École Française d’Extrême-Orient (EFEO), 118–19
École Pratique des Hautes Études, 87–8
Egypt, 63, 66, 103
Egyptian numbers, 25, 209, 213
Etruscan numbers, 82–3
Euclid (mathematician), 39, 56, 69
Evans, Arthur (archaeologist), 183–4, 186
Fermat’s Last Theorem, 83
Fibonacci sequence, 13, 19
Ford, Harrison, 179
fractions, 25, 147
French Colonial architecture, 115
French colonialism, 87, 114, 119, 133, 216
Fu Nan, Chinese Kingdom of, 91–3, 205
Galileo, 36, 194
Ganesha (Hindu god), 103
Garuda (mythical bird), 102, 139
Ghosh, B. K., 63
golden ratio, 12–13
Gothama, Siddhartha, 141–2. See also Buddha, Buddhism.
Greece and Greek culture
Acropolis, 10, 12, 30
alphabet, 13–14, 64, 65–6, 82
Diophantus (mathematician), 23
elements of the universe, 36
Euclid (mathematician), 39, 55–6, 69
and India, 65, 88–9
logic, 54
Parthenon, 12–13
Piraeus, 10–12, 14–15
Pythagoras (mathematician), 39, 54, 66
Tower of the Winds, 30
Grothendieck, Alexander (French mathematician), 58–60, 136, 158–62
Grothendieck’s prime, 161
Recoltes et Semailles (Reapings and Sowings), 158–9
Gwalior zero, 78–81, 83–4, 90, 93, 95, 107, 210
Hab Touch (Mr. Hab), 129, 165–7, 171–3, 191, 198, 201–8, 210, 216, 218–9
Hanoi, Vietnam, 87–8, 118–19, 121
Hardy, G. H., 70–1
Harihara (Hindu god), 36
Hayashi, Takao (mathematician), 42, 45–7
Hebrew alphabet, 14, 66
Heinzelin de Braucourt, Jean de (Belgian explorer), 20
Herzl, Theodor, 7
Hinduism, 34–6, 90, 144, 148–50, 204, 206, 214
Brahma, 35–6, 101–2, 129
Chandra, 103
Devi, 102
Durga, 35, 45
Ganesha, 102
Harihara, 36
Lakshmi, 35, 100–1
Parvati, 35, 45
Shiva, 35–6, 45, 102, 129, 138–9, 149
Surya, 103
Trimurti, 36
Ifrah, Georges (researcher), 32–3, 84
India and Indian culture
Ashoka, King, 64, 66–7, 223
Delhi, 38, 42, 72
Gwalior zero, 78–81, 83–4, 90, 93, 95, 107, 210
Khajuraho (group of temples), 42–5, 48–53, 72–3, 92
Khandela inscription, 72–3, 80
Madhya Pradesh, 42–3, 78
National Museum (New Delhi), 65
Nana Ghat cave inscriptions, 64–7, 223
New Delhi, 39, 43, 65
Rajasthan, 72–4
See also Buddhism; Hinduism
Indonesia, 90
Palembang, 96, 106–7, 210
Sumatra, 96–7
infinity, 35–7, 56–8, 78, 100–1, 103–4, 142, 144–50, 161, 212–13
Ishango bone, 20–2
Island of Meroe, 103
Israel, 1, 7, 18, 109
Jainism, 34, 36–7, 42, 44–7, 51, 78, 103–4, 145, 148, 150
Jantar Mantar (Jaipur observatory), 74
Jayavarman II, 93
Jayavarman VII, 103, 115, 207
Jean-Marc (Hindu from Chennai), 138–9, 141–2, 144, 148–50
K-127 (inscription artifact)
605 (Khmer numerals), 95, 107, 175, 177, 209–10
discovery of, 94
fear of removal to Italy, 188–92
loss of, 99, 107–8
museum display text, 208–11
rediscovery of, 174–87
search for, 116–17, 120, 126–7, 129, 134–5, 150–1, 163, 166–7
transfer to Cambodian National Museum, 218–19
See also Cœdès, Georges (archaeologist)
Kakutani, Shizuo (mathematician), 196–7
Karpinski, Louis C., 75–7
Kaye, G. R. (scholar), 77, 79, 88–9, 93, 96–7, 106
Keay, John, 63
Khajuraho (group of temples), 42–5, 48–53, 72–3, 92
Khandela inscription, 72–3, 80
Khmer empire, 92–3, 100–1, 103, 107
Cœdès on number system, 215
Old Khmer (language), 22, 87, 94–5, 114, 205, 208–9
zero, 150, 156, 175, 218, 219
Khmer Rouge, 98–9, 107–8, 125, 129, 150, 163, 166, 173, 176, 200, 202–3, 210, 219, 222
Killing Fields, 99
koppa (archaic Greek letter), 14
Kristof, Nicholas, 165, 197
Laci (Aczel’s childhood tutor), 2–16, 18–20, 30, 34, 52, 83–4, 110–12, 221–3
Lakshmi (Hindu goddess), 35, 100–1
Laos, 87, 93, 118, 127, 131–3, 135, 149–51, 197
Leclère, Adhémard (archaeologist), 94
Leonardo of Pisa (Fibonacci): Liber Abaci, 26. See also Fibonacci sequence
Linton, Fred, 56–7, 60–1, 105, 136
logic, 36–7, 39–41, 51–62, 71, 105–6, 109, 136–42, 148
Long Count (Mayan calendar), 31
Luang Prabang, Laos, 127, 131–3, 149–51
Kiridara Hotel, 133
M.V. Yaffo (Aczel’s father’s cargo ship), 18–19
Madhya Pradesh (Indian state), 42–3, 78
magic square, 45, 47–51
Mahabharata (Indian epic), 205
Majorana, Ettore (theoretical physicist), 157–8
Marcel, Pierre, 120
Maxwell, James Clerk, 24
Maya civilization
calendars, 31–2
number system, 31–4, 212, 215
zero, 31, 79–80, 203
McLeish, John, 63
Mersenne, Marin (mathematician), 194–6
Meskin, Jacob, 152–5, 162
Mexico City, 29–31
Miró, Joan, 223
Mohenjo Daro (first known city in India), 63
Monaco, 1–2, 4
Mouhot, Henri, 92, 207
Mula-madhyamaka-karikas (Fundamental middleway verses), 154
Naga (seven-headed cobra), 103, 129
Nagarjuna (philosopher), 39–40, 55, 57, 60–1, 105–6, 136–7, 139–40, 152–5
Nana Ghat cave inscriptions, 64–7, 223
Nandin (bull), 102, 129
Narayana. See Vishnu (also Narayana)
National Museum of Anthropology (Mexico City), 29–31
negative numbers, 23–4, 76, 106, 209
Neugebauer, Otto (mathematician), 25
Nhat Hanh, Thich, 106
Nicholas Bourbaki (pseudonymous mathematical association), 59
Nine Chapters on the Mathematical Art, 23
number systems
base–5, 215
base–8, 216
base–10, 25, 33, 65, 208, 210, 215
base–18, 31–2, 215
base–20, 22, 25, 31–2, 215
base–60, 24–5, 65, 75, 209
numbers
defined, 10
negative, 23–4, 76, 106, 209
prime, 6, 19, 22, 56, 160–1, 194–7, 201
very large, 6, 37, 78, 104, 144–5, 148
See also number systems
numerals
Arabic, 8–9, 26, 47–50, 95
Hindi, 47–51, 75
letters as, 13–15, 208, 223
Khandela inscription, 72–3, 80
magic square, 47–50
Mayan, 31
Nana Ghat cave inscriptions, 64–7, 223
Roman, 15–16, 23
See also number systems; K-127 (inscription artifact); zero
Oppenheimer, Robert, 139
Palembang (Indonesian province), 96, 106–7, 210
Parsvanatha temple (Khajuraho), 47–8
Parvati (Hindu goddess), 35, 45
Pellegrino, Lorella (archaeologist), 180–91, 193, 219
Penrose, Roger: The Road to Reality, 150
Phnom Penh (Cambodian capital), 94, 118, 120, 122, 165–6, 191–2, 194, 197
Bangkok contrasted with, 199–200
Cambodian National Museum, 102, 128–9, 165, 203–4, 218–19
FFC café and bar, 200
InterContinental Hotel, 122, 199
Wat Langka (Bangkok), 122–4
Wat Pho (Bangkok), 123, 164
Phoenicia, 14, 39
Phoenician alphabet, 14, 66
phonetic alphabet, 214–15
Piraeus (port of Athens), 10–12, 14–15
Pol Pot, 98, 108
pre-counting, 21–2
prime numbers, 6, 19, 22, 56, 160–1, 194–7, 201
Pythagoras (mathematician), 38–9, 54, 66
Raiders of the Lost Ark (film), 179–80
Rajasthan (Indian state), 72–4
Raju, C. K., 38–42, 61, 69
Ramanujan, Srinivasa (Indian mathematician), 70–1
Ramayana (Indian epic), 205
Rava, Antonio (conservator), 184–6
Rig-Veda (Sanskrit text), 62–3. See also Vedas
Roemer, Andres, 28–9
Rotanak Yang, 127–8, 163–6, 171
Russell, Bertrand (mathematician), 136, 147
Sambor on Mekong, Cambodia, 116, 120–1, 124, 206
art style, 94, 116
discovery of K-127, 94–5, 107, 166, 176, 208
Trapang Prei (temple), 94–5, 107, 208
Sambor Prei Kuk, 94, 102, 129, 206
sampi (archaic Greek letter), 14
Sanskrit, 57, 62, 66, 78, 87, 90, 114, 154, 204–5, 216
Sen, Suraj, 78
Shakti (female goddess), 35, 45
Shimura, Goro, 83
Shiva (Hindu god), 35–6, 45, 102, 129, 138–9, 149
Short Count (Mayan calendar), 31–2
Shunya (zero, Sanskrit), 40–1, 69
Shunyata (Buddhist void), 40–1, 78, 104, 105–6, 127, 135, 137, 142, 148, 152, 154–5, 161–2
Siem Reap, Cambodia, 91, 166, 168–9, 182, 193
Angkor Conservation, 163, 165–8, 172, 181, 210, 217
Siem Reap Museum, 116–17, 166, 171
sifr (zero, Arabic), 26
Sihanouk, Norodom, 200
Smith, David Eugene, 45, 64
Soviet Union, 6–7
SS Theodor Herzl (Aczel’s father’s cruise ship), 1–2, 7, 10, 15
Sumatra, 96–7
sun god, 103
Surya (Hindu god), 103
Suvarnabhumi International Airport (Bangkok), 113, 194
sva-bhava (unchanging nature), 154
Taormina, Francesca (archaeologist), 180–3
tetralemma (four corners/possibilities, also catuskoti), 57, 60–1, 105–6, 136, 139–42, 148, 152
Thailand, 9, 87, 93
Bangkok, 199–200
Galerie Mouhot, 115–16
Jim Thompson’s contributions to, 156–7
language, 93
Suvarnabhumi International Airport, 113, 194
Thai National Library, 114
Thompson, Jim, 156–8
Tonatiuh (Aztec sun god), 29–30
Tonle Sap River, 123, 200
topos (space), 59–61, 105, 160–1
Touch, Hab. See Hab Touch (Mr. Hab)
transistor radio, 14–15, 110
Trapang Prei (temple), 94–5, 107, 208. See also K-127 (inscription artifact)
Trimurti (Hindu triad), 36
tuk-tuk (auto rickshaw), 42
University of Alaska in Juneau, 28
University of California at Berkeley, 18, 20, 28
Varanasi (holy city), 42, 114, 127, 134, 168–72, 200–1
Vedas (Sanskrit texts), 48, 62–3, 214
very large numbers, 6, 37, 78, 104, 144–5, 148
vigesimal (base 20), 31–2. See also number systems: base–20
Vishnu (also Narayana), 35–6, 78, 91, 100–2, 115, 129, 139, 214
void, Buddhist (Shunyata), 40–1, 78, 104, 105–6, 127, 135, 137–8, 142, 148, 152–3, 154–5, 161–2, 223
Wats (temples)
Angkor Wat, 91–3, 123, 129, 139, 165, 169, 174, 176, 205
Wat Langka (Bangkok), 122–4
Wat Pho (Bangkok), 123, 164
Wat Xieng Thong (Luang Phrabang), 135
Weierstrass, Karl (mathematician), 146
Western Ghats (mountain range), 64
Yama (ruler of the departed), 102
Yasovarman I (Angkorian king), 101
Zegarelli, Mark, 54–5
zephirum (zero), 26
zero
Gwalior zero, 78–81, 83–4, 90, 93, 95, 107, 210
Khmer, 150, 156, 175, 218, 219
Mayan, 31, 79–80, 203
as number, 212–17
place-holding, 27, 34, 75, 78, 97, 152–3, 208–9, 212
Shunya (Sanskrit), 40–1, 69
sifr (Arabic), 26
See also K-127 (inscription artifact)
Zhou Daguan (also Chou Ta-kuan), 101–2
Zim Lines, 7, 18, 110, 223
Acknowledgments
I am extremely grateful to the Alfred P. Sloan Foundation in New York and to Doron Weber, the Foundation’s director of the Program for the Public Understanding of Science and Technology, as well as to members of the Foundation’s staff, for supporting me in writing this book. I can easily say that without the Sloan Foundation’s faith in me, and the research grant it generously provided, this book would not have been written, and the precious stone artifact known as K-127, which bears the earliest zero in our number system, would not have been rediscovered and brought to the attention of the world of science.
Others have helped me on this quest as well. I thank His Excellency Hab Touch, director general of the Cambodian Ministry of Culture and Fine Arts, for his invaluable assistance to me in rediscovering the stele he has since dubbed “Khmer Zero.” I thank Chamroeun Chhan, Rotanak Yang, Ty Sokheng, Sathal Khun, Darryl Collins, Takao Hayashi, C. K. Raju, Fred Linton, Jacob Meskin, Marina Ville, W. A. Casselman, Eric Dieu, and especially Andy Brouwer in Phnom Penh for their help.
I am grateful to my agent, Albert Zuckerman of Writers House in New York, for his enthusiasm for this project and his support of this book’s publication. Many thanks to my editor at Palgrave Macmillan, Karen Wolny, for believing in this book and for her thoughtful editing, comments, and suggestions, which greatly improved the manuscript. Warm thanks also to Lauren LoPinto for her editing of the book, to Carol McGillivray for her superb editing and insightful comments, to production manager Alan Bradshaw for handling the subtle complexities of producing it, and to copy editor Bill Warhop for his superb editing. Thanks also to designer Rachel Ake, to art director David Baldeosingh Rotstein, and to typesetter Letra Libre for their work in turning the manuscript into a complete bo
ok.
Finally, I am extremely grateful to my wife, Debra, for all her suggestions and help, for joining me on parts of the big adventure of searching for the first zero, and for taking some of the photographs in this book.
About the Author
Amir D. Aczel is the author of twenty books, including The Riddle of the Compass, The Mystery of the Aleph, and the international bestseller Fermat’s Last Theorem. An internationally known writer of mathematics and science, he is a Fellow of the John Simon Guggenheim Memorial Foundation, and his research for this book was supported by a grant from the Alfred P. Sloan Foundation.
FINDING ZERO. Copyright © Amir D. Aczel, 2015. All rights reserved. For information, address Palgrave Macmillan Trade, 175 Fifth Avenue, New York, N.Y. 10010.
us.macmillan.com
The Library of Congress has cataloged the print edition as follows:
Aczel, Amir D., author.
Finding zero : a mathematician’s odyssey to uncover the origins of numbers / Amir D. Aczel.
pages cm
Includes bibliographical references (pages ).
ISBN 978-1-137-27984-2 (hardback)
1. Numerals—History. 2. Zero (The number)—History. I. Title.
QA141.2.A29 2015
513.5—dc23
2014024462
e-ISBN 978-1-4668-7910-2
First Edition: January 2015
eBooks may be purchased for business or promotional use. For information on bulk purchases, please contact Macmillan Corporate and Premium Sales Department by writing to [email protected].
Finding Zero Page 19