X and the City: Modeling Aspects of Urban Life
Page 26
accidents, car, 12, 77
avoiding, 80–85
advection-diffusion, 183
air pollution, 179, 191
its effect on visibility, 191
air travel, 7–10, 13
analogies, 2–3
asteroids, 248
capture cross section for, 303
close encounters with, 303–7
collisions with, 249
Astronomy Picture of the Day (APOD), 248
bamboo, growth of, 42
bedbugs, 142
beltways, 133
M25 in UK, 125, 133
Brownian motion, 181
buses, bunching of, 64
cancer. See tumors
Central Park: jogging in, 49–51
squirrels in, 47
sunbathing in, 48–49
circumhorizontal arc, 301
cities: Bessel, 163–64
equilibrium models of, 159
scaling in, 171
Cobb-Douglas production function, 30
comet, Shoemaker-Levy, 247
concert halls, 19–21
dental offices, number of, 16
dewbows, 230
diamond dust, 229, 301
Dido, princess, 4–5
and the sinc function, 263–68
theorems for, 261–62
differential equations: Bernoulli, 172
delay-, 108
diffusion, 182–83, 188, 293–95
nonlinear, 137, 140
diffusion equation, 293
and diffusion coefficient, 294. See also differential equations
dimension: Hausdorff-Besicovitch, 288
topological, 288
distributions: exponential, 91
negative exponential, 91, 96
Poisson (see Poisson distribution)
doomsday, 136, 139, 176
earthquakes, in Virginia, 25
ellipses, 219–20
expectation value, 95
eruption of, 180
flashlight beams, 242–43
fractal, 3
box-, 289
Cantor dust, 290
and fractal dimension, 3, 169–70, 288–90
Koch snowflake, 289
Menger sponge, 290
Sierpinski triangle, 289
fruit, 36–37
gaps, at school crossings, 92–94
gasoline, consumption of, 74–76
geometrical optics, 193
Goldilocks principle, 11, 17, 39, 251
grass seed, 42–43
halos: ice crystal, 204, 224
in lighthouse beams, 239–240
harmonic mean, 74
headway, 88, 94
heat island, urban, 58, 61
houses of worship, 16–17
hyperbolic orbit, 304
isoperimetric inequality, 261
isoperimetric theorem, 261
jogging, 49–51
Lagrange multipliers, 31, 277
leaves, 45
light: and diffraction, 192–93
and light bulbs, 222
and light pillars, 222–23
and polarization, 198–200
reflection and refraction of, 192
and scattering of (see scattering)
wavelengths of, 192–93
London, population of, 144
M25 motorway, 105, 133
malls, 25–27
Malthusian growth, 137
Mandelbrot set, 287
Manhattan distance, 270
Manhattan metric, 26
maps, 10–11
mathematical models: continuum, 136, 284
deterministic vs. probabilistic, 89
incorporating reaction time, 117–20
kinematic vs. dynamic, 89–90, 99
metaphors, 2–3
meteor crater (Winslow, Arizona), 249–50, 252–53
Minnaert, Marcel, 216, 225
and “apples,” 224
and “cigars,” 224–25
moonbows, 230
mortgages, 32–33
museums, 17–19
normal distribution, 184
Old Dominion University, 253
orchids, maturation of, 43–44
overtaking (passing), 88
pedestrians, 72
petrol. See gasoline consumption
Poisson distribution, 28, 91, 273
derivation of, 274–75
population: of certain cities, 151–52
and population densities, 152
population growth: exponential, 136
Gompertz-like, 173
logistic, 140
Malthusian, 137
post office, 27–28
probability: density function of, 94
geometric, 29, 281–82
and probability distribution, 275
puddles: rain sparks in, 216
wave patterns on, 256–57
rain: and circles, 217
precipitation rate, 52
and reflections, 218–19
running or walking in, 51–56
and sparks, 216
speed of, 259
rainbows, 193, 203–4, 223–34
details of formation, 297–302
in lighthouse beams, 234–39
in searchlight beams, 240–42
tertiary and quaternary, 297
random walk, 291–93
rats, 142
reaction time, 80–83, 108, 113–14, 116, 117–20
refractive index, 204–5, 212–14, 299
Reynolds number, 23–24, 60, 181–82
ring roads. See beltways
risk: of dying while driving, 12
of dying while flying, 12
restaurants, number of, 16
roadbows, 203–7
scaling, 38, 167, 169–71
allometric, 168
geometrically similar, 168
isometric, 168
scattering, 192
Mie, 200
Rayleigh, 194, 198, 200–202
searchlights, 234
shadows, 58, 210
skyscrapers, 21–23, 309
slugs, 44–45
smog, 59, 179, 191
smokestacks, 185–87
speed, terminal, 180
Stirling’s approximation, 274
Stokes’ law, 180
stopping distance, 80–85
Strouhal number, 24
sunbathing, 48–49
taxicab geometry, 65
taxicab metric, 26, 270
taxis, 11–12
tomatoes, consumption of, 39–40
traffic: equation of continuity for, 99
flow of, 97
reaction time in, 117
shock waves in, 90, 101
and traffic jams, 97, 100–101, 126–27
at traffic signals, 100–103
in tunnels, 103–5
traffic signals: delays at, 101–3
legal maneuvers for, 76–80
transportation, public, 63–71
average speed of, 65–68
trips, average length of, 68–71
tumors, 1, 3
turkey, cooking, 37–39
visibility, as affected by air pollution, 191
watermelons. See fruit
weather, in cities, 56–58
wind: effects of on tall buildings, 21–24, 52, 58–59
effects of on pollution, 182–84, 186–88
windows, 211–16
worms, 44
Zipf’s law, 168
%); " class="sharethis-inline-share-buttons">share