31 Jul Some Favourite Mathematical Constants
Mathematical constant are really exciting and wonderful in the world of numbers. All numbers are not created equal; that certain constants appear at all and then echo throughout mathematics, in seemingly independent ways, is a source of fascination. Just as physical constants provide “boundary conditions” for the physical universe, mathematical constants somehow characterize the structure constants, the Archimedes’ constant $$pi$$ (pi) was once regarded as the king, and in modern times (after 1980), the Feigenbaum Universal constant $$delta=4.6692016091029dots$$ is regarded as the ‘Super King’ in this field on the basis of its increasing importance and tremendous uses in mathematical sciences, physics, chemistry, biosciences, economics, engineering etc.
The constants listed below are rather arbitrarily organized by topicwise. The concrete elaboration of how these constants are derived needs a long discussion, and hence detailed discussion is intentionally omitted. Interested readers are requested to contact the author for details of any constant. However, how the Feigenbaum Universal constant can be obtained is very briefly highlighted below with the help of a simple example.
Let $$f(x)=ax(1x),$$ where a is a constant. The interval $$[0,1]$$ is mapped into itself by $$f$$ for each value of $$ain [0,4].$$ This family of functions, parameterized by a, is known as the family of logistic maps. What are the 1cycles (i.e. fixed points) of f? Solving x=f(x) we obtain
x=0 (which attracts for a<1 and repels for a>1),
and $$x=frac{a1}{a}$$ (which attracts for 1<a<3 and repels for a>3).
What are the 2cycles of f? That is, what are the fixed points of the iterate $$f^{2}$$ which are not fixed points of f? Solving $$x=f^{2}(x), xneq f(x),$$ we obtain the 2cycle.
$$x=frac{a+1pm (a^{2}2a3)^{frac{1}{2}}}{2a}$$ (which attracts for $$3<a<1+sqrt{6}$$ and repels for $$ a>1+sqrt{6}$$).
For $$a>1+sqrt{6}=3.4495dots$$ an attracting 4cycle emerges. We can obtain the 4cycle by numerically solving $$x=f^{4},$$ and $$xneq f^{2}.$$ It can be shown that 4cycle attracts for 3.4495….<a<3.5441…., and repels for a>3.5441…..
For a>3.5441…., an attracting 8cycle emerges. We can obtain the 8cycle by numerically solving $$x=f^{8},$$ and $$xneq f^{4}.$$ It can be shown that 8cycle attracts for 3.5441….<a<3.5644…., and repels for a>3.5644…..
For how long does the sequence of period doubling bifurcations continue? It’s interesting that this behavior stops for short of 4. Setting $$a_{0}=1,a_{1}=3,a_{2}=1+sqrt{6}=3.4495dots , a_{3}=3.5441dots ,a_{4}=3.5644dots ,$$ and so forth denote the cascade of bifurcations, it can be proved that
$$a_{infty}=lim_{infty}a_{n}=3.5699dots <4.0$$
This point marks the separation between the “periodic regime” and the “chaotic regime” for this family of quadratic functions. The sequence $${a_{n}}$$ behaves in a universal manner such that the ratio $$frac{a_{n}a_{n1}}{a_{n+1a_{n}}}$$ tends to a universal constant $$delta =4.6692016091029dots .$$
The elementary particle theorist, Mitchell J. Feigenbaum working in the University of Princeton, U.S.A, has explain in details the creation of this constant in his two marvelous papers.
(i) Quantitative Universality for a class of Nonlinear Transformations, J. of Statistical Physics, 19:1(1978), 2552.
(ii) The Universal Metric Properties of Nonlinear Transformations, J. of Statistical Physics, 21:6(1979), 669706.
Favourite Mathematical constants so far we know are listed bellow with their approximate numerical values:
Wellknown constants 
Numerical Value 

1  Zero  0 
2  One  1 
3  Imaginary unit  $$sqrt{1}$$ 
4  Pythagoras’ constant  $$sqrt{2}$$ 
5  Golden mean  $$phi=1.6180339887dots$$ 
6  Natural logarithmic base  C=2.7182818285…. 
7  Archimedes’ constant  $$pi=3.14159265358979dots$$ 
8  EulerMascheroni constant  $$gamma=0.5772156649dots$$ 
9  Ape’’ry’s constant  $$zeta(3)=1.202056903dots$$ 
10  Catalan’s constant  G=0.915965594…. 
11  Khintchine’s constant  K=2.68545200…. 
12  Feigenbaum constant  $$delta=4.6692016091029dots$$ 
13  Madelung’s constant  $$M_{2}=1.6155426267dots$$ 
14  Chaiten’s constant  Not available 
Constants associate with Number Theory
15  HardyLittlewood constant  $$C_{twin}=0.6601618158dots$$ 
16  Hadamardde la Valle’e Poussin constant  $$C_{1}=0.2614972128dots$$ 
17  LandauRamanujan constant  K=0.764223653…. 
18  Brun’s constant  B=1.90211605778…. 
19  Artin’s constant  $$C_{Artin}=0.3739558136dots$$ 
20  Linnik’s constant  Not available 
21  HafnerSarnakMeCurley constant  $$D_{infty}=0.3532363719dots$$ 
22  GaussKuzminWirsing constant  $$lambda=0.3036630029dots$$ 
23  StolarskyHarborth constant  $$Theta=1.58496dots$$ 
24  Porter’s constant  C=1.4670780794…. 
25  GlaisherKinkelin constant  A=1.28242713…. 
26  Franse’nRobinson constant  2.8077702420…. 
27  AllodiGrnstead constant  0.809394020534…. 
28  Niven’s constant constant  C=1.705211…. 
29  Backhouse’s constant  1.456074485826…. 
30  Mill’s constant  C=1.3064…. 
31  Stieltjes constant  $$gamma_{0}=0.5772156649dots$$ 
32  LiouvilleRoth constant  0.0110001000…. 
33  Diophantine approximation constant  $$gamma_{1}=0.4472135955dots$$ 
34  Erdos reciprocal sum constant  3.0089 
35  Abundant number density constant  0.2441<A<0.2909 
36  Selfnumber density constant  $$lambda=0.252660259dots$$ 
37  Cameron’s sumfree set constant  0.21759<c<0.21862 
38  Euler totient function asymptotic constant  A=1.9435964368….B=0.0595536246…. 
39  NielsonRamanujan constant  Not available 
40  Triplefree set constant  0.6135752692…. 
41  DeBruijnNewman constant  Not yet available 
42  Freiman’s constant  Not yet available 
43  Cahen’s constant  Not yet available 
Constants associate with Analytic Inequalities
44  Shapiro’s cycle sum constant  0.4945668…. 
45  CarlsonLevin constant  $$Gammafrac{1/4}{pi ^{1/4}}$$ 
46  LondauKolmogorov constant  $$C(3,1)=(frac{243}{8})^{1/3}$$ 
47  Hilbert’s constant  Not available 
48  CopsondeBruijn constant  C=1.1064957714…. 
49  WirtingerSobolev isoperimetric constant  Not available 
50  WhitneyMikhlin extension constant  2.05003 
Constants associate with the Approximation of Functions
51  Wilbraham Gibbs constant  G=1.851937052…. 
52  Lebesgue constant  C=0.9894312738…. 
53  Favard constant  Not available 
54  Bernstein’s constant  $$beta=0.2801694990dots$$ 
55  The “oneninth” constant  0.1076539192…. 
56  Laplace limit constant  $$lambda=0.6627434193dots$$ 
Constants associate with Enumerating Discrete structures
57  Abelian group enumeration constant  A=2.2948566…. , B=1.3297682…. 
58  R’enyi’s parking constant  0.7475979203…. 
59  Golomb Dickman constant  $$lambda=0.6243299885dots$$ 
60  Lengyel’s constant  $$lambda=1.0986858055dots$$ 
61  Otter’s tree enumeration constant  $$beta=0.5349485dots$$ 
62  Polya’s random walk constant  $$rho=0.3405373296dots$$ 
63  Selfavoidingwalk connective constant  2.6381585…. 
64  Feller’s coin tossing constant  $$alpha=1.087378025dots ,beta=1.236839845dots$$ 
65  Har square entropy constant  k=1.503048082…. 
66  Binary search tree constant  Precise numerical value not available 
67  Digital search tree constant  c=0.3720486812…. 
68  Quardtree constant  C=4.3110704070…. 
69  Extreme value constant  $$mu_{6}=1.2672063606dots$$ 
70  Patternfree word constant  $$varsigma=1.302dots$$ 
71  TakeuchiPrellberg constant  c=2.239433104…. 
72  Random percolation constant  $$K_{B}=0.0355762113dots$$ 
73  LenzIsing constant  $$rho=0.218094dots$$ 
74  2D Monomerdimer constant  1.338515152…. 
75  3D Dimer constant  $$lambda=0.209174dots$$ 
76  Lieb’s square ice constant  Not yet available 
Constants associate with Functional Iteration
77  Gauss’s lemniscate constant  0.83462684167…. 
78  Grossman’s constant  Not available 
79  Plouffe’s constant  0.4756260767…. 
80  Lehmer’s constant  0.5926327182…. 
81  Iterated exponential constant  0.7666646959…. 
82  Continued fraction constant  0.76519769…., 0.7098034428…. 
83  Infinite product constant  2.0742250447…. 
84  Quadratic recurrence constant  C=1.502836801…. 
85  Conway’s constant  $$lambda=1.303577269dots$$ 
Constants associate with Complex Analysis
86  BlochLandau constant  B=0.4718617….L=0.5432588…. 
87  MasserGramain constant  C=0.6462454398…. 
88  John constant  4.810477…. 
Constants associate with Geometry
89  Geometric probability constant  4.2472965…. 
90  Circular coverage constant  0.8269933431…. 
91  Universal coverage constant  Not yet available 
92  Hermite’s constant  0.7404804897…. 
93  Tammes’ constant  Not yet available 
94  Calabi’s triangle constant  1.5513875245…. 
95  Graham’s hexagon constant  0.6495190528…. 
96  Traveling salesman constant  0.521…. 
97  Moving sofa constant  A=0.09442656084…. 
98  Beam detection constant  5.1415926536…. 
99  Heilbronn Triangle constant  H=0.1924500897…. 
100  Moser’s worm constant  Not yet available 
101  Rectilinear crossing constant  0.70449881…. 
102  Maximum irrodius constant  0.2041241452…. 
103  Magic geometric constant  0.6675276<m<0.6675284 
Almost all the constants seem to be irrationals although rigorous proofs are not available. All the constants have numerous fascinating applications, and thus irrational numbers play very important role in studying modern number theory. Details of some interesting constants and some of their applications will be highlighted in ‘Ganit Bikash’ in near future.
Author Prof. Tarini Kumar Dutta is a Professor in Mathematics Department, Gauhati University.
[This article was first published in Ganit Bikash, Volume 27, JuneDecember, 2000.]
Managing Editor of the English Section, Gonit Sora and Research Associate, Cardiff University, UK.