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The spiral of Theodorus up to the triangle with a hypotenuse of √17.

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.

Contents

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.

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, due to the fact that 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]

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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

The spiral extended to three windings.

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:

\tan\left(\varphi_n\right)=\frac{1}{\sqrt{n}}.

Therefore, the growth of the angle φn of the next triangle n is:[1]

\varphi_n=\arctan\left(\frac{1}{\sqrt{n}}\right)

The total angle φ(k) for the kth triangle is:[1]

\varphi\left (k\right)=\sum_{n=1}^k\varphi_n
A triangle or section of spiral

Radius

The growth of the radius of the spiral at a certain triangle n is

\Delta r=\sqrt{n+1}-\sqrt{n}

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]

References


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