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) was once regarded as the king, and in modern times (after 1980), the Feigenbaum Universal constant 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 where a is a constant. The interval is mapped into itself by for each value of 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 (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 which are not fixed points of f? Solving we obtain the 2cycle.
(which attracts for and repels for 1+sqrt{6}" />).
For 1+sqrt{6}=3.4495dots" /> an attracting 4cycle emerges. We can obtain the 4cycle by numerically solving and 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 and 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 and so forth denote the cascade of bifurcations, it can be proved that
This point marks the separation between the “periodic regime” and the “chaotic regime” for this family of quadratic functions. The sequence behaves in a universal manner such that the ratio tends to a universal constant
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  
2  One  1 
3  Imaginary unit  
4  Pythagoras’ constant  
5  Golden mean  
6  Natural logarithmic base  C=2.7182818285…. 
7  Archimedes’ constant  
8  EulerMascheroni constant  
9  Ape’’ry’s constant  
10  Catalan’s constant  G=0.915965594…. 
11  Khintchine’s constant  K=2.68545200…. 
12  Feigenbaum constant  
13  Madelung’s constant  
14  Chaiten’s constant  Not available 
Constants associate with Number Theory
15  HardyLittlewood constant  
16  Hadamardde la Valle’e Poussin constant  
17  LandauRamanujan constant  K=0.764223653…. 
18  Brun’s constant  B=1.90211605778…. 
19  Artin’s constant  
20  Linnik’s constant  Not available 
21  HafnerSarnakMeCurley constant  
22  GaussKuzminWirsing constant  
23  StolarskyHarborth constant  
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  
32  LiouvilleRoth constant  0.0110001000…. 
33  Diophantine approximation constant  
34  Erdos reciprocal sum constant  3.0089 
35  Abundant number density constant  0.2441<A<0.2909 
36  Selfnumber density constant  
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  
46  LondauKolmogorov constant  
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  
55  The “oneninth” constant  0.1076539192…. 
56  Laplace limit constant 
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  
60  Lengyel’s constant  
61  Otter’s tree enumeration constant  
62  Polya’s random walk constant  
63  Selfavoidingwalk connective constant  2.6381585…. 
64  Feller’s coin tossing constant  
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  
70  Patternfree word constant  
71  TakeuchiPrellberg constant  c=2.239433104…. 
72  Random percolation constant  
73  LenzIsing constant  
74  2D Monomerdimer constant  1.338515152…. 
75  3D Dimer constant  
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 
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.]
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ignacitum
Posted at 00:03h, 28 JanuaryThe growing collection of mathematical constants:
https://en.wikipedia.org/wiki/Mathematical_constants_and_functions
Gonit Sora
Posted at 01:23h, 28 JanuaryThank you for the comment.