions to the right of the equals signs are the “prime decompositions” of the numbers 21 and 260, respectively. Thus, we can express Euclid’s result by saying that every counting number bigger than 1 is either prime or else has a unique (up to changing the order) prime decomposition.

This fact, called the fundamental theorem of arithmetic, tells us that the primes are like the chemist’s atoms—the fundamental building blocks out of which all numbers are constructed. Just as an understanding of the unique molecular structure of a substance can tell us a lot about its properties, knowing the unique prime decomposition of a number can tell us a lot about its mathematical properties.

Finally one of the deepest observations about the pattern of the primes was made by Gauss, Riemann’s Ph.D. advisor. In 1791, when he was just 14 years old, Gauss noticed that the prime density DN = P (N) / N is approximately equal to 1/ln (N), where ln (N) is the natural logarithm of N. As far as Gauss could tell, the bigger N got, the better this approximation became. He conjectured that this was not just an accident, and that by making N sufficiently large, the density Dn could be made as close as you want to 1/ln (N). Gauss was never able to prove his conjecture. This was finally achieved—using some very powerful math—in 1896 by the Frenchman Jacques Hadamard and the Belgian Charles de la Vallée Poussin, working independently. Their result is known today as the Prime Number Theorem.

There are at least two amazing aspects of this result. First, it demonstrates that, despite the seemingly random way that the primes crop up, there is a systematic pattern to the way they thin out. The pattern is not apparent if you look at an arbitrary finite stretch of the numbers. No matter how far out along the numbers you go, you can find clusters of several primes close together as well as stretches as long as you like in which there are no primes at all. Nevertheless, when you step back and look at the entire sequence of counting numbers, you see that there is a very definite pattern:  The larger N becomes, the closer the density DN gets to 1/ln (N).

The second and far more important feature of the Prime Number Theorem is the nature of the pattern of the primes that is uncovers. The counting numbers are discrete objects, invented by our ancestors some 8,000 years ago as a basis for trading. The natural logarithm function was invented by sophisticated mathematicians a mere two hundred years ago. It is not discrete; rather, its definition depends upon a detailed analysis of infinite processes, and forms part of the discipline sometimes called advanced calculus and sometimes called real analysis. One of several equivalent definitions of ln(x) is as the inverse to the exponential function ex.

If you wanted to represent the prime numbers on a graph, the most obvious way would be to mark a point at each prime number on the x-axis, as shown in Figure 1 below.

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The graph of the function ln(x), on the other hand, is a smooth, continuous curve, as shown in Figure 2 below. The question is this:  Why is there a connection between the irregularly spaced points on the x-axis in in Figure 1 and the smooth curve shown in Figure 2? How is it that the function ln(x) can tell us something about the pattern of the primes?

To sum up, a proof of the Riemann Hypothesis would add to our understanding of the prime numbers and the way they are distributed. This would do far more than satisfy the curiosity of mathematicians. Besides having implications in mathematics well beyond the patterns of the primes, it would have ramifications in physics and modern communications technology specifically Internet security.

Footnotes:

(1) Encyclopaedia Britannica, The 100 Most Influential Scientists (Philadelphia, 2008), p. 1239.

(2) J. Carlson, A. Jaffe, and A. Wiles, Editors, The Millennium Prize Problems (Cambridge, 2006), p. 108.

(3) Lloyd Motz and Jefferson Have Weaver, The Story of Mathematics (New York, 1993), p. 227.

(4) Ibid., p. 227.

(5) Ibid., p. 230.

(6) Ibid., p. 230.

(7) Ibid., p. 230.

(8) Ibid., p. 230.

(9) Ibid., p. 231.

(10) Ibid., p. 231.

(11) Ibid., p. 232.

(12) Ibid., p. 232.

(13) Ibid., p. 234.

(14) Ibid., p. 234.

(15) Ibid., p. 234.

(16) Encyclopaedia Britannica, Micropaedia, Volume 4, 1993, 15th Edition, p. 590.

(17) Ibid., p. 590.

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