The strange attractors discussed here result from iteratingequations of the form xnext=Poly(xprev). First the iteration runs itscourse for some 1000 turns, then the present value is plotted againstthe fifth previous value. The plotting process is repeated until 11,000iterations have been reached. When computing the iterations, the strangeattractor cases (frequently considerably fewer than 10 percent) arefound by computing an average Lyapunov exponent (or, for higherdimensions than 1, exponents), positive values indicating strangeattractors. In this way, fixed point and limit cycle cases are discardedduring the first few iterations. Initial values of x (and y,z,…) are fixed at 0.05, which seems limited.
The bulk of the book uses polynomials of degree from 2 to 5, fromone dimension up to four dimensions. The book’s purpose is to depict notso much the mathematical side of the subject, but a gradually built-upcomputer program that allows the user to generate, select, and inspectstrange attractor pictures of artistic interest. It contains manyaesthetic results, presented in various forms such as monochrome,grayscale, color, shadowed, sliced, and stereoscopic pictures.
In computer science and programming circles, one often finds peoplemaking these kinds of pictures who know little about the mathematicalbackground. This author, being a physicist, knows very well what he isdoing, although he sometimes expresses it in oblique references withoutexplaining in detail what is behind it.
A certain dullness pervades the book. The author uses randomlychosen coefficients in the range (-1.2,+1.2) in steps of 0.1. Since atwo-dimensional polynomial of degree 2 has 16 coefficients, this casealready presents 25 possibilities, but he does not stop here. Equationswith approximately 500 coefficients are used. Sprott talks of havingvisually inspected about 100,000 cases in order to determine theiresthetic appeal and decide whether they qualified for presentation inthe book (selections were iterated from 500,000 to 10 million times).The mind boggles at such a Herculean task, yet readers sometimes cannothelp feeling bored by the succession of polynomials trying to attracttheir attention. Did this huge time investment on the part of the authoronly produce nice pictures?
For reasons I find hard to understand, the author has chosen toencode the coefficients as ASCII characters. Although at the beginningof the book he presents a table for all 95 displayable representations,corresponding to a coefficient range (-4.5,+5.0) in steps of 0.1, heonly uses the 25 upper-case characters, so he might as well leave outthe complete table. I admit that ordinary numerical notation would takemore space, but it would certainly be more self-explanatory. Degree anddimension are also encoded as ASCII characters, again leaving me towonder why.
Appendices list the codes (ASCII sequences) of all interestingstrange attractors. The book discusses at length the gradualconstruction of a menu-driven computer program for a PC that searchesfor strange attractors, allowing the user to select (regular) dimension,display mode, and so on, and also allowing users to enter codesmanually. Since most sequences are long, this option is a pain in theneck, and should be replaced by some means to browse through the casescomputed by the author, which are present on a disk accompanying thebook. This disk also contains various versions of the original BASICprogram (not particularly exemplary for students of programming),including C-versions.
The book is certainly not a scientific work, yet titles of somesections are of a forced popular form (for example, the title of thepreface is “Why This Book Is for You”). If a new edition ofthe book appears, I would also recommend that X’ appear with a prime instead of an apostrophe. Theauthor should also discuss other initial values for X and the effect (if any) of the choice of the previousiterate on the result, and he should explain the selection procedure forstrange attractors in one place. A better computer program, with morestructure and more comments, including an explanation of the role of allvariables, preceded by its technical design (possibly inpseudocode), would be appreciated.
