Mathematicians with special interest in biology, physics, geography, astronomy, architecture, design, etc. , and being prepared to take pictures at any time, might try to answer unusual questions like the followings: What do a zebra, a tiger shark, and a hard coral have in common? How is this with drying mud, wings of dragon flies, and the structures of leaves? What is the "snail king" and is there also a "worm king"? Which curves stay of the same type after being photographed? Do fishes see like we do if we look through a fisheye lens? Which geometric properties of an object have physical consequences? Which kinds of geometric patterns appear when waves are interfering? In this book you can find 180 double pages with at least as many questions of this kind. The principle to attack a problem is often similar: It starts with a photo that is for some reasons remarkable. In a short description an explanation is offered, including relevant Internet links. Additionally one can frequently find computer simulations in order to illustrate and confirm.
Inhaltsverzeichnis
1;Preface;5 1.1;Preface;6 1.2;Maths and nature photography;8 2;1 Mathematical interplay;17 2.1;Zebra stripes and number codes;18 2.2;How a number becomes a zebra;20 2.3;The chicken and the egg;22 2.4;The tortoise paradox;24 2.5;Discerning information from photos;26 2.6;Repeatability of experiments;28 2.7;Reproduction of water lilies;30 2.8;Transitivity and combinatorics;32 2.9;Cameras and hand luggage;34 2.10;Beyond the limits of microscopy;36 2.11;Endless loops;38 2.12;Mathematical crochet work;40 2.13;Ispiration through fascination;42 3;2 The mathematical point of view;45 3.1;Remarkably similar;46 3.2;Associations;48 3.3;Similar, but not by accident;50 3.4;Iterative shape approximation;52 3.5;Rhombic zones;54 3.6;Nets of skew rhombuses;56 3.7;Oblique parallel projections;58 3.8;Fibonacci and growth;60 3.9;Different scales;62 3.10;The volume of a wine barrel;64 3.11;Three simple rules;66 4;3 Stereopsis or spatial vision;69 4.1;Depth perception;70 4.2;Two projections in one image;72 4.3;Compound eyes;74 4.4;Distance tables;76 4.5;Lens eyes;78 4.6;Eyes with mirror optics;80 4.7;Using antennae for accuracy;82 4.8;Intersecting the viewing rays;84 4.9;Natural impressions;86 4.10;Photo stitching;88 4.11;Impossibles;90 4.12;Cuboid or truncated pyramid?;92 5;4 Astronomical vision;95 5.1;Sunset;96 5.2;Solar eclipse;98 5.3;When the sun is very low;100 5.4;Fata Morgana;102 5.5;The scarab and the sun;104 5.6;The law of Right Angles;106 5.7;The beginning of spring;108 5.8;The wrong tilt of the moon;110 5.9;The sun at the zenith;112 5.10;Central american pyramids;114 5.11;The arctic circle;116 5.12;The southern sky;118 6;5 Helical and spiral motion;121 6.1;Helicoid;122 6.2;Thrust or lift?;124 6.3;The spiral;126 6.4;Of king snails and king worms;128 6.5;Exponential growth;130 6.6;Helispirals;132 6.7;From formulas to animal horns;134 6.8;Millipedes and pipe surfaces;136 6.9;Scope of intelligence;138 7;6 Special curves;141 7.1;The catenary;142 7.2;Invariance under central projectio
n;144 7.3;The parabola;146 7.4;Knots;148 7.5;Contours with cusps;150 7.6;Geodesic gifts;152 8;7 Special surfaces;155 8.1;The sphere;156 8.2;The spheres silhouette;158 8.3;Approximating curved surfaces;160 8.4;Flexible and versatile;162 8.5;Development;164 8.6;Puristic beauty;166 8.7;Stable and simple construction;168 8.8;Minimized surface tension;170 8.9;Minimal surfaces;172 8.10;Soap bubbles;174 9;8 Reflection and refraction;177 9.1;The spherical refl ection;178 9.2;Il Carnevale & geometry;180 9.3;Mirror symmetry;182 9.4;The planar refl ection;184 9.5;Starfi sh and radial symmetry;186 9.6;The pentaprism;188 9.7;The billiard effect;190 9.8;Sound absorption;192 9.9;The optical prism;194 9.10;Rainbow theory;196 9.11;At the foot of the rainbow;198 9.12;Above the clouds;200 9.13;Spectral colours underwater;202 9.14;Colour pigments or iridescence?;204 9.15;Fish-eye perspective;206 9.16;Snells window;208 9.17;Total refl ection and image raising;210 9.18;A fi sh-eye roundtrip;212 10;9 Distribution problems;215 10.1;Even distribution on surfaces;216 10.2;Distribution of dew;218 10.3;Contact problems;220 10.4;A platonic solution;222 10.5;Spiky equal distribution;224 10.6;Elastic surfaces;226 10.7;Quite dangerous;228 10.8;Pressure distribution;230 10.9;Fluctuations of weight;232 10.10;Kissing numbers;234 11;10 Simple physical phenomena;237 11.1;Newtons laws of motion;238 11.2;Jet propulsion and suction;240 11.3;Selective light absorption;242 11.4;Relative velocities;244 11.5;The aerodynamical paradox;246 11.6;Flying in formation;248 11.7;Form follows function;252 11.8;Offspring on parachutes;254 11.9;The fastest track;256 11.10;Manoeuvering through curves;258 11.11;Mathematics and bees;260 11.12;Interferences;262 11.13;Doppler effect and the Mach cone;264 11.14;Sonic waves on strange paths;266 11.15;Acceleration vs. constant speed;268 11.16;Stroboscopic effect;270 12;11 Cell arrangements;273 12.1;Reproduction of daisies;274 12.2;Spirals or no spirals?;276 12.3;Calculating rot
ation;278 12.4;Voronoi diagrams;280 12.5;Iterative Voronoi structures;282 12.6;Columnar basalt;284 12.7;3D cells;286 12.8;Random paths;288 12.9;Winding curves;290 12.10;Fractal sphere packing;292 13;12 The difference between big and small;295 13.1;Decimal powers among animals;296 13.2;150 million years without change;298 13.3;Legendary strength;300 13.4;Where is gravity?;302 13.5;Threads from protein;304 13.6;Dangerous glue;306 13.7;Giant elephant ears;308 13.8;Floating coins;310 13.9;Model and reality;312 13.10;Scale-independent depth of fi eld;314 13.11;Blurry decisions;316 13.12;Fluids;318 13.13;Fractions of a millisecond;320 13.14;Flexible straws;322 13.15;Biomass;324 13.16;Like an oil bath;326 13.17;Survival of the fi ttest;328 14;13 Tree structures and fractals;331 14.1;Sum of cross-sections;332 14.2;Systematic chaos;334 14.3;Branching;336 14.4;Wonderful coincidences;338 14.5;Fractal contours;340 14.6;Fractal pyramids;342 14.7;Animals or plants?;344 14.8;Mathematical ferns;346 14.9;Fractal expansion;348 14.10;Level curves;350 14.11;From octahedrons to snowfl akes;352 15;14 Directed motion;355 15.1;Non-round gears;356 15.2;Transmission matters;358 15.3;Robust and effi cient;360 15.4;Light-footedness and reaction time;362 15.5;Throwing parabola;364 15.6;Jumping up high;366 15.7;With a club and cavitation;368 15.8;Toys of changing colour;370 15.9;Flight acrobatics;372
