Mathematics Is The Collection of All Possible Patterns

Mathematics Is The Collection of All Possible Patterns

Mathematics Is The Collection of All Possible Patterns Spirals Leonardo: freehand octagonal chapel

Strasbourg cathedral Complexity that is neither too simple nor to difficult Complexity with reductionism is science Complexity without reductionism is art Art is I. Science is We

Arts in Space and Time L N T

M Dimension L1 L2

L3 T0 frieze painting

sculpture T1 music film

theatre Four-dimensional geometry Charles Hinton The Fourth Dimension, (1904)

Salvador Dali, Corpus Hypercubus, (1954) Optimal Viewing Distance S x2 = T(S+T) T = 50m

S = 5.5m x = 52.6m a T b x

tan(a) = Sx/[x2 + T(S+T)] sec2(a)da/dx = -S[x2 T(S+T)]/ [x2 + T(S+T)]2 Catherine Opie, Twelve Miles to the Horizon D2 2HR D2 + R2 = (H + R)2 = H2 + R2 + 2HR R2 + 2HR

D = 1600 (5H) metres (5H) miles H = 1.8m D = 4800m or 3 miles Sunset #8 (2009) Opies 12 miles is OK for a cruise ship with H = 29m The Tunnel of Eupalinos (530-520 BC)

10 yrs of tunneling 170m mean depth 1036m The Rough and the Smooth

Self-similarity Helge von Kochs Snowflake, 1904 Mandelbrots set (1980): the set of start points that stay at finite distances form the black region with its infinitely intricate boundary x x2 y2 + a ; y 2xy + b

Any part of the boundary contains copies of the whole set Jack the Dripper Jackson Pollock's Convergence (1952), 237.5cm393.7cm, oil on canvas 2009 The Pollock-Krasner Foundation/Artists Rights Society (ARS), New York.

Fract(ion)al Dimension Cover with N(r) squares of side r N(r) r D 1D2 Large D means greater intricacy: the line behaves like an area

Pollocks complexity increases with t Fractal signature ? Can you tell a Fake Pollock ?

Can a fractal analysis distinguish true from fake? YES Richard Taylor et al NO Kate Jones-Smith et al Controversial! String surface model: hyperbolic paraboloid

by Fabre de Lagrange, 1872 Two bars equally spaced, each turns on an arm perpendicular to itself and one arm swings on a pillar; these arms can be ranged in one plane, and also turned end for end. Henry Moore, Bird Basket, 1939

Jerusalem Chords Suspension Bridge Bzier-du Casteljau Curves Smooth curves in car design (Renault), VW, Mercedes computer graphics and font creation Postscript, Adobe illustrator, True Type, CorelDRAW Bzier-du Casteljau Curves

Linear B(t) = P0 +(P1 P0) t = (1-t) P0 + tP1 Quadratic B(t) = (1-t) [(1-t) P0 + tP1] + t[(1-t) P1 + tP2] Cubic

B(t) = (1-t)B(quadratic) + tB(quadratic) B(t) = (1-t)3P0 + 3(1-t)2tP1 + 3(1-t)t2P2 + t3P3 Quartic Etc.. 0

curve drawn The Gallery Problem camera How many cameras are needed to watch a gallery and where should they be placed?

Simple Polygonal Galleries Regions with holes are not allowed and no self intersections convex polygon one camera is enough an arbitrary n-gon (n corners)

? cameras might be needed 3-Colouring the Gallery Assign each corner a colour: pink, green, or yellow. Any two corners connected by an edge or a diagonal must have

different colours. Thus the vertices of every triangle will be in three different colors. A 3-colouring is always possible. Put cameras at corners of same colour Pick the smallest of the coloured corner groupings to locate the cameras. You will need at most [n/3] = [19/3] = [6.33] = 6 cameras (yellow or pink). Chvtal (1972)

Maths and Poetry Andrei Markovs invention of mathematical textual analysis (1913) The statistical theory of non-independent sequences of events Markov Processes ual Analysis of Textual Interconnectiv Interconnecti

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