Spiral of Theodorus
In geometry, the spiral of Theodorus (also called square root spiral, Einstein spiral or Pythagorean spiral)[1] is a spiral composed of contiguous right triangles. It was first constructed by Theodorus of Cyrene.
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Construction
The spiral is started with an isosceles right triangle, with each leg having a length of 1. Another right triangle is formed, an automedian right triangle with one leg being the hypotenuse of the prior triangle (with length √2) and the other leg having length of 1; the length of the hypotenuse of this second triangle is √3. The process then repeats; the ith triangle in the sequence is a right triangle with side lengths √i and 1, and with hypotenuse √(i + 1).
History
Although all of Theodorus' work has been lost, Plato put Theodorus into his dialogue Theaetetus, which tells the reader of his achievements. It is assumed that Theodorus had proved that all of the square roots of non-square integers from 3 to 17 are irrational by means of the Spiral of Theodorus.[2] Plato quoted Theaetetus speaking to Socrates:
It was about the nature of roots. Theodorus was describing them to us and showing that the third root and the fifth root, represented by the sides of squares, had no common measure. He took them up one by one until he reached the seventeenth, when he stopped. It occurred to us, since the number of roots appeared to be infinite, to try to bring them all under one denomination.
Plato does not attribute the irrationality of the square root of 2 to Theodorus, because it was well known before him. Theodorus and Theaetetus split the rational numbers and irrational numbers into different categories.[3]
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.[4]
Overlapping
In 1958, 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.[4]
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.
Growth rate
Angle
If φn is the angle of the nth triangle (or spiral segment), then:
Therefore, the growth of the angle φn of the next triangle n is:[1]
The sum of the angles of the first k triangles is called the total angle φ(k) for the kth triangle, and it equals:[1]
with
Radius
The growth of the radius of the spiral at a certain triangle n is
Archimedean spiral
The Spiral of Theodorus approximates the Archimedean spiral.[1] Just as the distance between two windings of the Archimedean spiral equals mathematical constant pi, as the number of spins of the spiral of Theodorus approaches infinity, the distance between two consecutive windings quickly approaches π.[5]
The following is a table showing the distance of two windings 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% |
| → ∞ | → π | → 100% |
As shown, after only the fifth winding, the distance is a 99.97% accurate approximation to π.[1]
Continuous curve
The question of how to interpolate the discrete points of the spiral of Theodorus by a smooth curve (not merely piecewise continuous), similar to the way in which the Gamma function interpolates the factorial function, was proposed and answered in (Davis 1993, pp. 37–38) by analogy with Euler's formula for the gamma function. He found the function:
An axiomatic characterization of this function is given in (Gronau 2004) as the unique function that satisfies the functional equation
the initial condition 
and monotonicity in both argument and modulus; alternative conditions and weakenings are also studied therein.
An alternative derivation is given in (Heuvers , Moak & Boursaw 2000).
Some suggested an alternative interpolation which connects the spiral and an alternative inner spiral, as in (Waldvogel 2009).
References
- ^ a b c d e Hahn, Harry K.. "The Ordered Distribution of Natural Numbers on the Square Root Spiral". arXiv:0712.2184.
- ^ Nahin, Paul J. (1998), An Imaginary Tale: The Story of [the Square Root of Minus One], Princeton University Press, p. 33, ISBN 0691027951, http://books.google.com/?id=WvcfqBgZDWQC&printsec=frontcover
- ^ Plato; Dyde, Samuel Walters (1899), The Theaetetus of Plato, J. Maclehose, pp. 86–87, http://books.google.com/?id=wt29k-Jz8pIC&printsec=titlepage
- ^ a b Long, Kate. "A Lesson on The Root Spiral". http://courses.wcupa.edu/jkerriga/Lessons/A%20Lesson%20on%20Spirals.html. Retrieved 2008-04-30.
- ^ Hahn, Harry K. (2008). "The distribution of natural numbers divisible by 2, 3, 5, 7, 11, 13, and 17 on the Square Root Spiral". arXiv:0801.4422.
- Davis, P. J. (1993), Spirals from Theodorus to Chaos
- Gronau, Detlef (March 2004), "The Spiral of Theodorus", The American Mathematical Monthly (Mathematical Association of America) 111 (3): 230–237, doi:10.2307/4145130, JSTOR 4145130
- Heuvers, J.; Moak, D. S.; Boursaw, B (2000), "The functional equation of the square root spiral", in T. M. Rassias, Functional Equations and Inequalities, pp. 111–117
- Waldvogel, Jörg (2009), Analytic Continuation of the Theodorus Spiral, http://www.math.ethz.ch/~waldvoge/Papers/theopaper.pdf
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