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Logarithmic Spirals

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I saw a video on YouTube by a guy who makes kinetic art mobiles. He had an interesting one made out of toothed spirals, so I had a go at recreating it in C4D.


I tried using a helix spline, but couldn't seem to get the objects to interact correctly, so I made a proper log spiral using tracer. The movement is generated using dynamics - there's one collider that's rotated in XPresso driving 2 rigid bodies.






make log spiral.c4d

Edited by jed
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Whoa-- amazing!    It was only when camera dollied back that I perceived what you had achieved.    Amazing and very tricky to do.     Is that a phi  (golden ratio) spiral?

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This is really awesome, nice job Jed. Keep them coming, its always interesting to see what you have to show and share. I would like to see those shared on the Market, just so they are not forgoten and burried inside forums. 

U Render Quality Assurance Specialist

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@rasputin I just used the general log spiral equation




where angle is in radians, e is the natural log base (aka Euler's number 2.71828) and tweaked the constants a + k until it looked right. I tried using a helix and radial bias but the gears were jumping. I guess it's the properties of logs that makes it work. Also, when I added the struts, with a helix I had to use a spline on the step effector to make things fit, but with the log curve the struts fitted without a spline.


I usually animate gears with XPresso, but with the varying radii that would be beyond my math skills. Dynamics to the rescue !


The video above is version 2 - I didn't like how dynamics had introduced a small collision gap, so after baking for TR I increased the teeth size. The dynamics is Compound Collision Shape, but with all the geometry it was too slow in the viewport. I added some switched off rigid body tags to parts that don't contribute to the dynamics calculation.


Here's the video I got the design from. The guy has some good stuff eg non-circular gears



I'm still trying to figure out square gears...


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Fantastic work Jed,

If you havn't already, check out a book called "The power of Limits" It's quite old, but talks about geometry of nature. 

I've found it very interesting, although the math stuff is way over my head, but I think you should be able to translate alot of the content into your tests judging by the work posted here.


There's also a Houdini artist named Serjan Burlak that does a lot of this kind of work, I found out about the book from a podcast he was a guest on.


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@Hrvoje can formula spline use polar coordinates ie radius + angle ? Spirals are easiest described (and understood) using polar vs cartesian.


TBH I've never really understood what 't' is in formula spline.


in math -

y = f(x) ie y is a function of x


in C4D

x = f(t) ie x is a function of t

y = f(t) ie y is a function of t


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@Hrvoje I understand how to represent polar coords in cartesian in order to use formula spline. By 'differ' I meant you find it a good way to get a log spiral and I don't. Converting 1 equation with 2 variables into 2 equations with 3 variables - an extra unnecessary variable t - makes things more complex IMHO. Those 2 equations do not shout out 'spiral' to me, but the polar version does.


I must admit that my approach to C4D is to use math, XPresso, Python etc and ignore built-in functions that I haven't bothered to learn about.


A frequent comment on my stuff is 'there's a field that can do that'. I have, of course, never tried to learn fields...

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