Harry K. Hahn
The Square Root Spiral ( or “Spiral of Theodorus” or “Einstein-Spiral” ) is a very interesting geometrical structure, in which the square roots of all natural numbers have a defined (spatial ) position to each other. The Square Root Spiral develops from a right angled base triangle with the two legs ( cathets ) having the length 1, and with the long side ( hypotenuse ) having a length which is equal to the square root of 2. The Square Root Spiral is formed by further adding right angled triangles on this base triangle. In this process the longer legs of the next triangles always attach to the hypotenuses of the previous triangles. And the longer leg of the next triangle always has the same length as the hypotenuse of the previous triangle, and the shorter leg always has the length 1. In this way a spiral structure is developing in which the spiral is created by the shorter legs of the triangles, which have the constant length of 1, and where the lengths of the radial rays ( or spokes ) coming from the centre of this spiral are the square roots of the natural numbers ( sqrt 2 , sqrt 3, sqrt 4, sqrt 5 …. ). The most striking property of the Square Root Spiral is surely the fact, that the distance between two successive winds of the Square Root Spiral quickly strives for the well known geometrical constant Pi (π) !! But there are many more interesting interdependencies between the natural numbers, which can be discovered in this amazing geometrical structure. For example all numbers divisible by the same prime factor always lie on defined spiral-graphs, which run in a clear order through the Square Root Spiral. And the square numbers 4, 9, 16, 25, ... form a highly three-symmetrical sytem of three spiral-graphs, which divide the Square Root Spiral into three equal areas. It is the same with Prime Numbers. Prime Numbers also clearly accumulate on defined spiral-graphs. A mathematical analysis shows, that all these spiral graphs represent special quadratic polynomials. E
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