No. 11, 1967 (Geometry, topology, and minimal surfaces)
India ink on paper, 21x32 cm.
In this cavernous space, the gallery of a great austere castle, three beings watch from above as other creatures pass time in a menagerie of magnificent mathematical forms, each a different perturbation of physical space. Above and to the right, an animated polyhedron comes to life and begins to decompose itselﬁ breaking down into its constituent parts, the scorpion-like shells of which it is made. Observe the tail of the seeming scorpion, arching upward and toward the shells head, revealing intuitively facets of the object's structure and form. See how the shells ultimately come together to create a single, infinite polyhedron. Meanwhile, in the center of the vast hall, a large torus, or donut-shaped object, is turning itself inside out, transforming itself and the space around it. Interestingly, even though the torus, which has been cut, or punctured, twists in space and turns inside out, the new object is still a torus, although the inside and outside surfaces have interchanged.
At lower left, bathed in the shadow of a great pillar, lies an object called Antoine's Necklace, quite familiar in topology. To its right, in the lighted area, rests a soap film, which stretches across a circular wire. Composed by joining together an ordinary Mobius strip with a triple Mobius strip, this minimal surface is remarkable in that it can be contracted continuously along its boundary without tearing. It can even be transformed to create another object known to topologists as a punctured Bing house. Finally, in the room's center, lies a large 2-adic solenoid.