he read the works of EUCLID in the original Greek, the
works of SIR ISAAC NEWTON in Latin, and the works of
PIERRE-SIMON DE LAPLACE in French. He found a subtle error in Laplace’s classic text Mécanique céleste, and
wrote a letter to the astronomer royal of Ireland, John
Brinkley, explaining the error and how it should be corrected. Brinkley immediately recognized Hamilton’s
genius as a rising mathematician and publicly dubbed
him the “first mathematician of his age.”
Hamilton entered Trinity College at the age of 18
to study optics and mathematics. His original work in
these fields as an undergraduate, which included two
papers “Systems of Right Lines in a Plane” and “Theory of Systems of Rays,” was regarded as so significant
and innovative as to warrant his immediate appointment as a professor of astronomy at the college before
the completion of his basic degree.
Along with his work in optics, Hamilton made significant contributions to the field of GRAPH THEORY
and to the algebra of COMPLEX NUMBERS, publishing
results on the latter topic in his 1837 paper “Preliminary and Elementary Essay on Algebra as the Science
of Pure Time.” In 1842 he took on the difficult challenge of trying to create an algebraic system for threedimensional space that had the same algebraic properties
as the complex numbers in two-dimensional space. [A
point (x,y) in the plane can be matched with the complex number x + iy. This thus provides a means to “multiply” to two points in space: (x 1 , y 1 ) · (x 2 , y 2 ) = (x 1 x 2 –
y 1 y 2 , x 1 y 2 + x 2 y 1 ).] Although he was never able to find a
solution to this “multiplication of triples” problem, his
efforts did lead him to the remarkable discovery of a different type of number system suitable for four-dimensional space. He called this system the quarternions, and
found some surprising connections to mathematical
physics. In particular he observed that each quaternion
corresponds naturally to a physical transformation in
three-dimensional space and that the multiplication of
two quaternions matches perfectly with the composition
of the two physical transformations they represent. Thus
the geometry of three-dimensional physical space can be
reduced to the algebraic study of the algebra of quaternions. Hamilton was convinced his work would revolutionize mathematical physics. Although it does have
applications to the field today, sadly, his work did not
have the impact he hoped.
Hamilton received many awards throughout his
life, most notably a knighthood in 1835 and election to
the National Academy of Sciences in the United States
as its first foreign member. He wrote poetry for solace
throughout his life, and argued publicly that the language of mathematics is just as artistic as the language
expressed through poetry. His close friend poet William
Wordsworth (1772–1834) did not agree. Hamilton
died near Dublin on September 2, 1865.
Hamiltonian path/circuit See GRAPH THEORY.
ham-sandwich theorem As a generalization of the
INTERMEDIATE-VALUE THEOREM and the two-pancake
theorem that follows from it, mathematicians have
proved the following result, called the ham-sandwich
theorem:
Given any three objects sitting in three-dimensional space, there exists a single plane that
simultaneously slices the volume of each object
exactly in half.
For example, there is a single plane that simultaneously
slices the Eiffel tower, the planet Neptune, and this
book each precisely in half by volume. The theorem
gains its name from the following interpretation:
It is possible, in a single planar cut, to divide
each of two pieces of bread and a slab of ham
into two pieces of equal volume. This is possible no matter the shape of the food pieces
and no matter where in space the three items
are placed.
The result generalizes to higher dimensions:
Given any N objects sitting in N-dimensional
space, it is always possible to find an (N – 1)dimensional “plane” that simultaneously slices
the “volume” of each object in half.
(With N = 2, this is the two-pancake theorem.)
handshake lemma This amusing result states that,
at any instant, the number of people on this planet, living or deceased, who have taken part in an odd total
number of handshakes is necessarily even. This lemma
can be proved with the aid of GRAPH THEORY.
handshake lemma 245
works of SIR ISAAC NEWTON in Latin, and the works of
PIERRE-SIMON DE LAPLACE in French. He found a subtle error in Laplace’s classic text Mécanique céleste, and
wrote a letter to the astronomer royal of Ireland, John
Brinkley, explaining the error and how it should be corrected. Brinkley immediately recognized Hamilton’s
genius as a rising mathematician and publicly dubbed
him the “first mathematician of his age.”
Hamilton entered Trinity College at the age of 18
to study optics and mathematics. His original work in
these fields as an undergraduate, which included two
papers “Systems of Right Lines in a Plane” and “Theory of Systems of Rays,” was regarded as so significant
and innovative as to warrant his immediate appointment as a professor of astronomy at the college before
the completion of his basic degree.
Along with his work in optics, Hamilton made significant contributions to the field of GRAPH THEORY
and to the algebra of COMPLEX NUMBERS, publishing
results on the latter topic in his 1837 paper “Preliminary and Elementary Essay on Algebra as the Science
of Pure Time.” In 1842 he took on the difficult challenge of trying to create an algebraic system for threedimensional space that had the same algebraic properties
as the complex numbers in two-dimensional space. [A
point (x,y) in the plane can be matched with the complex number x + iy. This thus provides a means to “multiply” to two points in space: (x 1 , y 1 ) · (x 2 , y 2 ) = (x 1 x 2 –
y 1 y 2 , x 1 y 2 + x 2 y 1 ).] Although he was never able to find a
solution to this “multiplication of triples” problem, his
efforts did lead him to the remarkable discovery of a different type of number system suitable for four-dimensional space. He called this system the quarternions, and
found some surprising connections to mathematical
physics. In particular he observed that each quaternion
corresponds naturally to a physical transformation in
three-dimensional space and that the multiplication of
two quaternions matches perfectly with the composition
of the two physical transformations they represent. Thus
the geometry of three-dimensional physical space can be
reduced to the algebraic study of the algebra of quaternions. Hamilton was convinced his work would revolutionize mathematical physics. Although it does have
applications to the field today, sadly, his work did not
have the impact he hoped.
Hamilton received many awards throughout his
life, most notably a knighthood in 1835 and election to
the National Academy of Sciences in the United States
as its first foreign member. He wrote poetry for solace
throughout his life, and argued publicly that the language of mathematics is just as artistic as the language
expressed through poetry. His close friend poet William
Wordsworth (1772–1834) did not agree. Hamilton
died near Dublin on September 2, 1865.
Hamiltonian path/circuit See GRAPH THEORY.
ham-sandwich theorem As a generalization of the
INTERMEDIATE-VALUE THEOREM and the two-pancake
theorem that follows from it, mathematicians have
proved the following result, called the ham-sandwich
theorem:
Given any three objects sitting in three-dimensional space, there exists a single plane that
simultaneously slices the volume of each object
exactly in half.
For example, there is a single plane that simultaneously
slices the Eiffel tower, the planet Neptune, and this
book each precisely in half by volume. The theorem
gains its name from the following interpretation:
It is possible, in a single planar cut, to divide
each of two pieces of bread and a slab of ham
into two pieces of equal volume. This is possible no matter the shape of the food pieces
and no matter where in space the three items
are placed.
The result generalizes to higher dimensions:
Given any N objects sitting in N-dimensional
space, it is always possible to find an (N – 1)dimensional “plane” that simultaneously slices
the “volume” of each object in half.
(With N = 2, this is the two-pancake theorem.)
handshake lemma This amusing result states that,
at any instant, the number of people on this planet, living or deceased, who have taken part in an odd total
number of handshakes is necessarily even. This lemma
can be proved with the aid of GRAPH THEORY.
handshake lemma 245