fractal

Pictures of Math VII

This post was inspired by an article I read on sciencealert.com.

Simon Beck’s Snow Art

(source)

Simon Beck is the world’s first snow artist. Each pattern takes him 11 hours, and he uses nothing more than a compass and his snowshoes. He chooses to draw maths due to the simplicity of the patterns.

Fabergé Fractals

UK physicist Tom Beddard decided to create digital renderings of 3D Fabergé eggs covered in fractal patterns.

“The formulae effectively fold, scale, rotate or flip space. They are truly fractal in the fact that more and more detail can be revealed the closer to the surface you travel.” – Beddard

For more read here.

3D Models by Henry Segerman

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Henry Segerman is an Australian mathematician who creates 3D printed models that express mathematical formulae and concepts.  This allows his students to better understand them.

“The language of mathematics is often less accessible than the language of art, but I can try to translate from one to the other, producing a picture or sculpture that expresses a mathematical idea.” 

Hevea Project

A French team of mathematicians called the Hevea Project, have created digital constructions of isometric embeddings.

“Take a sphere – say the surface of a tennis ball – and imagine shrinking it down to have a nanometre radius,” writes Daniel Matthews about isometric embedding. “Nash and Kuiper show that by ‘ruffling’ the surface sufficiently (but always smoothly; no creasing or folding or ripping or tearing allowed!) you can have an isometric copy of your original tennis ball, all contained within this nanometre radius.”

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Mandelbrot and Julia Sets

In today’s post I wanted to quickly highlight a cool relationship between Mandelbrot and Julia sets.

Consider the function, which depends of complex parameter z:

f(z) = x2 z 

Fixing this z, f(z) defines a map from the complex plane to itself. We can start from any value of x and apply this function over and over, which would give us a sequence of numbers. This sequence can either go off to infinity, or not. The boundary of the set of values of x where it doesn’t is the Julia set for this particular z, which we fixed initially.

Conversely, starting with x = 0, we can draw the set of numbers for which the resulting sequence does not go off to infinity. This is called the Mandelbrot set. (Note the subtle difference between the two).

Okay, so the cool relationship is that, near the number z, the Mandelbrot looks like the Julia set for the number z, or as Wikipedia describes:

“There is a close correspondence between the geometry of the Mandelbrot set at a given point and the structure of the corresponding Julia set.”

To illustrate this, consider the following Julia set:

relationship_between_mandelbrot_sets_and_julia_sets-1

Zooming into the Mandelbrot set at the same value of z gives us this image:

Relationship_between_Mandelbrot_sets_and_Julia_sets.PNG

They are extremely similar! So, essentially, the Mandelbrot sets looks like a lot of Julia sets! (Click here to explore this in more detail).

This amazing result is used in lots of results on the Mandelbrot set, for example, it was exploited by Shishikura to prove that “for a dense set of parameters in the boundary of the Mandelbrot set, the Julia set has Hausdorff dimension two, and then transfers this information to the parameter plane“.

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Dragon Curve

The dragon curve is a recursive non-intersecting curve whose name derives from its similarity to a dragon. There are many different types of dragon curves.

Heighway Dragon

The Heighway Dragon was first investigated by physicists at NASA John Heighway, Bruce Banks and William Harter.

The construction of the Heighway Dragon can be written as a Lindenmayer System with

  • Angle 45°
  • Axiom FX (Initial String)
  • String Rewriting rules
    • X —> X+YF+
    • Y —> −FX-Y

where  ‘F‘ means ‘draw forward’, ‘+’ means ‘turn 45° clockwise’, and ‘‘ means ‘turn 45° anticlockwise’. X and Y do not correspond to any drawing action and are only used to control the evolution of the curve. This notation will be used for the rest of the post.

In other words, the Heighway dragon is constructed by replacing a line segment with two segments “with a right angle and with a rotation of  45° alternatively to the right and to the left“.

Recursive construction of the curve

Twindragon

The twindragon can be constructed by joining two Heighway dragon curves back-to-back, so one is rotated by 180°.

It can also be written an a Lindenmayer system by adding another section in the initial string: FX+FX+.

Terdragon

The terdragon was first introduced by Davis and Knuth in a paper on dragon curves in 1970. It can be described by the following Lindenmayer system:

  • Angle 120°
  • Axiom F (Initial String)
  • String rewriting rules
    • FF+F−F

Lévy Dragon

Finally, the Lévy dragon is a curve studied by Paul Lévy as part of a general study of self-similar curves (see Fractals). This was motivated by the work of Helge von Koch and the Koch Curve. It is also known as the Lévy C curve.

Its Lindenmayer system can be described as follows:

  • Angle 120°
  • Axiom F (Initial String)
  • String rewriting rules
    • F ↦ +F–F+

Sources: 1 | 2 | 3 | 4

Happy New Year everyone!! M x

The Sierpinski Triangle

The Sierpinski Triangle, or Sierpinski Sieve, is a fractal described by Polish Mathematician Sierpinski in 1915, although it appeared in Italian art from the 13th century. It has an overall shape of an equilateral triangle, and is subdivided recursively into smaller equilateral triangles.

Sierpinski sieve from rule 90

Source: Wolfram Mathworld

Constructing a Sierpinski Triangle

STEP 1:

Start with an equilateral triangle.

STEP 2:

Connect the midpoints of each side, hence dividing it into 4 smaller congruent equilateral triangles.

Triangle

STEP 3:

Now cut out the triangle in the centre.

Step One

STEP 4:

Repeat steps 2 and 3 with each of the remaining smaller triangles.

[Sierpinski Triangle]

Properties

 

If we let N_n be the number of black triangles after iteration nL_n be the length of a side of a triangle, and A_n be the fractional area which is black after the nth iteration, then:

Screen Shot 2016-11-27 at 6.53.59 PM.png

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