Spiral of Theodorus

Summary

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.

The spiral of Theodorus up to the triangle with a hypotenuse of

Construction edit

The spiral is started with an isosceles right triangle, with each leg having unit length. Another right triangle is formed, an automedian right triangle with one leg being the hypotenuse of the prior triangle (with length the square root of 2) and the other leg having length of 1; the length of the hypotenuse of this second triangle is the square root of 3. The process then repeats; the  th triangle in the sequence is a right triangle with the side lengths   and 1, and with hypotenuse  . For example, the 16th triangle has sides measuring  , 1 and hypotenuse of  .

History and uses edit

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 edit

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 edit

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 edit

 
Colored extended spiral of Theodorus with 110 triangles

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 edit

Angle edit

If   is the angle of the  th triangle (or spiral segment), then:

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

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]

 
where
 
(OEISA105459).
 
A triangle or section of spiral

Radius edit

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

 

Archimedean spiral edit

The Spiral of Theodorus approximates the Archimedean spiral.[1] Just as the distance between two windings of the Archimedean spiral equals mathematical constant  , as the number of spins of the spiral of Theodorus approaches infinity, the distance between two consecutive windings quickly approaches  .[6]

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 edit

 
Philip J. Davis' analytic continuation of the Spiral of Theodorus, including extension in the opposite direction from the origin (negative nodes numbers).

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]

 
which was further studied by his student Leader[8] and by Iserles.[9] This function can be characterized axiomatically as the unique function that satisfies the functional equation
 
the initial condition   and monotonicity in both argument and modulus.[10]

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 edit

References edit

  1. ^ a b c d e Hahn, Harry K. (2007), The ordered distribution of natural numbers on the square root spiral, arXiv:0712.2184
  2. ^ Nahin, Paul J. (1998), An Imaginary Tale: The Story of  , Princeton University Press, p. 33, ISBN 0-691-02795-1
  3. ^ Plato; Dyde, Samuel Walters (1899), The Theaetetus of Plato, J. Maclehose, pp. 86–87
  4. ^ a b Long, Kate, A Lesson on The Root Spiral, archived from the original on 11 April 2013, retrieved 30 April 2008
  5. ^ Teuffel, Erich (1958), "Eine Eigenschaft der Quadratwurzelschnecke", Mathematisch-Physikalische Semesterberichte zur Pflege des Zusammenhangs von Schule und Universität, 6: 148–152, MR 0096160
  6. ^ 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
  7. ^ Davis (2001), pp. 37–38.
  8. ^ Leader, Jeffery James (1990), The generalized Theodorus iteration (PhD thesis), Brown University, p. 173, MR 2685516, ProQuest 303808219
  9. ^ In an appendix to (Davis 2001)
  10. ^ Gronau (2004). An alternative derivation is given in Heuvers, Moak & Boursaw (2000).
  11. ^ Waldvogel (2009).

Further reading edit

  • Davis, P. J. (2001), Spirals from Theodorus to Chaos, A K Peters/CRC Press
  • Gronau, Detlef (March 2004), "The Spiral of Theodorus", The American Mathematical Monthly, 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 (ed.), Functional Equations and Inequalities, pp. 111–117
  • Waldvogel, Jörg (2009), Analytic Continuation of the Theodorus Spiral (PDF)