Shadows of Reality by Tony Robbin
This book is marvelous, simply maaahvelous!
As promised on the front page, Robbin incorporates Picasso (Three Women of Avignon and Portrait of Henry Kanweiler) and “the 4th dimension in the popular imagination” into his sweeping portrayal of four-dimensional thinking. He also talks about quasicrystals.
On the subject of visualizing the fourth dimension, this author has much to say. There are two ways to picture four-dimensional objects: slicing, and projection. Slicing is level curves. Like imagine a pyramid being construed to a two-dimensional viewer (Flatland-style) as a succession of squares that get smaller and smaller at a linear rate.
Projection is Robbin’s favored means of visualisation. Projection yields quasicrystals and aperiodic tilings — so maybe there is a deeper truth there. Imagine a chair being construed to a two-dimensional viewer by drawing out the shadow of the chair. You could move the lamp around (in 3-D) and eventually the mathematically adept two-dimensional viewer could describe the whole chair.
(In slicing terms, you would first have four separated squares — chair-legs — and then a huge oblong — the seat — and finally some squares and circles in a row — the seat back.)