Apple Valley, MN
It has recently been found that a boundary of any shape can be completely filled with ever-smaller shapes in the limit of an infinite number of shapes. In two dimensions the areas of the shapes must obey a negative-exponent power law versus shape number i, with an exponent c such that 1 < c < c_max, with c_max depending upon the particular shape. Placement of the shapes is by nonoverlapping random search. Despite the randomness of the placement algorithm, available evidence indicates that the process does not halt.
Circles -- Random and Ordered
Structures involving randomness are unusual in mathematics, so it is of interest to make a connection with what is known. The statistical geometry algorithm works with power-law exponents c over a wide range (but c > 1). For triangles a central feature of increasing c is increasing order. The well-known Apollonian circles construction provides a natural end point for this sequence, with a fractal dimension D which fits nicely with D for the statistical geometry structures.