Mandelbrot and dimensionality
Benoit B Mandelbrot
aka
Snezana Lawrence
snezana@mathsisgoodforyou.com
s.lawrence2@bathspa.ac.uk
http://www.ted.com/talks/
benoit_mandelbrot_fractals_the_art_of_roughness?
language=en#t-71100
what does B stand for
in Benoit B Mandelbrot?
•
20th November 1924 - 14th
October 2010
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Polish-born Jewish, brought up in
France, lived his later years in
America. •
Inventor of ‘fractal’ geometry,
theory of roughness, and selfsimilarity in nature
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Discovery of the Mandelbrot set,
based on Julia set
•
An uncle who was a
mathematician, Szolem
Mandelbrojt (early member of
Bourbaki group, 1899-1983)
lessons
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I’m going to look up for some lessons that could be taken to
a maths classroom - about people and about mathematics:
new facts? new uses of old maths? new attitudes?
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a lesson from a young child, finding himself in Paris in 1940s
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Our constant fear was that a sufficiently determined foe might
report us to an authority and we would be sent to our deaths.
This happened to a close friend from Paris, Zina Morhange, a
physician in a nearby county seat. Simply to eliminate the
competition, another physician denounced her ... We
escaped this fate. Who knows why?
how do you come across
something interesting?
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Iteration, sets, and functions
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The paper describing the
iteration of rational function
f(x). •
Julia gave a precise description
of the set J(f) of those z in C for
which the nth iterate f^n(z)
stays bounded as n tends to
infinity.
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http://sites.mathdoc.fr/JMPA/
PDF/
JMPA_1918_8_1_A2_0.pdf
http://aleph0.clarku.edu/~djoyce/julia/explorer.html http://www.easyfractalgenerator.com/julia-set-generator.aspx http://paulbourke.net/fractals/
how long is the
coastline of Britain?
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self-similarity: a self-similar
object is exactly or approximately
similar to a part of itself
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scale invariance is a form of
exact self-similarity •
plane and space-filling •
Cantor’s ideas about cardinality
of infinities: in particular that the
cardinality of infinite number of
points in a unit interval is the
same cardinality as the infinite
number of points in any finitedimensional manifold, such as
the unit square
Sierpinski carpet and
Koch curve
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ai+1=8⁄9⋅ai. So ai=(8⁄9)^i, which tends to
0 as i goes to infinity
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Hausdorff dimension - defined by Felix
Hausdorff - is a measure of the local size
of a set of numbers, taking into account
the distance between each of its
members
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Hausdorff dimension of a single point is
0, of a line is 1, of a square is 2, of a cube
is 3
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Hausdorff dimension of fractals is
formalised by scale factor (3) and selfsimilar objects (4), after a first iteration the
dimension D will be D = (log N)/(log S) =
(log 4)/(log 3) ≈ 1.26
some more maths
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cardinality of sets, cardinality of infinite sets
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can the part be the same size as the whole? What about
Hilbert’s hotel?
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what kind of infinity is the one that we all strive for (being
megalomaniacs as we usually are) and that which we
don’t want to know about (not wanting to find out in
practice what area do our lung’s bronchi cover)?
between the friends
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is learning a dialogue? •
Plato
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his five elements
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his ideal world
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his way of teaching
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his academy
what is the world
like?
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a page from the 1519 edition of
Aristotle’s De Caelo (On the
heavens). Augsburth, Sigmund
Grimm and Max Wirsung.
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The Eart sits at the center of the
universe, composed of two
elements, earth and water (terra &
aqua). It is surrounded by two
other elements, air and fire (aer
and ignis), then the spheres which
carry the Sun, Moon, planets and
stars around the Earth.
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Outside the spheres is the realm
of the Prime Mover, represented
here by a winged figure.
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A magnitude if divisible one way is a line, if two ways a
surface, and if three a body. Beyond these there is no other
magnitude, because the three dimensions are all that there
are, and that which is divisible in three directions is divisible in
all’ (Aristotle 2012, 268a:10–15). But the possibility of an
extension of dimensions appeared to Aristotle, although he
rejected it little later: ‘All magnitudes, then, which are divisible
are also continuous. Whether we can also say that whatever is
continuous is divisible does not yet, on our present grounds,
appear. One thing, however, is clear. We cannot pass beyond
body to a further kind, as we passed from length to surface,
and from surface to body’ (Aristotle 2012, 268a:25–30).
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For further reference on Aristotle’s mention of the
dimensionality in other works, see (Cajori 1926)
others on dimensions
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Ptolemy denied and disproved it but nevertheless mentioned and contemplated
upon it (Cajori, 1926: 397; Heiberg 1893: 7a, 33)
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John Wallis, although writing this whilst considering geometric interpretations of
quantities he was developing in the context of algebra, wrote (Wallis 1685:126):
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A Line drawn into a Line shall make a Plane or Surface; this drawn into a
Line, shall make a Solid: But if this Solid be drawn into a Line, or this Plane
into a Plane, what shall it make? A Plano-Plane? That is a Monster in
Nature, and less possible than a Chimaera or Centaure. For Length, Breadth
and Thickness, take up the whole of Space. Nor can our Fansie imagine
how there should be a Fourth Local Dimension beyond these Three.
Lagrange spoke of three coordinates to describe the space of three
dimensions, introducing time as the fourth, and denoting it t (Lagrange 1797:
223)
stairways to heaven
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Möbius (1827) first spoke
about an object getting out of a
dimension it belonged to in
order to perform a spatial
operation. •
If one had a crystal, structured
like a left-handed staircase,
how would one get its threedimensional reflection?
getting out of one
own’s plane
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Zöllner (Johann Friedrich, 1834-1882)
further simplified this in 1878.
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If one has a circle and a point outside
of it, how can one get the point into
the circle without cutting or crossing
over the circumference?
!
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His work Uber Wirkungen in die
Ferne (On effects at a distance)
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a happy coincidence for a cultural
reference (Goethe’s poem Effect at a
Distance https://en.wikisource.org/
wiki/
The_Works_of_J._W._von_Goethe/
Volume_9/Effect_at_a_Distance
the platonic thread
•
Elementa doctrinae solidorum
published in 1758, in which Euler
described for the first time what was to
become known as Euler’s characteristic
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the expression which conveys the
information that in all convex solid
bodies the sum of the solid angles and
the number of faces is equal to the
number of edges add 2
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We now usually denote Euler’s
characteristic by Greek letter chi and
describe it for convex polyhedra where V is the number of vertices, E is
the number of edges, and F is the number
of faces in a polyhedron.
looking at it from another perspective
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If we further analyse the formula we notice that we begin
from the first variable which counts points (point we
earlier took to represent 0th dimension); the second
variable which numbers the edges in a solid,
(representing line, 1st dimension) and the third variable,
numbering the faces of a solid, (polygon is bound part of
a plane, representing the 2nd dimension).
Schläfli, his graphs
and symbols
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In 1852, Ludwig Schläfli
(1814-1895), a Swiss
mathematician published a book
Theorie der vielfachen Kontinuität,
(Theory of Continuous Manifolds),
in which he wrote about the four
dimensions
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Schläfli looked at Elementa
doctrinae solidorum and !
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Schläfli (1852) showed that this
formula is also valid in four
dimensions or indeed any higher
dimension.
it’s elementary
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The mathematical description
of generating the fourth (and
higher dimensions) was first
given in an elegant way by
William Stringham (Stringham,
1880: 1):
!
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‘let such angle be called
elementary’…
back to school with the
schoolmaster Abbott
(1838-1926)
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Flatland
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Flatland is a land that is flat. It is
(Abbott, 1884: 2):
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like a vast sheet of paper on
which straight Lines, Triangles,
Squares, Pentagons, Hexagons,
and other figures, instead of
remaining fixed in their places,
move freely about, on or in the
surface, but without the power
of rising above or sinking below
it, very much like shadows –
only hard and with luminous
edges – and you will then have a
pretty correct notion of my
country and countrymen…
how to pose rhetorical questions?
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place of women in society
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nature of space and time
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can we visualise different dimensions?
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how can we understand them?
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are there any higher dimensional beings we can interact
with?
back to fractals
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Mandelbrot coined the term
‘fractal’ in 1977
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fractal relates to fractional
dimension
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the introduction of this notion
means the suggestion of the
consideration of a dimension
as a continuous quantity
ranging from 0 to infinity
some considerations on pedagogy
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history of mathematics is not something that is interesting
because you have the knowledge of detail, but because
you start understanding mathematics in more detail (a
process that may go on and on)
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some understanding of a mathematical process that
could get the students to speculate on its future
development, something that shows the process of
invention is a tool that develops creativity
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history of dimensionality can be accessed at any point
(literally and figuratively)
finally
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To contextualise and engage
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To make meaningful and build a network
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To learn new mathematics via an old thing
!
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Thank you!
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snezana@mathsisgoodforyou.com