Spiral of Theodorus

Spiral of Theodorus

In geometry, the spiral of Theodorus (also called "square root spiral" or "Einstein spiral") is a spiral composed of contiguous right triangles. It was first constructed by Theodorus of Cyrene.

Construction

The spiral is started with an isosceles right triangle, with each leg having a unit length of 1. Another right triangle is formed, with one leg being the hypotenuse of the prior triangle and the other with length of 1. The process then repeats.

Hypotenuse

Each of the triangle's hypotenuse "hi" gives the square root to a consecutive natural number, with "h"1 = √2

Plato, tutored by Theodorus, questioned why Theodorus stopped at √17. The reason is commonly believed to be that the √17 hypotenuse belongs to the last triangle that does not overlap the figure.citeweb
last=Long
first=Kate
title=A Lesson on The Root Spiral
url=http://courses.wcupa.edu/jkerriga/Lessons/A%20Lesson%20on%20Spirals.html
accessdate=2008-04-30]

Extension

Theodorus stopped his spiral at the triangle with a hypotenuse of √17. If the spiral continued to infinitely many triangles, many more interesting characteristics lie in the spiral.

Pi

As the number of spins of the spiral approaches infinity, the distance between two consecutive winds of the spiral approaches the mathematical constant π. [citation
last=Hahn
first=Harry K.
title=The distribution of natural numbers divisible by 2, 3, 5, 7, 11, 13, and 17 on the Square Root Spiral
publication-date=June 28, 2007
place=Ettlingen, Germany
accessdate=2008-04-30
url=http://arxiv.org/abs/0801.4422]

The following is a table showing the distance of two winds of the spiral approaching pi:

As shown, after only the fifth spiral, the distance is 99.97% accurate to π.citation
last=Hahn
first=Harry K.
title=The Ordered Distribution of Natural Numbers on the Square Root Spiral
publication-date=June 20, 2007
place=Ettlingen, Germany
accessdate=2008-05-02
url=http://arxiv.org/abs/0712.2184]

Overlapping

In 1958, Frage von E. Teuffel proved that no two hypotenuses will ever coincide, regardless of how far the spiral is continued. Also, if the sides of unit "one" length are extended into a line, they will never pass through any of the other vertices of the total figure.

Archimedean spiral

The Spiral of Theodorus approximates the Archimedean spiral.

References


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