Here's a piecewise linear spiral approximation to a nautilus (shown me by Jack Holloway): http://www.amazon.com/The-Irrationals-Story-Numbers-Count/dp/0691143420/ref=... --rwg Also, quarter-circles (in "four point ellipses") do interesting things<http://gosper.org/pumpstill.gif> . ----- Hi Jim, Yes, there is more to be said, but I had to use that curve (consisting of quarter circles) because the point I wanted to make is specifically about that curve. The curve is only a ruler-and-compass approximation to an exponential spiral and the differences may be important in some contexts. But it is a much larger error to claim that curve is a model for the nautilus, and that big error is continuously being copied and re-published. E.g., they both appear on the cover of Livio's popular book The Golden Ratio, and inside, his text about them confusingly does not clarify the difference. As to the curve not being a "true spiral" I don't know of a standard definition of "true spiral" that includes the many types of spiral that people often study, but doesn't include that assemblage of quarter circles. As far as I am concerned, it is a "true spiral" that happens not to be C2 continuous. Georgehttp://georgehart.com/ On 8/15/2012 11:02 PM, James Propp wrote:>>From 0:20 to 0:40 in his video, George shows a famous picture of the> "golden spiral". But as many of you probably know (no doubt including> George), this picture is bogus. Specifically, it conflates two different> curves: a curve made up of quarter-circular arcs (which isn't a true> spiral) and a curve made of true spiral arcs (which isn't tangent to the> rectangles).>> George doesn't discuss this in his video, since minor issues like this> would distract from his main point. But still, I wish the video hadn't> used the inaccurate picture.>> Yes, I'm being a bit picky. At times like this I am reminded of a motto I> saw on a tee-shirt once: "I'm not pompous. I'm pedantic. There's a> difference." :-)>> Jim>> On Tue, Aug 14, 2012 at 4:22 PM, George Hart <george@georgehart.com <http://gosper.org/webmail/src/compose.php?send_to=george%40georgehart.com>> wrote:>>> Hi Alex,>>>> Unfortunately, as a consequence of this being hosted by the Guardian, I>> expect all the folks who uncritically see the golden ratio everywhere will>> now be blathering about golden uteri. I've started making a series of math>> videos and here's one rebuttal to the golden ratio cult:>>>> http://www.youtube.com/watch?**v=_gxC8OjoQkQ<http://www.youtube.com/watch?v=_gxC8OjoQkQ>>>>> As to the image on that page, I hope you don't mind if I interpret your>> use of the middle finger as an editorial comment. I feel it's misleading>> to show precise metal calipers and print 1.618, as if finger measurements>> were precise enough to support four significant digits.>>>> George>> http://georgehart.com/>>>>>>>>>> On 8/14/2012 10:23 AM, Alex Bellos wrote:>>>>> I've just launched a maths blog for the Guardian>>>>>> Here's my first post: the research is serious, but any conclusions need>>> to be taken with a medium to large dosage of salt.>>>>>> http://www.guardian.co.uk/**science/alexs-adventures-in-**>>> numberland/2012/aug/14/golden-**ratio-uterus<http://www.guardian.co.uk/science/alexs-adventures-in-numberland/2012/aug/14/golden-ratio-uterus>>>>>>> alex>>> ______________________________**_________________