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The Ordered Distribution of Natural Numbers on the Square Root Spiral

By Hahn, Harry K.

Book Id:WPLBN0100301947 Format Type:PDF (eBook) File Size:2.18 MB. Reproduction Date:6/20/2007

Hahn, H. K. (2007). The Ordered Distribution of Natural Numbers on the Square Root Spiral. Retrieved from http://www.gutenberg.us/

Description
The Square Root Spiral ( or “Wheel of Theodorus” or “Einstein Spiral” or
“Root Snail” ) is a very interesting geometrical structure, in which the square roots of all natural numbers have a clear defined spatial orientation to each other. This enables the attentive viewer to find many interdependencies between natural numbers, by applying graphical analyses techniques. Therefore the square root spiral should be an important research object for all professionals working in the field of number theory !
The most amazing 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 !!
Mathematical proof that this statement is correct is shown in Chapter 1
“The correlation with Pi“ in the mathematical section.

Summary
Natural numbers divisible by the same prime factor lie on defined spiral graphs which are running through the “Square Root Spiral“ ( or “Spiral of Theodorus” or “Wurzel Spirale“ oder “Einstein-Spiral” ).
Prime Numbers also clearly accumulate on such spiral graphs.
And the square numbers 4, 9, 16, 25, 36 … form a highly three-symmetrical system of 3 spiral-graphs, which divide the square-root-spiral into 3 equal areas.
A mathematical analysis shows, that these spiral graphs are defined by quadratic polynomials.
The Square Root Spiral is a geometrical structure which is based on the 3 basic constants 1, sqrt2 and Pi , and the continuous application of the Pythagorean Theorem of the right angled triangle.
Fibonacci number sequences also play a part in the structure of the Square Root Spiral. Fibonacci-Numbers divide the Square Root Spiral into areas and angle sectors with constant proportions. These proportions are linked to the “golden mean” ( golden section ), which behaves as a self avoiding walk constant in the lattice-like structure of the square root spiral.
see FIG. 1 / 9 / 10 / 18 / 19

Excerpt
Another striking property of the Square Root Spiral is the fact, that the square roots of all square numbers ( 4, 9, 16, 25, 36… ) lie on 3 highly symmetrical spiral graphs which divide the square root spiral into 3 equal areas.
( see FIG.1 : graphs Q1, Q2 and Q3 drawn in green color ).