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Beautiful patterns can be created by drawing sets of nested polygons such that the incircle of the nth polygon is the circumcircle of the (n+1)st and successive polygons are ...

The nested radical constant is the constant defined by C = sqrt(1+sqrt(2+sqrt(3+sqrt(4+sqrt(5+...))))) (1) = 1.75793275... (2) (OEIS A072449). No closed-form expression is ...

A graph or directed graph together with a function which assigns a positive real number to each edge (Harary 1994, p. 52).

Polynomials O_n(x) that can be defined by the sum O_n(x)=1/4sum_(k=0)^(|_n/2_|)(n(n-k-1)!)/(k!)(1/2x)^(2k-n-1) (1) for n>=1, where |_x_| is the floor function. They obey the ...

A geometric construction, also called a verging construction, which allows the classical geometric construction rules to be bent in order to permit sliding of a marked ruler. ...

The functions theta_s(u) = (H(u))/(H^'(0)) (1) theta_d(u) = (Theta(u+K))/(Theta(k)) (2) theta_c(u) = (H(u))/(H(K)) (3) theta_n(u) = (Theta(u))/(Theta(0)), (4) where H(u) and ...

A paradox in decision theory. Given two boxes, B1 which contains $1000 and B2 which contains either nothing or a million dollars, you may pick either B2 or both. However, at ...

Samuel Pepys wrote Isaac Newton a long letter asking him to determine the probabilities for a set of dice rolls related to a wager he planned to make. Pepys asked which was ...

Curves with Cartesian equation ay^2=x(x^2-2bx+c) with a>0. The above equation represents the third class of Newton's classification of cubic curves, which Newton divided into ...

Let pi_n(x)=product_(k=0)^n(x-x_k), (1) then f(x)=f_0+sum_(k=1)^npi_(k-1)(x)[x_0,x_1,...,x_k]+R_n, (2) where [x_1,...] is a divided difference, and the remainder is ...