Logarithmic Spiral

By Rodrigo Afanador Carrasco on February 26, 1998 :

Subject: HELP CONSTRUCTING A LOGARITHMIC SPIRAL

I was told , as a homework , to draw a logarithmic spiral based on a hexagon constructed with golden mean sides. Since I am a Law student I can barely understand and need help badly.

Please Help Me .

Rodrigo Afanador C.

By David Sanders on April 7, 1998 :

From: David Sanders
On: 4/7/1998 at 15:40

Dear Rodrigo,

A logarithmic spiral is one which spirals out at a faster and faster rate as you go out from the starting point. It has the equation (in polar coordinates),

r = a e^{b θ}

. (Compare this to an Archimedes spiral of the form

r = a θ

, which goes out a constant distance each time round the spiral.)

The golden mean is the number $ϕ$ such that if you have a rectangle with sides 1 and $ϕ$ , and you cut off a square of side 1 from the rectangle, you are left with another rectangle which also has sides in the ratio $1 : ϕ$ , so that if you cut off another square then...
If you carry on this construction, cutting off squares over and over again, certain points will lie on a logarithmic spiral. Try drawing it to see what I mean. It is explained a bit more (along with a lot of other interesting information), on the `Eric's Treasure Trove of Mathematics' web-site, which there is a link to in the NRICH Archive. Look under `Golden Ratio', under the letter `G'.

However, I don't quite understand how the hexagon comes into your question.

I hope this has been helpful, sorry it's taken so long to reply.

David.

By The Editor :

You might like to look at this site as well.