INFINITY is a curious old bogeyman. It creeps up on us during childhood, and the best insights into it always seem to come in a glimpse, out of the corner of the eye.
Einstein once compared it to the situation of a small child in a huge library, surrounded by countless books in many tongues. "The child does not understand the languages in which they are written. He notes a definite plan in the arrangement of books, a mysterious order which he does not comprehend, but only dimly suspects.
Dr Clifford Pickover's off-beat inventive book about infinity has similar childhood resonances. It reaches back to a world of classroom doodlings, to those times of struggling to scribble bubbles in the spaces between other bubbles. Or of drawing lines which fork in two, and each fork branches off and each branch forks off again, to eternity (or till you run out of paper). Or of trying to sketch a pentangle inside another pentangle, ad infinitum (or ad that moment when your 2HB pencil finally snaps).
In his chapter on "infinite star chambers", or nested five-pointed stars, Pickover suggests a similar mindgame. Draw a pentangle with five line segments, each just one inch long. Next, draw a larger five-pointed star that just encompasses it. Then draw another one to encompass these first two. And so on, until after just 15 such "nestings" the cumulative length of all the lines of all IS stars is as long as ... the Panama Canal. By just 26 nestings, he says, the star is slightly bigger than our real star, the Sun. By 65 nestings, it would fill our entire universe.
These are mind-blowing moments, when infinity becomes a slippery monster with countless guises. It refuses to be pinned down or give up its secrets when we stare at it head-on. We are like the lost child in Einstein's library, mesmerised by the feedback loops and nested roots.
Pickover offers various helpful keys or "launchpads" for exploring infinity. The emphasis in these sideways glances is on fun and puzzles, sci-fi stories about Mandlebrot sets, and reflections about automated computer art. Sprinkled throughout are international discussions from the Internet - that strange electronic brain in which ongoing debates seem to tend towards the infinite. Above all, he asks plenty of "what if?" questions.
What if technologists could build an unbreakable ladder from the Earth to the Moon? How long would it take to reach the top? How many supply teams would be required?
Or what if we lived in an "infinity world", where the Earth's maps are repeated forever? Would the term "superpower" become meaningless? Would space travel evolve?
His universe is populated by strange frogs, wriggling worms with weird allergies, and infinite chessboards. This is a wonderland which recalls Borges and Carroll, or Martin Gardner and Douglas Hofstadter's sublime columns in Scientific American in the early 1980s.
Pickover's style is conversational and never condescending. The chapters are snappy and self contained, with several beautiful colour illustrations (his packed CV includes "award-winning computer graphics artist").
Mind you, he also expects his readers to do a fair amount of work - on scraps of paper, on their personal computers and (above all) in their heads - all processes which are increasingly bypassed in these instant-gratification, pill-popping, mouse-clicking times.
The book celebrates hands-on computer experimentation, so a modest amount of programming skill is handy, though not essential. An appendix includes over 68 pages of computer programs - in C code, REXX and a BASIC which generally avoids the dialect differences which spoil many a computer book.
For Pickover, PCs are the sketch pads and laboratories of our age, microscopes and spaceships for professionals and amateurs alike to explore the infinitely large and infinitely small.
Most of the maths (and physics) isn't too difficult; instead of a chore - an arid and useless jail sentence within some outdated school curriculum - it's an enchanting journey, revealing the beauty of patterns and numbers for artists and scientists alike.
This is mathematical tourism at its very best, a fine reminder that our universe is infinitely stranger than we can ever, ever imagine.