The
most beautiful
equation in mathematics
by Keith Devlin
Bertrand Russell, the famous English mathematician and philosopher, wrote
in his 1918 book Mysticism and Logic:
“Mathematics, rightly viewed, possesses not only truth, but supreme
beauty—a beauty cold and austere, like that of sculpture, without
appeal to any part of our weaker nature, without the gorgeous trappings
of painting or music, yet sublimely pure, and capable of a stern perfection
such as only the greatest art can show.”
Mathematics, the socalled science of patterns, is a way of looking at
the world, not only the physical, biological, and sociological world we
inhabit, but also the inner world of our minds and thoughts. Mathematics'
greatest success has undoubtedly been in the physical domain. Yet, as
an entirely human creation, the study of mathematics is ultimately a study
of humanity itself. For none of the entities that form the substrate of
mathematics exist in the physical world. The numbers, the points, the
lines and planes, the surfaces, the geometric figures, the functions,
and so forth are pure abstractions that exist only in the mind.
At the supreme level of abstraction where mathematical ideas may be found,
seemingly different concepts sometimes turn out to have surprisingly intimate
connections. There is, surely, no greater illustration of this than the
equation discovered in 1748 by the great Swiss mathematician Leonhard
Euler.
Euler’s equation
connects the five most significant and most ubiquitous constants in mathematics:
e, the base of the natural logarithms; i, the square root of –1;
þ, the ratio of the circumference of a circle to its diameter; 1, the
identity for multiplication; and 0, the identity for addition.
The number 1, that most concrete of numbers, is the beginning of counting
and the basis of all commerce, engineering, science, and music. The number
0 began life as a mere place holder in computation, a marker for something
that is absent, but eventually gained acceptance as a symbol for the ultimate
abstraction: nothingness. As 1 is to counting and 0 to arithmetic, þ is
to geometry, the measure of that most perfectly symmetrical of shapes,
the circle — though like an eager young debutante, þ has a habit
of showing up in the most unexpected of places. As for e, to lift her
veil you need to plunge into the depths of calculus — humankind’s
most successful attempt to grapple with the infinite. And i, that most
mysterious square root of –1, surely nothing in mathematics could
seem further removed from the familiar world around us.
Five different numbers, with different origins, built on very different
mental conceptions, invented to address very different issues. And yet
all come together in one glorious, intricate equation, each playing with
perfect pitch to blend and bind together to form a single whole that is
far greater than any of the parts. A perfect mathematical composition.
Like a Shakespearean sonnet that captures the very essence of love, or
a painting that brings out the beauty of the human form that is far more
than just skin deep, Euler’s equation reaches down into the very
depths of existence. It brings together mental abstractions having their
origins in very different aspects of our lives, reminding us once again
that things that connect and bind together are ultimately more important,
more valuable, and more beautiful than things that separate.
Keith Devlin, guest lecturer for the Wabash Center for Inquiry in the
Liberal Arts, is executive director of the Center for the Study of Language
and Information at Stanford University and is a contributor National Public
Radio’s Weekend Edition.
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