Art Meets Mathematics in the Fourth Dimension
There is little change with respect to the first edition of this book that was published in 2011 with the title Incipit. The main (mathematical) idea is to visualize the 3-sphere. That is the analogue of the familiar sphere but in a 4-dimensional space. It is a classical idea to help us imagine the imaginable by analogy, i.e., by thinking of how a person living in a 2-dimensional world would experience a 3-dimensional object. The book Flatland (1884) by Edward Abbott Abbott is a classical example of this idea. Lipscomb uses Freddy the penguin who has a 2-dimensional view (and sometimes even 1-dimensional), meaning that he can only observe slices of a 3-dimensional object. An object passing through his planar view would materialize out of the blue, morph into different shapes and disappear again into oblivion. So this is not a very good idea. The actual visualization here is by first sampling the 3-sphere. i.e., by taking the points where a particular grid, the 4-web, intersects the 3-sphere, and project these points on a 3-dimensional subspace which depend on the position of the camera. In fact, by placing the camera on one of the axes of the grid, some points will be hidden by others, basically eliminating one of the dimensions. The result is a point cloud in our 3-dimensional space that has a diamond-like shape and with some imagination, the density of the points gives the impression of a human head with a broad forehead and a pointy chin. The Blu-ray disk that accompanied the first edition is replaced by a link to the Springer website of the book where the videos can be viewed.
Since the 4-web has a fractal structure, Lipscomb places the result in the realm of fractal and mathematical art, and gives some more ideas of how art, mathematics and science although they have their own approach, do actually meet in this idea of visualizing the 4th dimension. To illustrate this, he devotes a chapter to the paper by Mark Peterson in the American Journal of Physics Dante and the 3-sphere (1979) in which it is explained how Dante in his Divine Comedy designs Heaven as a mirror image of Hell. The two "worlds" unite like a ball touching a mirror is glued to its mirror image and a 3-sphere is indeed obtained by gluing the 2-sphere boundaries of two 3-disks (two solid balls). Besides Dante's Art, Science is represented by Einstein's world view and the difficulty to visualize R × S². Einstein in his book Ideas and Opinions starts from the stereographic projection of a circle on a sphere to produce a "shadow" on the plane and lifted this idea to a higher dimension. Both discussions by Lipscomb are intended to stress the importance of the 4th dimension and the 3-sphere.
Lipscomb then moves on to the technical part of the visualization: the construction of the 4-web. This is a Sierpinski-like fractal subdivision of an "external" 4-simplex. Think of a 4-simplex as 3 points in a horizontal plane and one point above and one point below the plane, defining 5 points that are all pairwise connected. Thus one has a simplex above the plane and one below, glued together with one extra internal edge connecting the the upper and lowermost vertices. (The usual 5-cell places the point below the plane inside the top simplex above the plane.) The subdivision shrinks the edges to half their size in the direction of each of of the 5 points. This gives 5 smaller sub-simplexes leaving a hole in the middle. Subdividing each of them in a similar way gives the fractal 4-web with holes similar to a Sierpinski gasket. The mathematics and a Pov-Ray program are described to generate the 4-web and its intersections with the 3-sphere which eventually results in the sampled "images" of the 3-sphere.
The last two chapters of the book are more mathematical, making all the equations and transformations explicit. It is derived where to find some of the 2-sphere slices of the 3-sphere in the image produced with the 4-web. The 2-spheres are obtained by keeping one of the 4 coordinates constant on the 3-sphere. Think of the higher dimensional analogue of picturing great circles on a 2-sphere. This image is called God's Image? (with question mark). This name is inspired by Michelangelo's fresco on the ceiling of the Sistine Chapel called the Creation of Adam where God's finger touches the finger of Adam. If was Frank Meshberger who recognized in the picture of God, surrounded by angels, an astonishingly correct anatomical picture of the brain. This human ability of pattern recognition (Meshberger had a medical training), is also what we should use when looking at the point cloud: to imagine and recognize what is not actually there.
Several appendices collect extra material to accompany the chapters. For example, to go with the Einstein chapter, a short introduction to cosmology is included. There are also several color plates added, but the videos should of course be watched via the website. To be honest, I cannot really share the author's enthusiasm for these images and videos. Instead I find the (freely available) animated movie Dimensions by Jos Leys, Étienne Ghys, and Aurélien Alvarez, much more instructive and inspiring to picture the 4th dimension. There are of course similarities but the objectives are also different. The movie is made mainly for instructional reasons and considers mostly polytopes placed in a didactical mathematical framework. This book on the other hand focusses almost solely on the 3-sphere, but also wants to show the 4th dimension as a meeting point of art and mathematics, considering the visualization itself artwork, the computer being the instrument used to create a pointillistic 3-dimensional piece of art. It can even be seen as a "message from the other side" because that fact that the point cloud shapes this image of a human head, with eyes, cheekbones, chin, mouth, and even a moustache can be experienced as "creepy" if one is susceptible to esoteric experiences. When it comes to art, the fourth dimension of course has inspired several novelists, painters, musicians, and film directors, besides Dante, but a discussion of those would go far beyond the limitations of this review.