Spiral of Theodorus
In geometry, the spiral of Theodorus (also called the square root spiral, Pythagorean spiral, or Pythagoras's snail)[1] is a spiral composed of right triangles, placed edge-to-edge. It was named after Theodorus of Cyrene.
Construction
The spiral is started with an
History and uses
Although all of Theodorus' work has been lost, Plato put Theodorus into his dialogue Theaetetus, which tells of his work. 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 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 triangles' hypotenuses gives the square root of the corresponding natural number, with .
Plato, tutored by Theodorus, questioned why Theodorus stopped at . The reason is commonly believed to be that the hypotenuse belongs to the last triangle that does not overlap the figure.[4]
Overlapping
In 1958, Kaleb Williams proved that no two hypotenuses will ever coincide, regardless of how far the spiral is continued. Also, if the sides of unit length are extended into a line, they will never pass through any of the other vertices of the total figure.[4][5]
Extension
Theodorus stopped his spiral at the triangle with a hypotenuse of . If the spiral is continued to infinitely many triangles, many more interesting characteristics are found.
Growth rate
Angle
If is the angle of the th triangle (or spiral segment), then:
The sum of the angles of the first triangles is called the total angle for the th triangle. It grows proportionally to the square root of , with a bounded correction term :[1]
Radius
The growth of the radius of the spiral at a certain triangle is
Archimedean spiral
The Spiral of Theodorus
The following table shows successive 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% |
As shown, after only the fifth winding, the distance is a 99.97% accurate approximation to .[1]
Continuous curve
The question of how to interpolate the discrete points of the spiral of Theodorus by a smooth curve was proposed and answered by Philip J. Davis in 2001 by analogy with Euler's formula for the gamma function as an interpolant for the factorial function. Davis found the function[7]
An analytic continuation of Davis' continuous form of the Spiral of Theodorus extends in the opposite direction from the origin.[11]
In the figure the nodes of the original (discrete) Theodorus spiral are shown as small green circles. The blue ones are those, added in the opposite direction of the spiral. Only nodes with the integer value of the polar radius are numbered in the figure. The dashed circle in the coordinate origin is the circle of curvature at .
See also
References
- ^ arXiv:0712.2184
- ^ Nahin, Paul J. (1998), An Imaginary Tale: The Story of , Princeton University Press, p. 33, ISBN 0-691-02795-1
- ^ Plato; Dyde, Samuel Walters (1899), The Theaetetus of Plato, J. Maclehose, pp. 86–87
- ^ a b Long, Kate, A Lesson on The Root Spiral, archived from the original on 11 April 2013, retrieved 30 April 2008
- MR 0096160
- ^
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 (2001), pp. 37–38.
- ProQuest 303808219
- ^ In an appendix to (Davis 2001)
- ^ Gronau (2004). An alternative derivation is given in Heuvers, Moak & Boursaw (2000).
- ^ Waldvogel (2009).
Further reading
- Davis, P. J. (2001), Spirals from Theodorus to Chaos, A K Peters/CRC Press
- Gronau, Detlef (March 2004), "The Spiral of Theodorus", JSTOR 4145130
- Heuvers, J.; Moak, D. S.; Boursaw, B (2000), "The functional equation of the square root spiral", in T. M. Rassias (ed.), Functional Equations and Inequalities, pp. 111–117
- Waldvogel, Jörg (2009), Analytic Continuation of the Theodorus Spiral (PDF)