General References to Appropriate Literature

The great choice of books on beautiful mathematics makes it diffcult to list an appropriate selection. The collections of mathematical problems originally published in monthly columns of various journals can be stimulating. Here I would like to mention in particular the fol- lowing authors who have published a great number of books:

• Martin Gardner (among others Mathematical puzzles and diversions) • Theoni Pappas (among others The Joy of Mathematics) • Ian Stewart (among others Professor Stewart’s Cabinet of Mathematical Curiosities) • Heinrich Hemme (among others Heureka!, in German)

Beautiful and exciting mathematics is the focus of the books by:

• Hans Walser (among others Der Goldene Schnitt, Geometrische Miniaturen, Symmetrie in Raum und Zeit, DIN A4 in Raum und Zeit – in German) • Roger B. Nelsen (among others Proofs without Words I, II, III) • Claudi Alsina and Roger B. Nelsen (among others Charming proofs, Icons of mathematics) • Albrecht Beutelspacher (among others Wie man in eine Seifenblase schlüpft – in German) • Julian Havil (among others Gamma, Nonplussed!, Impossible?) • George G. Szpiro (among others A mathematical medley: Fifty easy pieces on mathematics) • Eli Maor and Eugen Jost (Beautiful Geometry) • Alfred S. Posamentier (among others Mathematical Amazements and Surprises) • Martin Erickson (among others Beautiful mathematics)

Among the numerous websites dealing with beautiful mathematics and mathematical puzzles, only the most important ones can be mentioned here:

• www.cut-the-knot.org • www.gogeometry.com • www.mathsisfun.com • www.mathpuzzle.com • www.recreomath.qc.ca • www.walser-h-m.ch/hans (mainly in German) • www.mathematische-basteleien.de (mainly in German)

A collection of very interesting articles on various mathematical topics can be found on the website of Mathematical Association of America (MAA). They are especially worth reading because of their historical reference and the question of practicability in school lessons:

• www.maa.org/press/periodicals/convergence

If you want to know more about those personalities who over the centuries have contrib- uted to the development of mathematical theories or have deepened the knowledge about them, I recommend in the frst place the website of MacTutor History of Mathematics archive the School of Mathematics & Statistics at the University of St Andrews, Scotland:

• https://mathshistory.st-andrews.ac.uk/Biographies • For those who would like to get a summary of the life and work of individual selected personalities, we refer to Heinz Klaus Strick’s histories (https://mathshistory.st-andrews.ac.uk/Strick/) – original version (in German): www.spektrum.de/mathematik/ monatskalender/index/ Index

A Bhaskara, 337 Addition method, 303 Billiard tournament, 22 Al-Buzjani, Abu'l-Wafa, 339 Binet, Jacques Philippe Marie, 65 Algorithm, Euclidean, 68 Binomial coeffcient, 305, 317 Al-Haitham, Abu Ali al-Hasan ibn, 38, 306 Binomial theorem, 305, 317 Alhazen, 38, 306 Bisection, continued, 139 Al-Karaji, Abu Bakr ibn Muhammad ibn Boundary points, 194 al-Husayn, 42 Bouwkamp, Christoffel Jacob, 260 Al Kashi, Jamshid, 80 Bouwkamp notation, 260 Altitude theorem, 238, 327 Braided bands, 87 Angle sum formulae, 112 Brooks, Rowland Leonard, 266 Annulus, 81 Aperiodic tile patterns, 164 Apollonios of Perge, 274 C Application of areas, 244 Cardioid, 117 Arbelos, 344 CAS, 305 Archimedes, 79, 344 Cassini, Giovanni Domenico, 237 Shoemaker’s knife, 344 Cassini identity, 237 Area of a circle, 80 Central angle, 11 Area of polygons, 193 Central limit theorem, 227 Argand diagram, 16 Chains, Markov, 229 Arithmetic sequence, 282 Cheney, Fitch, 354 Arithmetic sequences of higher order, 298 Chessboard coloring, 92 Aryabhata, 41 Chou Pei Suan Ching, 336 Astroid, 117 Circuits, electrical, 266 Austin, David, 278, 280 Circular rings, 81 Coeffcients, comparison of, 300 Coin, 218 B Combination table, 206 Babylonian mathematicians, 52 Comparison of probability distributions, 225 Balanced ternary system, 162 Complex number, 16 Beam balance, 155 Complex plane, 16, 17 Bell-shaped curve, 209, 228 Congruences, 133 Bernoulli, Jacob, 317 Construction with a compass and a ruler, 20 Bernoulli number, 318 Continued fraction, 60

Converse of the Pythagorean theorem, 324, 338 Euclidean algorithm, 68 Coprime, 6 Euclidʼs theorem, 243, 325 Cosines, law of, 241, 271, 346 Euclid’s right triangle altitude theorem, 327 Cube numbers, sum of, 41, 134, 302, 306, 312 Euler, Leonhard, 74, 132, 218 Curry, Paul, 233 Expected value, 212, 222 Curryʼs triangle paradox, 233, 246 Curvature, integer, 274 Curve of pursuit, 113 F Curve stitching, 103 Factorization of the generating function, 220 Cycles, 121 Faulhaber, Johannes, 295 Cycle length, 130 Faulhaberʼs formulas, 295 Cycles, periodic, 131 Fermat, Pierre de, 314 Cycloid, 115 Fibonacci, 64, 237 Cyclotomic polynomial equation, 17 Fibonacci number, 64 Fibonacci rectangle, 63 Fibonacci sequence, 64, 237 D Figurate numbers, 309 Da Vinci, Leonardo, 328 Final digit, 121 Decagram, 4 Formula, Moivre–Binet, 65 Decision procedure, 212 Fortune, wheels of, 215 De Moivre, Abraham, 17, 65, 318 Four color theorem, 254 Density function, 210, 228 Fraction, continued, 60 Descartes, René, 274 Function Descartesʼ four-circle theorem, 274 generating, 218 Diagonal, number of, 9, 103 polynomial, 301 length, 10 square, 109 Diamonds, 164 Fundamental theorem of algebra, 17 Diophant, 27 Dissection of rectangles, 60, 254 Distribution, probability, 207 G Division of terms, 18 Galilei, Galileo, 31, 210 Divisor, greatest common, 68 Game, fair, 21, 213 Dobriner, Hermann, 334 Gardner, Martin, 214, 233 Dodecagram, 5 Gauss, Carl Friedrich, 16, 24, 132, 210 Dodecahedron, 217 Gaussian density function, 209 Domino, 91 Generating function, 218 Duijvestijn, Adrianus Johannes Wilhelmus, Geometric mean theorem, 327 254, 263 Geometric sequence, 148 Geometric series, 148 Gnomon, 28 E Goldbach, Christian, 132 Electrical circuits, 266 Golden number, 67 Elementary row operations, 303 Golden section, 293 Enneagram, 3 Golden triangle, 15 Envelope, 109 Golomb, Solomon W., 91 Epicycloid, 115, 117, 118 Göpel, Adolph, 331 Epstein, Paul, 333 Greatest common divisor, 68 Equation system, linear, 303 Grid paper, 179 Euclid, 52, 68, 124, 238, 243, 324 Gutheil, Benjir from, 332 Index 363

H Limit of geometric series, 148 Harriot, Thomas, 309 Logarithmic spiral, 113 Heart Figure, 343 Loyd, Sam, 233, 247 Hendecagram, 4 L-shaped form, 28, 44, 48 Heptagram, 2 Lune of Hippocrates, 342 Heronian triangle, 352 Heron of Alexandria, 352 Hexagonal grid, 203 M Hexagram, 2 Markov chains, 229 Hexahedron, 205 Mental arithmetic, 121, 128 Hexomino, 91 Mersenne, Marin, 314 Hippocrates of Chios, 341 Missing square, 233 Histogram, 207 Modulo, 133 Hypocycloid, 118 Moivre, Abraham de, 17, 65, 318 Moroń, Zbigniew, 253, 256

I ICM, 253 N Icosahedron, 223 Nautilus, 67 Identity of dʼOcagne, 245 Nephroid, 117 Imaginary part, 16 Network of electrical currents, 267 Induction Nicomachus of Gerasa, 41 proof by, 309 Nielsen, Jacob, 333 Inscribed angle theorem, 11 Normal distribution, 227 Interior points, 194 Number Interpolation, 319 complex, 16 fgurate, 309 golden, 65 J pentagonal, 310, 341 Jigsaw pieces, 91 polygonal, 309 tetrahedral, 310 triangular, 26, 33, 298, 309 K Number chains, 129 Kirchhoff's laws, 267 Kissing circles, 277 Knuth, Donald E., 295 O Ocagne, Philbert Maurice d, 245 Octagram, 3 L Octahedron, 223 Lagrange, Joseph-Louis, 319 Ornament, 89 Lagrange interpolation, 320 Orthodiagonal quadrilaterals, 346 Laplace, Pierre-Simon, 218 Law of cosines, 241, 271, 346 Law of sines, 271 P Leibniz, Gottfried Wilhelm, 301 Pair of divisors, 181 Length of a cycle, 131 Pappos chain, 282 Leonardo da Vinci, 328 Pappos of Alexandria, 282 Leonardo of Pisa, 65 Parabola, 106 Limit, 72 Paradox, Curryʼs triangle, 233, 246 364 Index

Pascal, Blaise, 316 R Pascalʼs triangle, 305, 316 Rabbit problem, 65 Pattern, small and large squares, 335 Radius, integer, 105, 278 Pattern of stones, 23 Random output, 215 Penrose, Roger, 164 Real part, 16 Penrose tiling, 67 Reciprocal function, 110 Pentagon, 1 Rectangle Pentagon number, 310, 341 golden, 72 Pentagram, 1 maximum area, 185 Pentomino, 91, 191 Rectangles with a given area, 182 Perigal, Henry, 330 Rectangles with a given perimeter, 185 Perimeter, 180 Recursive determination of coeffcients, 304 Peripheral angle, 11 Regular n-sided fgure, 7, 103 Π, 79 Regular stars, 1 Pick, Georg Alexander, 193 Representations as sums of natural numbers, 34 Pick’s theorem, 193 Rhombus, 164 Platonic solids, 223 Right angle altitude theorem, 238 Point, interior, 194 Rotation of a point around the origin, 112 Pólya, George, 308 Row operations, 303 Polygon, 1 Rules of the game Polygonal number, 309 fair, 21, 214 Polynomial function, 301 Polynomials, Decomposition of, 218 Polyomino, 91 S Power of two, 38, 46 Salinon, 345 Powers, sums of, 295 Salt cellar, 345 Primitive Pythagorean triples, 51 Sangaku, 271, 286 Probability distribution, 207 Schedule, tournament, 21 Proclus, 336 Schierscher, Georg, 67 Proofs by dissection, 330 Schläfi symbol, 6 Puzzle proof, 335 Schneider, Ivo, 297 Pyramidal number, 310 Scientifc American, 214, 233 Pythagoras of Samos, 323 Section, golden, 15, 67, 72 Pythagoras spiral, 339 Seki Kowa, 286 Pythagorean Partners, 347 Semicircle, shifted, 84 Pythagorean theorem, 16, 47, 88, 240, 249 Sequence Pythagorean theorem, Generalization, 341 arithmetic, 282, 298 Pythagorean tree, 340 geometric, 148 Pythagorean triple, 47, 52, 324 Sequence of differences, 298 Series, geometric, 148 Shape, L-shaped form, 28 Q Shape, triangular, 23 Quadratic equation, 275 Sicherman, George, 214 Quadratic function, 109 Sigma rule, 228 Quadrilateral, 346 Similarity, 150, 245 Quadrisection, continued, 143 Simple Perfect Squared Rectangles, 253 Index 365

Simple Perfect Squared Squares, 253 Tetrahedral numbers, 310 Smith, Cedric Austin Bardell, 266 Tetrahedron, 216 Soddy, Frederick, 274 Tetromino, 91 Solid, platonic, 223 Thabit ibn Qurra, 338 Spiral curve, 64, 113 Thales, 88 Spiral, logarithmic, 113 Theorem, Descartesʼ four-circle, 274 Spiral, Pythagorean, 339 Theorem, four color, 254 Spiral-like decomposition of regulars polygons, Theorem, fundamental, 17 147 Theorem, general binomial, 305 Sprague, Roland P., 253 Theorem, Geometric mean, 327 Square, largest possible, 58 Theorem, inscribed angle, 11 Square grid, 194 Theorem, Pick’s, 193 Square number, 121, 310 Theorem, Pythagorean, 16, 47, 88, 240, 249 Square roots, continued fraction of, 74 Theorem, right angle altitude, 238, 327 Squared paper, 253, 263 Theorem, Sylvester’s, 34 Squared rectangles, simple perfect, 253 Theorem, Thales', 88 Squared square, 263 Thieme, Karl Gustav Hermann, 334 Squared squares, simple perfect, 253 Thread patterns, 103 Squaring the square, 253 Tile patterns, 164, 334 Standard deviation, 222 Tournament, 21 Star fgure, 1, 105, 174 Tracking curve, 118 States of a system, 229 Transition, 129, 229 Stone, Arthur Harold, 266 Trapezoidal shape, 224 Sum, consecutive natural numbers, 34 Tree, Pythagorean, 340 Sum, consecutive triangular numbers, 27 Tree-like tessellation, 169 Sum, frst cube numbers, 41 Triangle, golden, 15 Sum, frst natural numbers, 24 Triangle Sum, frst odd natural numbers, 29 almost right-angled, 239 Sum, frst square numbers, 39 Triangles, Heronian, 352 Sum of products of consecutive natural num- Triangular fgure, 24, 93, 99 bers, 315 Triangular grid, 202 Sum of spots, 205 Triangular number, 26, 33, 298, 309 Sum symbol, 301 Triangular shape, 23, 224 Sylvester, James Joseph, 34 Triple, Pythagorean, 47, 52, 324 Symmetrical tessellation, 172 Trisection, continued, 141 System of equations, 265, 301 Tromino, 92 Tutte, William Thomas, 266 Twin Circles, 344 T Table of combinations, 206 Tangents, 103 U Tangents, construction of crossing, 89 Unit circle, 16 Tangents to circles, 88 Ternary numeral system, 159 Tessellation of rectangles, 59, 94, 253 V Tessellation of regular 10-sided fgures, 164 Value, critical, 212 Tessellation of triangular fgures, 99 Variance, 223 Tessellation with pentominoes, 94 Vertex angle, 15 366 Index

Vietaʼs formula, 276 Wheatstone, Charles, 44 Viète, François, 276 Wheels of fortune, 215

W Z Walser, Hans, 77, 345, 348 Zu Chongzhi, 80 Weighing of objects, 155