The suffix ‘n’ indicates a note. Italic page references refer to Figures.
A
abstract algebra 101, 103, 276, 279–80
acceleration, defined 39–40
aerodynamics 86, 168, 170, 172
air travel 167
aircraft design 172
(Taqi) al-Din Muhammad ibn Ma’ruf al-Shami al-Asadi 199
al-Katibi, Najm al-Din al-Qazwini 57
Alexander, James Waddell 101, 103–4
algebra
abstract 101, 103, 276, 279–80
notation of 78
relation to geometry 9
Alhazen 57
‘alternate worlds’ 258
American Psychological Association 125
analysis 51, 82 see also calculus
analysis of variance method 125
angular momentum 49
animal population dynamics 286–7, 291–2, 330n
Arago, François 233
arbitrage pricing theory 304, 306
Aristarchus of Samos 56
Aristotle 46–7
arithmetic, rules of 77
Arnold, Vladimir 289–90
Arp, Halton 241
arrow of time 208, 211, 213, 318
Aryabhata 56
astronomy
chaotic dynamics in 290–1
geocentric universe 8
interpretations of the night sky 55–6
lasers in 263
tests of relativity 233–5
use of logarithms 29
atomic bomb 229
atomic theory. see molecules
‘average man’ 116
B
Babylonian mathematics 6–7, 80
Bachelier, Louis 302–4
banking system
acceptance of derivatives 297
modularity and 314
Bell, Alexander Graham 183
‘bell curve’ see normal distribution
The Bell Curve (book) 123–5, 127
Berkeley, George, Bishop of Cloyne 44–6, 50–1
Bernoulli, Jacob 112–14
Bernoulli, John 137–9, 141–3, 145, 154
Bessel functions 146
binary notation 271–3
binomial coefficients 113–14
binomial distribution 120
bioinformatics 281
black body radiation 248
black holes 235–6
Black–Scholes Equation 295, 304–6, 314, 318, 320
critique of 310–11
‘black swan events’ 303
blip-like Fourier analysis 161–2
blood flow 174–5
Boltzmann, Ludwig 206–9, 247–9, 281
Bolzano, Bernard 51
Boulliau, Ismaël (Bullialdus) 59, 61
boundary conditions 212, 214, 318
Boyle, Robert 200
Boyle’s law 205–6
braids 103
Branca, Giovanni 199
Brownian motion (Robert Brown) 208
as financial system model 301, 303, 304, 307
bubonic plague 37–8
C
calculation, Napier on 24–5
calculus
history 38–42
partial derivatives 139–40
Cantor, Georg 156
Cardano, Girolamo 75, 77–80, 109–10, 323n–324n
Carnot cycle (Nicolas Léonard Sadi Carnot) 202–4
Cartesian coordinates (René Descartes) 13
cellular automata 320
central limit theorem 119–20, 324n
CFD (computational fluid dynamics) 170, 173
breadmaking analogy 289–90
deterministic chaos 283, 285, 291, 293
population dynamics and 286–7, 291–2, 330n
sensitivity to initial conditions 290
Charles, Jacques Alexandre César 200
circles
aerofoils from 86
equations for 14
knots as 100
lines of force as 85
Clapeyron, Émile 201
clustering and statistics 127
coding theory 276–9
coin tosses 110, 112–16, 272–4, 288
collateralised debt obligations 309, 312
colouring, knot diagrams 102
comets
Giacobini–Zinner 69
Oterma 66–7
commodity prices 298, 304, 310–12
communication channels
capacity 275
noise 269
complex equations 86
complex functions 85–6
definition of 87
logarithms of 29
see also imaginary numbers
complexity science and economic models 310
compressible fluid, air as 168
compression algorithms 157–60
computational fluid dynamics (CFD) 170, 173
computers, market growth 271
confidence, statistical 122
conic sections 57
conservation of energy
first law of thermodynamics and 200, 201–2
kinetic theory and 205
special relativity and 225
conservation of momentum 171, 225
continuity in topology 92, 96–7
continuous functions 46, 156, 274
continuum mathematics 310, 320
continuum mechanics 174
continuum models 171
convergence issues 155–6
Copenhagen interpretation 252–62, 328n
Copernicus, Nicolaus 57–8
‘cosine rule’ 11
cosines 10, 83–4, 142–3, 152–4
cosmological constant 236, 237
cosmological microwave background 236, 241
credit default swaps 308–9, 312
Crick, Francis 105
crown and anchor game 111
cubes, and topology 91
cubic equations 79–80, 323n–324n
curl, Maxwell’s equations 187–8, 326n
curved surface geometry 15–19, 231–2
D
d’Alembert, Jean Le Rond 139–42, 145
dark energy 237, 238, 239, 240, 242
dark matter 237–8, 239, 240, 242
data compression
digital photography 157–60
fingerprints 162
medical imaging 163
Daubechies, Ingrid 161, 162
Davy, Humphry 182
de Broglie, Louis-Victor 249
decibel unit 34
Dellnitz, Michael 72
derivatives (calculus) 43–5, 47, 50–2
partial derivatives 139–40, 170
derivatives (financial instruments) 297–8, 308–9
determinism
distinguished from predictability 290–1, 293
distinguished from randomness 288
deterministic chaos 283, 285, 291, 293
complex functions and 85–6
entropy 204
inevitability of chaos and 286–7, 289
Schrödinger’s wave equation 250
see also partial differential equations
differentiation, introduced 43
digital photography 157–60
dimensions
heat equation 152
multidimensional geometry 276, 278–9
Navier-Stokes equation 171
Solar System description 66, 96
topology 100
vectors 186
wave equations 144–5
Diophantus 78
discontinuous functions 143, 154, 155
discrete cosine transform 159
disease clusters 128
divergence (Maxwell’s equations) 187
DNA
encoded information 281–2, 319
Fourier transforms 157
topology and 105–6
dodecahedra 91
drums 144–6
Earth interior of 147
Moon effects on 291
shape of 8–9
views of 167–8
earthquakes 30, 144–7, 157, 221
economics, classical 310
ecosystems
modelling, finance analogy 312–14
Edison, Thomas 183
efficiency, in error detection 270, 279
eggs
assembling 214
unboiling 211–12
Einstein, Albert 219–20
and atomic weapons 229–30
Brownian motion 208
field equations 231, 235, 240–1
photoelectric effect 249
special relativity 224–5
electromagnetic field 185–6, 221
electromagnetic induction 186
electromagnetic spectrum 189, 192–3
electromagnetism
impact on everyday life 181
magnetohydrodynamics and 318
Hohmann transfer ellipses 65, 69
energy
conservation of 48, 68, 200–2, 205, 225
dark energy 237
defined 48–9
kinetic energy 48–9, 68, 202, 203, 205, 229
renewable resources 209–10
energy landscapes 68–72
engineers’ use of logarithms 29, 30
Enron Corporation 308
entropy 203–4, 206, 209, 211, 213–14
as ‘missing information’ 281
enzymes 106
equals sign vii
equations
alternatives to 319–20
describing lines on planes 13–14
E=mc2 as archetype 219, 227–9, 327n–328n
‘most beautiful’ 84
Pythagoras’s Theorem 5
quadratic 78–9
symmetry of 211
types of vi–vii, 318
wider influence of 317–18
see also differential equations
Eratosthenes Batavus 12
Eratosthenes (of Cyrene) 11
error-detection and -correction codes 267–8, 270, 276, 278, 280
errors
channel capacity and 275
normal distribution 115, 116, 119
Euclid of Alexandria
on conic sections 57
fifth axiom 14–15
proof of Pythagoras’s Theorem 4
eugenics movement 124
Euler, Leonhard 69, 78, 84, 160
formula for polyhedra 89, 92–4, 97
Euler characteristic 95, 97, 101
Everett, Hugh, Jr. 258–60
evolution of sensory perception 33–4
evolutionary computing 319
exchange traded funds (ETFs) 311–12
‘expectation’ and probability 111
exponential functions 30
complex numbers 83–4
F
faces, regular solids 58
Faraday, Michael vii, 181–6
Fechner, Gustav 33
Feynman diagrams 105
fields, in electromagnetism 85–6, 96, 182, 185–7
‘financial ecosystems’ 313–14
financial instruments 297, 299, 306, 308
see also derivatives; futures and options
financial transactions tax 313
fingerprints 162
Fisher, Ronald Aylmer 122, 128
five-body dynamics 71
fluid dynamics
before Euler 169
magnetohydrodynamics 318
fluids
application of Newton’s laws of motion 170
mathematics of magnetism and 185
Food and Drugs Administration (US) 174
forces
see also gravity
Fourier, Joseph 151
Fourier analysis 155
Fourier transform 149
blips 160
data compression 159
error distributions and 119
Schrödinger’s equation 251
free will 116
friction 47, 170, 201–2, 212–13
Fukushima Dai-ichi power plant 31
functions
complex functions 85–6
exponential functions 30, 83–4, 152, 154
futures and options 299, 305, 308, 330n
fuzzy boundary theory 69
G
galactic rotation curves 237–8, 241
Galileo Galilei 39–42, 47–8, 224, 230
Galois fields (Évariste Galois) 280
Galton, Francis 119–21
games of chance 109–11
Gauss, Carl Friedrich
electromagnetism 183
imaginary numbers 81–2
least squares method 118
magnetism 96
and non-Euclidean space 16–17, 18–19
notation 78
‘Gaussian curvature’ 18
general relativity 220, 231, 240
geometry
coordinate geometry 13
curved surface geometry 12, 15–19, 231–2
non-Euclidean 15–16
relation to algebra 9
global warming 175–7
GPS (Global Positioning System) 13, 219, 242–3
gravitational constant 59, 323n
gravity
Galileo’s investigations 39–41
general relativity and 220, 231–2
Newton’s law of 53, 55, 60–1, 65, 230
three-body problem 63
greenhouse gases 176
Greenspan, Alan 314
groups (algebra) 101
H
Haldane, Andrew 312–14
half-lives, radioisotopes 31–2
Hamilton, William Rowan 86–7
Hamiltonian operators 250, 251
Hamming, Richard 276, 277, 279
Hardy, Godfrey Harold 184
harmonics 143
harmonies 134–5, 137, 139, 143–4
hearing, human 33–4
heat
and temperature 203–4
time-reversal scenario 212
heat equation (Fourier) 151, 156
heat flow 198
heights
of children 121
of structures 10–11
Heisenberg, Werner 250
heliocentric theories 56–8
Henry, Joseph 182
herd instinct 311
heredity 120–1
Hertz, Heinrich 190
Hitler, Adolph 259–61
Hohmann transfer ellipses 65, 69
hole-through-a-hole-in-a-hole 95–6, 97
HOMFLY polynomial 104
Hooke, Robert 59–61, 137, 322n
Hubble, Edwin 236
Huffman code 159
Huisman, Jef 291–2
hyperbolic space 16–17
hypercubes 276–7
hypotenuse, defined 5
hypothesis testing 122–3
I
i (imaginary unit). see square root of minus one
icosahedra 91
images. see data compression; photography
see also complex numbers
India
astronomy 56
Great Trigonometric Survey 12–13
infinite series 83–4, 142–3, 145, 153–4, 161
inflation 237, 238, 239–40, 242
information
as a measurable quantity 268, 269–70
as negative entropy 213
redundant 158
information content, H 265, 273–4, 329n
instantaneous rates of change 41, 43
integral calculus
entropy and 204
introduced 43
magnetism and 96
integration theory 156
intelligence and IQ testing 124–7
International Monetary Fund 307, 311
Interplanetary Superhighway 72, 105
intervals 134–6
invariants, combinatorial 96
invariants, topological 89, 93–4, 96–7, 101, 103–4
inverse square laws
expected for electromagnetism 185
IQ testing 124–7
irrational numbers 155
ISEE (International Sun-Earth Explorer)-3 69
J
Japanese earthquake, 2011 30–1
Jones polynomial (Vaughan Jones) 103–4, 105, 106
Joukowski transformation 86
JPEG (Joint Photographic Experts Group) 158
Jupiter
K
Kelvin, William Thomson, Lord 247
Kepler, Johannes 39, 41, 58–9, 63
kinetic energy 48–9, 68, 202–3, 205, 229
kinetic theory of gases 200, 202, 204–6
Klein bottle (Felix Klein) 98
knots 100–5
Krönig, August 205
L
Lagrange, Joseph-Louis 69
Laplace, Pierre Simon de 29, 119–20
Large Hadron Collider 219, 242
law of gravity (Newton) 53, 60–1, 65, 230
laws of conservation. see conservation laws
laws of motion (Newton) 45, 46–7, 60
application to fluids 170
MOND and 240
laws of planetary motion (Kepler) 39, 58–9, 61
laws of thermodynamics 200, 210–11
second law 197, 204, 209–10, 213, 326n–327n
zeroth law 203
Le Verrier, Urbain 233–4
least squares method 117–18, 125
Lebesgue, Henri 156
Legendre, Adrien-Marie 117–18
Leibniz, Gottfried Wilhelm 39, 42–5, 50, 81
Lemaître, Georges 236
life, and the second law of thermodynamics 210
light
as an electromagnetic wave 189, 220
particulate behaviour 227–8, 249
wave-particle duality 249–50
see also speed of light
light bulbs 247
light cones 227
linear equations 141
linking numbers (electromagnetism) 96
logarithmic multiplication equation 21, 29
logarithms 24–9
of complex functions 86
human perception and 33–4
natural 30
in nature 33–4
slide rules and 30
of trigonometric functions 29, 31
Long Term Capital Management (LTCM) 306–7
Lorenz, Hendrik 223–4
Lorenz-FitzGerald contraction 223–6
M
MacKay, Robert 241
magnetohydrodynamics 318
manned flight 167
many-worlds interpretation 258–61
maps
disc-shaped Earth 9
Marconi, Guglielmo 191
‘mark to market accounting’ 308, 309
mathematical models
climate forecasting 176
gas laws 200
vibrating strings 137–8
Maxwell, James Clerk 181, 183, 206, 208
Einstein and 224
Maxwell-Boltzmann distribution 206, 247
May, Robert (Lord May of Oxford) 286, 288, 291, 312–14
mean, defined 115
mean free paths 205
measurement, quantum mechanics 252, 256
medical imaging 162–3
medical research 174
medical use of lasers 263
Meitner, Lise 230
memory 213–14
Mercury 233
‘Midas formula’ 305
Milgrom, Mordehai 240
Millikan, Robert 248
mines 199–200
modes, vibrational 139, 145, 147, 248
molecules, in thermodynamics 200, 205–6
MOND (modified Newtonian dynamics) 240
Moon
effects on Earth 291
origins 49, 323n Morley, Edward 222–3, 224
Morse, Samuel 183
mortgages betting on defaults 309
self-certified 301
motion, Newton’s laws of 45, 46–7, 60, 170, 240
multidimensional geometry 276, 278–9
multiplication
equation for logarithmic 21
relative complexity of 25–6, 27–8
music 133–6
N
Napier, John 24–9
natural logarithms 30
nature
as essentially mathematical 38
instances of chaos in 285
logarithms in 33–4
Navier–Stokes Equation 165, 169, 171, 175, 177
negative entropy 213
Neugebauer, Otto 7
Newcomen, Thomas 199
Newton, Isaac
physics of 220
night sky, interpretation 55–6
Nobel Prize in Economics 305, 330n nodes 143
noise measurement 34
non-Euclidean geometries 15–16
normal distribution (bell curve)
definition 115
effect of combining 120
financial models and 301–2
formula for 107
kinetic theory of gases 206
null hypotheses and 123
normal modes 145, 147, 248, 251, 325n
notation
algebraic 78
origin of number symbols 23
origin of the equals sign vii
null hypotheses 122–3
number symbols, origin 23
O
observational limitations, quantum mechanics 252, 256
octahedra 91
odds, statistical 110–11
oil price 306
oil prospecting 148
operator algebras 103
options, futures and 299, 305, 308, 330n
orbital resonances, Jupiter 67, 71
Ordnance Survey of Great Britain 12
Ørsted, Hans Christian 182
orthogonality 160
P
parallel universes 258–61
partial derivatives 139–40, 170
partial differential equations 141, 151, 169, 172, 320
philosophical questions
action at a distance 221
heliocentrism 57
mathematics and reality vii–viii, 82, 127
quantum mechanics 252, 254, 318
status of relativity 224
photoelectric effect, Einstein and 249
photography 157–60
pianos 136
picture frame topology 93–4
Planck, Max (and Planck’s constant) 247–9, 250
planetary motion laws 39, 58–9, 61
playing cards illustrations 147, 206–7, 211, 214
Plimpton, George Arthur 7
chaos theory and 63–4, 105, 285, 289
polyhedra
see also regular solids
polynomials
calculus application to 44, 323n
HOMFLY polynomial 104
Jones polynomial 103–4, 105, 106
Reed-Solomon encoding 280
population dynamics and chaos 286–7, 291–2, 330n
positive feedback 298
power, ‘new axis of’ 307
pressure, in ideal gases 205
Principia (Newton)
astronomical significance 55, 59
priority disputes 42, 44, 60–1, 118
probability and wave functions 253
probability theory
crown and anchor game 111
drug testing and 127
financial models and 301
message encoding and 272–4
origins 111
risk and 128–9
see also normal distribution
projective plane 98
property, as security 297–8, 301, 311
Ptolemy (Claudius Ptolemaeus) 42, 56–7, 134
pumps, steam 199–200
Pythagoras’s Theorem
coding theory and 276
consequences 9
distance calculations from 14
special relativity and 225–7
Pythagorean triples 5
Pythagorean world view 3, 56, 58, 134–7, 143
Q
quadratic equations 78–80
quantum computers 264
quantum dots 264
quantum field theory and topology 92
quantum mechanics
and the classical world 255–9, 260–1
imaginary numbers in 76, 250–1
many-worlds interpretation 258–61
observational limitations 252, 256
practical utility 261–4
Schrödinger’s wave equation 250
vacuum energy discrepancy 328n
quartic equations 79–80
Quetelet, Adolphe 116–17, 119, 121, 124
quincunxes 120
R
race and IQ 124
radar 192
radio 190
random walks 301–2
randomness
chaos distinguished from 285
distinguished from determinism 288
patterns in 109
Recorde, Robert vii
Reed-Solomon codes 280
regression to the mean 121
regular solids
Kepler and 58
Reidemeister moves (Kurt Reidemeister) 101–3
relativity
general relativity 220, 231, 233–5, 240, 242–3
misleadingly named 219–20
special relativity 220, 224–9, 242–3
renewable energy 209–10
rice trading, Dojima exchange 298–9, 312
Riemann, Georg Bernhard 18–20, 95, 156, 231, 276
Riemannian metric 19, 231, 277
risk
financial, assessment 309, 314
probability theory and 128–9
Rourke, Colin 241
Royal Institution, London 181–2
Rule 110
automaton 320
Russian financial crisis, 1998 306–7
S
Sachs, Abraham 7
satnavs and relativity 219, 242
Schrödinger, Erwin 213, 250–1, 253–4
Schrödinger’s cat 253–6, 258–60, 318
Schwarzschild, Karl 235
sensory perception
data compression and 159–60
logarithms and 33–4
Shannon, Claude 269–70, 272–4, 279, 329n
ship design 172
short-selling 304
simplification, in topology 93, 101
sine curves 137–8
slide rules 30
Smale, Stephen 289–90
Smoluchowski, Marian 208
Snellius, Willebrord 12
Snow C(harles) P(ercy) 197–8, 209
social sciences
mathematical models 127
Solar System prediction horizons 290–1
sound levels 34
space missions
Hiten probe 69
interplanetary travel 67–8, 71–2
Newton’s law of gravity 65
view of Earth and 167–8
space-time
curvature 231–2
inflation and 240–1
Lorenz-FitzGerald contraction 225
topology and 105
Spearman, Charles 125–7
special relativity 220, 224–9, 242–3
speculation, financial 300, 302, 309, 312
speed of light
the ether and 222
Gran Sasso neutrinos 327n
Maxwell’s equations 188–9
special relativity and 224–6
spheres
from deformation of solids 92–4, 97
hypercubes and 278–9
square root of minus one 73, 77, 250–1
see also complex numbers; imaginary numbers
square roots
in Babylonian mathematics 7
duality of 76–7
solving cubic equations 79–80
using logarithms 29
squaw on the hippopotamus joke 3–4, 321n
standard deviation
defined 115
and extreme events 302–3
statistical mechanics 206
statistics
of randomness 109
statistical significance 122
steady state (populations) 287
steady state (universe) 241
steam engines 198–200
stents 174
stock market fluctuations 302
Stokes, George Gabriel 169, 170
subduction 147
subprime mortgage market 298, 311–12
Sundman, Karl Fritiof 64–5
superposition principle
applied to cats 253–5, 258, 328n
applied to the universe 258
Schrödinger’s equation 251–2, 257
supersonic flight 173–4
surfaces, in topology 98–100
surveying 11–12
symbolic dynamics 288
symmetry, of equations and solutions 211
T
T-waves 193
Taleb, Nassim Nicholas 303
tangents 10
‘tangles’ 106
Taqi al-Din Muhammad ibn Ma’ruf al-Shami al-Asadi 199
telephony
Fourier transforms 156–7
progress in 268
temperature
definition and measurement 202–3
in ideal gases 205–6
Tesla, Nikola 191
tetrahedra 91
Theory of Everything 319
thermodynamics
see also laws of thermodynamics
three-body problem 63–5, 69, 320
time, arrow of 208, 211, 213, 318
time-reversal symmetry 211, 214–15
triangles
triangulation 12–13
trigonometric functions
heat equation 152–4
vibrational frequencies 146
see also cosines; sines
trigonometry 9–12
Trojan points 69
U
ultraviolet catastrophe 248–9
see also probability theory
Unified Field Theory 319
universal law of gravity 61–2
universe
age of 241
‘heat death’ 209
matter distribution 210, 240–1
parallel universes 258
unknot, the 100–5
V
‘value at risk’ calculations 309
vector calculus 188, 193, 326n
vector fields 187
vectors defined 186
velocity
defined 39
as a vector 186
velocity fields 170
Venus missions 72
vertices
hypercubes 276
regular solids 91–6
vibrating strings 134–5, 137–8, 140, 143, 325n
and time intervals 40
vibrational frequencies 145–6
vibrational modes (black body radiation) 248
violin string vibration 133–4, 137
vision, human 159–60
Vulcan (hypothetical planet) 233–4, 328n
W
Warson, James 105
water, and fluid dynamics 168
Watt, James 198
wave equation 131, 139, 141–2, 144
animations 325n
heat equation compared 152–3
Maxwell and 188–9
multi-dimensional 144
wave functions 250–1, 252–3, 255–6, 258–60
wave number 143
wave-particle duality/wavicles 249–50
wave phenomena 133
wavelet/scalar quantization (WSQ) method 162
waves, analysis 86
weather forecasting 289, 292–3
Weber-Fechner law 33–4
Weierstrass, Karl 51
Weissing, Franz 291, 292
wind tunnels 172–3
Wolfram, Stephen 320
X
X-events (extreme events) 302–4, 306–7
X-rays 192
Z