A 2-adic solenoid
No. 195, 1977
(Topology, differential equations, Hamiltonian mechanics, symplectic geometry)
India ink on paper, 46.5x57 cm.
This 2-adic solenoid, a contortion of space, where surfaces fold in on themselves around a central axis, is known in topology as an object rich in complexity and unusual properties, useful in verifying many geometric conjectures. Though not obvious from its final appearance, this object comes about when many tori, or donut shaped objects, are embedded in each other. To build such a shape, one begins with a single torus, copies it, and then winds the copy around the original. This process is then repeated over and over, moving toward infinity. Eventually, one creates a set of tori inside of tori, unwrapped donuts twisting inside themselves, like snakes coiled into rings. And the process goes on to its very limit. In each case too, part of each torus has been stripped away, opening up its interior, to reveal the creation of this space.
Mathematicians and physicists have become more interested in this shape in recent years as they find it arising in an area of study called Hamiltonian mechanics. These winding tori, so to speak, turn out to describe important characteristics of certain equations of motion, particularly integrable differential equations and their solutions. So we have a case where a well understood, classical object in mathematics has been, in a sense, reborn, owing to recent discoveries in mathematical physics.
