2015 Joint Mathematics Meetings
John Shier
Artists
Statement
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
Artworks
![Image for entry 'Circles -- Random and Ordered'](/_next/image?url=https%3A%2F%2Fsubmit.bridgesmathart.org%2Frails%2Factive_storage%2Fblobs%2Fproxy%2FeyJfcmFpbHMiOnsiZGF0YSI6NjQyNiwicHVyIjoiYmxvYl9pZCJ9fQ%3D%3D--aff5835ff3c0530bf6c2bc7ce8010d4b82d4482c%2Fcirc_vs_c_jmm_sm.jpg&w=1536&q=75)
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.