The Golden Ratio

The Golden Ratio was known to the ancients, and is the next most significant irrational number after pi. Its value (to the first 10 decimal places) is 1.6180339887... and its property is best described as a simple geometrical relationship, which Isotiles™ capture perfectly. As you can see from the two pentagons shown here, the smaller is made from one acute isosceles triangle with an obtuse isosceles triangle on either side. If we describe the shorter length as 1, the longer length is 1.6180339887... or 'g' for short (actually it is now conventional to call it by another Greek letter - phi).

When a larger pentagon is made with Isotiles, with its side now g, it is clear that the longer length (diagonally from each base corner to the apex) is g+1. This is the golden ratio: g is such a value that 1/g=g/(g+1) and g/1=(g+1)/g.

This is the relationship that makes so many symmetrical tessellations possibile with Isotiles, and gives them a unique quality.