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
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
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:
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]
The growth of the radius of the spiral at a certain triangle 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, 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
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
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 .
^Teuffel, Erich (1958), "Eine Eigenschaft der Quadratwurzelschnecke", Mathematisch-Physikalische Semesterberichte zur Pflege des Zusammenhangs von Schule und Universität, 6: 148–152,
MR0096160
^
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
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