Mathematics
Mathematics
Isis
The
Collection
of
All
The Collection of All
Possible
Patterns
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
N
L

M
T
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)
Opie’s 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 Koch’s
‘Snowflake’, 1904
Mandelbrot’s 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), 237.5cm×393.7cm, 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
1D2
Large D means greater intricacy:
the line behaves like an area
Pollock’s complexity increases with time
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
Bézier-du Casteljau Curves
Smooth curves in car design (Renault), VW, Mercedes
computer graphics and font creation
Postscript, Adobe illustrator, True Type, CorelDRAW
Bézier-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<t<1
Fraction of
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).
Chvátal (1972)
Maths and Poetry
Andrei Markov’s invention of mathematical textual analysis (1913)
The statistical theory of non-independent sequences of events
‘Markov Processes’
Visual Analysis of Textual Interconnectivity