Mandelbrot and dimensionality. Benoit B Mandelbrot aka Snezana Lawrence

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1 Mandelbrot and dimensionality Benoit B Mandelbrot aka Snezana Lawrence snezana@mathsisgoodforyou.com s.lawrence2@bathspa.ac.uk

2 benoit_mandelbrot_fractals_the_art_of_roughness? language=en#t-71100

$Inventor of fractal geometry, theory of roughness, and selfsimilarity in nature Discovery of the Mandelbrot set, based on Julia set An uncle who was a mathematician, Szolem Mandelbrojt (early$

3 what does B stand for in Benoit B Mandelbrot? 20th November th October 2010 Polish-born Jewish, brought up in France, lived his later years in America. Inventor of fractal geometry, theory of roughness, and selfsimilarity in nature Discovery of the Mandelbrot set, based on Julia set An uncle who was a mathematician, Szolem Mandelbrojt (early member of Bourbaki group, )

4 lessons 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? a lesson from a young child, finding himself in Paris in 1940s 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?

5 how do you come across something interesting? Iteration, sets, and functions 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. PDF/ JMPA_1918_8_1_A2_0.pdf

7 how long is the coastline of Britain? self-similarity: a self-similar object is exactly or approximately similar to a part of itself 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

$is 1, of a square is 2, of a cube is 3 Hausdorff dimension of fractals is formalised by scale factor (3) and selfsimilar objects (4),$

8 Sierpinski carpet and Koch curve ai+1=8 9 ai. So ai=(8 9)^i, which tends to 0 as i goes to infinity 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 Hausdorff dimension of a single point is 0, of a line is 1, of a square is 2, of a cube is 3 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

9 some more maths cardinality of sets, cardinality of infinite sets can the part be the same size as the whole? What about Hilbert s hotel? 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)?

10 between the friends is learning a dialogue? Plato his five elements his ideal world his way of teaching his academy

11 what is the world like? a page from the 1519 edition of Aristotle s De Caelo (On the heavens). Augsburth, Sigmund Grimm and Max Wirsung. 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. Outside the spheres is the realm of the Prime Mover, represented here by a winged figure.

12 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). For further reference on Aristotle s mention of the dimensionality in other works, see (Cajori 1926)

13 others on dimensions Ptolemy denied and disproved it but nevertheless mentioned and contemplated upon it (Cajori, 1926: 397; Heiberg 1893: 7a, 33) John Wallis, although writing this whilst considering geometric interpretations of quantities he was developing in the context of algebra, wrote (Wallis 1685:126): 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)

14 stairways to heaven 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?

15 getting out of one own s plane Zöllner (Johann Friedrich, ) further simplified this in 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?! His work Uber Wirkungen in die Ferne (On effects at a distance) a happy coincidence for a cultural reference (Goethe s poem Effect at a Distance wiki/ The_Works_of_J._W._von_Goethe/ Volume_9/Effect_at_a_Distance

angles and the number of faces is equal to the number of edges add 2 We now usually denote Euler s characteristic by Greek letter chi

16 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 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 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.

17 looking at it from another perspective 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 In 1852, Ludwig Schläfli (1814-1895), a Swiss mathematician

he wrote about the four dimensions Schläfli looked at Elementa doctrinae solidorum and!

18 Schläfli, his graphs and symbols In 1852, Ludwig Schläfli ( ), a Swiss mathematician published a book Theorie der vielfachen Kontinuität, (Theory of Continuous Manifolds), in which he wrote about the four dimensions Schläfli looked at Elementa doctrinae solidorum and! Schläfli (1852) showed that this formula is also valid in four dimensions or indeed any higher dimension.

23 it s elementary The mathematical description of generating the fourth (and higher dimensions) was first given in an elegant way by William Stringham (Stringham, 1880: 1):! let such angle be called elementary

back to school with the schoolmaster Abbott (1838-1926) Flatland Flatland is a land that is flat.

figures, instead of remaining fixed in their places, move freely about, on or in the surface, but without the power of rising

25 back to school with the schoolmaster Abbott ( ) Flatland Flatland is a land that is flat. It is (Abbott, 1884: 2): 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

27 how to pose rhetorical questions? place of women in society nature of space and time can we visualise different dimensions? how can we understand them? are there any higher dimensional beings we can interact with?

$back to fractals Mandelbrot coined the term fractal in 1977 fractal relates to fractional dimension the introduction of$

28 back to fractals Mandelbrot coined the term fractal in 1977 fractal relates to fractional dimension the introduction of this notion means the suggestion of the consideration of a dimension as a continuous quantity ranging from 0 to infinity

29 some considerations on pedagogy 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) 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 history of dimensionality can be accessed at any point (literally and figuratively)

30 finally To contextualise and engage To make meaningful and build a network To learn new mathematics via an old thing! Thank you! snezana@mathsisgoodforyou.com

Fractals. Justin Stevens. Lecture 12. Justin Stevens Fractals (Lecture 12) 1 / 14

Fractals. Justin Stevens. Lecture 12. Justin Stevens Fractals (Lecture 12) 1 / 14 Fractals Lecture 12 Justin Stevens Justin Stevens Fractals (Lecture 12) 1 / 14 Outline 1 Fractals Koch Snowflake Hausdorff Dimension Sierpinski Triangle Mandelbrot Set Justin Stevens Fractals (Lecture