Tuesday, 16 February 2010

Five-point star alternative of first method

After a little review of my method I decided to use the finished star and see how it might relate to the rectangle in other ways. The images below show the most simple explanation I got.

I started by mapping out the simple golden rectangle with itself reduced three times in golden proportion. You have to do this all four ways (as in the second image) because the original shape has four edges. This is was is slightly misleading when you read about the golden section – the first image here is usually what you’ll find but the second image is the proper division of the rectangle.

The third image shows our key points. Those darkest rectangles indicate the edges of the pentagram (its width). So, we know already how long one line in the pentagram is: the distance from the edge of one dark rectangle to the edge of the other. With that, we know the length of every line in the pentagram because they must all be the same.

All we need, then, is a direction for the line to go from our red mark on the left in the third image. The answer is through the cross-over point of the two diagonals of the darkest rectangles. From there on in it’s just a case of using the same line length to get the remaining edges. If you want to be really geeky you can divide your first lines of the star by the golden ratio and you’ll have an intersection point for the other lines.

Incidentally, I did say in the other post that the size (size, not proportion) of the star did not seem to relate to the rectangle but, in fact, it does. If we say the length of the rectangle is x and the golden section is φ then the length of one line in the pentagram is (x/φ² + x/φ⁴). In other words, it’s the length of two of the smaller rectangles added together: so it does relate!…just really subtly…..

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