ActiveState®, Komodo®, ActiveState Perl Dev Kit®, at the bottom of the page there is a link to Curt McMullen's software called "lim" that draws fractals in postscript format. However, there is one equation I have not understand. In this definition we have used the Anemone plugin to model the Apolloian fractal. An Apollonian gasket can also be constructed by replacing one of the generating circles by a straight line, which can be regarded as a circle passing through the point at infinity. That equation comes from that. An Apollonian gasket can be constructed as follows. It simply randomly finds 3 tangent circles at each iteration and tries to add new circles. This is not the standard way though. The good thing is it can start w/ any arbitrary configuration of main circles, unlike the standard way. The code is split into the following files: apollon.py contains all the pure math stuff; coloring.py contains helpers for color mapping
But I think my idea was instead of starting with a unit circle and trying to figure out how to pack N circles inside of it, I realized the problem would be simpler if the N circles all have centers on the circumference of unit circle (not inside of it). I noticed you assign colors (or images) to different circles, is there any special tips here?I think that paper you found has the solutions for both creating generations and coloring. I am trying to find a way, but cannot figure it out now.Since there are no 3d extension of complex numbers, I am guessing 3d solution must use 4d numbers: That is why I gave up on this method and switched to a much better way later.
Do you have any idea on generating circles by generations, which means generating circles from big to small?I don't think I can explain better without drawings etc also it was years ago for me. © 2020 ActiveState Software Inc. All rights reserved. Alternatively, two of the generating circles may be replaced by parallel straight lines, which can be regarded as being tangent to one another at infinity. But generating them randomly of course is not a good efficient way. Could you explain the equation? In mathematics, an Apollonian gasket or Apollonian net is a fractal generated. Integral Apollonian circle packing defined by circle curvatures of (−6, 10, 15, 19) Named for the Greek mathematician Apollonius of Perga, this type of fractal can be drawn (by … So I use either randomly assigned colors or just create colors using mod operation:a,b,c values selected from 4,8,16,32,64,128 and a<>b,a<>c,b<>cThank you so much for your patience! 3The curvature of a circle (bend) is defined to be the inverse of its radius. Since there is a The Apollonian gasket is the limit set of a group of Möbius transformations known as a Integral Apollonian circle packing defined by circle Integral Apollonian circle packing defined by circle curvatures of (−3, 5, 8, 8) An Apollonian Gasket is a type of fractal image that is formed from a collection of ever-shrinking circles contained within a single large circle. ActiveState Tcl Dev Kit®, ActivePerl®, ActivePython®, These lines are perpendicular to one another, so the Apollonian gasket also has rotational symmetry of degree 2; the symmetry group of this gasket is If all three of the original generating circles have the same radius then the Apollonian gasket has three lines of reflective symmetry; these lines are the mutual tangents of each pair of circles.
If any four mutually tangent circles in an Apollonian gasket all have integer curvature then all circles in the gasket will have integer curvature.it follows that one may move from one quadruple of curvatures to another by If none of the curvatures are repeated within the first five, the gasket contains no symmetry, which is represented by symmetry group Whenever two of the largest five circles in the gasket have the same curvature, that gasket will have If two different curvatures are repeated within the first five, the gasket will have DIf the three circles with smallest positive curvature have the same curvature, the gasket will have The figure at left is an integral Apollonian gasket that appears to have
But I think my idea was instead of starting with a unit circle and trying to figure out how to pack N circles inside of it, I realized the problem would be simpler if the N circles all have centers on the circumference of unit circle (not inside of it). I noticed you assign colors (or images) to different circles, is there any special tips here?I think that paper you found has the solutions for both creating generations and coloring. I am trying to find a way, but cannot figure it out now.Since there are no 3d extension of complex numbers, I am guessing 3d solution must use 4d numbers: That is why I gave up on this method and switched to a much better way later.
Do you have any idea on generating circles by generations, which means generating circles from big to small?I don't think I can explain better without drawings etc also it was years ago for me. © 2020 ActiveState Software Inc. All rights reserved. Alternatively, two of the generating circles may be replaced by parallel straight lines, which can be regarded as being tangent to one another at infinity. But generating them randomly of course is not a good efficient way. Could you explain the equation? In mathematics, an Apollonian gasket or Apollonian net is a fractal generated. Integral Apollonian circle packing defined by circle curvatures of (−6, 10, 15, 19) Named for the Greek mathematician Apollonius of Perga, this type of fractal can be drawn (by … So I use either randomly assigned colors or just create colors using mod operation:a,b,c values selected from 4,8,16,32,64,128 and a<>b,a<>c,b<>cThank you so much for your patience! 3The curvature of a circle (bend) is defined to be the inverse of its radius. Since there is a The Apollonian gasket is the limit set of a group of Möbius transformations known as a Integral Apollonian circle packing defined by circle Integral Apollonian circle packing defined by circle curvatures of (−3, 5, 8, 8) An Apollonian Gasket is a type of fractal image that is formed from a collection of ever-shrinking circles contained within a single large circle. ActiveState Tcl Dev Kit®, ActivePerl®, ActivePython®, These lines are perpendicular to one another, so the Apollonian gasket also has rotational symmetry of degree 2; the symmetry group of this gasket is If all three of the original generating circles have the same radius then the Apollonian gasket has three lines of reflective symmetry; these lines are the mutual tangents of each pair of circles.
If any four mutually tangent circles in an Apollonian gasket all have integer curvature then all circles in the gasket will have integer curvature.it follows that one may move from one quadruple of curvatures to another by If none of the curvatures are repeated within the first five, the gasket contains no symmetry, which is represented by symmetry group Whenever two of the largest five circles in the gasket have the same curvature, that gasket will have If two different curvatures are repeated within the first five, the gasket will have DIf the three circles with smallest positive curvature have the same curvature, the gasket will have The figure at left is an integral Apollonian gasket that appears to have