Apollonian-gasket

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The Apollonian gasket is a fractal constructed from a triple of circles, where each circle is tangent to the other two. Each level continues this pattern, adding 2·3more circles on the nth level of the gasket, for a total of 3n+1 + 2 circles after n stages. Repeating this process and taking the limit gives an object like the gasket pictured above on the left.

The Apollonian gasket is also closely related to the undirected graph known as the Apollonian network. The network can be created by first taking three tangent circles, inscribing a circle in the gap created by the three circles, and continuing this process, and then giving each circle a vertex and each pair of tangent circles an edge. This process is seen in the second picture above which shows how it is related to the gasket, and the construction leads to the object pictured on the right.

Pretty interesting relation between the continuous fractal and the discrete graph!

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Apollonian Gasket Variations by Fdecomite

In mathematics, an Apollonian gasket or Apollonian net is a fractal generated from triples of circles, where each circle is tangent to the other two. It is named after Greek mathematician Apollonius of Perga.

An example of a simple Apollonian Gasket:

External image

By viewing the example, it may be simple to deduce the construction of an Apollonian Gasket, however, instructions can be found here in this wonderful Vihart video. Have fun creating your own Apollonian Gaskets!

There’s another fractal hiding in one of these gaskets. Can you spot it?

Oh my God, I finally did it! Using Sage, I made an IFS to make the Apollonian gasket with 2 circles.

Program:

import random

image = Graphics()

pts = 1

its = 5000

a = matrix([[1,0],[-2j,1]])

b = matrix([[1-1j,1],[1,1+1j]])

A = a^-1

B = b^-1

z0 = 1

for x in range(pts):

    z = z0

    for y in range(its):

        r = random.random()

        if r <= 0.25:

            z = (a[0,0]*z+a[0,1])/(a[1,0]*z+a[1,1])

        elif 0.25<r<=0.5:

            z = (b[0,0]*z+b[0,1])/(b[1,0]*z+b[1,1])

        elif 0.5<r<=0.75:

            z = (A[0,0]*z+A[0,1])/(A[1,0]*z+A[1,1])

        elif 0.75<r<=1:

            z = (B[0,0]*z+B[0,1])/(B[1,0]*z+B[1,1])

        image += point((real(z), imag(z)), rgbcolor=(0,0,1), size = 1)         

show (image, aspect_ratio = 1)

I will warn you: If you use that, change ‘its’ to a number comparable to your computer’s power.

WHAT IT DOES:

We have two matrices, a and b. We have their inverses, respectively- A and B. These have complex entries.

For a matrix

[a,b]

[c,d]

we transform a complex z to az+b/cz+d. Each time we choose to transform by a random matrix, and plot the point. Thank you, Indra’s Pearls, I finally understand.

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Coincidence

Out for a walk (and to see if I could find any more interesting signs to acquire) and I saw this beam probably cast iron, it is remarkable how much of a beam can be cut away and thus reduce its weight without much decreasing its strength. so we have this with circles and triangle spaces cast in it.  Form follows function but it may still have aesthetics included. 

Second photograph there is another sort of gasket (there are actually dozens) like the Sierpinski gasket I so love this one is called the Apollonian gasket and is formed from packing a circle with smaller ones, to some rule, in my case largest three similar you can fit in.

Relatively easy to draw or program on a computer but long winded to make. These are 10 cms across not particularly complex and have 40+ elements in each, need to make a third and then group them and infill the gaps.

I do see this as a maquette perhaps each smaller element getting taller so as to produce a 3D tower, may try it in copper sheet, but would need to practise my soldering.

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I’ve just been doodling with Apollonian gasket fractals, and I’ve ended up with these on my psychology case study notes.

If I hadn’t finished studying for the day, I would probably consider this a testament to my lack of productivity. But seeing as I have finished studying for the day, I consider it a testament to my lack of productivity in general!

If you take the Elder Wand out of the Deathly Hallows, you get an incomplete Apollonian Gasket

An Apollonian Gasket is a fractal generated from triples of circles (or triangles, or even elephants) which, when complete, represents the smallest countable form of infinity. 

So I suppose you could say that the Deathly Hallows is a form of infinity.

It goes on forever.

Always. 

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Apollonian Gasket

or sometimes you just have to do something mindless and then you wonder why you ever started it, because you feel obliged to finish it.

The Apollonian gasket is a fractal way of filling a circle with smaller ones to some rule I don’t know if there is a preferred rule but mine was fill the circles with the three largest that will fit inside and then put the largest one you can to infill and so on.  I have seen them done with the two largest and with four…

Drawing such a gasket is quite time consuming even if you know the Descartes theorem that tells you the circumferences of the filler circles so these days most of those you see were done on a computer.  You do occasionally see hand cut paper ones, this one is made from the bottom up start with sub-circles in threes and then put the circumferential circle round them and so on. Obviously cut paper ones are even more time consuming than drawn ones.

This one comprises about 150 elements the next level up would be three of these plus circumferential circle plus all the infill so you will understand that I am reluctant to commit myself to it.

Learning points

  • I still have in mind that this might translate to another material and three dimensional, not spherical, that would be lovely but too difficult to do with the materials and equipment at my disposal; I think columnar with the smaller circles being proportionately taller and possibly different material or colour.
  • I have not played with Google Sketch up for a bit and it or 3DS Max might lend itself to such design.
  • Of course rolling paper round a dowel would also work 
  • If all the elements were the same height from one angle you would see a paper column and from the other the gasket.

I have some things to think about you could say I have some thingking to do. 

Thingking (Noun, neologism, portmanteau word from things and thinking) 

Isn’t English a lovely language, you can invent words if you want to and if enough people use it then the OED will remember it for ever.

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I want her in my math class!!