by Carl Sagan
Every edge of a regular solid is shared by the sides of two adjacent polygons. Think again of the cube, where every edge is a boundary between two squares. If we count up all the sides of all the faces of a polyhedron, n F, we will have counted every edge twice. So
n F = 2 E (Equation 3)
Let r represent how many edges meet at each vertex. For a cube, r = 3. Also, every edge connects two vertices. If we count up all the vertices, r V, we will similarly have counted every edge twice. So
r V = 2 E (Equation 4)
Substituting for V and F in Equation (2) from Equations (3) and (4), we find
If we divide both sides of this equation by 2 E, we have
We know that n is 3 or more, since the simplest polygon is the triangle, with three sides. We also know that r is 3 or more, since at least 3 faces meet at a given vertex in a polyhedron. If both n and r were simultaneously more than 3, the left-hand side of Equation (5) would be less than ⅔ and the equation could not be satisfied for any positive value of E. Thus, by another reductio ad absurdum argument, either n = 3 and r is 3 or more, or r = 3 and n is 3 or more.
If n = 3, Equation (5) becomes (1/3) + (1/r) = (1/2) + (1/E), or
So in this case r can equal 3, 4, or 5 only. (If E were 6 or more, the equation would be violated.) Now n = 3, r = 3 designates a solid in which 3 triangles meet at each vertex. By Equation (6) it has 6 edges; by Equation (3) it has 4 faces; by Equation (4) it has 4 vertices. Clearly it is the pyramid or tetrahedron; n = 3, r = 4 is a solid with 8 faces in which 4 triangles meet at each vertex, the octahedron; and n = 3, r = 5 represents a solid with 20 faces in which 5 triangles meet at each vertex, the icosahedron (see figures on this page).
If r = 3, Equation (5) becomes
and by similar arguments n can equal 3, 4, or 5 only, n = 3 is the tetrahedron again; n = 4 is a solid whose faces are 6 squares, the cube; and n = 5 corresponds to a solid whose faces are 12 pentagons, the dodecahedron.
There are no other integer values of n and r possible, and therefore there are only 5 regular solids, a conclusion from abstract and beautiful mathematics that has had, as we have seen, the most profound impact on practical human affairs.
FOR FURTHER READING
(The more technical scientific works are asterisked.)
CHAPTER 1
Boeke, Kees. Cosmic View: The Universe in Forty Jumps. New York: John Day, 1957.
Fraser, Peter Marshall. Ptolemaic Alexandria. Three volumes. Oxford: Clarendon Press, 1972.
Morison, Samuel Eliot. Admiral of the Ocean Sea: A Life of Christopher Columbus. Boston: Little, Brown, 1942.
Sagan, Carl. Broca’s Brain: Reflections on the Romance of Science. New York: Random House, 1979.
CHAPTER 2
Attenborough, David. Life on Earth: A Natural History. London: British Broadcasting Corporation, 1979.
*Dobzhansky, Theodosius, Ayala, Francisco J., Stebbins, G. Ledyard and Valentine, James. Evolution. San Francisco: W.H. Freeman, 1978.
Evolution. A Scientific American Book. San Francisco: W.H. Freeman, 1978.
Gould, Stephen Jay. Ever Since Darwin: Reflections on Natural History. New York: W.W. Norton, 1977.
Handler, Philip (ed.). Biology and the Future of Man. Committee on Science and Public Policy, National Academy of Sciences. New York: Oxford University Press, 1970.
Huxley, Julian. New Bottles for New Wine: Essays. London: Chatto and Windus, 1957.
Kennedy, D. (ed.). Cellular and Organismal Biology. A Scientific American Book. San Francisco: W.H. Freeman, 1974.
*Kornberg, A. DNA Replication. San Francisco: W.H. Freeman, 1980.
*Miller, S.L. and Orgel, L. The Origins of Life on Earth. Englewood Cliffs, N.J.: Prentice-Hall, 1974.
Orgel, L. Origins of Life. New York: Wiley, 1973.
*Roemer, A.S. “Major Steps in Vertebrate Evolution.” Science, Vol. 158, p. 1629, 1967.
*Roland, Jean Claude. Atlas of Cell Biology. Boston: Little, Brown, 1977.
Sagan, Carl. “Life.” Encyclopaedia Britannica, 1970 and later printings.
*Sagan, Carl and Salpeter, E.E. “Particles, Environments and Hypothetical Ecologies in the Jovian Atmosphere.” Astrophysical Journal Supplement, Vol. 32, p. 737, 1976.
Simpson, G.G. The Meaning of Evolution. New Haven: Yale University Press, 1960.
Thomas, Lewis. Lives of a Cell: Notes of a Biology Watcher. New York: Bantam Books, 1974.
*Watson, J.D. Molecular Biology of the Gene. New York: W.A. Benjamin, 1965.
Wilson, E.O., Eisner, T., Briggs, W.R., Dickerson, R.E., Metzenberg, R.L., O’Brien, R.D., Susman, M., and Boggs, W.E. Life on Earth. Stamford: Sinauer Associates, 1973.
CHAPTER 3
Abell, George and Singer, B. (eds.) Science and the Paranormal. New York: Scribner’s, 1980.
*Beer, A. (ed.). Vistas in Astronomy: Kepler, Vol. 18. London: Pergamon Press, 1975.
Caspar, Max. Kepler. London: Abelard-Schuman, 1959.
Cumont, Franz. Astrology and Religion Among the Greeks and Romans. New York: Dover, 1960.
Koestler, Arthur. The Sleepwalkers. New York: Grosset and Dunlap, 1963.
Krupp, E.C. (ed.). In Search of Ancient Astronomies. New York: Doubleday, 1978.
Pannekoek, Anton. A History of Astronomy. London: George Allen, 1961.
Rey, H.A. The Stars: A New Way to See Them, third edition. Boston: Houghton Mifflin, 1970.
Rosen, Edward. Kepler’s Somnium. Madison, Wis.: University of Wisconsin Press, 1967.
Standen, A. Forget Your Sun Sign. Baton Rouge: Legacy, 1977.
Vivian, Gordon and Raiter, Paul. The Great Kivas of Chaco Canyon. Albuquerque: University of New Mexico Press, 1965.
CHAPTER 4
Chapman, C. The Inner Planets. New York: Scribner’s, 1977.
Charney, J.G. (ed.). Carbon Dioxide and Climate: A Scientific Assessment. Washington, D.C.: National Academy of Sciences, 1979.
Cross, Charles A. and Moore, Patrick. The Atlas of Mercury. New York: Crown Publishers, 1977.
*Delsemme, A.H. (ed.). Comets, Asteroids, Meteorites. Toledo: University of Ohio Press, 1977.
Ehrlich, Paul R., Ehrlich, Anne H. and Holden, John P. Ecoscience: Population, Resources, Environment. San Francisco: W.H. Freeman, 1977.
*Dunne, James A. and Burgess, Eric. The Voyage of Mariner 10. NASA SP-424. Washington, D.C.: U.S. Government Printing Office, 1978.
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Goldsmith, Donald (ed.). Scientists Confront Velikovsky. Ithaca: Cornell University Press, 1977.
Kaufmann, William J. Planets and Moons. San Francisco: W.H. Freeman, 1979.
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Krinov, E.L. Giant Meteorites. New York: Pergamon Press, 1966.
Lovelock, L. Gaia. Oxford: Oxford University Press, 1979.
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Masursky, Harold, Colton, C.W. and El-Baz, Farouk (eds.). Apollo Over the Moon: A View from Orbit. NASA SP-362. Washington, D.C.: U.S. Government Printing Office, 1978.
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CHAPTER 5
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Batson, R.M., Bridges, T.M. and Inge, J.L. Atlas of Mars: The 1:5,000,000 Map Series. NASA SP-438. Washington, D.C.: U.S. Government Printing Office, 1979.
Bradbury, Ray, Clarke, Arthur C., Murray, Bruce, Sagan, Carl, and Sullivan, Walter. Mars and the Mind of Man. New York: Harper and Row, 1973.
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Hartmann, W.H. and Raper, O. The New Mars: The Discoveries of Mariner 9. NASA SP-337. Washington, D.C.: U.S. Government Printing Office, 1974.
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Lowell, Percival. Mars and Its Canals. New York: Macmillan, 1906.
Lowell, Percival. Mars as an Abode of Life. New York: Macmillan, 1908.
Mars as Viewed by Mariner 9. NASA SP-329. Washington, D.C.: U.S. Government Printing Office, 1974.
Morowitz, Harold. The Wine of Life. New York: St. Martin’s, 1979.
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The Martian Landscape. Viking Lander Imaging Team, NASA SP-425. Washington, D.C.: U.S. Government Printing Office, 1978.
*Viking 1 Mission Results. Science, Vol 193, No. 4255, August 1976.
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*Viking 2 Mission Results. Science, Vol. 194, No. 4271, December 1976.
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Wallace, Alfred Russel. Is Mars Habitable? London: Macmillan, 1907.
Washburn, Mark. Mars At Last! New York: G.P. Putnam, 1977.
CHAPTER 6
*Alexander, A.F.O. The Planet Saturn. New York: Dover, 1980.
Bell, Arthur E. Christiaan Huygens and the Development of Science in the Seventeenth Century. New York: Longman’s Green, 1947.
Dobell, Clifford. Anton Van Leeuwenhoek and His “Little Animals.” New York: Russell and Russell, 1958.
Duyvendak, J.J.L. China’s Discovery of Africa. London: Probsthain, 1949.
*Gehrels, T. (ed.). Jupiter: Studies of the Interior, Atmosphere, Magnetosphere and Satellites. Tucson: University of Arizona Press, 1976.
Haley, K.H. The Dutch in the Seventeenth Century. New York: Harcourt Brace, 1972.
Huizinga, Johan. Dutch Civilization in the Seventeenth Century. New York: F. Ungar, 1968.
*Hunten, Donald (ed.). The Atmosphere of Titan. NASA SP-340. Washington, D.C.: U.S. Government Printing Office, 1973.
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*“First Scientific Results from Voyager 2.” Science, Vol. 206, No. 4421, p. 927, November 23, 1979.
Manuel, Frank E. A Portrait of Isaac Newton. Washington: New Republic Books, 1968.
Morrison, David and Samz, Jane. Voyager to Jupiter. NASA SP-439. Washington, D.C.: U.S. Government Printing Office, 1980.
Needham, Joseph. Science and Civilization in China, Vol. 4, Part 3, pp. 468–553. New York Cambridge University Press, 1970.
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Wilson, Charles H. The Dutch Republic and the Civilization of the Seventeenth Century. London: Weidenfeld and Nicolson, 1968.
Zumthor, Paul. Daily Life in Rembrandt’s Holland. London: Weidenfeld and Nicolson, 1962.
CHAPTER 7
Baker, Howard. Persephone’s Cave. Athens: University of Georgia Press, 1979.
Berendzen, Richard, Hart, Richard and Seeley, Daniel. Man Discovers the Galaxies. New York: Science History Publications, 1977.
Farrington, Benjamin. Greek Science. London: Penguin, 1953.
Finley, M.I. Ancient Slavery and Modern Ideology. London: Chatto, 1980.
Frankfort, H., Frankfort, H.A., Wilson, J.A. and Jacobsen, T. Before Philosophy: The Intellectual Adventure of Ancient Man. Chicago: University of Chicago Press, 1946.
Heath, T. Aristarchus of Samos. Cambridge: Cambridge University Press, 1913.
Heidel, Alexander. The Babylonian Genesis. Chicago: University of Chicago Press, 1942.
Hodges, Henry. Technology in the Ancient World. London: Allan Lane, 1970.
Jeans, James. The Growth of Physical Science, second edition. Cambridge: Cambridge University Press, 1951.
Lucretius. The Nature of the Universe. New York: Penguin, 1951.
Murray, Gilbert. Five Stages of Greek Religion. New York: Anchor Books, 1952.
Russell, Bertrand. A History of Western Philosophy. New York: Simon and Schuster, 1945.
Sarton, George. A History of Science, Vols. 1 and 2. Cambridge: Harvard University Press, 1952, 1959.
Schrödinger, Erwin. Nature and the Greeks. Cambridge: Cambridge University Press, 1954.
Vlastos, Gregory. Plato’s Universe. Seattle: University of Washington Press, 1975.
CHAPTER 8
Barnett, Lincoln. The Universe and Dr. Einstein. New York: Sloane, 1956.
Bernstein, Jeremy. Einstein. New York: Viking, 1973.
Borden, M. and Graham, O.L. Speculations on American History. Lexington, Mass.: D.C. Heath, 1977.
*Bussard, R.W. “Galactic Matter and Interstellar Flight.” Astronautica Acta, Vol. 6, p. 179, 1960.
Cooper, Margaret. The Inventions of Leonardo Da Vinci. New York: Macmillan, 1965.
*Dole, S.H. “Formation of Planetary Systems by Aggregatio
n: A Computer Simulation.” Icarus, Vol. 13, p. 494, 1970.
Dyson, F.J. “Death of a Project.” [Orion.] Science, Vol. 149, p. 141, 1965.
Gamow, George. Mr. Tompkins in Paperback. Cambridge: Cambridge University Press, 1965.
Hart, Ivor B. Mechanical Investigations of Leonardo Da Vinci. Berkeley: University of California Press, 1963.
Hoffman, Banesh. Albert Einstein: Creator and Rebel. New York: New American Library, 1972.
*Isaacman, R. and Sagan, Carl. “Computer Simulation of Planetary Accretion Dynamics: Sensitivity to Initial Conditions.”Icarus, Vol. 31, p. 510, 1977.
Lieber, Lillian R. and Lieber, Hugh Gray. The Einstein Theory of Relativity. New York: Holt, Rinehart and Winston, 1961.
MacCurdy, Edward (ed.). Notebooks of Leonardo. Two volumes. New York: Reynal and Hitchcock, 1938.
*Martin, A.R. (ed.). “Project Daedalus: Final Report of the British Interplanetary Society Starship Study.” Journal of the British Interplanetary Society, Supplement, 1978.
McPhee, John A. The Curve of Binding Energy. New York: Farrar, Straus and Giroux, 1974.
*Mermin, David. Space and Time and Special Relativity. New York: McGraw-Hill, 1968.
Richter, Jean-Paul. Notebooks of Leonardo Da Vinci. New York: Dover, 1970.
Schlipp, Paul A. (ed.). Albert Einstein: Philosopher-Scientist, third edition. Two volumes. La Salle, Ill: Open Court, 1970.
CHAPTER 9
Eddy, John A. The New Sun: The Solar Results from Skylab. NASA SP-402. Washington, D.C.: U.S. Government Printing Office, 1979.
*Feynman, R.P., Leighton, R.B. and Sands, M. The Feynman Lectures on Physics. Reading, Mass.: Addison-Wesley, 1963.
Gamow, George. One, Two, Three … Infinity. New York: Bantam Books, 1971.