Sandpile geometry: surfaces and skeleton | Tangente

The delicate geometry of sandpiles

There are a thousand ways to keep your mind active while soaking up the sun on the beach. Let's set aside those—not so foolish after all—that involve not tanning at all. Instead, let's borrow our young nephew's bucket, spade and sieve, and embark on the great geometric adventure of sandpiles!

What is a sandpile? The question may sound innocuous, but we have to start somewhere! For our purposes, we gently pour fine, dry, uniform sand onto a piece of rigid cardboard propped up so that it is level (the base) until the pile is saturated. In other words, not a single grain can be added to the pile without rolling down the slope. The slope has then reached a critical value, the same at every point on the pile, which is why it is called a constant-slope surface.

A keen sense of geometry ----------------------------

Let us begin with some preparatory hands-on work. Cut a variety of bases from cardboard—a circle, square, rectangle, convex or non-convex polygon, or free-form shape—and observe what happens. As expected, the circle produces a cone of revolution, the square a pyramid, and the rectangle a roof whose four faces all have the same pitch. But here is the surprise: if we try to reproduce exactly a roof made entirely of flat surfaces over an L-shaped building, the sand refuses to cooperate! In the re-entrant corner formed by two roof faces, the sand continues to accumulate until it forms a conical surface (part of a cone of revolution), and a curved ridge line appears, comprising line segments and two elliptical arcs.