### Infinite Series

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Dimensions: 18” H X 24” W X 1.5” D. Landscape orientation.

Acrylic paint, open-sourced printed materials overlaid with high gloss varnish.

Original Artwork $750 + shipping.

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Srinivasa Ramanujan

This collage celebrates Srinivasa Ramanujan's life and his breakthroughs in pure mathematics. Specifically, his contributions to the infinite series formulas he and Hardy created for Pi, prime numbers (recognize them in the Infinite Series collage?), and mock theta functions. In addition, Ramanujan's research led to the infinite Farey series and their graphic representations in Ford Circles. He was born on December 22, 1887, in Madras, India and was the mathematician depicted in the movie “The Man Who Knew Infinity.” He had little formal training in pure mathematics. However, his brilliance enabled him to make substantial contributions to mathematical analysis, number theory, infinite series, and continued fractions. Ramanujan corresponded with a well-known mathematician G.H. Hardy at Trinity College, Cambridge, England. Hardy was impressed by Ramanujan’s work and invited him to England in order to advance Ramanujan’s work as well as to collaborate. Once in England, Ramanujan’s work earned him the prestigious Fellowship of the Royal Society. He became one of the youngest Fellows of the Royal Society and the first Indian to be elected a Fellow of Trinity College, Cambridge. He tragically became sick in 1919, traveled home to India, where he died in 1920. His work is still being researched today and has been used to understand black hole physics and string theory.

Elements of Art

A Farey Sequence from order 1 to 8

Portrait of G.H. Hardy

Ford circle diagram

Farey diagram

2

17

“A Formula of Ramanujan in the Theory of Primes”

Illustration of a black hole

Black hole physics

Infinity symbol

3

Quote from Ramanujan

A letter which was sent from Ramanujan to G.H. Hardy

A Farey sunburst

Illustration of Ford circles

Infinite Series

5 and 7

Pi

First 500 Digits of Pi

Portrait of Ramanujan

Signature of Ramanujan

The formula that relates Pi to a circle

Ramanujan’s infinite series formula for Pi

Om symbol

Explanation of Elements

A Farey sequence of order n is the sequence of completely reduced fractions, usually are between 0 and 1, which have denominators less than or equal to n, arranged in order of increasing size. Each Farey sequence starts with the value 0, denoted by the fraction 0/1, and ends with the value 1, denoted by the fraction 1/1.

Portrait of G.H. Hardy. Hardy was the Trinity College mathematics professor that Ramanujan corresponded with and who invited him to come to Cambridge to study and publish.

Ford circle diagrams are made up of individual Ford circles, a special case of mutually tangent circles. Each circle has its center positioned and radius as an irreducible fraction. Each circle is tangent to the horizontal axis, and any two Ford circles are tangent or disjointed to each other.

Farey diagram to F9 represented with circular arcs.

2, 3, 5, 7, 17 are prime numbers. Prime numbers are numbers that have only two factors: 1 and themselves.

“A Formula of Ramanujan in the Theory of Primes” is a paper that was published in April 1937 and was authored by G.H. Hardy using Ramanujan’s theories and formulas for Primes. Hardy gave Ramanujan full credit for his contribution to the content of the paper.

Illustration of a black hole from Wikipedia

Black hole physics:

The study of the nature and properties of the matter and energy of black holes. A black hole is a region of spacetime where gravity is so strong that nothing, even light, can escape once it has passed its event horizon.

The infinity symbol is called a lemniscate and represents something that is boundless or endless and was introduced in the 17th century as mathematicians developed and worked with calculus, infinite series, and infinitely small quantities.

Quote from Ramanujan. “An equation means nothing to me unless it expresses a thought of God.” Ramanujan was a deeply religious Hindu and followed the traditions of the Brahmin culture, including ceremonial worship called Puja and vegetarianism. He believed that the family goddess Namagiri Thayer endowed him with mathematical knowledge, revealed formulas to him and that all his mathematical skills were God speaking through him.

A letter which was sent from Ramanujan to G.H. Hardy. In this letter, Ramanujan outlined his summation formula for Hardy.

A Farey Sunburst is a graphic way to represent the relationship between the fractions in this series. Technically, a Farey sequence of order n connects the visible integer grid points from the origin in the square of side 2n, centered at the origin. Using Pick’s theorem, the sunburst area is 4(|Fn|-1), where |Fn| is the number of fractions in Fn.

Illustration of Ford circles:

Ford circles are a special case of mutually tangent circles based on the Farey sequence of irreducible fractions, usually from 0 to 1.

Infinite Series is the sum of infinitely many numbers related in a given way and listed in a given order.

Pi is the ratio of the circumference of any circle to the diameter of that circle.

First 500 Digits of Pi. Since Pi is an irrational number, it has an infinite number of digits in its decimal representation and does not settle into an infinitely repeating pattern of digits.

Portrait of Ramanujan before 1920. Unknown photographer.

Ramanujan’s signature from Wikimedia.

The formula that relates Pi to a circle.

The first calculation of Pi was done by Archimedes of Syracuse (287 -212 BC), one of the greatest mathematicians of the ancient world. Pi is the ratio of the circumference of any circle to the diameter of that circle.

Ramanujan’s Infinite series formula for Pi.

The accuracy of Pi improves by increasing the number of digits for calculation. In 1914, Ramanujan discovered the formula for computing Pi that converges rapidly.

The Om symbol is considered the highest sacred symbol of Hinduism. It refers to Atman (soul) and Brahman (ultimate reality, entirety of the universe, truth, divine, supreme spirit, and knowledge). Ramanujan was a devout Hindu and greatly valued this symbol.

References:

“Black Hole .” Wikipedia, Wikimedia Foundation, 4 June 2021, en.wikipedia.org/wiki/Black_hole.

Bogart, Steven. “What Is Pi, and How Did It Originate?” Scientific American, Scientific American, 17 May 1999, www.scientificamerican.com/article/what-is-pi-and-how-did-it-originate/.

A Brief History of π - Exploratorium. www.exploratorium.edu/sites/default/files/pdfs/history_of_pi.pdf.

“Farey Sequence.” Wikipedia, Wikimedia Foundation, 25 May 2021, en.wikipedia.org/wiki/Farey_sequence.

“File:Farey Diagram Horizontal Arc 9.Svg.” Wikipedia, Wikimedia Foundation, en.wikipedia.org/wiki/File:Farey_diagram_horizontal_arc_9.svg.

“Ford Circle.” Wikipedia, Wikimedia Foundation, 23 Mar. 2021, en.wikipedia.org/wiki/Ford_circle.

Hardy, G. H. “A Formula of Ramanujan in the Theory of Primes.” Journal of the London Mathematical Society, vol. s1-12, no. 2, 1937, pp. 94–98., doi:10.1112/jlms/s1-12.1.94.

“Infinite Series.” Encyclopædia Britannica, Encyclopædia Britannica, Inc., www.britannica.com/science/infinite-series.

“Infinity.” Wikipedia, Wikimedia Foundation, 5 June 2021, en.wikipedia.org/wiki/Infinity.

“Irrational Number.” Wikipedia, Wikimedia Foundation, 25 May 2021, en.wikipedia.org/wiki/Irrational_number.

“Om.” Wikipedia, Wikimedia Foundation, 7 June 2021, en.wikipedia.org/wiki/Om.

“Prime Number.” Wikipedia, Wikimedia Foundation, 30 May 2021, en.wikipedia.org/wiki/Prime_number.

“QUOTES BY SRINIVASA RAMANUJAN: A-Z Quotes.” A, www.azquotes.com/author/21463-Srinivasa_Ramanujan.

“Srinivasa Ramanujan.” Wikipedia, Wikimedia Foundation, 31 May 2021, en.wikipedia.org/wiki/Srinivasa_Ramanujan.