Common mathematical constants (some of which are also ubiquitous in science)

Ubiquitous in many different fields of science, such recurring constants include π, e and the Feigenbaum constants which are linked to the mathematical models used to describe physical phenomena, Euclidean geometry, analysis and logistic maps respectively. However, mathematical constants such as Apéry's constant and the Golden ratio occur unexpectedly outside of mathematics.

Archimedes' constant π

The circumference of a circle with diameter 1 is π.

Pi, though having a natural definition in Euclidean geometry (the circumference of a circle of diameter 1), may be found in many different places in mathematics. Pi is also the circumference of any given circle divided by its diameter.

Key examples include the Gaussian integral in complex analysis, n^th roots of unity in number theory and Cauchy distributions in probability. However, its universality is not limited to mathematics. Indeed, various formulas in physics, such as Heisenberg's uncertainty principle, and constants such as the cosmological constant bear the constant pi. The presence of pi in physical principles , laws and formulas can have very simple explanations. For example, Coulomb's law, describing the inverse square proportionality of the magnitude of the electrostatic force between two electric charges and their distance, states that, in SI units,

^[1]

Euler's number e

Exponential growth (green) describes many physical phenomena.

Euler's Number – also known as the exponential growth constant – e appears in many areas of mathematics, and is defined as the value of the following expression:

For example, the Swiss mathematician Jacob Bernoulli discovered that e arises in compound interest: An account that starts at $1, and yields 1+R dollars at simple interest, will yield e^R dollars with continuous compounding. The constant e also has applications to probability theory, where it arises in a way not obviously related to exponential growth. Suppose that a gambler plays a slot machine with a one in n probability and plays it n times. Then, for large n (such as a million) the probability that the gambler will win nothing at all is (approximately) 1/e.

Another application of e, discovered in part by Jacob Bernoulli along with French mathematician Pierre Raymond de Montmort, is in the problem of derangements, also known as the hat check problem.^[2] Here n guests are invited to a party, and at the door each guest checks his hat with the butler who then places them into labelled boxes. But the butler does not know the name of the guests, and so must put them into boxes selected at random. The problem of de Montmort is: what is the probability that none of the hats gets put into the right box. The answer is

and as n tends to infinity, p_n approaches 1/e.

 

The Feigenbaum constants α and δ

Bifurcation diagram of the logistic map.

Iterations of continuous maps serve as the simplest examples of models for dynamical systems.^[3] Named after mathematical physicist Mitchell Feigenbaum, the two Feigenbaum constants appear in such iterative processes: they are mathematical invariants of logistic maps with quadratic maximum points^[4] and their bifurcation diagrams.

The logistic map is a polynomial mapping, often cited as an archetypal example of how chaotic behaviour can arise from very simple non-linear dynamical equations. The map was popularized in a seminal 1976 paper by the English biologist Robert May,^[5] in part as a discrete-time demographic model analogous to the logistic equation first created by Pierre François Verhulst. The difference equation is intended to capture the two effects of reproduction and starvation.

Apéry's constant ζ(3)

 

Despite being a special value of the Riemann zeta function, Apéry's constant arises naturally in a number of physical problems, including in the second- and third-order terms of the electron's gyromagnetic ratio, computed using quantum electrodynamics.^[6] Also, the neuroscientist David Eagleman noted that

^[7]

where are the neutron mass, the electron mass and the Golden ratio respectively.

 

The golden ratio φ

Golden rectangles in an icosahedron

An explicit formula for the n^th Fibonacci number involving the golden ratio.

The number turns up frequently in geometry, particularly in figures with pentagonal symmetry. Indeed, the length of a regular pentagon's diagonal is times its side. The vertices of a regular icosahedron are those of three mutually orthogonal golden rectangles. Also, it appears in the Fibonacci sequence, related to growth by recursion.^[8] It is approximately equal to 1.61803398874, or, more precisely

Adolf Zeising, whose main interests were mathematics and philosophy, found the golden ratio expressed in the arrangement of branches along the stems of plants and of veins in leaves. He extended his research to the skeletons of animals and the branchings of their veins and nerves, to the proportions of chemical compounds and the geometry of crystals, even to the use of proportion in artistic endeavours. In these phenomena he saw the golden ratio operating as a universal law.^[9] Zeising wrote in 1854:

[The Golden Ratio is a universal law] in which is contained the ground-principle of all formative striving for beauty and completeness in the realms of both nature and art, and which permeates, as a paramount spiritual ideal, all structures, forms and proportions, whether cosmic or individual, organic or inorganic, acoustic or optical; which finds its fullest realization, however, in the human form.^[10]

 

The Euler–Mascheroni constant γ

The area between the two curves (red) tends to a limit.

The Euler–Mascheroni constant is a recurring constant in number theory. The French mathematician Charles Jean de la Vallée-Poussin proved in 1898 that when taking any positive integer n and dividing it by each positive integer m less than n, the average fraction by which the quotient n/m falls short of the next integer tends to γ as n tends to infinity. Surprisingly, this average doesn't tend to one half. The Euler–Mascheroni constant also appears in Merten's third theorem and has relations to the gamma function, the zeta function and many different integrals and series. The definition of the Euler–Mascheroni constant exhibits a close link between the discrete and the continuous (see curves on the left).

 

Conway's constant λ

Conway's look-and-say   sequence

Conway's constant is the invariant growth rate of all derived strings similar to the look-and-say sequence (except two trivial ones).^[11] It is given by the unique positive real root of a polynomial of degree 71 with integer coefficients.^[12]

 

Khinchin's constant K

If a real number r is written as a simple continued fraction:

then, as Russian mathematician Aleksandr Khinchin proved in 1934, the limit as n tends to infinity of the geometric mean: (a₁a₂...a_n)^1/n exists and is a constant, Khinchin's constant, except for a set of measure 0.^[13][14]