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A New Appreciation for the Irish Mathematician William Hamilton

Besides pure mathematics, the fields of mechanics, topology, string theory, and mirror symmetry owe much to William Hamilton.

September 2, 2020, will mark the 155th anniversary of the death of the Irish mathematician William Rowan Hamilton. Today, Hamilton's work is proving central to the areas of field theories such as electromagnetism, and quantum mechanics.

Hamilton was born in Dublin, Ireland to an Irish solicitor, and at the age of three, he was sent to live with an uncle who ran a school. There, Hamilton displayed an uncanny ability to learn languages, becoming fluent in Hebrew, Persian, Arabic, Hindustani, Sanskrit, and Malay.

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At age 18, Hamilton entered Trinity College, Dublin where he received first a BA, then an MA in mathematics in 1837. While still an undergraduate, Hamilton was appointed to the post of Royal Astronomer of Ireland, and he moved to the Dunsink Observatory where he spent the remainder of his life.

Dunsink Observatory
Dunsink Observatory, Source: Dwmalone/Wikimedia Commons

Hamilton is most famous for reformulating Newtonian mechanics into Hamiltonian mechanics. He also created the Hamiltonian path, which is a traceable path that visits each vertex of a polyhedron exactly once.

Hamiltonian path around a dodecahedron
Hamiltonian path around a dodecahedron, Source: Arne Nordmann/Wikimedia Commons

A polyhedron is a three-dimensional shape with flat polygonal (multi-sided) faces, straight edges, and sharp corners or vertices. Hamiltonian paths in polyhedra had also been studied during the 18th-century mathematicians Abraham de Moivre and Leonhard Euler, by the 9th-century Indian mathematician Rudrata, and by Islamic mathematician al-Adli ar-Rumi.

Symplectic Geometry

Hamilton's work led to a new field called symplectic geometry, which is the study of geometric spaces having a symplectic structure. Eh, what's a symplectic structure?

A symplectic structure provides a way of measuring the area of a space. Hamilton discovered these spaces while analyzing the motion of the planets, and in such a space, you can change the space’s shape only if its area stays the same.

As a planet moves through space, its position in three-dimensional space is described by three coordinates along x, y, and z axes. Hamilton saw that at each point in three-dimensional space, you could also assign three additional coordinates: xm, ym, and zm that specify the planet’s momentum along each axis.

Symplectic space
A symplectic space, Source: Marcia Wendorf

Thus, each point in three-dimensional space can be assigned six coordinates, three that specify its position, and three that specify its momentum. This becomes a six-dimensional symplectic space. The word "symplectic" comes from the Greek word sumplektikós, which means "braided together". This reflects the way symplectic structure and complex numbers are intertwined.

If you remember from your high school algebra class, complex numbers include i which is the square root of -1. Complex numbers can be written in the form:
a + bi 
where a reflects the real part, and b is the imaginary part. You can define a six-dimensional symplectic space using three complex numbers. Today, symplectic geometry is used in the fields of string theory, topology, and mirror symmetry.

Besides inventing symplectic geometry, Hamilton also made breakthroughs in areas of conjugate algebraic couple functions (complex numbers are constructed as ordered pairs of real numbers), the solvability of polynomial equations, and the theory of Fluctuating Functions, which is used in Fourier analysis.

Quaternions

Hamilton is also the discoverer of quaternions, which are a number system that extends the complex numbers. An odd feature of quaternions is that multiplication of two quaternions is noncommutative. Commutative means that if we change the order of the operands, the result does not change.

In the real number system, "3 + 4 = 4 + 3" and "2 × 5 = 5 × 2", however, division and subtraction are noncommutative. For example, "3 − 5 ≠ 5 − 3".

Quaternions are expressed as:
a + bi + cj + dk
where a, b, c, and d are real numbers, and i, j, and k are quaternions. Quaternions were instrumental in putting the first man on the moon, and they are used for computer-generated graphics in movies.

Hamilton's legacy

On October 16th, 1843, Hamilton and his wife were walking along the bank of the Royal Canal when at Broome Bridge, Hamilton had a Eureka moment. He hastily scratched onto the bridge his formula for Quaternion algebra:
i² = j² = k² = ijk = -1.

Hamilton died in September 1865, and he is buried in Dublin's Mount Jerome Cemetery. In 2018, Ireland's National Transport Authority marked Hamilton's "graffiti" on Broome Bridge by commissioning an artwork for the space.

In 2005, the 200th anniversary of Hamilton's birth, Trinity College Dublin launched the Hamilton Mathematics Institute. The Royal Irish Academy holds an annual public Hamilton Lecture, and in 1943, two commemorative stamps were issued by Ireland in honor of William Hamilton.

Hamilton coin
Hamilton coin, Source: Wikimedia Commons

In 2005, a 10 Euros commemorative silver proof coin was issued by the Central Bank of Ireland to commemorate the 200 years since Hamilton's birth.

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