golden rectangle and golden spiral
A golden rectangle is a rectangle with ratio .
A real golden spiral isn't a spiral consisting of quarter circles with increasing radius, but a logaritmic spiral. In such a spiral every 90° turn the distance between the pole and the points on the spiral increases wih factor .
Starting from some nice properties you can combine such a logaritmic spiral with a golden rectangle. Even if you are not familiar with parametric curves, you can follow these properties in following applet.
- You can find all points in which the tangents to the curve are vertical as the intersection points of the line through the pole P with slope the arctangens of the parameter b.
- You can find the points in which the tangents are horizontal on the line through P, perpendicular to the first line.
- You can control that the radii at the consecutive intersection points with the spiral indeed are increasing with factor every 90°.
- Drawing the tangents in four consecutive tangent points you get a golden rectangle.
- The 4th point (below left), in which the tangent is vertical, doesn't lie in this golden rectangle.
- If you draw a spiral using circular arcs in this golden rectangle, you notice that this approaches the real logaritmic spiral. In this spiral the point below left (with the biggest radius) is a corner of the golden rectangle. Always at the start of a new circular arc the distortion is clearly visible.