The Beauty of
Mathematics
Equations
Please study the following equations, which each have a short description, but which
are in no particular order. If you want to research any of them further, please go
ahead. You should rate the beauty of each equation on a scale from -5 (ugly),
through zero (neutral) up to +5 (beautiful).
Office use only
Subject ID
Date returned
The Beauty of Mathematics - Equations
1
#
Equation
Description
Rating
1
Euler's identity links 5 fundamental
mathematical constants with three
basic arithmetic operations each
occurring once.
2
The Pythagorean identity, which
states that for any angle, the square
of the sine plus the square of the
cosine is 1.
3
Euler's formula for triangulation of
a polyhedron, where V is the
number of vertices, E edges and F
faces.
4
The Gauss-Bonnet theorem
connects the geometry of surfaces
(curvature) to their topology (Euler
characteristic).
5
Identity between exponential and
trigonometric functions derivable
from Euler's formula for complex
analysis.
6
Definite Gaussian integral -
ubiquitous in mathematical physics.
7
The reciprocal of the zeta function
can be expressed as a Dirichlet
series over the Möbius function
μ(n)
8
Series expansion for the
exponential function.
9
The Fourier transform of a
Gaussian is a Gaussian.
10
An identity for Euler's number e.
The Beauty of Mathematics - Equations
2
#
Equation
Description
Rating
11
The cardinality of the continuum
has the same number of elements
as the power set of the integers
and is greater that the cardinality
of the integers.
12
Defines the Mandelbrot set, that
is a complex number c is part of
the Mandelbrot set if, when
starting with z
0
= 0 and applying
the iteration repeatedly, the
absolute value of z
n
remains
bounded however large n gets.
13
Identity making use of the Dirac
delta function.
14
Equation expressing the inverse
value of π as an infinite sum.
15
The smallest number expressible
as the sum of two cubes in two
different ways.
16
Pythagoras' theorem: In any
right-angled triangle, the area of
the square whose side is the
hypotenuse (the side opposite the
right angle) is equal to the sum of
the areas of the squares whose
sides are the two legs (the two
sides that meet at a right angle).
17
The first fundamental theorem of
calculus guarantees the existence
of antiderivatives for continuous
functions.
18
Cauchy's residue theorem is a
powerful tool to evaluate line
integrals of analytic functions
over closed curves.
19
Lotka-Volterra equations
(predator-prey equations), are a
pair of first-order, non-linear,
differential equations used to
describe the dynamics of
biological systems in which two
species interact, one a predator
and one its prey.
20
Diffusion equation, a partial
differential equation which
describes density dynamics in a
material undergoing diffusion.
The Beauty of Mathematics - Equations
3
#
Equation
Description
Rating
21
π is the ratio of a circle's
circumference with its diameter.
22
The derivative of an exponential
is an exponential.
23
Taylor expansion of an analytic
function.
24
Eigenvalue equation for an
operator A.
25
Triangle inequality for a normed
vector space, the norm of the sum
of two vectors is at most as large
as the sum of the norms of the
two vectors.
26
Prime number theorem: If x is any
positive real number, then π(x),
which is the number of primes
less than x, is approximately x/log
x.
27
The Euler product formula for the
Riemann zeta function where the
left hand side equals the Riemann
zeta function and the product on
the right hand side extends over
all prime
numbers p.
28
Pythagoras' theorem for a 3:4:5
triangle.
29
Cauchy's integral formula.
30
Equation expressing the
connection between π and odd
numbers.
The Beauty of Mathematics - Equations
4
#
Equation
Description
Rating
31
The Basel problem, set as a
challenge by Jakob Bernoulli in
1689 and triumphantly solved by
Euler in 1735.
32
Sum of an infinite geometric series
33
Birkhoff's ergodic theorem.
34
Stoke's theorem states that the
integral of a differential form ω
over the boundary of some
orientable manifold M is equal to
the integral of its exterior
derivative dω over the whole of M
35
A special case of Poisson
summation.
36
The quadratic variation of a one
dimensional Brownian motion, up
to time t is t. This is basic to both
Feynman's path integral approach
to quantum mechanics and to
stochastic
analysis.
37
Another equation due to Euler: on
the left hand side we have an
infinite product and on the right
hand side appears a power series
where p(n) denotes the number of
all possible representations of n as
a sum of natural numbers.
38
An inequality due to Cauchy
expressing a relation between a
product and a sum.
39
The cardinality of the empty set is
zero.
40
Cartan structural equations, where
e is the frame field, omega the
connection form, T the torsion
form and R the curvature form.
The Beauty of Mathematics - Equations
5
#
Equation
Description
Rating
41
Stirling's approximation to n!.
42
Integral formula for a character of
an irreducible representation of a
Lie group corresponding to the co-
adjoint orbit Ω.
43
Volume of an n-dimensional
sphere.
44
Relation between the sphere, the
complex projective line and the
special orthogonal groups SO(3)
and SO(2).
45
An example of an exact sequence
where the image of one morphism
equals the kernel of the next.
46
Second Bianchi identity of the
Riemann tensor.
47
Definition of a Möbius
transformation.
48
Clifford algebra.
49
The integer 1 expressed as an
infinite series involving the
Riemann zeta function
50
The intersection of a set with its
complement is empty.
2.
The Beauty of Mathematics - Equations
6
#
Equation
Description
Rating
51
The complement of the universal set
is the empty set.
52
A formulation of the spectral
theorem expresses the operator as
an integral of the coordinate
function over the operator's
spectrum with respect to a
projection-valued measure.
53
Properties defining Berezin
integration, where theta is an
anticommuting variable over the
complex numbers.
54
Cauchy-Riemann equations are a
system of two partial differential
equations which must be satisfied if
a complex function is complex
differentiable.
55
Laplace equation.
56
Pell's diophantine equation.
57
Sine-Gordon equation, a nonlinear
hyperbolic partial differential
equation in 1 + 1 dimensions.
58
Fermat's Last Theorem states that
no three positive integers a, b, and c
can satisfy this equation for any
integer value of n greater than two.
59
Riemann's functional equation,
where Γ(s) is the gamma function
and ζ(s) is the Riemann zeta-
function
60
Contracted Bianchi identity where
R
μν
is the Ricci tensor and R is the
scalar curvature