The Hofstadter butterfly, a quantum fractal made up of integers describing quantum Hall states, is shown to be related to an integral Apollonian gasket with D3 symmetry. This mapping unfolds as the self-similar butterfly landscape is found to describe a close packing of (Ford) circles that represent rational flux values and is characterized in terms of an old ((300BC) problem of mutually tangen...

A Review of Michael Trimble, Why humans like to cry: Tragedy, evolution, and the brain. Crying is an important human behavior, and we are all aware of the impact that our tears can have on others, and theirs on us. From an ethological perspective, the survival value of crying is clear, particularly in the case of infants and small children. However, much less is known about the evolutionary mea...

In a letter to the Editor of Science (176, 966, 1972), Professor Szent-Gyorgyi described his approach to science as Dionysian. This little book provides some further insight into the meaning of this statement. In 114 pp. he writes about some of his research interests over the past half century. This account does not describe the work in much experimental detail, but rather tries to give the mot...

In two dimensions, start with three mutually tangent circles, with disjoint interiors (a circle with negative radius has the point at infinity in its interior). We can draw two new circles that touch these three, and then six more in the gaps that are formed, and so on. This procedure generates an (infinite) Apollonian packing of circles. We show that the sum of the bends (curvatures) of the ci...

We introduce an orthoplicial Apollonian sphere packing, which is a sphere packing obtained by successively inverting a configuration of 8 spheres with 4-orthplicial tangency graph. We will show that there are such packings in which the bends of all constituent spheres are integral, and establish the asymptotic local-global principle for the set of bends in these packings.