Africa (Fractal)

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]

Definition

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

$A = (a_{1}, a_{2}) = (0, 0)$	$B = (b_{1}, b_{2}) = (2, 0)$
$C = (c_{1}, c_{2}) = (3, \frac{16}{3})$	$D = (d_{1}, d_{2}) = (4 - \sqrt{5}, \frac{16}{3})$
$E = (e_{1}, e_{2}) = (\frac{9 - 3 \sqrt{5}}{2}, \frac{8 \sqrt{5} + 8}{3})$	$F = (f_{1}, f_{2}) = (\frac{- 3 + \sqrt{5}}{2} - \frac{\sqrt{5} - 3}{2 - \sqrt{5}}, \frac{8 \sqrt{5} + 8}{3})$
$G = (g_{1}, g_{2}) = (- 1 - \frac{\sqrt{5} - 3}{2 - \sqrt{5}}, \frac{16}{3})$	$H = (h_{1}, h_{2}) = (- 1, \frac{16}{3})$

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

$c_{1} - b_{1} = a_{1} - h_{1}$

$d_{1} - e_{1} = f_{1} - g_{1}$

$c_{2} - b_{2} = h_{2} - a_{2}$

$e_{2} - d_{2} = f_{2} - g_{2}$

$\frac{b_{1} - a_{1}}{c_{1} - d_{1}} = \frac{c_{1} - b_{1}}{d_{1} - e_{1}} = \frac{c_{2} - b_{2}}{e_{2} - d_{2}} = \frac{1 + \sqrt{5}}{2} = φ$

$\frac{2 (h_{1} - g_{1})}{φ} + \frac{b_{1} - a_{1}}{φ^{2}} = e_{1} - f_{1}$

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

↑ ^1.0 ^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.
↑ 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/.
↑ 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/.
↑ "Fractal Africa". The De Morgan Forum – London Mathematical Society. September 21, 2016. http://education.lms.ac.uk/2016/08/hamid-naderi-yeganeh-fractal-africa/.
↑ "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.
↑ Antonick, Gary (10 August 2015). "John Conway’s Wizard Puzzle". The New York Times. https://wordplay.blogs.nytimes.com/2015/08/10/feiveson-1/.
↑ Alexandre Borovik (2016-08-25). "Hamid Naderi Yeganeh: Fractal Africa" (in en). https://demorgangazette.wordpress.com/2016/08/25/hamid-naderi-yeganeh-fractal-africa/.
↑ Hamid Naderi Yeganeh, Africa-Fractal, http://archive.org/details/Africa-Fractal, retrieved 2022-06-23

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[articleguardian-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/.

[London-Mathematical-Society-ref-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

v t e 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