Women seem to be described very harshly by Abbott, starting with their lower status as less than complete polygons. This was Abbott’s way of satirizing the ridiculous discrimination against women common in his day.

By combining a fourth dimension, time, with the familiar three dimensions of space, Einstein’s equations could describe laws of physics that did not depend on how an observer was moving. That was a good thing, because it would be a pretty messy universe if the math to describe it changed whenever you moved. With relativity theory, any observer can establish a frame of reference to describe what goes on, giving an event a set of four coordinates, specifying its time and location in the three dimensions of space. Observers moving in different directions and at different speeds will generally describe things with a different set of coordinates. Einstein’s equations make it possible for the laws of nature to stay the same for these different observers, or when the frame of reference of any particular observer changes. Einstein was not explicit about this in his original papers; the first clear enunciation of time as the fourth dimension in relativity theory came from the mathematician Hermann Minkowski.

Rosie is a medical writer for the Los Angeles Times, where she writes the funniest yet most intelligent health column in the nation.

Wigner, E. 1960. The unreasonable effectiveness of mathematics in the natural sciences. Communications in Pure and Applied Mathematics. P. 534 in The World Treasury of Physics, Astronomy, and Mathematics, T. Ferris, ed. Boston: Little, Brown, 1991.

Wigner, p. 536.

Lakoff, G., and R. Núñez. 2000. Where Mathematics Comes From. New York: Basic Books, p. 3.

Lakoff and Núñez, p. 344.

Jaffe, R. L., W. Busza, F. Wilczek, and J. Sandweiss. 2000. Review of speculative “disaster scenarios” at RHIC. Reviews of Modern Physics 72:1126. Also available at xxx.lanl.gov/abs/hep-ph/9910333.

Leon Lederman, interview with the author at Fermilab, June 16, 1997.

Gell-Mann proposed the strangeness idea in 1953; later that year the same idea was developed independently by Tadao Nakano and Kazuhiko Nishijima in Japan. A good, brief but more technically detailed account of the origins of strangeness is given in Pais, A. 1986. Inward Bound. New York: Oxford University Press, pp. 519-521.

Yuval Ne’eman, an Israeli physicist, independently proposed the same basic idea at about the same time.

Group theory was not obscure to mathematicians, of course. More details on group theory will appear in Chapter 3.

Murray Gell-Mann, lecture in Santa Fe, N.M., September 23, 1999.

Pais, A. 1952. Some remarks on the V-particles. Physical Review 86:672.

Gell-Mann, interview by the author in Santa Fe, N.M., September 16, 1997.

An electron volt is a unit of energy equal to the amount of energy it takes to boost an electron through a potential of 1 volt. But it is used as a convenient unit of mass in particle physics, reflecting the interchangeability of mass and energy. The mass of a proton is a little less than 1 billion electron volts, or 1 GeV.

Barnes, V. E., et al. 1964. Observation of a hyperon with strangeness number three. Physical Review Letters 12(February 24):206.

Gell-Mann, interview by the author in Santa Fe, N.M., September 16, 1997.

Willy Fischler, lecture at Southern Methodist University in Dallas, TX, February 8, 1999.

Brian Greene, conversation at dinner with the author in Ann Arbor, Michigan, July 11, 2000.

Edward Witten, interview by the author in Ann Arbor, Mich., July 10, 2000.

Witten points out that it’s possible, perhaps, that current ideas about the big bang will turn out to be wrong. Strictly speaking, the astronomical evidence indicates that the universe was very hot and dense in its youth. It’s conceivable that at the very beginning it was cold—conditions under which it might have been possible to create strange quark matter. “This would be a good idea if the big bang were really cold and the heating occurred later, after the quark matter was formed,” Witten said. In fact, some scientists have speculated on the possibility of a cold big bang, notably Harvard astrophysicist David Layzer. But the overwhelming consensus of cosmologists remains otherwise. “Quark matter in the early universe is very hard to make, if it’s true that the early

universe was hot,” Witten says. “What we really know is that today the ratio of photons to baryons (heavy particles such as protons and neutrons) is very high. If that was true in the early universe, then the early universe was a very bad place to make quark matter.”

Witten, interview in Ann Arbor, Mich., July 10, 2000.

At the moment (February 2002) Teplitz is on leave from Southern Methodist University to serve as a senior policy analyst with the White House Office of Science and Technology Policy.

Strange quark nuggets might be stable over a wide range of sizes, from baseballsized chunks weighing a trillion tons down to the size of an ordinary atomic nucleus. Lightweight nuggets might burn up in the atmosphere, displaying themselves as meteors. See Crawford, H., and C. Greiner. 1994. The search for strange matter. Scientific American 270(January):72-77.

In these calculations, the authors suggest that an initially lightweight strangelet, smaller than a uranium nucleus, actually grows in mass by absorbing neutrons (and maybe some protons as well) from atoms in the air on its way down. See Banerjee, S., et al. 2000. Can cosmic strangelets reach the earth? Physical Review Letters 85(August 14):1384-1387.

Broderick, J., et al. 1997. Millimeter-wave signature of strange matter stars. xxx.lanl.gov/abs/astro-ph/9706094, June 10.

Jaffe, et al., p. 1136.

Pais, A. 1986. Inward Bound. New York: Oxford University Press, p. 286.

Gamow, G. 1961. Biography of Physics. New York: Harper & Row, p. 262. While this story seems consistent with Dirac’s character, it’s worth keeping in mind that Gamow’s anecdotes are not always easy to verify.

Heilbron, J., and T. Kuhn. 1969. The genesis of the Bohr atom. Historical Studies in the Physical Sciences 1:257.

Hermann, A. 1971. The Genesis of Quantum Theory, translated by C. Nash. Cambridge, Mass.: MIT Press, p. 157.

Dirac, P. 1978. Directions in Physics. New York: John Wiley & Sons, p. 4.

When Heisenberg showed his math to Max Born, Born informed him that it was merely a reinvention of matrix algebra, a development dating to the 1850s.

Dirac, P. 1983. The origin of quantum field theory. P. 44 in The Birth of Particle Physics, L. Brown and L. Hoddeson, eds. Cambridge: Cambridge University Press.

Dirac, Directions in Physics, p. 14.

Dirac, Directions in Physics, p. 15.

For a good explanation in more depth, see Dirac’s own discussion in Directions in Physics, pp. 11 ff.

Dirac, P. 1930. A theory of electrons and protons. Proceedings of the Royal Society (London), Series A, 128. P. 1195 in The World of the Atom, H. Boorse and L. Motz, eds. New York: Basic Books, 1966.

Dirac, Directions in Physics, p. 17.

Oppenheimer, J. R. 1930. On the theory of electrons and protons. Physical Review 35. P. 1205 in Boorse and Motz.

Pais, A. 2000. The Genius of Science. New York: Oxford University Press, p. 59.

Gordon Kane, conversation with the author in Ann Arbor, Mich., July 11, 2000.

Dirac, P. 1971. The Development of Quantum Theory. New York: Gordon and Breach, p. 56.

Dirac, Directions in Physics, p. 17.

Anderson, C. 1999. The Discovery of Anti-matter. Singapore: World Scientific, p. 25.

Anderson, C. with H. Anderson. 1983. Unraveling the particle content of cosmic rays. P. 140 in Brown and Hoddeson.

This is not true of all particles without charge. The neutron, for instance, has no net electrical charge. But its component quarks do. An antineutron comprises the antimatter counterparts of its three quarks. Instead of two downs and an up, as in the neutron, the antineutron is made of two anti-downs and one anti-up. The charges still add up to zero.

Weyl, H. 1949. Philosophy of Mathematics and Natural Science. Princeton: Princeton University Press, p. 208.

Crease, R. P., and C. C. Mann. 1986. The Second Creation. New York: Macmillan, p. 209.

Leon Lederman, interview with the author at Fermilab, 1997.

Yang, C. N., and T. D. Lee. 1956. Physical Review 104 (October):258.

There is a possibility that some very slight electromagnetic interaction might be possible between mirror matter and ordinary matter. Quantum effects could permit an ordinary photon to convert itself into a mirror photon on very rare occasions. If so, mirror matter would appear to have a small electrical charge.

Some physicists insist that if the masses aren’t identical, you shouldn’t call it “mirror matter” but rather “shadow matter.” The use of mirror matter seems to have become common in either case, however.

Mohapatra, R. N., and V. Teplitz. 1996. Structures in the mirror universe. xxx.lanl.gov/abs/astro-ph/9603049, March 12.

Astronomers have searched for MACHOs by training telescopes on the Magellanic Clouds, small satellite galaxies to the Milky Way. A Magellanic star brightens for a while when a MACHO passes in front of it (because the MACHO’s gravity distorts the starlight). MACHO hunters have recorded about 20 cases of such Magellanic star brightenings. If the population of MACHOs in the Magellanic direction is typical, then they cannot account for all the dark matter estimated to lurk in the Milky Way’s halo. By some estimates, MACHOs could make up half the invisible halo mass, but maybe a lot less, and certainly not all of it.

Mohapatra, R. N., and V. Teplitz. 1999. Mirror matter MACHOs. xxx.lanl.gov/ abs/astro-ph/9902085, February 4.

M. Zapatero Osorio, et al. 2000. Discovery of young, isolated planetary mass objects in the σ Orionis star cluster. Science 290:103.

Foot, R., et al. 2000. Do “isolated” planetary mass objects orbit mirror stars? xxx.lanl.gov/abs/astro-ph/0010502, October 25.

Hill, C., and L. Lederman. 2000. Teaching symmetry in the introductory physics curriculum. xxx.lanl.gov/abs/physics/0001061, version 2, February 7, pp. 1-2. See also www.emmynoether.com.

Neal Lane, conversation with the author at Fermilab, June 14, 1999.

McGrayne, S. B. 2001. Nobel Prize Women in Science, 2nd ed. Washington, D.C.: Joseph Henry Press, p. 72.

Actually, the issue of conservation of energy in general relativity is more complicated than this; in different situations the very notions of energy and conservation are not easily defined.

Technically, Noether showed that a conservation law is linked to a continuous symmetry. A sphere possesses continuous symmetry with respect to rotation, because it stays the same no matter how small a turn you give it. A snowflake has discrete symmetry, because you must turn it in increments of 60º to make it look the same. For more on this, see Hill and Lederman, pp. 6 ff.

Another way of explaining it was suggested to me by Rabindra Mohapatra. If you rotate a triangle, all the points are changed at the same time, so the symmetry is “global.” A gauge symmetry, on the other hand, allows changing a system at one point independently of other points. In a moving system where all points are connected, information about the change at one point must then be communicated to the other points; that communication is accomplished by the transmission of a force.

Steven Weinberg, interview with the author in Austin, TX, November 21, 1997.

Wilczek, F. 2001. Future summary. International Journal of Modern Physics A 16:1653-1678. Available at xxx.lanl.gov/abs/hep-ph/0101187.

Edward Witten, interview with the author in Princeton, N.J., April 6, 1995.

Ramond showed how fermions could be incorporated into string theory, paving the way for work showing the connection between string theory and supersymmetry.

See Kane, G., and M. Shifman. 2000. Foreword. P. ix in The Supersymmetric World: The Beginnings of the Theory. Singapore: World Scientific. Also available at xxx.lanl.gov/abs/hep-ph/0102298.

Savas Dimopoulos, conversation with the author in Houston, TX, November 1, 2000.

Rita Bernabei, lecture in Austin, TX, December 11, 2000.

Blas Cabrera, lecture in Austin, TX, December 11, 2000.

van den Bergh, S. 1999. The early history of dark matter. xxx.lanl.gov/abs/ astro-ph/9904251, April 19, pp. 2-3.

Schucking, E. 2001. A personal memoir of 1958. Physics Today 54 (February):47.

Rosenfeld, L. 1967. Niels Bohr in the thirties. P. 127 in Niels Bohr, S. Rozental, ed. New York: John Wiley & Sons.

Pais, A. 2000. The Genius of Science. New York: Oxford University Press, p. 244.

Solomey, N. 1997. The Elusive Neutrino. New York: Scientific American Library, p. 16.

Reines, F. 1996. The neutrino: from poltergeist to particle. Reviews of Modern Physics 68(April):318.

Reines, p. 318.

The origin of the Q-ball idea seems to be a paper from 1985 by the Harvard physicist Sidney Coleman. The Q refers to a standard symbol that physicists use to denote a conserved quantity, or “charge.” In this case the charge is a special symmetry property related to the particle number of the balls.

Kusenko, A. 1997. Q-balls in the MSSM. xxx.lanl.gov/abs/hep-ph/9707306, July 10.

Kusenko, A., and M. Shaposhnikov. 1997. Supersymmetric Q-balls as dark matter. xxx.lanl.gov/abs/hep-ph/9709492, version 3, October 30.

Dvali, G., et al. 1997. New physics in a nutshell, or Q-ball as a power plant. xxx.lanl.gov/abs/hep-ph/9707423, version 2, October 30.

Kolb, E., et al. 1998. WIMPZILLAS! xxx.lanl.gov/abs/hep-ph/9810361, October 14.

Rocky Kolb, interview with the author at Fermilab, May 8, 2001.

They are not necessarily colliding in the literal sense. After all, it is hard to say what it would mean for subatomic particles to collide, since they are not themselves like billiard balls but are rather fuzzy and wavy. By collision physicists mean that two particle come close enough to each other to cause some effect, such as a change in direction, which is pretty much what happens with real collisions, too.

Andrei Linde, interview by the author at The Woodlands, TX, January 7, 1991.

Rees, M. 1997. Before the Beginning. Reading, Mass.: Perseus Books, p. 3.

Tropp, E. A., et al. 1993. Alexander A. Friedmann: The Man who Made the Universe Expand. Cambridge: Cambridge University Press, p. 255.

Tropp et al, p. 73.

K. C. Cole and Rosie Mestel of the Los Angeles Times.

Tropp et al, p. 37.

The Dutch astronomer Willem de Sitter is sometimes credited with forecasting the expansion of the universe in a paper in 1917. While expansion may be implicit in de Sitter’s work, he did not discuss it explicitly, as Friedmann did.

Friedmann, A. 1922. On the curvature of space. Zeitschrift für Physik 10:377-386. P. 49 in Cosmological Constants, J. Bernstein and G. Feinberg, eds. New York: Columbia University Press, 1986.

Friedmann, in Bernstein and Feinberg, p. 58.

Kragh, H. 1996. Cosmology and Controversy. Princeton: Princeton University Press, p. 27.

Hubble, E. 1929. A relation between distance and radial velocity among extra-galactic nebulae. Proceedings of the National Academy of Sciences. 15(March 15):168-173. P. 81 in Bernstein and Feinberg.

At least it has always been interpreted as a slur. Hoyle apparently told a journalist years later that he merely meant to make the idea sound dramatic.

Gott, J. R. 2001. Time Travel in Einstein’s Universe. Boston: Houghton Mifflin, p. 160.

Even before Guth’s original version, a similar idea had been proposed in the Soviet Union by Alexei Starobinsky. But it was Guth’s version that started the inflation bandwagon.

Alan Guth, lecture in Washington, D.C., April 14, 1999.

Rees, M. 2001. Concluding perspective. xxx.lanl.gov/abs/astro-ph/0101268, January 16, p. 6.

Rees, Concluding perspective, p. 8.

Lawrence Krauss, interview by the author in Washington, D.C., April 28, 2001.

Its official name is the Einstein field equation. Various textbooks present this equation in a wide range of different forms. This form is from one of Einstein’s early papers: Einstein, A. 1915. On the general theory of relativity (addendum). Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin. P. 109 in The Collected Papers, vol. 6. Princeton: Princeton University Press, 1997.

Wheeler, J. 1990. A Journey into Gravity and Spacetime. New York: Scientific American Library, pp. 11-12.

The radiation density drops off more rapidly because it depends on the fourth power of the universe’s radius. The matter density depends only on the cube of the radius—in other words, it’s proportional to the volume. So radiation density drops faster than matter density and at some point will fall below the matter density.

“Initial conditions” are a necessary part of applying the laws of physics. The laws simply state what will happen, given such-and-such a situation—the position and velocities of particles, any forces in the neighborhood, etc. Those are the initial conditions.

Einstein, A. 1917. Cosmological considerations on the general theory of relativity. Sitzungsberichte der Preussischen Akademie der Wissenschaften. P. 26 in Cosmological Constants, J. Bernstein and G. Feinberg, eds. New York: Columbia University Press, 1986.

Gamow, G. 1970. My World Line. New York: Viking Press, p. 44.

In 1923, Hermann Weyl showed how de Sitter’s approach to Einstein’s theory could describe an expanding universe. “If there is no quasi-static world, then away with the cosmological term,” Einstein replied to Weyl in a postcard. But Einstein continued to reject the idea of an expanding universe as physically real, as he told Lemaître at a meeting in 1927. Only after Hubble’s analysis did Einstein explicitly disavow the cosmological constant, in an obscure journal in 1931. See Straumann, N. 2000. On the mystery of the cosmic vacuum energy density. xxx.lanl.gov/abs/astro-ph/0009386, September 25, pp. 3-4.

Einstein, A. 1956. The Meaning of Relativity, 5th ed. Princeton: Princeton University Press, p. 127.

Josh Frieman, discussion with reporters at Fermilab, May 1, 1998.

Siegfried, T. 1992. Einstein buried his “mistake,” but it’s still haunting scientists. Dallas Morning News, January 20, p. 7D.

Discussion with reporters at Fermilab, May 1, 1998.

Michael Turner, lecture at Fermilab, May 1, 1998.

In case you missed it, these were figure skater Nancy Kerrigan’s comments after she was bashed in the knee shortly before the 1994 Winter Olympics.

Other precursor papers can be traced back to the early 1980s. Frieman, by the way, does not like the name quintessence, arguing tongue-in-cheek that pentessence would make more sense, because Aristotle was Greek, not Roman.

Robert Caldwell, talk at Fermilab, May 1, 1998.

Michael Turner, interview by the author at Fermilab, May 26, 1999.

Gu, J.-A., and W.-Y. P. Hwang. 2001. The fate of the accelerating universe. xxx.lanl.gov/abs/astro-ph/0106387, June 21.

Krauss, L. 2001. Atom. Boston: Little, Brown, p. 275.

Lawrence Krauss, interview by the author in Washington, D.C., April 28, 2001.

Newman, J. 1955. James Clerk Maxwell. Scientific American 192(June). P. 156 in Lives in Science. New York: Simon & Schuster.

Siegel, D. 1981. Thomson, Maxwell, and the universal ether in Victorian physics. P. 249 in Conceptions of Ether, G. N. Cantor and M. J. S. Hodge, eds., Cambridge: Cambridge University Press.

Siegel, p. 254.

Maxwell, J. C. 1864. A dynamical theory of the electromagnetic field. Philosophical Transactions of the Royal Society of London 155. P. 857 in The World of Physics, vol. 1, by J. H. Weaver. New York: Simon & Schuster, 1987.

Holton, G. 1971-1972. On trying to understand scientific genius. American Scholar 41(Winter):102.

Stachel, J. 1998. Einstein’s Miraculous Year. Princeton: Princeton University Press, p. 15.

Schwarz, J. H. 2000. String theory: the early years. xxx.lanl.gov/abs/hep-th/ 0007118 version 3, July 26, p. 3.

Murray Gell-Mann, interview by the author in Santa Fe, N.M., September 16, 1997.

Schwarz, J. H. 2000. Reminiscences of collaborations with Joël Scherk. xxx.lanl.gov/abs/hep-th/0007117, July 14, p. 3.

Schwarz, Reminiscences, p. 4.

Both correctly point out that such a theory would not explain all the events that depend on historical contingency or any of a number of complicated things. But, while acknowledging their legitimate objections, I use the phrase occasionally as a convenient shorthand.

Sullivan, W. 1985. Is absolutely everything made of string? New York Times, May 7, p. 1C.

Siegfried, T. 1985. Superstring: theory ties forces together in major physics breakthrough, Dallas Morning News, April 22, p. 7D.

To write that number out, you’d put 30 zeroes to the right of the decimal point, then the 1.

Murray Gell-Mann, interview in Santa Fe, N.M., September 16, 1997.

Mach, E. 1960. The Science of Mechanics. LaSalle, Ill.: Open Court Publishing, pp. 588-589.

Mach, E. 1960. Space and Geometry. LaSalle, Ill.: Open Court Publishing, p. 138.

Rocky Kolb, interview with the author at Fermilab, June 16, 1999.

Joe Lykken, conversation with the author at Fermilab, June 15, 1999.

Israel, W. 1987. Dark stars: the evolution of an idea. P. 201 in 300 Years of Gravitation, S. W. Hawking and W. Israel, eds. Cambridge: Cambridge University Press. Michell’s paper was communicated by his friend Cavendish to the Royal Society on November 27, 1783.

Israel, p. 203.

Thorne, K. S. 1994. Black Holes and Time Warps. New York: W. W. Norton, p. 124.

Bernstein, J. 1996. The reluctant father of black holes. Scientific American 274(June):83.

Schwarzschild, K. 1916. On the gravitational field of a sphere of incompressible fluid according to Einstein’s theory. Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin (1916), translated by S. Antoci. xxx.lanl.gov/abs/physics/9912033, December 16, 1999, p. 9. To be precise, Schwarzschild wrote, “For an observer measuring from the outside . . . a sphere of given gravitational mass α/2K² can not have a radius measured from the outside smaller than P_o = α.” For a sphere of incompressible fluid, the limit is 9/8 times α.

Bernstein, p. 84.

Oppenheimer, J. R., and H. Snyder. 1939. On continued gravitational contraction. Physical Review 56(September 1):457.

Oppenheimer and Snyder, p. 456.

Oppenheimer and Snyder, p. 459.

The Oppenheimer-Snyder paper was in the same issue as the famous Bohr-Wheeler paper describing the basic physics of nuclear fission.

Thorne, pp. 210-211.

Siegfried, T. 1998. Black hole was catchy for Wheeler, Dallas Morning News, October 19, p. 4F.

Newcomb, S. 1894. Modern mathematical thought. Nature 49:325-329. P. 386 in Time Machines, by P. Nahin. 2nd ed. New York: Springer-Verlag.

Isaksson, E. Gunnar Nordström (1881-1923): on gravitation and relativity. www.helsinki.fi/~eisaksso/nordstrom/nordstrom.html.

Kaluza, T. 1921. On the unification problem of physics. Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin. P. 53 in The Dawning of Gauge Theory, L. O’Raifeartaigh, ed. Princeton: Princeton University Press.

Klein, O. 1926. Quantum theory and five-dimensional relativity. Zeitschrift für Physik 37. P. 68 in O’Raifeartaigh.

Instead of obeying the inverse-square law, for instance, the strength of gravity would diminish as the cube of the distance between two bodies, assuming one additional dimension.

Andy Strominger, telephone interview by the author, 1995.

Later, Duff moved to the University of Michigan.

Siegfried, T. 1990. Superstrings snap back, Dallas Morning News, March 19, p. 6D.

I encountered the triangle-cone example in Durham, I. T. 2000. A historical perspective on the topology and physics of hyperspace. xxx.lanl.gov/abs/ physics/0011042, November 18.

Savas Dimopoulos, interview by the author in Palo Alto, Calif., February 20, 2001.

In that view, the boundary is 10-dimensional, but maybe only three of the space dimensions are big, so that our universe appears to us to be a three-brane.

These parallel worlds are not the same thing as the multiverse, the multiple bubbles of spacetime inflating out of a common vacuum. The multiple bubbles we met before would all be just parts of our own familiar three-dimensional space—too far away to communicate with, but part of our same fabric. They would be very, very distant—too far away for light to ever travel from there to here. In other words, there’s no need to worry about what’s going on in them. But the parallel brane worlds could literally be less than a silly millimeter away.

Joe Lykken, telephone interview by the author, July 1, 1999.

Lisa Randall, interview by the author in Ann Arbor, Mich., July 13, 2000.

Joe Lykken, interview by the author in Lake Tahoe, Calif., December 11, 1999.

Rocky Kolb, interview by the author at Fermilab, June 16, 1999.

Lisa Randall, talk in San Francisco at the annual meeting of the American Association for the Advancement of Science, February 16, 2001.

Joe Lykken, interview by the author in Lake Tahoe, Calif., December 11, 1999.

One modern commentator’s assessment: “Kant’s belief that Euclidean geometry was true, because our intuitions tell us so, seems to me to be either unintelligible, or wrong.” P. 85 in Gray, J. 1989. Ideas of Space. Oxford: Clarendon Press.

Kline, M. 1985. Mathematics and the Search for Knowledge. New York: Oxford University Press, p. 152.

Kline, p. 152.

Kline, p. 152.

Bell, E. T. 1937. Men of Mathematics. New York: Simon & Schuster, p. 297.

Weber, A. S., ed. 2000. 19th Century Science: An Anthology. Peterborough, Canada: Broadview Press, p. 138.

Bolyai’s comment came in a letter to his father in 1823, quoted (with a slightly different translation) on p. 107 of Ideas of Space, by J. Gray. Oxford: Clarendon Press, 1989.

Bell, p. 490.

Riemann, B. 1959. On the hypotheses which lie at the foundations of geometry. P. 411 in A Source Book in Mathematics, by D. E. Smith. New York: Dover Publications.

Smith, p. 424.

Smith, p. 425.

Einstein’s main problem in formulating general relativity was to find a mathematical way of expressing “general covariance”—the equivalence of all accelerating systems regardless of the coordinate system you used to keep track of their motion. But Einstein could not reconcile general covariance with Euclidean geometry. Grossmann showed Einstein that Riemannian geometry could describe general covariance consistently.

Einstein, A. 1915. On the general theory of relativity (addendum). Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin. P. 108 in The Collected Papers, vol. 6. Princeton: Princeton University Press, 1997.

Keep in mind, Einstein’s general relativity describes the curvature of space and time combined, as spacetime. But talking only about the curvature of space is a shorthand approach that usually does no damage.

Luminet, J.-P., and B. Roukema. 1999. Topology of the universe: theory and observation. xxx.lanl.gov/abs/astro-ph/9901364, January 26, p. 2.

Luminet and Roukema, pp. 2-4.

News conference, American Physical Society meeting, Columbus, Ohio, April 17, 1998.

Levin, J. 2001. Topology and the cosmic microwave background. xxx.lanl.gov/ abs/gr-qc/0108043, August 16, p. 3.

David Spergel, interview by the author in Chapel Hill, N.C., April 11, 2001.

Luminet, J.-P., et al. 1999. Is space finite? Scientific American 280(April):92.

Barrow, J., and J. Levin. 1999. Chaos and order in a finite universe. xxx.lanl.gov/ abs/astro-ph/9907288, July 21, p. 2.

Levin, J., and I. Heard. 1999. Topological pattern formation. xxx.lanl.gov/abs/ astro-ph/9907166, July 13, p. 1.

Clark, R.W. 1971. Einstein: The Life and Times. New York: World Publishing, p. 10.

Einstein, A. 1951. Autobiographical notes. P. 5 in Albert Einstein: Philosopher-Scientist, P. Schilpp, ed., vol. 1. New York: Harper and Row.

Einstein, p. 16.

Einstein, p. 53.

If you are moving in a straight line at constant speed, you occupy a legitimate frame of reference, or “inertial frame,” for making observations and measurements. Your conclusions about the laws of physics should be the same as those of anyone else in any other inertial frame. Therefore there must be some symmetry group—a group of operations—that can reorient your inertial frame to make it identical to any other inertial frame. The mathematical operations describing that symmetry are known as the Lorentz group.

If an object somehow begins life at faster-than-light speeds, special relativity’s rules would not be broken. Such particles, known as tachyons, would be “legal,” but there is no solid evidence that they actually exist. There is another possible loophole to the speed-of-light limit, known as the Scharnhorst effect, in which light can go slightly faster than its usual speed in a vacuum. The Scharnhorst effect achieves this trick by putting two metal plates close enough together to restrict the wavelengths of photons that can pop into existence out of the vacuum. This effect “clears out” some of the quantum clutter in the vacuum, enabling light to zip through more rapidly. The extra speed is far too small to measure, though. If you started a race between a Scharnhorst photon and an ordinary photon at the time of the universe’s birth, by now the faster photon would be ahead by less than the width of an atom.

Einstein, A. 1905. On the electrodynamics of moving bodies. Annalen der Physik 17. P. 139 in Einstein’s Miraculous Year, by J. Stachel. Princeton: Princeton University Press, 1998.

Einstein, A. 1911. The Theory of Relativity, lecture in Zurich, January 16, 1911. Naturforschende Gesellschaft in Zürich. Vierteljahrsschrift 56. Pp. 348-349 in The Collected Papers, vol. 3. Translated by A. Beck. Princeton: Princeton University Press, 1993.

Several books offer good in-depth explanations of the twin paradox. You might try Davies, P. 1995. About Time. New York: Simon & Schuster; Greene, B. 1999. The Elegant Universe. New York: W. W. Norton; or pp. 462 ff in Nahin, P. 1999. Time Machines. 2nd ed. New York: Springer-Verlag.

Minkowski, H. 1908. Space and Time, address delivered at the 80th Assembly of German Natural Scientists and Physicians, at Cologne, September 21, 1908. P.75 in The Principle of Relativity, by A. Einstein, et al., translated by W. Perrett and G. B. Jeffery. New York: Dover Publications, 1952.

Nahin, P. 1999. Time Machines. 2nd ed. New York: Springer-Verlag, pp. 140 ff.

Cumrun Vafa, e-mail correspondence with the author, October 23, 1996.

Andy Strominger, telephone interview with the author, October 22, 1996.

Cumrun Vafa, e-mail correspondence with the author, October 23, 1996.

Tegmark, M. 1997. On the dimensionality of spacetime. xxx.lanl.gov/abs/gr-qc/9702052, version 2, April 4, p. 2.

Tegmark, p. 3.

Bars, I., and C. Kounnas. 1997. Theories with two times. xxx.lanl.gov/abs/ hep-th/9703060, March 7.

Hull, C. M. 1999. Duality and strings, space, and time. xxx.lanl.gov/abs/ hep-th/9911080, November 11, pp. 3-4.

Hull, p. 12.

Hull, p. 14.

Hull, p. 14.

Lippmann, W. 1936. Public Opinion. New York: Macmillan, p. 25.

Lippmann, p. 29.

Wigner, E. 1991. The unreasonable effectiveness of mathematics in the natural sciences. Communications in Pure and Applied Mathematics. P. 527 in The World Treasury of Physics, Astronomy, and Mathematics, T. Ferris, ed. Boston: Little, Brown.

Lakoff, G., and R. Núñez. 2000. Where Mathematics Comes From. New York: Basic Books, p. 9.

George Lakoff, talk in San Francisco at the annual meeting of the American Association for the Advancement of Science, February 17, 2001.

Hertz, H. 1945. On the relations between light and electricity. P. 459 in The Autobiography of Science, F. R. Moulton and J. J. Schifferes, eds. Garden City, N.Y.: Doubleday, Doran.

Murray Gell-Mann, dinner talk in Pasadena, Calif., January 14, 2000.

Rocky Kolb, interview by the author in Chapel Hill, N.C., April 13, 2001.

Hertz, p. 460.

Holton, G. 1971-1972. On trying to understand scientific genius. American Scholar 41(Winter):102.

Cumrun Vafa, e-mail correspondence with the author, October 23, 1996.

For an elaboration on this view with regard to Spinoza, see Zimmermann, R. E. 2000. Loops and knots as topoi of substance. Spinoza revisited. xxx.lanl.gov/ abs/gr-qc/0004077, version 2, May 23.

Poincaré, H. 1952. Science and Hypothesis. New York: Dover Publications, p. 158.

Poincaré, p. 159.

Poincaré, p. 161.

Poincaré, H. 1963. Mathematics and Science: Last Essays, translated by J. W. Bolduc. New York: Dover Publications, p. 14.

Planck, M. 1949. Scientific Autobiography and Other Papers. New York: Philosophical Library, p. 58.

Poincaré, Mathematics and Science, p. 13.

Poincaré, Science and Hypothesis, p. 161.

As I was reading the proofs for this book, I was also reading the proofs of a new book by Stephen Wolfram, the physicist-turned-entrepreneur famous for developing the computer program Mathematica. Wolfram’s book, called A New Kind of Science, offers some interesting insights into the relationship of math to reality. By studying computer programs called cellular automata, Wolfram demonstrates that very simple rules can produce structures of great complexity. In fact, he shows, programs that exhibit behavior beyond some minimum threshold of complexity, while still fairly simple, can emulate any other computing system of whatever complexity. He therefore deduces a “principle of computational equivalence,” declaring that any programs or natural processes exceeding that threshold are ultimately equivalent in their computational sophistication.

In Wolfram’s view all natural processes can be considered to be, in essence, computations. And obviously mathematics can also be regarded as computation as well. Wolfram therefore concludes that there is an intrinsic equivalence between nature and mathematics, as all computation that is not trivially simple possesses equivalent sophistication. The power of math to represent reality is therefore merely a reflection of the intrinsic equivalence of both math and reality as equally powerful forms of computation.

Lippmann, p. 16.

Lippmann, p. 29.