Calabi-Yau Cross Sections:

I (A.J. Hanson) have also created a variety of graphics images derived from the Fermat Equation (see below) that are relevant to the Calabi-Yau spaces that may lie at the smallest scales of the unseen dimensions in String Theory; these have appeared in Brian Greene's books, The Elegant Universe and The Fabric of the Cosmos,, and in the book by Callender and Huggins, Physics Meets Philosophy at the Planck Scale. The writhing purple shapes in the October/ November 2003 NOVA production Elegant Universe, as well as the cover of the November 2003 Scientific American, were derived from software models I supplied to the NOVA graphics providers.


These images show equivalent renderings of a 2D cross-section of the 6D manifold embedded in CP4 described in string theory calculations by the homogeneous equation in five complex variables:
z15 + z25 + z35 + z45 + z55 = 0
The surface is computed by assuming that some pair of complex inhomogenous variables, say z3/z5 and z4/z5, are constant (thus defining a 2-manifold slice of the 6-manifold), normalizing the resulting inhomogeneous equations a second time, and plotting the solutions to
z15 + z25 = 1
The resulting surface is embedded in 4D and projected to 3D using Mathematica (left image) and our own interactive MeshView 4D viewer (right image). If you have CosmoPlayer, you can also interact with this VRML version of the quintic Calabi-Yau cross-section.

In the right-hand image, each point on the surface where five different-colored patches come together is a fixed point of a complex phase transformation; the colors are weighted by the amount of the phase displacement in z1 (red) and in z2 (green) from the fundamental domain, which is drawn in blue and is partially visible in the background. Thus the fact that there are five regions fanning out from each fixed point clearly emphasizes the quintic nature of this surface.

For further information, see: A.J. Hanson. A construction for computer visualization of certain complex curves. Notices of the Amer.Math.Soc., 41(9):1156-1163, November/December 1994.

Arbitrary Genus Surfaces:

This image shows my computer graphics construction of a four-hole torus described by an equation in complex two-space given by H. Blaine Lawson, "Complete Minimal Surfaces in S3," Ann. of Math. 92, pp.~335--374 (1970), with m = n = 2,

Im z1(m + 1) + |z2|(m-n) Im z2(n+1) = 0


|z1|2 + |z2|2 = 1

In general, the genus is m*n, and this surface is not actually minimal in S3 except for
m = n = 0 and m = n = 1.

Review article

Cover picture: IEEE Computer 27 (July 1994)

For more information about mathematical visualization in general, see the Web version of the review article Interactive Methods for Visualizable Geometry, by A.J. Hanson, T. Munzner, and G. Francis, published in IEEE Computer 27 , No. 7, pp. 73--83 (IEEE Computer Society Press, Los Alamitos, CA, July, 1994).

This page is taken from A.J. Hanson's web page