Spiral of Theodorus
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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.
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[edit] 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.
[edit] Hypotenuse
Each of the triangle's hypotenuse hi gives the square root to a consecutive natural number, with h1 = √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.[1]
[edit] Extention
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.
[edit] 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.[2]
The following is a table showing the distance of two winds of the spiral approaching pi:
| Winding No.: | Calculated average winding-distance | Accuracy of average winding-distance in comparison to π |
|---|---|---|
| 2 | 3.1592037 | 99.44255% |
| 3 | 3.1443455 | 99.91245% |
| 4 | 3.14428 | 99.91453% |
| 5 | 3.142395 | 99.97447% |
As shown, after only the fifth spiral, the distance is 99.97% accurate to π.[3]
[edit] 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.[1]
[edit] Archimedean spiral
The Spiral of Theodorus approximates the Archimedean spiral.[3]
[edit] References
- ^ a b Long, Kate. A Lesson on The Root Spiral. Retrieved on 2008-04-30.
- ^ Hahn, Harry K. (June 28, 2007), The distribution of natural numbers divisible by 2, 3, 5, 7, 11, 13, and 17 on the Square Root Spiral, Ettlingen, Germany, <http://arxiv.org/ftp/arxiv/papers/0801/0801.4422.pdf>. Retrieved on 30 April 2008
- ^ a b Hahn, Harry K. (June 20, 2007), The Ordered Distribution of Natural Numbers on the Square Root Spiral, Ettlingen, Germany, <http://arxiv.org/ftp/arxiv/papers/0712/0712.2184.pdf>. Retrieved on 2 May 2008
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