Tuesday, 16 February 2010

Five-point star: second method

In the post called Golden Pentagon Construction I make a five-point star with the method most often demonstrated by writers on the subject. The method below is another I have drawn up myself and is a little more direct than the aforementioned – though the shape will be the same size, either way it is drawn.

The first image shows the start of the construction. I begin with a golden rectangle and divide it on one side into smaller golden rectangles. One length in the five-point star is the right hand vertical edge of the largest rectangle. We know, then, how long each line in the shape will be.

The difficult part of making a pentagram is figuring out the angle of the points. The angle, from the top right corner, happens to be one which will lead the line to the corner of the smallest rectangle in the image – it is reduced by 1.618 four times from the largest one. How long you draw the line toward that corner is already clear – the length of the largest rectangle. However, it happens to be a the horizontal halfway point of the second largest rectangular section.

The process is repeated, as in the second image, for the upper half of the shape to get the black lines in the third image. Then, taking ignoring most of our divisions of the largest rectangle, we divide the right vertical edge by the golden section (the square inside the golden rectangle does this immediately). This 8:13 (golden ratio) point on the right edge is our intersection point for the other lines in the pentagram – shown in light grey. The lines can simply be extended until they meet or drawn until they match the length of the vertical edge of the rectangle.

The points are joined very easily, as in the last image, and the outer lines of the pentagon can be formed as a result. This is perhaps even easier a construction than the that of the other post but lacks the advantage of creating an object which is in parallel with the original rectangle.

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