Setting it straight with curved space

Georg Friedrich Bernhard Riemann ushered in a new era of geometry on June 10, 1854, when he delivered his classic lecture “On the Hypotheses Which Lie at the Foundations of Geometry”. Join A.S.Ganesh as he unravels the tale of how a shy and timid person with fantastic mathematical ability laid the groundwork for curved surfaces and higher dimensions...

June 09, 2019 01:23 am | Updated November 10, 2021 12:17 pm IST

Our understanding of the universe was overhauled in a period of 10 years from 1905-1915 when German-born theoretical physicist Albert Einstein gave us his special theory of relativity and the general theory of relativity. His description of gravity introduced the notion that a four-dimensional space-time warps and curves in response to mass or energy, changing centuries-old wisdom forever.

Bernhard Riemann.

Bernhard Riemann.

The geometric foundation for Einstein’s seminal work was laid a little over half-a-century before special relativity made its way. For, in 1854, Georg Friedrich Bernhard Riemann threw the doors of geometry open to curved spaces and higher dimensions, breaking away definitively from Euclidean geometry, which had dominated mathematical thought for over two millennia.

Born in 1826 in Hannover, Germany, Riemann was the second of six children born into a poor family. He started exhibiting traits for which he is now well-known from a very young age, be it his shyness, timidity or a fear of speaking in public, or his remarkable calculational capability and a natural flair for mathematics.

Taught by his father till the age of 10, his knowledge of mathematics soon surpassed those of his instructors when he attended a high school at Hannover. Realising that his faculty members were unable to keep up with Riemann, the school’s principal handed the youngster French mathematician Adrien-Marie Legendre’s “Theory of Numbers”.

“How far did you read?”

The principal was of the opinion that the colossal work, considered an authority in the subject of number theory, would consume Riemann and keep him occupied for long. Riemann, however, devoured it in days. It is believed that when the principal asked him days later “How far did you read?”, Riemann replied saying it is “a wonderful book” and that he had “mastered it”. Not convinced, the principal questioned Riemann from the book months later, only to find that Riemann really had all the answers.

Despite their poverty, Riemann’s father ensured that they had enough funds to send him to the University of Gottingen. Even though Riemann initially planned to study theology in the hope of shouldering the family’s financial burden, he swayed towards mathematics. Listening to the lectures of German mathematician Carl Friedrich Gauss inspired him and he soon switched his subjects.

Under Gauss’ tutelage, Riemann pursued his Ph.D. and completed his thesis by 1851. Gauss often remarked on Riemann having “a gloriously fertile imagination” and he put it to the test two years later when Riemann had to give an oral presentation to land a teaching position at Gottingen. The topic that Gauss hand-picked for Riemann for the lecture was the foundations of geometry.

A definitive breakthrough

Not one to disappoint his mentor, Riemann pored over the subject, developing a highly original theory of higher dimensions in the process and steering away from Euclidean geometry, which believed in only three dimensions or fewer. Despite his phobia for public speaking, he successfully presented these in a lecture titled “On the Hypotheses Which Lie at the Foundations of Geometry” on June 10, 1854. Some of his thoughts were so advanced that they were beyond many who had gathered on the day and only Gauss was able to fully appreciate his star pupil’s breakthrough.

The lecture, which consisted of two parts, included a working definition on calculating the curvature of space. Whereas the first part catered to defining an n-dimensional space, laying the foundation for Riemannian geometry, the second part discussed the dimensions of real space and the kind of geometry that could be used to describe it.

Even though the lecture was an emphatic triumph, and Riemann also made telling contributions to analysis and number theory, among others, his life was rather short-lived. Plagued by ill health and nervous breakdowns for much of the time he lived, he died prematurely in 1866, aged just 39. His works, however, have had a large influence on both mathematics and physics, immortalising him in more ways than one.


A million dollars and more

In 1859, Riemann formulated a hypothesis, which now bears his name – Riemann hypothesis. While grasping it might not be that easy, it should be noted that this problem is one of the seven problems in mathematics stated by the Clay Mathematics Institute as the Millennium Problems in 2000. Attempting any of these seven problems and providing a correct solution carries with it a prize money of $1 million, which will be awarded to the discoverers by the institute. Till date, only one of the seven problems have been solved. Riemann hypothesis is among the remaining six.

It is believed that Riemann’s work likely influenced Oxford Mathematics professor Charles Dodgson. Dodgson, who realised that children were much more receptive and open to certain seemingly bizarre concepts than adults, wove strange mathematical ideas into his novel. If you are wondering who this Dodgson really is, he went under the pseudonym Lewis Carroll. Yes, the same Carroll who gave us Alice’s Adventures in Wonderland and its sequel Through the Looking-Glass. Even though Dodgson was a traditional Euclidean, the preposterous world he constructed for Alice is said to symbolise the intellectual upheaval that mathematics was going through late in the 19th Century.

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