References
References |
|
Cohen, D. (1983) Waiting for the Apocalypse. Prometheus Books: New York.
Davies, P. (1992) The Mind of God. Touchstone/Simon and Schuster: New York.
Davis, P. and Hersh, R. (1981) The Mathematical Experience. Houghton Mifflin Company: Boston.
Davis, P. and Hersh, R. (1986) Descartes’ Dream. Harcourt Brace Jovanovich: New York.
Ellis, K. (1978) Number Power. St. Martin’s Press: New York.
Gardner, M. (1992) A Walk on the Wild Side. Prometheus Books: New York.
Gorman, P. (1979) Pythagoras: A Life. Routledge and Kegan Paul: Boston.
Guthrie, K. (1991) The Pythagorean Sourcebook and Library. Phanes Press: Grand Rapids, Michigan.
Kappraff, J. (1991) Connections. McGraw-Hill: New York.
McLeish, J. (1991) The Story of Numbers. Fawcette: New York.
Russell, B. (1945) A History of Western Philosophy. Simon and Schuster: New York.
Schimmel, A. (1993) The Mystery of Numbers. Oxford University Press: New York.
Belier, A. (1966) Recreations in the Theory of Numbers. Dover: New York.
Davis, P. and Hersh, R. (1981) The Mathematical Experience. Houghton Mifflin Company: Boston.
Ellis, K. (1978) Number Power. St. Martin’s Press: New York.
Guy, R. (1994) Every number is expressible as the sum of how many polygonal numbers? American Mathematics Monthly. 101(2): 169–172.
Heath, T. (1963) Greek Mathematics. Dover: New York.
Kordemsky, B. (1972) The Moscow Puzzles. Dover: New York. McLeish, J. (1991) The Story of Numbers. Fawcette: New York.
Nelsen, R. (1991) Proof without words: Sum of reciprocals of triangular numbers. Mathematics Magazine 64(3): 167.
Nelsen, R. (1993) Proofs without Words: Exercises in Visual Thinking. Mathematics Association of America: New York.
Nelsen, R. (1994) Proof without words: A triangular identity. Mathematics Magazine. 67(4): 293.
Sarton, G. (1952) A History of Science. Harvard University Press: Massachusetts.
Schimmel, A. (1993) The Mystery of Numbers. Oxford University Press: New York.
Schroeder, M. (1984) Number Theory in Science and Communication. Springer: New York.
Wells, D. (1987) The Penguin Dictionary of Curious and Interesting Numbers. Penguin: New York. (Many of the interesting triangular number formulas come from this book.)
Deakin, M. (1994) Hypatia and her mathematics. American Mathematical Monthly. 101(3): 234–243.
Gorman, P. (1979) Pythagoras: A Life. Routledge and Kegan Paul: Boston.
Miller, W. (1993) Proof without words: Sum of pentagonal numbers. Mathematics Magazine. 66(5): 325.
Pickover, C. (1992) Chapter 47, Undulating undecamorphic and undulating pseudofareymorphic integers. In Computers and the Imagination. St. Martin’s Press: New York. Also: Chapter 37, On the existence of cakemorphic integers (describes square pyramidal numbers). Also: Chapter 36, Infinite sequences in centered hexamorphic numbers (describes hexagonal numbers and centered hexagonal numbers).
Schimmel, A. (1993) The Mystery of Numbers. Oxford University Press: New York.
Kroon, R., Sancton, T. and Scott, G. (1994) In the reign of fire. Time. October 17. 59–60. (Describes the Order of the Solar Temple.)
Pearl, R. and Reed, L. (1920) On the rate of growth of the population of the United States since 1790 and its mathematical representation. Proceedings of the National Academy of Sciences. 6(6): 275–288.
von Foerster, H., Mora, P., and Amiot, L. (1960) Doomsday: Friday 13 November A.D. 2026. Science. 132: 1291–1295.
The following are controversial responses to the von Foerster paper:
Robertson, J., Bond, V., and Cronkite, E. (1961) Doomsday (Letter to the Editor) Science. 133: 936–937.
Hutton, W. (1961) Doomsday (Letter to the Editor) Science. 133: 937–939.
Howland, W. (1961) Doomsday (Letter to the Editor) Science. 133: 939–940.
Shinbrot, M. (1961) Doomsday (Letter to the Editor) Science. 133: 940–941.
von Foerster, H., Mora, P., and Amiot, L. (1961) Doomsday (Letter to the Editor) Science. 133: 941–952.
Serrin, J. (1975) Is “Doomsday” on Target? Science. 189: 86–88.
Beiler, A. (1964) Recreations in the Theory of Numbers. Dover: New York. (Contains much information on Pythagorean triangles.)
Dudley, U. (1992) Mathematical Cranks. Mathematical Association of America: Washington, D.C.
Schimmel, A. (1993) The Mystery of Numbers. Oxford University Press: New York.
Ellis, K. (1978) Number Power. St. Martin’s Press: New York.
Wells, D. (1987) The Penguin Dictionary of Curious and Interesting Numbers. Penguin: New York.
Beiler, A. (1964) Recreations in the Theory of Numbers. Dover: New York.
Spencer, D. (1982) Computers in Number Theory. Computer Science Press: New York.
Mott-Smith, G. (1954) Turks and Christians. In Mathematical Puzzles. Dover: New York. (p. 94)
Gardner, M. (1981) Science: Good, Bad and Bogus. Prometheus Books: Buffalo, New York. (Chapter 3 describes the work of Ramon Lull in fascinating detail and is the primary source for my information on Lull in this chapter. Gardner’s book is highly recommended.)
Reichardt, J. (1969) Cybernetic Serendipity: The Computer and the Arts. Praeger: New York. (This book has a chapter on Japanese haiku, computer texts, high-entropy essays, fairytales, and fake physics essays.)
Kress, N. (1994) An Untitled Column. Writer’s Digest. December, 8–10. (Describes the Lullian generation of book titles.)
Kurzweil, R. (1990) The Age of Intelligent Machines. MIT Press: Cambridge, Massachusetts. (This book contains information on pattern recognition, the science of art, computer-generated poetry, and artificial intelligence.)
Pickover, C. (1992) Computers and the Imagination. St. Martin’s Press: New York. (Describes computer poetry.)
Pickover, C. (1994) Mazes for the Mind: Computers and the Unexpected. St. Martin’s Press: New York. (Describes the Lullian approach for creating inventions.)
Pickover, C. (1995) Chaos in Wonderland: Visual Adventures in a Fractal World. St. Martin’s Press: New York.
Alvarez, L.W., Alvarez, W., Asaro, F., Michel, H.V. (1980) Extraterrestrial cause for the Cretaceous-Tertiary extinction. Science. 208, 1095–1108.
Alvarez, W., Asaro, F., Michel, H.V., Alvarez, L.W. (1982) Iridium anomaly approximately synchronous with terminal Eocene extinctions. Science. 216, 886–888.
Beatty, J. (1994) Secret impacts revealed. Sky and Telescope. February. 26–27’. (Describes atmospheric impacts recently revealed by the U.S. Department of Defense.)
Cohen, D. (1983) Waiting for the Apocalypse. Prometheus Books: New York.
Davies, P. (1994) The Last Three Minutes. Basic Books: New York. (Discusses the Swift-Tuttle comet and Nemesis or the Death Star.)
Davis, M., Hut, P. and Muller, R. (1984) Extinction of species by periodic comet showers. Nature. 308: 715–717.
Hoffman, A. (1985) Patterns of family extinction depend on definition and geological timescale. Nature. 315: 659–662.
Moore, P., Hunt, G., Nicolson, I., and Cattermole, P. (1990) The Atlas of the Solar System. Crescent Books/Crown Publishers: New York.
Rampino, M. and Stothers, R. (1984) Terrestrial mass extinctions, cometary impacts and the Sun’s motion perpendicular to the galactic plane. Nature. 308: 709–712.
Raup, D.M. and Sepkoski, J. (1984) Periodicity of extinctions in the geological past. Proceedings of the National Academy of Science. 81: 801–805.
Schwartz, R.D. and James, P.B. (1984) Periodic mass extinctions and the Sun’s oscillation about the galactic plane. Nature. 308, 712–713.
Whitmire, D. and Jackson, A. (1984) Are periodic mass extinctions driven by a distant solar companion? Nature. 308: 713–715.
Daniel, G. (1980) Megalithic monuments. Scientific American. 243(1): 78–90.
Hawkins, G. (1964) Stonehenge—a neolithic computer. Nature 202: 1258–1261.
Hawkins, G. (1973) Beyond Stonehenge. Harper & Row: New York.
Hoyle, F. (1966) Stonehenge—an eclipse predictor. Nature. 211(5048): 454–458.
Gardner, M. (1992) The great Urantia Book mystery. In A Walk on the Wild Side, Prometheus Books: New York. (Most of my information on Urantia comes from this wonderful book.)
Gardner, M. (1995) Urantia: The Great Cult Mystery. Prometheus Books: New York.
Penrose, R. (1989) The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press: Oxford.
Yates, F. (1964) Giordano Bruno and the Hermetic Tradition. University of Chicago: Chicago.
The following is a list of fractal resources sorted into various categories.
Books
1. Anthony, P. (1992) Fractal Mode. Ace: New York. (Science fiction.)
2. Barnsley, M. (1988) Fractals Everywhere. Academic Press: New York.
3. Batty, M., and Longley, P. (1994) Fractals Cities: A Geometry of Form and Function. Academic Press: California. (Topic: the development and use of fractal geometry for understanding and planning the physical form of cities; shows how fractals enable cities to be simulated through computer graphics.)
4. Briggs, J. (1992) Fractals. Simon and Schuster: New York.
5. Falconer, K. (1990) Fractal Geometry. Wiley: New York.
6. Feder, J. (1988) Fractals. Plenum: New York.
7. Fischer, P., Smith, W. (1985) Chaos, Fractals, and Dynamics. Marcel Dekker, Inc.: New York.
8. Glass, L., Mackey, M. (1988) From Clocks to Chaos: The Rhythms of Life. Princeton University Press: New Jersey.
9. Gleick, J. (1987) Chaos. Viking: New York.
10. Kaye, B. (1989) A Random Walk Through Fractal Dimensions. VCH Publishers: New York.
11. Lauwerier, H. (1990) Fractals. Princeton University Press, Princeton, New Jersey. Keterences
12. Mandelbrot, B. (1984) The Fractal Geometry of Nature. Freeman: New York.
13. Moon, F. (1987) Chaotic Vibrations. Wiley: New York.
14. Peak, D. and Frame, M. (1994) Chaos Under Control: The Art and Science of Complexity. Freeman: New York.
15. Peitgen, H., Richter, P. (1986) The Beauty of Fractals. Springer: Berlin.
16. Peitgen, H., Saupe, D. (1988) The Science of Fractal Images. Springer: Berlin.
17. Pickover, C. (1990) Computers, Pattern, Chaos and Beauty. St. Martin’s Press: New York.
18. Pickover, C. (1991) Computers and the Imagination. St. Martin’s Press: New York.
19. Pickover, C. (1992) Mazes for the Mind: Computers and the Unexpected. St. Martin’s Press: New York.
20. Pickover, C. (1994) Chaos in Wonderland: Visual Adventures in a Fractal World. St. Martin’s Press: New York.
21. Pickover, C. (1995) Keys to Infinity. Wiley: New York.
22. Pickover, C. (1995) The Pattern Book: Fractals, Art and Nature. World Scientific: New Jersey.
23. Pickover, C. (1996) Fractal Horizons: The Future Use of Fractals. St. Martin’s Press: New York.
24. Peterson, I. (1990) Islands of Truth. Freeman: New York.
25. Rietman, E. (1989) Exploring the Geometry of Nature. Windcrest: Pennsylvania.
26. Sprott, C. (1993) Strange Attractors: Creating Patterns in Chaos. M&T Books: New York. (A book by the guru of strange attractors and their graphics.)
27. Schroeder, M. (1984) Number Theory in Science and Communication. Springer: New York. (This book is recommended highly. An interesting book by a fascinating author.)
28. Schroeder, M. (1991) Fractals, Chaos, Power Laws. Freeman: New York.
29. Stewart, I. (1980) Does God Play Dice? Basil Blackwell: New York.
30. Stevens, C. (1989) Fractal Programming in C. M&T Books: New York.
31. Wegner, T., Peterson. M. (1991) Fractal Creations. Waite Group Press: California.
32. Wegner, T, Tyler, B., Peterson, M. and Branderhorst, P. (1992) Fractals for Windows. Waite Group Press: California.
33. West, B. (1990) Fractal Physiology and Chaos in Medicine. World Scientific: New Jersey.
1. AMYGDALA, a newsletter on fractals. Write to AMYGDALA, Box 219, San Cristobal, New Mexico 87564 for more information.
2. ART MATRIX, creator of beautiful postcards and videotapes of exciting mathematical shapes. Write to ART MATRIX, PO Box 880, Ithaca, New York 14851 for more information.
3. Bourbaki Software: FracTools (fractal generation software), A Touch of Chaos (fractal screen-saver for windows), FracShow CD (fractal slide shows on CD-ROM for DOS and Windows). Contact: Bourbaki Inc, PO Box 2867, Boise, ID 83701.
4. Chaos Demonstrations, a program which contains examples of Julia sets and over twenty other types of chaotic systems. This peer-reviewed program won the first annual Computers in Physics contest for innovative educational physics software. It is published by Physics Academic Software and is available from The Academic Software Library, Box 8202, North Carolina State University, Raleigh, NC 27695–8202. Chaos Demonstrations is by J. C. Sprott and G. Rowlands, and it requires an IBM PC or compatible computer.
5. “Chaos and Graphics Section” of Computers and Graphics. (This section is devoted to fractals and chaos in art and science.) Publishing, subscription, and advertising office for Computers and Graphics: Elsevier Science, Inc., 660 White Plains Road, Tarrytown, New York 10591–5153. Email: ESUK.USA@ELSEVIER.COM
6. Fractal Calendar. Address inquiries to J. Loyless, 5185 Ashford Court, Lilburn, Georgia 30247.
7. Fractal Discovery Laboratory. Designed for a science museum or school setting. Entertaining for a 4-year-old, and fascinating for the mathematician. Earl Glynn, Glynn Function Study Center, 10808 West 105th Street, Overland Park, KS 66214–3057.
8. Fractal Postcards. The Mathematical Association, 259 L Leicester, LE2 3BE U.K. Fractal Report, a newsletter on fractals. Published by J. de Rivaz, Reeves Telecommunications Lab. West Towan House, Porthtowan, Cornwall TR4 8AX, U.K.
9. HOP — Fractals in Motion. Software which produces a large variety of novel images and animations. HOP features Fractint-like parameter files, GIF read/write, MAP palette editor, a Screensaver for DOS, Windows, and OS/2, and more. Math coprocessor (386 and above) and SuperVGA required. The program is written by Michael Peters and Randy Scott. $30 shareware. Download locations: CompuServe GRAPHDEV Forum, Lib 4 (HOPZIP.EXE), The WELL ibmpc/graphics (HOPZIP.EXE), rever.nmsu.edu under/pub/hop, slopoke.mlb.semi.harris.com, ftp.uni-heidelberg.deunder/pub/msdos/graphics, spanky.triumf.ca [128.189.128.27] under [pub.fractals.programs.ibmpc], HOP WWW page: http://rever.nmsu.edu/~ras/hop. Subscriptions and information requests for the HOP mailing list should be sent to: hop-request@acca. nmsu.edu. To subscribe to the HOP mailing list, simply send a message with the word “subscribe” in the subject: field. For information, send a message with the word “INFO” in the subject: field.
10. The Great Media Company, PO Box 598, Nicasio, California 94946. This fine company distributes books, videos, prints, and calendars.
11. Recreational and Educational Computing Newsletter. Dr. Michael Ecker, 909 Violet Terrace, Clarks Summit, PA 18411.
12. Strange Attractions. A store devoted to chaos and fractals (fractal art work, cards, shirts, puzzles, and books). For more information, contact: Strange Attractions, 204 Kensington Park Road, London Wll INR England.
13. StarMakers Rising (Fractal posters), 6801 Lakeworth Road, Suite 201, Lakeworth, FL 33467.
14. YLEM — Artists using science and technology. This newsletter is published by an organization of artists who work with video, ionized gases, computers, lasers, holograms, robotics, and other nontraditional media. It also includes artists who use traditional media but who are inspired by images of electromagnetic phenomena, biological self-replication, and fractals. Contact: YLEM, Box 749, Orinda, CA 94563.
(This list is not meant to be comprehensive. It is included for researchers in the field who may not be aware of some of the following unusual applications. Some of the more “offbeat” topics are listed first.)
1. Taylor, C. (1990) Condoms and cosmology: The “fractal” person and sexual risk in Rwanda. Social Science and Medicine. 31(9): 1023–1028.
2. Entsminger, G. (1989) Stochastic fiction—fiction from fractals. Micro Cornucopia. 49: 96.
3. Pickover, C. (1993) Fractal fantasies. BYTE. March, p. 256. (Describes a game played on a fractal playing board.)
4. Lakhtakia, A. (1990) Fractals and The Cat in the Hat. Journal of Recreational Math. 23(3): 161–164.
5. Nottale, L. (1991) The fractal structure of the quantum space-time. In Heck, A. and Perdang, J. Applying Fractals in Astronomy. Springer: New York.
6. Landini, G. (1991) A fractal model for periodontal breakdown in periodontal disease. J. Periodontal Res. 26: 176–179.
7. Cutting, J., and Garvin, J. (1987) Fractal curves and complexity. Perception and Psychophysics. 42: 365–370.
8. Batty, M. and Longley, P. (1987) Urban shapes as fractals. Area. 19: 215–221.
9. Fogg, L. (1989) PostScriptals: Ultimate fractals via postscript. Micro Cornucopia. 49: 16–22. (Discusses the “ultimate” fractal at a resolution of 2540 dots per inch. Also challenges the reader to beat this “world’s highest resolution fractal.”)
10. Dixon, R. (1992) The pentasnow gasket and its fractal dimension. In Fivefold Symmetry. Hargittai, I., ed. World Scientific: New Jersey.
11. Clarke, A. (1989) Ghost from the Grand Banks. Bantam: New York. (A female character becomes insane after exploring the Mandelbrot set.)
12. Pickover, C. (1995) Is the fractal golden curlicue cold? The Visual Computer. 11(6): 309–312.
13. Lakhtakia, A., Messier, R., Vasandara, V., Varadan, V. (1988) Incommensurate numbers, continued fractions, and fractal immittances. Zeitschrift für Naturforschung 43 A: 943–955.
14. Pool, R. (1990) Fractal Fracas. Science. 249: 363–364. (“The math community is in a flap over the question of whether fractals are just pretty pictures or more substantial tools.”)
15. Batty, M., and Longley, P., Fotheringham, A. (1989) Urban growth and form: scaling, fractal geometry and diffusion-limited aggregation. Environment and Planning 21:1447–1472.
16. Hsu, K., Hsu A. (1990) Fractal geometry of music. Proceedings of the National Academy of Science. 87(3):938–941.
1. Aqvist, J., Tapia, O. (1987) Surface fractality as a guide for studying protein-protein interactions. Journal of Molecular Graphics. 5(1): 30–34.
2. Barnsley, M., Sloan, A. (1987) Chaotic compression (a new twist on fractal theory speeds complex image transmission to video rates). Computer Graphics World. November. 107–108.
3. Batty, M. (1985) Fractals—geometry between dimensions. New Scientist. April. 31–40.
4. Boyd, D. (1973) The residual set dimensions of the Apollonian packing. Mathematika 20: 170–74. (Very technical reading)
5. Brooks, R., Matelski, J. P. (1981) The dynamics of 2-generator subgroups of PSL(2, C). In Riemann Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference. Kyra, I. and Maskit, B. (eds.) Princeton University Press: New Jersey. (Note: This 1978 paper contains computer graphics and mathematical descriptions of both Julia and Mandelbrot sets.)
6. Casey, S. (1987) Formulating fractals. Computer Language. 4(4): 28–38.
7. Collins, J.J. and De Luca, C.J. (1993) Open-loop and closed-loop control of posture: A random-walk analysis of center-of-pressure trajectories. Experimental Brain Research. 95: 308–318.
8. Collins, J.J. and De Luca, C.J. (1994) Random walking during quiet standing. Physical Review Letters. 73: 764–767. (The Collins papers deal with fractal-like measures and the postural control system.)
9. Devaney, R., Krych, M. (1984) Dynamics of exp(z). Ergodic Theory & Dynamical Systems. 4: 35–52.
10. Dewdney, A. K. (1985) Computer Recreations. Scientic American. 253: 16–24.
11. Douady, A., Hubbard, J. (1982) Iteration des polynomes quadratiques complexes. Comptes Rendus (Paris) 2941: 123–126.
12. Family, F. (1988) Introduction to droplet growth processes: Simulation theory and experiments. In Random Fluctuations and Pattern Growth: Experiments and Models. Stanley, H., Ostrowsky, N. (eds.) Kluwer: Boston.
13. Fatou, P. (1919/1920) Sur les equations fonctionelles. Bull. Soc. Math. Fr. 47: 161–271.
14. Fournier, A., Fussel, D., Carpenter, L. (1982) Computer rendering of stochastic models. Communications of the ACM 25: 371–378.(How to create natural irregular objects.)
15. Gardner, M. (1978) White and brown music, fractal curves, and 1/f fluctuations. Scientific American. April 16–31.
16. Gardner, M. (1976) In which monster curves force redefinition of the word curve. Scientific American. December 235: 124–133.
17. Gordon, J., Goldman, A., Maps, J. (1986) Superconducting-normal phase boundary of a fractal network in a magnetic field. Phys. Rev. Let. 56: 2280–2283.
18. Grebogi, C, Ott, E., Yorke, J. (1985) Chaos, strange attractors, and fractal basin boundaries in nonlinear dynamics. Science 238: 632–637. (A great overview with definitions of terms used in the chaos literature.)
19. Grebogi, C., Ott, E., Yorke, J. (1985) Attractors on an N-Torus: Quasiperiodicity versus chaos. Physica 15D: 354–373. (Contains some gorgeous diagrams of dynamical systems).
20. Hirsch, M. (1989) Chaos, rigor, and hype. Mathematical Intelligencer. 11(3): 6–9. (Pages 8 and 9 include James Gleick’s response to the article.)
21. Holter, N., Lakhtakia, A., Varadan, V., Vasundara, V., Messier, R. (1986) On a new class of planar fractals: the Pascal-Sierpinski gaskets. J. Phys. A: Math. Gen. 19: 1753–1759.
22. Julia, G. (1918) Memoire sur l’iteration des fonctions rationnelles. J. Math. Pure Appl. 4: 47–245.
23. Kadanoff, L. (1986) Fractals: Where’s the physics? Physics Today. February: 6–7.
24. La Brecque, M. (1985) Fractal Symmetry. Mosaic. 16: 10–23.
25. Lakhtakia, A., Vasundara, V., Messier, R., Varadan, V. (1987) On the symmetries of the Julia sets for the process z → zPc. J. Phys. A: Math. Gen. 20: 3533–3535.
26. Lakhtakia, A., Vasundara, V, Messier, R., Varadan, V (1987) The generalized Koch Curve. J. Phys. A: Math. Gen. 20: 3537–3541.
27. Lakhtakia, A., Vasundara, V, Messier, R., Varadan, V (1986) Self-similarity versus self-affinity: the Sierpiñski gasket revisited. J. Phys. A: Math. Gen. 19: L985-L989.
28. Lakhtakia, A., Vasundara, V, Messier, R., Varadan, V. (1988) Fractal sequences derived from the self-similar extensions of the Sierpinski gasket. J. Phys. A: Math. Gen. 21: 1925–1928.
29. Mandelbrot, B. (1983) On the quadratic mapping z + z2 – μ for complex μ and z: The fractal structure of its M set, and scaling. Physica. 17D: 224–239.
30. Musgrave, K. (1989) The synthesis and rendering of eroded fractal terrains. Computer Graphics (ACM-SIGGRAPH). July 23(3): 41–50.
31. Norton, A. (1982) Generation and display of geometric fractals in 3-D. Computer Graphics (ACM-SIGGRAPH). 16: 61–67.
32. Peterson, I. (1984) Ants in the labyrinth and other fractal excursions. Science News. 125: 42–43.
33. Pickover, C. (1988) Symmetry, beauty and chaos in Chebyshev’s Paradise. The Visual Computer: An International Journal of Computer Graphics. 4:142–147.
34. Pickover, C. (1987) Biomorphs: computer displays of biological forms generated from mathematical feed back loops, Computer Graphics Forum. 5(4): 313–316.
35. Pickover, C. (1995) Synthesizing extraterrestrial terrain, IEEE Computer Graphics and Applications, 15: 18–21.
36. Phipps, T. (1985) Enhanced fractals. Byte. March 21–23.
37. Robinson, A. (1985) Fractal fingers in viscous fluids. Science. 228: 1077–1080.
38. Schroeder, P. (1986) Plotting the Mandelbrot Set. Byte. December 207–211.
39. Schroeder, M. (1982) A simple function. Mathematical Intelligencer. 4: 158–161.
40. Sorenson, P. (1984) Fractals. Byte. Sept. 9: 157–172 (A fascinating introduction to the subject.)
41. Symmetries and Asymmetries. (1985) Mosaic. Volume 16, Number 1, January/February. (An entire issue on the subject of fractals, symmetry, and chaos. Mosaic is published six times a year as a source of information for scientific and educational communities served by the National Science Foundation, Washington, D.C. 20550).
42. Thomsen, D. (1982) A place in the sun for fractals. Science News. 121: 28–32.
43. Thomsen, D. (1980) Making music-fractally. Science News March 117:189–190.
44. Ushiki, S. (1988) Phoenix. IEEE Transactions on Circuits and Systems. July 35(7): 788–789.
45. Voss, R. (1985) Random fractal forgeries. In Fundamental Algorithms in Computer Graphics. R. Earnshaw, ed. Springer-Verlag: Berlin, pp. 805–835.
46. West, S. (1984) The new realism. Science. 84, 5: 31–39.
47. West, B., Goldberger, A. (1987) Physiology in fractal dimensions. American Scientist. 75: 354–283 365. (This article describes the fractal characterization of the lungs’ bronchial tree, the Weierstrass function, the fractal geometry of the heart, and “fractal time.”)
Ascher, M., and Ascher, R. (1981) The Code of the Quipu: A Study in Media, Mathematics and Culture. Ann Arbor, MI: University of Michigan Press.
Ascher, M. (1991) Ethnomathematics: A Multicultural View of Mathematical Ideas. Pacific Grove, CA: Brooks/Cole Publishing Co.
Ascher, M. (1992) Before the conquest. Mathematics Magazine. Oct 65(4): 211–218.
Conklin, W. (1982) The information system of middle horizon quipus. In Ethnoastronomy and Archaeoastronomy in the American Tropics. Aveni, A. and Urton, G., eds. The New York Academy of Sciences: New York, pp. 261–281.
Hawkins, G. (1973) Beyond Stonehenge. Harper and Row: New York.
McLeish, J. (1991) The Story of Numbers. Fawcette: New York.
Zaslavsky, C. (1973) Africa Counts: Number and Pattern and African Culture. Brooklyn, NY: Lawrence Hill Press.
Barret, A., Mackay, A. (1987) Spatial Structure and the Microcomputer. Macmillan: New York.
Boles, M., Newman, R. (1990) Universal Patterns. Pythagorean Press: Bradford, Massachusetts.
Gardner, M. (1994) The cult of the Golden Ratio. Skeptical Inquirer. Spring 18(3): 243–247.
MacEoin, D. (1992) The Sources for Early Babi Doctrine and History. E. J. Brill: New York.
Markowsky, G. (1992) Misconceptions about the Golden Ratio. College Mathematics Journal. January. 23: 2–19.
Schroeder, M. (1986) Number Theory in Science and Communication. Springer: Berlin. (A goldmine of valuable information.)
Stewart, I. (1995) Daisy, daisy, give me your answer do. Scientific American. January. 272(1): 96–99.
Wells, D. (1987) The Penguin Dictionary of Curious and Interesting Numbers. Penguin: New York.
Hawkins, G. (1973) Beyond Stonehenge. Harper and Row: New York.
Pickover, C. (1992) Mazes for the Mind: Computers and the Unexpected. St. Martin’s Press: New York.
Pickover, C. (1993) Lava lamps in the 21st century. Visual Computer. Dec 10(3): 173–177.
Vichniac, G. (1986) Cellular automata models of disorder and organization. In Disordered Systems and Biological Organization. Bienenstock., E., Soulie, F., and Weisbuch, G., eds. Springer: New York.
Hawkins, G. (1973) Beyond Stonehenge. Harper and Row: New York.
Ellis, K. (1978) Number Power. St. Martin’s Press: New York.
Flam, F. (1994) Theorists make a bid to eliminate black holes. Science. 266(5193): 1945.
Horgan, J. (1995) Bashing black holes: theorists twist relativity to eradicate an astronomical anomaly. Scientific American. July 273(1): 16.
Lemonick, M. and Nash, M. (1995) Unraveling universe. Time, March 6, 145(9): 77–84.
Pynchon, T. (1973) Gravity’s Rainbow. Bantam: New York.
Cherrington, E. (1969) Exploring the Moon through Binoculars and Small Telescopes. Dover: New York.
Cohen, D. (1983) Waiting for the Apocalypse. Prometheus Books: New York.
Dyson, F. (1979) Time without end: physics and biology in an open universe. Reviews of Modern Physics. 51(3).
Gallant, R. (1994) Journey to Tunguska. Sky & Telescope. June. 87: 38–43.
Marsden, B. (1993) Comet Swift-Tuttle: Does it threaten Earth? Sky & Telescope. January. 85:16–19.
Morrison, D. and Chapman, C. (1990) Target Earth: It will happen. Sky & Telescope. March. 79: 261–262.
Weissman, R. (1990) Are periodic bombardments real? Sky & Telescope. March. 79: 266–270.
Dauben, J. (1979) Georg Cantor. Harvard University Press: Cambridge (Chapter 6. pp. 246, 295).
Gunn, J. (1988) The New Encyclopedia of Science Fiction. Viking: New York.
Hick, J. (1978) Evil and the God of Love. Harper and Row: New York.
Pickover, C. and Webb, D. (1995) To the Valley of the Seahorses. In Keys to Infinity (Pickover). Wiley: New York (Chapter 6, pp. 47–58).
Tipler, F (1994) The Physics of Immortality. Doubleday: New York.
Wang, Hao (1987) Reflections on Kurt Goedel. MIT Press: Mass. (p. 195)