Africa (fractal)

Fractal Africa

Africa is a fractal made of a set of octagons that have some resemblance to the shape of Africa.[1][2][3] The number of octagons of different sizes in the fractal is related to the Fibonacci sequence: 1, 2, 3, 5, 8, 13, 21, .... The height of the largest octagon of the fractal is φ times longer than that of the second octagon; where φ is the golden ratio.[4][5][6]

## Contents

* 1 Definition
* 2 Tessellation
* 3 Properties
* 4 References

## Definition

The original version of the fractal was defined by the octagon with the following vertices:[7]

A=(a1,a2)=(0,0)

B=(b1,b2)=(2,0)

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C=(c1,c2)=(3,163)

D=(d1,d2)=(4−5,163)

E=(e1,e2)=(9−352,85+83)

F=(f1,f2)=(−3+52−5−32−5,85+83)

G=(g1,g2)=(−1−5−32−5,163)

H=(h1,h2)=(−1,163)

The following relations are obvious and necessary based on the shape of fractal:

c1−b1=a1−h1

d1−e1=f1−g1

c2−b2=h2−a2

e2−d2=f2−g2

b1−a1c1−d1=c1−b1d1−e1=c2−b2e2−d2=1+52=φ

2(h1−g1)φ+b1−a1φ2=e1−f1

where φ is the golden ratio. The fractal is made up of a countable number of copies of the octagon and its lateral inversion. The octagon has some resemblance to the shape of Africa:

## Tessellation

The shape of the fractal can form the following tessellations:[8]

*   *

## Properties

In the fractal, the number of octagons of each size (in order of size) is the Fibonacci sequence from the second term: 1, 2, 3, 5, 8, 13, 21, .... The number of isosceles triangles of each size (in order of size) is the Fibonacci sequence from the first term: 1, 1, 2, 3, 5, 8, 13, 21, ....[1]

*   *

## References

1. ↑ 1.0 1.1 Bellos, Alex (February 24, 2015). "Catch of the day: mathematician nets weird, complex fish". The Guardian. https://www.theguardian.com/science/alexs-adventures-in-numberland/2015/feb/24/catch-of-the-day-mathematician-nets-weird-complex-fish.
2. ↑ Ouellette, Jennifer (February 28, 2015). "Physics Week in Review: February 28, 2015". Scientific American. https://blogs.scientificamerican.com/cocktail-party-physics/physics-week-in-review-february-28-2015/.
3. ↑ Chung, Stephy (September 18, 2015). "Next da Vinci? Math genius using formulas to create fantastical works of art". CNN. http://www.cnn.com/2015/09/17/arts/math-art/.
4. ↑ "Fractal Africa". The De Morgan Forum – London Mathematical Society. September 21, 2016. http://education.lms.ac.uk/2016/08/hamid-naderi-yeganeh-fractal-africa/.
5. ↑ "Importing Things From the Real World Into the Territory of Mathematics!". HuffPost (blog). http://www.huffingtonpost.com/hamid-naderi-yeganeh/importing-things-from-the_b_8111912.html.
6. ↑ Antonick, Gary (10 August 2015). "John Conway’s Wizard Puzzle". The New York Times. https://wordplay.blogs.nytimes.com/2015/08/10/feiveson-1/.
7. ↑ Alexandre Borovik (2016-08-25). "Hamid Naderi Yeganeh: Fractal Africa" (in en). https://demorgangazette.wordpress.com/2016/08/25/hamid-naderi-yeganeh-fractal-africa/.
8. ↑ Hamid Naderi Yeganeh, Africa-Fractal, http://archive.org/details/Africa-Fractal, retrieved 2022-06-23

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Fractals

Characteristics|

* Fractal dimensions
* Assouad
* Box-counting
* Correlation
* Hausdorff
* Packing
* Topological
* Recursion
* Self-similarity

Iterated function
system|

* Barnsley fern
* Cantor set
* Koch snowflake
* Menger sponge
* Sierpinski carpet
* Sierpinski triangle
* Space-filling curve
* Blancmange curve
* De Rham curve
* Dragon curve
* Koch curve
* Lévy C curve
* Peano curve
* Sierpiński curve
* T-square
* n-flake

Strange attractor|

* Multifractal system

L-system|

* Fractal canopy
* Space-filling curve
* H tree

Escape-time
fractals|

* Burning Ship fractal
* Julia set
* Filled
* Newton fractal
* Lyapunov fractal
* Mandelbrot set
* Misiurewicz point
* Newton fractal
* Tricorn
* Mandelbox
* Mandelbulb

Rendering techniques|

* Buddhabrot
* Orbit trap
* Pickover stalk

Random fractals|

* Brownian motion
* Brownian tree
* Fractal landscape
* Lévy flight
* Percolation theory
* Self-avoiding walk

People|

* Georg Cantor
* Felix Hausdorff
* Gaston Julia
* Helge von Koch
* Paul Lévy
* Aleksandr Lyapunov
* Benoit Mandelbrot
* Lewis Fry Richardson
* Wacław Sierpiński

Other|

* "How Long Is the Coast of Britain?"
* Coastline paradox
* List of fractals by Hausdorff dimension
* The Beauty of Fractals (1986 book)
* Fractal art
* Making a New Science (1987 book)
* The Fractal Geometry of Nature (1982 book)
* Kaleidoscope
* Chaos theory

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