In which the chambered nautilus flunks the test (a math lesson for Crayons)

Fifth grader Crayons/Dollygirl has been learning a bit of math history from John Tiner's Exploring the World of Mathematics.  In this week's lesson I had just intended to finish up the chapter on Number Patterns, but the Fibonacci business got away from me a bit.  But that's a good thing.

Here's the lesson as I plan to present it tomorrow, making use of online resources (including one with an unexpected surprise).

1.  Review what we have learned so far about Fibonacci numbers:  that they run in the sequence 1, 1, 2, 3, 5, 8, 13 and so on, with each pair of numbers adding up to the number following right after.

2.  Construct squares following the sequence from graph paper--that is, two with sides 1 unit long, one with sides 2 units long, and so on.  Colour them and cut them out.

3.  Arrange them as if you were packing them in a box, starting with the smallest ones in the centre.  See diagram in the book if you're not sure. Then watch this online animation.  At the end of the animation, watch the drawing of the spiral.  Can you see how that works?  This is called a Fibonacci spiral.

4.  Places in nature where Fibonacci spirals occur:  in spiral galaxies, in your inner ear, in pine cones, in cauliflower.  (I'm thinking maybe also in fiddleheads? I'm not sure about those.)  But not, according to this blog post and its accompanying slide show, in that classic example (cited in Tiner's book), the chambered nautilus.  So much for that. (Maybe it works for some people?)

5.  Another use of Fibonacci spirals:  in art, what is referred to as the Golden Mean or Golden Ratio.  Apparently our eyes just like to follow things that move around in those proportions. Here's a neat blog post showing photographic examples.  (According to the post, the photos were not deliberately planned to match up with the spiral: they just do because they're good photos.  Or they're good photos because they just do.)

6.  So a fun followup might be to find other examples of paintings or photographs that follow these proportions.  Or to deliberately create a drawing--or maybe just a colour pattern--that follows it, and see how that works.  Do you like the way it turned out--why or why not?