Friday, October 30, 2009



This is an open question: What is this fractal? It's a method for filling a 2D plane with circles in an orderly way - circles made of circles, all the way down. There are published examples of similar systems, like the Apollonian Gasket, the Kleinian Groups, Indra's Pearls, but I've never seen this particular arrangement before, and I've been looking for over ten years. Is this a trivial variation on something already known? Or a new and undiscovered thing? I have no idea. I found it while doodling in math class.

fractal sketch

As a teenager in the 90s, my brain was warped and wrinkled by rave flyers, computer graphics, and pop science. I remember zooming in and out of the Julia Set on an Amiga 500 while still in middle school. In high school, these patterns were reinforced with heavy doses of techno music, Mondo 2000, and chaos theory. Hence the embarassing caption to the drawing above, which was made in college calculus class. At the University of Maryland, they wouldn't let anyone enter the architecture major without a B or higher in calculus. It took me three tries. If someone had sat me down and said "look, calculus is all about the always-imossible reconciliation between the grid and the curve, a constant becoming that has to get infinitismally small before its realization," then, I might've got it the first time round.


When the architecture faculty introduced us to Autocad, one of the first things I did with it was to try drawing this thing accurately. The basic rule is to draw a circle around every intersection, which creates four more intersections, and so four more circles, always smaller than the last round. What I couldn't figure out in sketchbooks was whether or not the space left over would be circular too, and whether the centers of the circles were really precisely at the intersections, or shifted slightly. These questions were related - shifting the centers made for circular voids, but I couldn't decide if that was breaking too many rules, or if the whole thing really hung together like it seemed to.

FRACTAL cropped2

The trick that made it work was an inversion - in Autocad, it's impossible to draw the filled circles around the intersections first, without knowing where the centers really are, but it's possible to draw the voids first and work backwards from there. One of Autocad's many methods for drawing circles is as a three point snap. For any three points, there is one and only one circle that hits all of them - circumcircles, again. And luckily, one of the Object Snap (OSNAP!) settings is tangent to a curve; this means that you can easily draw a circle that's tangent to three other curves, without doing the math. Not that this drawing was easy, the image at the head of this post is a print from a .dwg with over 10,000 objects, the iterations are 9 levels deep.


This view of a hand drawn version shows the two types of circles, filled and void. The whole pattern could be made from either all filled, or all empty; the other kind is emergent from the constant reiteration of the one. The trick of drafting the thing by hand is again all about finding the right tools, and then reckoning. For the larger circles here, I improvised with a thumbtack and a piece of wire, the intermediate were drawn with a compass, the smaller were done with a circle template (always tricky to find one at the right radius), and the very smallest were filled in freehand.

rose cross process drawing 1

The hardest part of building the thing by hand is to find a center and a radius that will hit all of the adjacent circles at a tangent. In these drawings, I'm trying again and again to find some kind of geometric or proportional rule that will locate the next center point, and while there were a few promising patterns at the larger scale, they all broke down eventually, leaving trial and error as the only way through, but stll it works. That's why this is partially a plea to anyone else who might have more quantitative information about these geometries: help put this in context. What kind of thing is it?

2009-09-08 13.26.39

(for more process drawings, click here)


Dave said...

I'm afraid I can't help with the math, but I think Rhino's Circle from 3 Tangents command would help to construct one of these fairly simply.

Dave said...

Sorry, didn't read carefully enough to realize that you had already done exactly that.

Anonymous said...

Have you looked at the book Indras Pearls? I don't have it handy right now, but the fractals from it that are on the web seem awfully similar to yours.
For creating such fractals, I highly recommend Cinderella, here is a page about Indra's Pearls made with Cinderella.

Y said...

An Apollonian gasket?

Anonymous said...

If you rotate this by 45°, you are getting close.

sevensixfive said...

It's related, for sure, but I don't think it's the apollonian. I still haven't found an apollonian that displays the filled circles as well as the circular voids. Also, most of the apollonians show threefold symmetry, this has fourfold. Again, it might be just a trivial transformation, but I have no way of knowing.

Some of the Indra's Pearls fractals are really nice, and maybe even closer tot he mark.

fake is the new real said...

Can you do you have drawings showing three or four iterations of the figure? Right now it's hard to tell what circles belong to what step, so I'm having trouble calculating the fractal dimension.
I think that the tip that it's a form of Apollonian is on point. You might want to look at Ford Circles, for an example of how the geometry can work within non-circular bounds.

Anonymous said...

It seems that your rules are not consistent, which is probably what is leading to your confusion.
Initially you have 4 curves, 0 intersections, so the rule of "draw a circle around every intersection" can't be applied to your starting conditions.
If you can figure your initial objects and define the next step in terms of those, then you should be able to iterate multiple levels and see if it matches.
Hope, that helps.

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David Dumas said...

Your image is the orbit of a circle under a Kleinian group generated by two Mobius transformations (one elliptic of order 4, one parabolic).

I wrote up a short explanation that includes a version of your image with more circles. (You can also get just the image.)

sevensixfive said...

Hey David, thanks a lot, the image on that PDF is really deeply iterated.

I have to admit that the math is over my head, but I'm glad to know that it does rationalize into something algorithmically reproducible. Very cool.

Dimensions said...

I can show you a very easy way to draw this arrangement of circles.
Please contact me privately ( see the contact page on )

Anonymous said...

Nice hand drawing, sevesixfive.

One of the answers is that with two generator kleinian group as Dave posted.
The other easy one to understand is that with five circle inversions as Dimension posted.
You can see the five circle and its limit set below.

You may try this configuration with the tool below.

Hope this help.

Anonymous said...

Not five circles, but six circles.

BTW, Limit set of five circle inversions can be found below.

sevensixfive said...

Is this Jos? I really dig your site, hadn't seen that you have an example of this thing, too - cool to see someone else get to the same spot from a different direction.

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Anonymous said...

BTW, same image can be found at Fig 11.5 "Three pairs of tangent circles" in page 365 of "Indra's Pearl" as an example of limit sets of three generator kleinian group.

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Also, most of the apollonians show threefold symmetry, this has fourfold. Again, it might be just a trivial transformation, but I have no way of knowing. tava tea error fix Internet Hosting jump higher Electronic Cigarette

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I was really into fractals in college (I know...) when I was making rave flyers (I know!) for a friend's parties in Iowa (I know! I know! Shut up already!). Anyway, the thing that I really used to love doing with this fractal application that I had on my computer was zooming in to different parts of the familiar Mandelbrot set as far as I could. I never got very far...between 5 or 6 zooms in, my Packard Bell 486/66 (running Windows 3.11) would buckle under the computational pressure and hang.

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Rafael Miranda Molina said...

This is a very impressive construction, i'm wondering how exactly did you finally construct this circles, since you talk about 9 levels of iteration.

I know almost nothing about CAD applications, so maybe it is posible to iterate geometric procedures there... is it?

Anyway, you leave a open question, and my answer might sound rather simple, since you seem to know a lot about this design. I think it's an extension of the Apollonian Gasket, i the sense that it seems equivalent in it's formation, except that you start from four tangent circles.

You could keep extending this construction, to create other fractals, given 5, 6, etc .... tangent circles.

Anyway, you are in the right track when you mention and Kleinian Groups, though it seems to me that the Kleinian aproach is more related to packing problems, than the description of this shape itself. Actually, Poincare (who named these groups) and Klein, did not get to know fractal geometry.

Anyway, i would say that this construction and the Apollonian gasket, are part of a fractal family which has one thing in common: you start with "n" circles, tangent externally with each other, and internally to a bigger one, and then you recursively pack circles in the space they leave. That shape IS a fractal indeed.

Finally (this is a large comment, sorry), i find very, VERY interesting what you did with this post. You present this interesting construction and talk about how you did it and such things... that kind of mathematical blogging, to me, is the gold.

I try to do the same, every now and then, in my site: , i'm a chilean math teacher.

Anyway, congratulations for this and keep geo-blogging, it's really something.

Greetings from Chile

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