Sand Castles

The Postulates of the Science of Space
by William Kingdon Clifford

IN MY FIRST LECTURE I said that, out of the pictures which are all that we can really see, we imagine a world of solid things; and that this world is constructed so as to fulfil a certain code of rules, some called axioms, and some called definitions, and some called postulates, and some assumed in the course of demonstration, but all laid down in one form or another in Euclid's Elements of Geometry. It is this code of rules that we have to consider to-day. I do not, however, propose to take this book that I have mentioned, and to examine one after another the rules as Euclid has laid them down or unconsciously assumed them; notwithstanding that many things might be said in favour of such a course.

The Story of Louis Pósa
by Ross Honsberger

I WOULD LIKE TO TELL YOU something of the life and works of a remarkable young Hungarian named Louis Pósa (pronounced pO.sha), who was born in the late 1940's. When quite young he attracted the attention of the eminent Hungarian mathematician Paul Erdös (pronounced air.dish), who did much to help him develop. Erdös has recently written and spoken about some of the child prodigies he has known and I would like to tell you his story of Pósa.

Squaring the Square
by Ross Honsberger

IN 1936, FOUR STUDENTS AT TRINITY COLLEGE, Cambridge-Brooks, Smith, Stone, and Tutte--considered the problem of cutting up a rectangle into squares of unequal size (no two alike). It was known, at that time, that a rectangle 32 by 33 could be "squared". Stone became particularly interested in trying to prove that it was impossible to cut a given square into unequal squares. While not able to do this, he did discover a squaring of another rectangle.

The Mandelbrot Set
by A. K. Dewdney

THE MANDELBROT SET BROODS in silent complexity at the center of a vast two-dimensional sheet of numbers called the complex plane. When a certain operation is applied repeatedly to the numbers, the ones outside the set flee to infinity. The numbers inside remain to drift or dance about. Close to the boundary minutely choreographed wanderings mark the onset of the instability. Here is an infinite regress of detail that astonishes us with its variety, its complexity, and its strange beauty.