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5. Periodic Minimal Surfaces

These surfaces are two-dimensional universes or manifolds which are everywhere saddle-shaped, with two principal curvatures which are everywhere equal and opposite. The examples shown have no arbitrary parameters but represent cubic building blocks which can be stacked together indefinitely to separate the world into two sub-spaces, which in some cases may be the same and in some cases different,

The two-dimensional manifolds are non-Euclidean in that the perimeter of a small circle of given radius on the surface is greater than that of one of the same radius on the plane (which in turn is greater than one of the same radius on the surface of a sphere). A beetle exploring the surface would find its world to be complex in its labyrinthine interconnections.

Beyond the representation of the human figure, 3-D sculpture is disappointing. Cezanne said: "Treat nature in terms of the cylinder, the sphere, the cone, all in perspective". Barbara Hepworth was renowned for being the first sculptor to make a hole through a piece of stone in an abstract design, but artists now have many more interesting geometrical objects than the torus.