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The mathematical study of knots. Knot theory considers questions such as the following: 1. Given a tangled loop of string, is it really knotted or can it, with enough ...

The numbers defined by the recurrence relation K_(n+1)=1+min(2K_(|_n/2_|),3K_(|_n/3_|)), with K_0=1. The first few values for n=0, 1, 2, ... are 1, 3, 3, 4, 7, 7, 7, 9, 9, ...

Knuth's up-arrow notation is a notation invented by Knuth (1976) to represent large numbers in which evaluation proceeds from the right (Conway and Guy 1996, p. 60): m^n ...

Knuth's series is given by S = sum_(k=1)^(infty)((k^k)/(k!e^k)-1/(sqrt(2pik))) (1) = -2/3-1/(sqrt(2pi))zeta(1/2) (2) = -0.08406950872765599646... (3) (OEIS A096616), where ...

A fractal derived from the Koch snowflake. The base curve and motif for the fractal are illustrated below. The area enclosed by pieces of the curve after the nth iteration is ...

The approximation for pi given by pi approx sqrt((40)/3-2sqrt(3)) (1) = 1/3sqrt(120-18sqrt(3)) (2) = 3.141533.... (3) In the above figure, let OA=OF=1, and construct the ...

The function f_theta(z)=z/((1+e^(itheta)z)^2) (1) defined on the unit disk |z|<1. For theta in [0,2pi), the Köbe function is a schlicht function f(z)=z+sum_(j=2)^inftya_jz^j ...

If f is a schlicht function and D(z_0,r) is the open disk of radius r centered at z_0, then f(D(0,1)) superset= D(0,1/4), where superset= denotes a (not necessarily proper) ...

A tree with a finite number of branches at each fork and with a finite number of leaves at the end of each branch is called a finitely branching tree. König's lemma states ...

For every ring containing p spheres, there exists a ring of q spheres, each touching each of the p spheres, where 1/p+1/q=1/2, (1) which can also be written (p-2)(q-2)=4. (2) ...