Showing posts with label golden ratio. Show all posts
Showing posts with label golden ratio. Show all posts

## Wednesday, May 22, 2013

### The Golden Triangle in Scratch Project

In a recent post I described the Golden Rectangle in Scratch project.
To review, a Golden Rectangle is a rectangle with a length to width ratio of 1.61803… on out to an infinite number of decimal places (like Pi, Phi is an irrational number).
The number 1.61803… is called the Phi, Golden Ratio, symbol (F).
If, in an isosceles triangle, the ratio between the long side and the short side is 1.61803…, that is Phi, then the triangle is called a golden triangle.

A bit of geometry and algebraic manipulation shows that the isosceles triangle with base angles of 72º and an apex angle of 36º is a golden triangle.

Bisect one of the 72º angles and the result is again, a golden triangle. Half of 72º is 36º, which leaves a value of 72º for the third angle in the smaller isosceles triangle. Therefore, the smaller triangle is also a golden triangle. And so on, and so on. I love mathematics for the simple fact that even as those golden triangles get infinitesimally small, the angle values remain the same!

Here’s the next iteration.

You can view this project by clicking on this link,

http://scratch.mit.edu/projects/10133856/

and watch the program draw six iterations of the smaller golden triangles and then draw the spiral connecting all six of the triangles.

## Wednesday, May 8, 2013

### The Golden Rectangle in Scratch Project

Several months ago I completed a Scratch project on the golden rectangle of mathematical, architectural, and artistic fame. Below is a screenshot at the completion of a run of the scripts.
The first script draws a series of squares, starting with the largest square on the left. It continues drawing successively smaller squares in a clockwise spiral.  Each square cuts the larger golden rectangle into a a smaller square and a smaller golden rectangle.

The second script starts in the lower left corner and draws (in yellow) the spiral, in one smooth motion.

I have now written a pdf file that describes a bit of the history of the golden rectangle, its relationship to the Fibonacci numbers, and the cool script that draws the spiral. When given the radius and arc length, the script draws the arc with that radius. Scratch (unlike Logo) doesn’t have any circle tools so if Scratch 2.0 really lets us create our own “blocks” then I will make this script into a block and write other circle tools.