The mathematical physicist and cosmologist Roger Penrose, now professor emeritus at Oxford University, is best known to mathematicians for his discovery of Penrose tiles. These are two four-sided polygons that tile the plane only in a nonperiodic way, that is, without a fundamental region that repeats periodically like the hexagonal tiling of a bathroom floor, or the amazing tesselations of the Dutch artist M. C. Escher. To everyone’s surprise, including Penrose’s, his whimsical tiling turned out to underlie a previously unknown type of crystal. You can read all about this in my book Penrose Tiles to Trapdoor Ciphers.

Computers of the sort we know how to build obviously are no more aware of what they do than a typewriter knows it is typing.

Penrose’s two best sellers, The Emperor’s New Mind and its sequel, Shadows of the Mind, were slashing attacks on the opinions of a few artificial intelligence mavens that in just a few decades computers made with wires and switches will be able to do everything a human mind can do. Advanced computers, it was said, will some day replace the human race and colonize the cosmos! Penrose disagrees. Not until we know more about laws below the level of quantum mechanics, he argues, can computers cross that mysterious threshold separating our self-awareness from the unconscious networks of computers. Maybe the threshold will never be crossed. Computers of the sort we know how to build obviously are no more aware of what they do than a typewriter knows it is typing.

Penrose’s new book, The Road to Reality, is a monumental work of more than a thousand pages. Not since the publication of the redbound volumes of Richard Feynman’s Lectures on Physics has anyone covered in such awesome detail the struggles of today’s physicists to unravel the fundamental laws of our fantastic universe—to find the Holy Grail that has been called a TOE, or Theory of Everything.

Most of Penrose’s masterpiece, on which he labored for eight years, is on a technical level far beyond the reach of readers unable, as Penrose warns, to handle simple fractions. The book’s first half is a masterful survey of the mathematics essential for comprehending modern physics. Chapters cover hyperbolic geometry (illustrated with Escher’s models of the hyperbolic plane), complex numbers (so essential in quantum mechanics), Riemann surfaces, quaternions, n-dimensional manifolds, fibre bundles, Fourier analysis, Gödel’s theorem, Minkowski space, Lagrangians, Hamiltonians, and other terrifying topics. Later chapters are crisp introductions to relativity, quantum theory, the Big Bang, black holes, time travel, and many other areas of active research. I will skip over these densely packed chapters to focus on a few aspects of Penrose’s book that can be at least partly understood without mathematical fluency.

Penrose opens his mammoth treatise with a vigorous defense of Platonic realism. This is the view of almost all mathematicians and physicists. They take for granted that the objects and theorems of mathematics are timeless truths that have a strange existence independent of human minds and cultures. There is no galaxy in which two plus two is not four.

Penrose calls attention to an intricate pattern known as the Mandelbrot set. Generated on computer screens by an absurdly simple formula, this swirling pattern is so complex that successive magnifications of its parts always disclose totally unexpected properties. It is impossible, Penrose insists, to regard this mysterious pattern as something cobbled up by our minds. It existed timelessly as an abstract object, “out there,” before Benoît Mandelbrot discovered it. Perhaps it exists on extraterrestrial computer printouts, perhaps in the Mind of God. Exploring it is like exploring a vast jungle.

After his sweeping survey of mathematics, Penrose takes on the daunting task of explaining quantum theory, with emphasis on its bewildering paradoxes. Consider, for example, the mind-boggling EPR paradox. Its letters are the initials of Einstein and two associates who wrote a famous paper in which they maintained that their thought experiment proves that quantum mechanics is incomplete, a view shared by Penrose.

In its simplest form, the EPR paradox imagines a quantum reaction that sends two identical particles, A and B, flying apart in opposite directions. Particle A is measured to determine if its spin is right- or left-handed. In quantum theory a particle does not have a definite spin until it is measured. Its wave function is then said to “collapse,” and it acquires at random, like the heads and tails of a flipped coin, a precise handedness. Amazingly, particle B, which may be light years from A, instantly undergoes a similar wave collapse that gives it a spin opposite the spin of A. (The conservation of momentum requires that A and B have opposite spins.) Now according to relativity theory no information can travel faster than light. How then does B instantly “know” the outcome of a measurement of A?

The paradox is not resolved by saying that A and B are “entangled” in a single system with a single wave function. The problem is to explain how the two particles manage to stay connected. Einstein called it a “spooky action at a distance.” The EPR is only the most dramatic of many paradoxes of entanglement that have now been confirmed in laboratories. Like Einstein, Penrose believes that such paradoxes will not be resolved until quantum mechanics is found resting on a deeper theory.

Penrose is frank in admitting that he has “prejudices” which other physicists reject. For another instance, he is not impressed by the “many-worlds interpretation” of quantum phenomena. According to this eccentric view, every time a quantum event takes place the entire universe splits into two or more parallel universes, each containing a possible outcome of the event!

Take the notorious case known as “Schrödinger’s cat.” Imagine a cat inside a closed box along with a Geiger counter that emits random clicks. The first click triggers a device that kills the cat. Some quantum experts, notably Eugene Wigner, believed that no quantum event is real until it is observed by a conscious mind. Until someone opens the box and looks, the poor cat is a “superposition” of two quantum states, dead and alive. In the many-worlds interpretation the cat remains alive in one world, dies in the other. This proliferation of new universes, like the forking branches of a rapidly growing tree, naturally must include duplicates of you and me!

If these billions upon billions of sprouting universes are not “real” in the same way our universe is real, but only imaginary artifacts, then the many-worlds interpretation is just another way of talking about quantum events. Yes, the talk erases some of the bizarre concepts of quantum theory, but with such an enormous violation of Occam’s razor.

Quantum teleportation is another wild field of active research. Is it possible to scan an object, say an apple, and transmit to another spot its atomic structure? Will it be possible some day to teleport humans, the way they are beamed down to a planet from Star Trek’s spaceship? Penrose shows that such teleportations are not possible unless the original object is totally destroyed. If a person is teleported, will he be the same person after he is reconstructed? Or only a detailed copy that is a different person? Here we plunge into profound questions about human identity—questions that long bemused science-fiction writers, as well as philosophers going back to John Locke and earlier.

Superstrings were believed to be inconceivably minute loops the vibrations of which generated all the basic particles. In recent years superstring theory has been absorbed into a broader conjecture called membrane theory or M-theory. Although Penrose admires M-theory’s mathematical elegance, he suspects it has little relevance to the actual world. So far no way has been found to test it. Will it lead to the next great revolution in physics, as its enthusiasts hope, or will it prove to be a fad destined to go nowhere? Penrose cites numerous past conjectures that proponents thought much too beautiful not to be true, but which soon bit the dust.

The chief rival to M-theory, albeit having fewer disciples, is twistor theory. It was invented by Penrose who, along with his colleagues, has for decades been elaborating the theory. Twistors, deriving from what are known as spinors, are abstract entities which may provide the structure of spacetime. They have a permanent “chirality” (handedness). Penrose is rare among physicists in believing that the universe is fundamentally asymmetric with respect to time and chirality. There have been feeble efforts to combine parts of twistor theory and M-theory, but as things now stand, Penrose finds the two conjectures incompatible. If one is true, the other is false.

This review has given only a few fleeting glimpses into the rich abundance of Penrose’s book. Not only is it admirably written, but it is also cleverly illustrated by Penrose himself. Preceding each exercise is one of three tiny icons. A smiling face tells you that the exercise is “very straightforward.” A solemn faces means “needs a bit of thought.” And a frowning face suggests “not to be taken seriously.” One drawing, which Penrose repeats twice with subtle variations, shows three spheres at the corners of a triangle. Each represents a “form of existence.” One form is the Platonic realm of mathematics. Another is the physical world, and the third is the mental world. Tiny arrows show how the three worlds are clockwise connected.

The Road to Reality loops through a luxurious landscape suffused with the beauty, magic, and mystery of Being. “Why,” Penrose’s friend Stephen Hawking recently asked, “does the universe go to all the bother of existing?” To an atheist, G. K. Chesterton somewhere remarked, the universe is the most exquisite mechanism ever constructed by nobody.

For Penrose, science is a neverending effort to penetrate the secrets of what Einstein liked to call the Old One. He has no sympathy for those who think that all underlying principles of physics have now, or soon will be, discovered. (See John Horgan’s book The End of Science.) For all we know, the universe may have infinite levels of sub-basements and infinite levels of attics in the opposite direction.

Penrose opens and closes his book with two lovely parables about how intuitive insights can ignite scientific revolutions. His prologue is a tribute to Pythagoras for his discovery of irrationals, and the essential role of numbers in understanding how the Old One behaves. His postscript tells of Alícia, a postdoctoral student of physics at the Albert Einstein Institute in Golm, Germany. Alícia had been struggling with difficulties involving quantum gravity, black holes, and the monstrous explosion that created the universe. She had stayed up all night gazing at the stars through a large window. As dawn was about to break, she observed for the first time a rare event known as the green flash. “This experience mingled with some puzzling mathematics thoughts that had been troubling her throughout the night.”

The parable’s final sentence: “Then an odd thought overtook her …”