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Art Meets Mathematics in the Fourth Dimension

Stephen Leon Lipscomb
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
P. N. Ruane
, on

Stephen Lipscomb begins by declaring that: ‘Here begins new art created by capturing pictures of objects that live in the fourth dimension’

The principal 4-dimensional object is the 3-sphere (hypersphere), and the ‘new art’ arises from computer-derived fractal imagery representing the 4-web, which is analogous to a spider’s web but of ever-increasing density. The 4-web is the means by which the reader is led to visualise aspects of the hypersphere.

But the hypersphere is also deemed to be of theological significance because it is said to contain the Empyrean — the angelic sphere of Dante’s heaven, home of God and angels. This is idealised in a painting by Gustav Doré (1832–1883), and the similarity between fractal depictions of the hypersphere and this painting is remarkable given that Doré was born 100 years before Mandelbrot’s innovative work.

The geometrical model of Dante’s 3-sphere (God) is obtained by a process of gluing each point p on the boundary of the Aristotelian universe (a solid sphere) to a unique point q on the boundary of the Empyrean (another solid sphere). This enables those seeking a place in heaven to cross over from the Aristotle universe (our mundane world) to the Empyrean at every glued point. The result of all this gluing is the hypersphere, S3, which is referred to here as the harbinger for Einstein’s ideas on the shape of the universe.

Although the hypersphere is an abstraction beyond our perception, the goal is to create means that allow various aspects to be illustrated via the 4-web. For example, the essential nature of the circle, S1, is captured by a simple web in the form of a square grid as shown here.

This method is extended (in the illustration below) to show how points on S2 are captured by a cubic grid.

Obviously, as the grid becomes finer, more points will be captured and the resulting picture will be more accurate.

But what of the hypersphere? Grids (4-webs) that capture points on the hypersphere also live in 4-dimensional space; but how can they be visualised? One clue relates to the fact that the hypercube can be represented three dimensionally as below.

The construction of 4-webs is similarly displayed, where the basic cells are hexahedra placed vertex-to-vertex in the fashion below:

The fineness of this hexahedronal 4-web increases in the manner shown by the next diagram:

The 8th subdivision of the 4-web consists of 3,906,250 line segments capturing a total of 2,712 points of the hypersphere in 4-space. Arising from this process is the image below, referred to as the ‘face of God’. To see the ‘face’, imagine a toroidal hat hiding the eyes of the wearer. The nose has a triangular tip, and the top lip is more pronounced than the bottom lip, and the being itself could be androgynous.

There are many other coloured versions of this image contained within 184 pages of this book, which is lavishly illustrated in so many different respects. But where is the mathematics?

Embedded within other fractal images of the hypersphere are ellipsoids that are transformed images of the great spheres formed by the intersection of S3 with hyperplanes. Two chapters of the book take the reader through the underlying algebraic analysis (linear algebra mainly). Another chapter (Generating Hypersphere Art) explains the mathematical basis of, and provides computer codes for, the construction of the images.

For myself, I’m somewhat flummoxed by these snapshots of the inner life of the hypersphere. I’m strongly addicted to the idea of it being a 4-dimensional version of our everyday S2 (a compact surface of positive constant Gaussian curvature). But this book is a colourful presentation of big ideas and it’s the only mathematics book I know that has references to ‘hell’ in its index. To my knowledge, it’s the first book to be published on this theme.


All the images are from the book under review; used with permission.

While the author makes many references to a CD or DVD included with the book, the review copy we received did not include a disk.

Peter Ruane gives credence to Bernard Shaw’s idea that, in hell, one would meet more interesting characters, because heaven is occupied by conforming goody-goodies.