Math Forum Discussions

Jon Miller

Posts: 2
Registered: 2/18/05

Re: Trying to get a golden spiral from an overhead view of a helix
Posted: Feb 19, 2005 8:26 AM

Plain Text

Thank you both for your thoughts. The reason I thought that an
overhead perspective view of a helix would resemble a logarithmic
spiral is that I've seen side-by-side comparisons, for example in
National Geographic, of a spiral galaxy and then the DNA double helix
from the view I'm describing. The curves appear to be very similar,
and I've read that the arms of a spiral galaxy are basically
logarithmic spirals. The drawing I have in mind is sort of inspired by
that fact. The idea is to have a ton of stylized stars spiraling at
the viewer. In 3d, they'd be following a helical path, and in 2d, they
would at least come close to forming a golden spiral. That's part of
the drawing, anyway.

However, after gleaning what I could from Google searches for "spiral"
and "helix," it seems to me that a top-down perspective view of a
helix actually gives you a different kind of spiral: a hyperbolic
spiral. I'm not positive, and if someone could confirm or deny that it
would be a huge help. But the thing about logarithmic spirals is that
they continue to circle the origin even as the loops get infinitely
farther out. The rules of perspective make this impossible. I'm not
sure I can explain why, but I'm pretty sure that the rules of
perspective dictate that as the 3d helix comes closer and closer to
the viewer, the 2d representation of it comes closer and closer to a
ray shooting straight out from the origin. I'm not so hot on
perspective either, but I know that weird things start happening when
you try drawing stuff that would be outside of the field of vision in
real life, and I believe that's what the result would be. And a
hyperbolic spiral seems to do that. It's also what I think I saw when
I looked down the axis of a helix in my 3d modeling program. Here's
MathWorld's entry, by the way:

http://mathworld.wolfram.com/HyperbolicSpiral.html

So I may have to work with that shape, though it's kind of a shame,
because I think the golden spiral is especially beautiful. On the
other hand, I think that fact about a logarithmic spiral being formed
when you look down the central axis of a conical helix could prove
helpful. I believe that truly happens only in an orthographic
projection, not in perspective, but the perspective drawing would
resemble the orthographic projection close to the center. Maybe I can
work with a conical helix instead of a cylindrical one. I also think
it will be helpful as I continue to experiment to consider only
corresponding (similar? not sure what the word would be) points on
succesive loops, and the intervals between them.

Thanks again to both of you for helping, and sorry for the long post,
but any other insights would be greatly appreciated.

On Fri, 18 Feb 2005 14:47:51 -0500, Ed Wall wrote:
>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:
>>
>><a
href="http://www.levitated.net/daily/levGoldenSpiral.html">http://www.levitated.net/daily/levGoldenSpiral.html</a>
>>
>>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...
>>
>><a
href="http://mathworld.wolfram.com/LogarithmicSpiral.html">http://mathworld.wolfram.com/LogarithmicSpiral.html</a>
>>
>>...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!