 
Artwork from "Beyond the Third Dimension"
These diagrams are from Thomas Banchoff's volume in the Scientific
American Science Library series, Beyond the Third Dimension:
Geometry, Computer Graphics, and Higher Dimensions, (W. H.
Freeman and Co., New York, 1990). While a graduate student at Brown
University, I produced most of the line art for this book (more than 200
images in all) using the 2D drawing program Aldus Freehand
(now Macromedia Freehand ). This was the first time Freeman had
used computergenerated artwork, and they were surprized at how quickly we
could produce updates and corrections to the diagrams.
In this selection of images from three chapters of the book, we see
the following: at the upper left, an unfolded hyperpyramid having a
cubical base with apex directly over (in fourspace) one of the corners
of the base cube; at the upper right, an unfolded and exploded
hypercube; at lower left, an image of the {\it stella octangula}, the
stellation of an octahedron that fits within a cubical box; at the
center, purple pyramids illustrating how the volume of a pyramid does
not depend on the position of the apex (since the slabs are all the
same volume in either case); and finally, at the lower right, a
pyramid indicating how the volume of a trianglebased pyramid (e.g. a
tetrahedron) is half the volume of the corresponding squarebased one.
The production of the artwork for this book is the subject of the
article "Illustrating Beyond the Third Dimension"
that appears in my publications list.
