Suppose R is any ring with an identity and Jacobson radical rad R. An m by n matrix A over R is called von Neumann (vN) regular iff AXA = A for some X. This X is called a (1,2)-inverse iff in addition XAX = A. Moreover, it is the Moore-Penrose (or MP) inverse iff in addtion (AX)* = AX and (XA)* = XA for some involution * (such as conjugate-transpose). See paper 1 or paper 2.

Named after the English mathematician John Wallis (1616 –1703), this formula is popular in many calculus courses. It is a slowly convergent product, but its importance is historic and aesthetic.

I always found this a simple yet beautiful formula.

I generalized it, obtaining the given expressions, using e, Euler's number, gamma the Euler-Mascheroni constant and A the Glaisher-Kinkelin constant.

See my paper in "The American Mathematical Monthly".

Here is a new regular polyhedron. It is of Kepler-Poinsot type and infinite, but regular indeed.
It is not a simple compound of 4 Petrie-Coxeter {4,6} copies since 8 hexagons meet in each vertex, nand ot 4.
My paper discussing this result appeared in "The American Mathematical Monthly".