Rolling fractals

[Tglad]

A guy called Matthen made a nice fractal by rolling circles inside circles: http://blog.matthen.com/post/15481855128/if-you-roll-a-circle-inside-one-3-times-its-size

However I think he got the maths a bit wrong and so the circles are sliding, not rolling. If you make them roll, and you make them all roll at the same speed^* then the resulting inner circle just carves out a circle.

However, if you make each child circle roll the opposite direction to its parent you end up with a fractal similar to Matthens, but different (circles half in radius):



I'm trying to work out if the curve self intersects or not... a close up of the suspect region doesn't make it clear...


^* the most logical choice since half size circles spin twice the angular speed and so cover the same distance in a given time

[Tglad]

You can also stack the circles on the outside (rather than nested), which gives the below fractal when the circles halve in radius. In this case reversing the direction of child compared to parent produces a plane circle, and keeping the direction the same sign produces this cauliflower-like fractal.. so its the opposite of the internal circle one.

[eiffie]

There is also this thread (although the rotations/speed are again different).
http://www.fractalforums.com/non-fractal-related-chit-chat/fun-with-dynamic-spirograph-circle-rolling-around-rolling-circle

[Tglad]

It looks like Benice has got the maths right... I was aiming for
http://1.bp.blogspot.com/-0ADHLWarHII/T-gD_OgEejI/AAAAAAAAA1Q/1avr4AybIY8/s1600/fractal_spirograph_003.gif
and
http://4.bp.blogspot.com/-uwu4QSRQk7M/T-gEJbl5m4I/AAAAAAAAA1o/pR9gW6xy-tc/s1600/fractal_spirograph_006.gif

Perhaps my images above were bad maths... I'll have to take a look.

[Tglad]

Hmm... I looked again at my maths and it still seems correct, now I think that Benice's examples are possibly 'wrong' in the sense of not being a scale symmetric rolling motion.... they're still nice shapes though. If would be great if someone else could try... basically each circle is half the parent's size and each circle rolls at an angular velocity inversely proportional to its radius, so the contact points all move at the same speed.

[eiffie]

Sorry Tglad I have no idea what the right math would be but they are fun in 3d too!
<a href="http://www.youtube.com/v/zn0C80TeMG4&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/zn0C80TeMG4&rel=1&fs=1&hd=1</a>

[kram1032]

Those are beautiful.
I can't say for sure but as far as I can tell, you both got your maths right and he simply did different examples than you did.

[knighty]

Have you considered changing the phases? The initial positions of of the circles haven't to be aligned

BTW, they are related to polynomials P(z) where z=exp(i*t). One can also use "polynomials" whith non integer exponents.

[Tglad]

Yes, you can get twisted results by for example adding a phase offset to each circle.

The first fractal in this post comes in a family, a bit like the multi-brots:

You define them by the ratio of child curvature to parent, in the pic it goes -4, -3, -2,   2, 3, 4.     (-1, 0 and 1 are not fractals)

It looks like all the members of this family don't self intersect (so are a proper embedding on the 2d plane)... probably any curvature ratio with integer magnitude is like this.

[knighty]

Interesting!

It looks like all the members of this family don't self intersect (so are a proper embedding on the 2d plane)... probably any curvature ratio with integer magnitude is like this.

IMHO you could check for self intersection by using the derivative of the associated polynomial along the curve. If it is zero then there is an intersection. This is not easy though because if the curves nearly intersct the magnitude of the derivative would be very small.

[eiffie]

Any ideas on a distance estimate to these?

[cKleinhuis]

i have to interrupt, how qualify those curves as fractal? they might be complex, but self similarity i dont encounter, and infinite lengthes neither nevertheless an interesting method

[Tglad]

Look at the first image, there is self similarity up the end of the stalks... like mandelbrots etc it isn't exactly scale symmetric but definitely similar at each scale.
The 2nd image is like 4 leaves... which are self similar looking (see pic)
I'm pretty sure they are infinite in length too... but I haven't proved it.

Effie, no idea.. but a good question... hmmm.

Knighty, I'd like to try something like you say to see if they self intersect... it would be nice if they didn't... but they get close. May be very hard to prove.

[kram1032]

There is very clear self-similarity similar to what a De Rham curve might do. There probably are version of this that have an infinite boundary too but others may be bounded. Still, the circumference versus area should be a substatial ratio in any case.

Note that there are infinitely many cricles rotating inside each other at all scales. That this produces fractals is not surprsing at all, but it's quite beautiful to look at and closely related to fourier analysis. Very nice.

[eiffie]

If you reduce the radius by half each iteration and perform infinite iterations how long is the path after the 1st circle finishes its rotation?