B3D Chapter 3

Lisa Eckstein

Further Reading

I found a short biography of Alicia Boole Stott, the "genius at predicting the slicing sequences of four-dimensional polyhedra". (p. 50)


Think about how contours are important in visual art. An artist representing a three-dimensional figure, such as a person, in a two-dimensional medium, such as drawing or painting, must create the illusion of depth on a flat surface. Often this is accomplished by showing the pattern of light and shadow on the figure. Shadows most frequently appear where contours change - at places which would be critical levels if the object were sliced. Look at the drawing of Friedrich Froebel on page 42. There are shadows on the side of his nose, which is a "peak" on the landscape of his face, and in the wrinkles ("craters") at the side of his mouth. Although this picture of Froebel is flat, we interpret the pattern of the shadows and understand that Froebel has depth. By contrast, the drawing of two-dimensional Yendred (page 41) does not rely on any illusion or artistic convention - it presents Yendred as he actually is.

Imagine a set of geometrical models for a fourth-dimension kindergarten class. What objects would make up the set? How many eyelets would each object need so that every view and slicing sequence (through the third dimension) could be studied?

Put together the fold-out model shown on page 50 and try the puzzle on your friends. You'll be amazed how difficult people find it!

The use of light and shading in painting to accentuate a three-dimesional impression is a fascinating chapter in the study of dimensionality. Some of my friends are quite expert in the mathematics of the question--compare "A Topological Picturebook" of George Francis for example. Another friend, Andrew Hanson, specializes in lighting effects for semi-transparent four-d objects, but his work is only at a rudimentary stage in many ways.

So, what kinds of Kindergarten gifts would be good for leading students into the fourth dimension?

Have you tried that tetrahedron puzzle on your friends? It is a striking exercise. Some can't do it at all, and others do it so fast they can't imagine that there is a problem. The record in my experience belongs to a third grader in Santa Monica, Sergio Martinez, who had it together just about as fast as he caught the pieces. Then when none of the others could assemble it, he showed them how, "Just put the squares together and twist." I tried to contact him through the school a couple of years later but the family had moved. I often wonder if he'll turn up sometime.