Geometry Project

From: Marilyn S. Heath
Subject: Geometry Project
Date: Tue, 1 Nov 1994 22:27:48 -0500 (EST)

Dear Dr. Math,

I am a ninth grader at the Upper Darby High School.  I have been 
assigned a Geometry project and there are no restrictions on ways of 
obtaining answers.  Could you please give me some information for 
this project?

Project:
Geometry In The Real World

"This project consists of thinking of three examples of geometry in the 
real world.  One example must be non man-made (found in nature).  Each 
example must have a written explanation of how geometry relates to it.  
Each example also must be visually displayed."

Thank you for your time

Dan Heath

From: Dr. Ethan
Subject: Re: Geometry Project
Date: Wed, 2 Nov 1994 01:17:36 -0500 (EST)

Dan, this is a great problem.  I must say that if I were doing it, 
I would try to make them all be natural.  I could give you a huge 
list but I think you'd have more fun thinking of a lot of them yourself.  
So here are some questions to get you going. The first ones start 
easier and then they get harder. At the end I will give you some great 
examples that are a little tougher.

   What things are shaped like circles? (Think Big)
   What are some objects that are symmetrical along a line?
     (Bugs, for example)
   What are some objects symmetrical around a point?
     (Flowers)
   What do bees build? 
   (And on the same shape theme think winter)
   Think about the faces of gemstones or quartz.
   Think about birds in flight.

Now to get a little harder.

There are some interesting relations beween Fibonacci numbers
(0,1,1,2,3,5,8,13,21,34,...), rectangles, and the spiraling of 
things like conch shells. To help you see this take a sheet of paper 
and in the middle draw a square 1cm by 1cm. Then lengthen one side 
to 2 cm and draw the new rectangle. Then lengthen the shorter side 
to 3 and draw the new rectangle. Then lengthen the new shorter side 
to 5 and draw the new rectangle. Keep drawing the new rectangles on 
top of each other and a beautiful spiral should appear.  

My last two examples are pretty intense geometrical shapes. The first 
is the shape that a rope or chain makes when hung from its end points.  
It is called a caternary curve and is a really interesting shape.  
The St. Louis arch is also made in this shape. You may want to think 
about why this is a good shape to build an arch in. I'm sure that you 
could find more information about this shape if you looked. If you want 
some ideas of where to look write back and I'll give you some.

The last shape is one that I am just starting to understand. (In fact
I have to do a presentation on it on Friday.)  The shapes are called 
minimal surfaces but you can just think of them as soap bubbles. (Which 
by the way have great geometrical properties - spheres when by 
themselves, but what do they do when they touch?) Anyway when you put 
a shape in soap and a bubble forms on it the soap will connect the 
edges in the shortest way possible. When the shape is just a ring 
it will of course go straight across.  

What would happen if the shape were more interesting?  Try bending 
a coat hanger, dipping it into soap, and then seeing how the soap 
connects all the edges. It can be pretty fascinating.

Hope this gives you some stuff to think about. Write back if you need 
more.

Ethan - Doctor On Call