Many of the designs on this site contain spirals and self-similar shapes derived from the golden ratio and other geometrical constructions.


Golden Rectangle

The golden ratio has a natural self-similarity. It is defined by that rectangle which can be partitioned into a square plus a remaining area having the same proportions as the original rectangle. There is only one rectangle that satisfies this requirement and the proportions of the rectangle are the golden ratio φ =1.618... The smaller rectangle can be further subdivided in the same way, over and over again, to create a self-similar object. The inscribed curve (which passes through the intersection of the straight lines), is called the "golden spiral". This is a logarithmic spiral defined by r∝exp(bθ) where b = (2/π)ln(φ) . This construction is also related to the Fibonacci numbers.



√2 Rectangle

A similar construction can be made by defining the rectangle that can be partitioned into exactly two smaller copies of itself. The proportions of this rectangle are √2. Again, this subdivision can be repeated indefinitely. The corresponding logarithmic spiral is specified by r∝exp(bθ) with b=(2/π)ln(√2).



Silver Rectangle.

Yet another construction can be made by considering a rectangle which is replicated when a square on each end leaves a remaining area that has the same proportions as the original shape. This proportion is known as the silver ratio , δ=(1+√2). As suggested in the figure, the ratios of the sides of the squares leads to the Pell numbers. The inscribed spiral is another logarithmic spiral with b=(2/π)ln(δ).



√3 Rectangle

Here is a construction that divides the rectangle into three equal areas all the same shape as the original. The ratio of the sides of this rectangle is √3, and the logarithmic spiral is specified by b=(2/π)ln(√3).




"Golden Triangles"

Finally, an analogous construction based on equilateral triangles instead of rectangles. The ratio of one triangle to the next is the golden ratio, φ . But, the inscribed spiral is different from the golden spiral. Here, r∝exp(bθ) with b = 3/(2π)ln(φ).




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