Geometry, most high school students will attest, is a dull subject. This dullness, however, is not only inescapable but essential. Memorizing theorems and deriving proofs is no fun, but doing such tasks teaches us—as “relevant” and “creative” courses in “communication” or “personal development” do not—that the mind must submit to truth, not the other way around. The only alternative to the tedium of geometry, Latin, physics, and other rigorous subjects is a radical subjectivism in which everyone designs his own curriculum—and universe. Imposed discipline of some sort or licensed solipsism are the only available pedagogical options.

But if discipline begins in drudgery, that is not where it need end. In geometry, for instance, once the essentials have been mastered, a philosophical appraisal of Euclid’s founding premises and of conceivable alternatives may lead to exciting and profound insights. The Greeks and the Scholastic& alike recognized that contemplation of geometry leads finally to the divine source of all form. “God,” said Plato, “geometrizes always.” Unfortunately, most high school and even college teachers never open such horizons to their students. Geometry 1 & 2 lead only to Trigonometry and Computer Science.

Rudy Rucker is a rare mathematician not afraid to pursue the speculative implications of his discipline. In particular, Rucker is intrigued by the possibility of viewing our three-dimensional space as the “surface” of a four dimensional reality. To explore this notion, Rucker renews an analogy first proposed in 1884 by another mathematician, Edwin Abbott Abbott. First, imagine two-dimensional beings living in a hypothetical Flatland; then, imagine how these beings might perceive three-dimensional beings such as a sphere or a human. Because of new discoveries in astronomy, relativity, and quantum mechanics, Rucker can now expand the analogy in ways inconceivable to Abbott. Considering our uni verse as the surface of hyperspace is easier now that we’ve discovered that space has curvature and a “shape.” And since *The Fourth Dimension* is written for laymen and not just mathematicians, it affords the average reader a rare intellectual flight out of the mundane.

Yet in another sense, Rucker’s book is a step backward from Abbott’s *Flatland* of a century ago. For Abbott, a clergy man, a journey into the fourth dimension suggested new ways of thinking about God and His angels. Many of Abbott’s most appreciative readers were Christians (as well as a sprinkling of spiritualists) whose faith chafed within the 3-D rigidities of most late-Victorian philosophies. Rucker acknowledges the possible metaphysical interpretations of his hyperdimensional vision (“Just to have something to think about, we might think of heaven as lying *ana* above space, and hell as lying kata below”). But theology is at most a diversion for Rucker. When he gets serious, he explicitly repudiates belief in God along with all other appeals to “occult powers.” Like many other science-fiction writers (White Light is his best known SF novel), Rucker is bored by the logocentrism of Christianity and prefers dabbling in quasi Oriental mysticism. But without some principle of ontological hierarchy, rest ing firmly on some irreducible Rock of self-existing reality, all figures, geometric and human, are reduced to shadows.

Perhaps it is the rootlessness of his intellectualizing that induces Rucker’s confessed fear of death, which he tries to assuage through provocative but in conclusive theorizing on the nature of time. (This desperate ransacking of mathematics for some defense against mortality compares unfavorably with the remarkable absorption in mathematics that made Archimedes heedless of his own life: the Syracusan geometer’s last words, said to an enemy soldier who interrupted his work on problems sketched in the sand, were “Don’t disturb my figures!”) Repudiating any ultimate basis for meaning, Rucker can push back the circumference of every day thinking, but he cannot provide thought with any stable center. When he finally ends up proposing not a four-dimensional but an infinite dimensional universe, his cosmology turns out to be as boundless as it is empty.

[*The Fourth Dimension: Toward a Geometry of Higher Reality*, by Rudy Rucker (Boston: Houghton Mifflin) $17.95]

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