Math Forum Discussions

Ed Wall

Posts: 837
Registered: 12/3/04

Re: Trying to get a golden spiral from an overhead view of a helix
Posted: Feb 18, 2005 2:56 PM

Plain Text

Interesting question. Perhaps you might say something in a bit more
detail on how other logarithmic spirals are approximated by helix
projections. Enneper, I think, showed that the projection of a helix
on a cone was a logarithmic spiral, but I'm not sure of the cite.

Ed Wall

>Hi, I wonder if someone could help me with a project I'm working on.
>It's a perspective drawing with what I hope are interesting geometric
>underpinnings, but which are a bit beyond me, unfortunately. Here's
>the situation. The viewer or camera or whatever is looking right down
>the central axis of a helix. It's my understanding that if you're
>looking right down the central axis, a 2d representation of the helix
>would approximate a logarithmic spiral. If it's a perspective and not
>an orthographic drawing, that is. I'm going for one logarithmic spiral
>in particular, what I think is called a golden spiral. The one shown
>here:
>
>http://www.levitated.net/daily/levGoldenSpiral.html
>
>Now, it seems to me that the two properties of the helix that I can
>adjust to make it appear that way are its radius and the distance
>between its loops or coils. (Sorry, part of the problem is that I
>don't really know the vocabulary.) My question is, can anyone help me
>figure out what those two attributes of the helix should be, relative
>to each other, for the view I'm describing to come as close as
>possible to a golden spiral? Would it be the golden ratio or
>something?
>
>Finally, the entry for logarithmic spiral on MathWorld...
>
>http://mathworld.wolfram.com/LogarithmicSpiral.html
>
>...has something about approximating a logarithmic spiral by starting
>with equally spaced rays and drawing a perpendicular from one to the
>next. That would seem to relate, but I just can't get my head around
>it. Thanks so much for any help you can give me!