Golden Rectangle & Equiangular Spiral
Given a Rectangle having sides in the ratio , the Golden Ratio is defined such that partitioning the original Rectangle into a Square and new Rectangle results in a new Rectangle having sides with a ratio . Such a Rectangle is called a golden rectangle, and successive points dividing a golden rectangle into Squares lie on a Logarithmic Spiral. The spiral is not actually tangent at these points, however, but passes through them and intersects the adjacent side, as illustrated below.
Euclid used the following construction for a golden rectangle and .
Draw the square ABCD, call E the midpoint of AC, so that AE = EC = x. Now draw the segment BE which has lenght
and construct EF with this length. Now construct FG=EF, then
The length and width of the nth Golden rectangle can be written as linear expressions where the coefficients a and b are always Fibonacci numbers! These Golden rectangles can be inscribed in a logarithmic spiral as pictured. Assume that the lower left corner of the first rectangle is the origin of an xy-coordinate system. Question: what is the accumulation point for the spiral? Answer:
Such logarithmic spirals are "equiangular" in the sense that every line through cuts across the spiral at a constant angle . In this way, logarithmic spirals generalize ordinary circles (for which =90 deg). The logarithmic spiral pictured gives rise to the constant angle
In sum, the following features of interest in the figure below should be noted:
1. The limiting point O is called the pole of the equiangular spiral which passes through the golden cuts D, E, G, J, ... (The sides of the rectangle are nearly but not quite tangential to the curve.)
2. Alternate golden cuts on the rectangular spiral ABCFH... lie on the diagonals AC and BF. This suggests a convenient method of constructing the figure.
3. The diagonals AC and BF are mutually perpendicular
4. The points E, O, J are collinear, as also are the points G, O, D.
5. The four right angles at O are bisected by EJ and DG so that these lines are mutually perpendicular.
6. AO/OB = OB/OC = OC/OF= ... There si an infinite number of similar triangles, each being one-half of a golden rectangle.
The first of the six features shows the connection between the logarithmic spiral and the golden section.
An interesting property of the spiral is
worth noting. However different two segments of the curve may
be in size they are not different in shape. Suppose a photograph
were taken with the aid of a microscope of the convolutions near
the pole O, too small to be visible to the unaided eye. If such
a copy were suitably enlarged it could be made to fit exactly
on a spiral of the size of the above figure. The spiral is without
a terminal point: it may grow outwards (or inwards) indefinitely,
but its shape remains unchanged.