Presentation about Fractal for PON course "Looking into fashion: from fractals to tailoring" Liceo Scienticio Siciliani 3 Maggio 2018

1. Anna Alfieri
Rosalia Imeno
FRACTALS1
Looking into fashion:
from fractals to tailoring

2. What are Fractals?
 Fractal definition from MathWorld
– A fractal is a geometrical object or quantity that
displays self-similarity, in a somewhat technical
sense, on all scales.
– Fractals don’t need to exhibit exactly the same
structure at all scales, but the same "type" of
structures can appear on all scales.
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3. Koch snowflake
Given an equilateral triangle, we divide each side into three equal
parts, we eliminate the central part and on it we build an equilateral
triangle.

4. Self-Similarity Property of Fractal
 Self similarity across scales
– As one zooms in or out the geometry/image has a
similar (sometimes exact) appearance
– Types of self-similarity
 Exact self similarity
 Approximate self similarity
 Statistical self similarity
4

5. Exact Self-Similarity
Koch snowflake
5

6. Exact Self-Similarity
6

7. Approximate Self-Similarity
 Structures that are recognizably similar but not
exactly
– More common type of
self-similarity
– Example: Mandelbrot set
ITEPC 06 - Workshop
on Fractal Creation7

8. Statistical Self-Similarity
 Irregularity is the same on the average
– Example: coastline
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9. Self-Similarity in Real World
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10. Task (in small group)
Build a powerpoint where you describe:
1.What a fractal is
2.A geometrical example
3. Five examples of fractals in the world around us
Which are the most important properties of a fractal?

11. A fractal has the following
features:
1. It has a fine structure at small
scales.
2. It is too irregular to be easily
described in traditional
Euclidean geometric language.
3. It is self-similar (at least
approximately)
4. It has a Hausdorff dimension
which is greater than its
topological dimension
5. It has a simple and recursive
definition.
The term "fractal" was coined by Benoît Mandelbrot
1975 and is derived from the Latin fractus meaning
"broken" or "fractured."

12. ferns, trees
Fractal shape in the world…
cancer cells
lungs and nervous
system heart

13. From traditional geometry
To fractal geometry

14. [1] M.Barnsley, Fractal Everywhere, AP Professional (1988)
[2] S.Bercia – G.Dragoni – G.Gottardi, Dizionario biografico degli Scienziati,
Zanichelli – Le Scienze (1999) CD-ROM
[3] P.Brandi – R.Ceppitelli – A.Salvadori, Un’introduzione Elementare alla
Modelliz-zazione Matematica, Università degli Studi di Perugia (2000)
[4] P.Brandi - L.Lotti – A.Salvadori, Un'introduzione elementare alla
modellizzazione frattale, Atti Convegno Internazionale Gian Carlo Rota Memorial
Conference, Barisciano (AQ) (2002) 21-34
[5] P.Brandi – A.Salvadori, (a) Sull’istituzione di percorsi multidisciplinari di
approfondimento per il conseguimento di crediti formativi, Atti Convegno Nazio-nale
Mathesis, L’Aquila (1998) 75-78
(b) Un approccio alla modellizzazione matematica: i problemi di ottimizzazione, Atti XX
Convegno Nazionale UMI-CIM, Orvieto (1998)
(c) Una proposta concreta di innovazione didattica tra Scuola ed Università, Atti II
Convegno Nazionale ADT (2000) CD-ROM
I parte, Lettera Matematica PRISTEM, 43 (2002) 17-23
II parte, Lettera Matematica PRISTEM, 44 (2002) 55-61
(e) Modelli Matematici Elementari, I&II, Università degli Studi di Perugia (2002)
Frattali Usati:
Edgar “ Measure,Topology, and Fractal Goemetry” pag 19,30,164
Kevin Lee “ Fractal Attraction”
Gary Flake “ the computational Beauty of nature”pag 109,110