In geometry, the spiral of Theodorus (also called the square root spiral, Pythagorean spiral, or Pythagoras's snail) 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 (angled) triangle, with each leg having unit length. It has a hypotenuse with length equal to the square root of 2. Then, another right triangle (which is the only automedian right triangle) is formed, with one leg being the hypotenuse of the prior right triangle and the other leg having a length of 1; the length of the hypotenuse of this second right triangle is the square root of 3. The process then repeats; the n {\displaystyle n} th triangle in the sequence is a right triangle with the side lengths n {\displaystyle {\sqrt {n}}} , 1 {\displaystyle 1} , and n + 1 {\displaystyle {\sqrt {n+1}}} . For example, the 22nd triangle has sides measuring 1 {\displaystyle 1} , 22 {\displaystyle {\sqrt {22}}} , and a hypotenuse of 22 + 1 = 23 {\displaystyle {\sqrt {22+1}}={\sqrt {23}}} .
History and uses Although all of Theodorus' work has been lost, Plato quoted some of his writing into the dialogue Theaetetus, which tells of Theodorus' ideas. It is believed that Theodorus had proved that all the square roots of integers from 3 to 17, excepting 4, 9, and 16, are irrational by means of the Spiral of Theodorus. Plato does not attribute the proof of the irrationality of the square root of 2 to Theodorus, as it was well known before him. Theodorus and Theaetetus split the rational numbers and irrational numbers into different categories.
Hypotenuse Each of the triangles' hypotenuses h n {\displaystyle h_{n}} gives the square root of the corresponding natural number plus one, with h 1 = 2 {\displaystyle h_{1}={\sqrt {2}}} and h n = n + 1 {\displaystyle h_{n}={\sqrt {n+1}}} . Plato, tutored by Theodorus, questioned why Theodorus stopped at 17 {\displaystyle {\sqrt {17}}} . The reason is commonly believed to be that the 17 {\displaystyle {\sqrt {17}}} hypotenuse belongs to the last triangle which does not overlap the original unit triangle.
Overlapping In 1958, Kaleb Williams proved that two hypotenuses will never overlap, no matter how far the spiral is continued. In addition, if a random side of unit length is extended into an infinite line, the line will never pass through any of the vertices in the spiral (excepting the two defining the line) .
Extension
Theodorus stopped his spiral at the 16th triangle with a hypotenuse of 17 {\displaystyle {\sqrt {17}}} . However, if the spiral is continued indefinitely by adding triangles, more interesting characteristics can be found:
Growth rate
Angle
If φ n {\displaystyle \varphi _{n}} is the angle of the n {\displaystyle n} th triangle (or spiral segment), then:
tan φ n = 1 n {\displaystyle \tan \varphi _{n}={\frac {1}{\sqrt {n}}}}
Therefore, the subtended angle φ n {\displaystyle \varphi _{n}} of the next triangle n {\displaystyle n} is exactly
φ n = arctan 1 n {\displaystyle \varphi _{n}=\arctan {\frac {1}{\sqrt {n}}}}
For the 16-segment portion of the spiral constructed by Theodorus, the sum of the subtended angles is
arctan 1 1 + arctan 1 2 + ⋯ + arctan 1 16 ≈ 6.128731 rad ≈ 351.1504 ∘ {\displaystyle \arctan {\frac {1}{\sqrt {1}}}+\arctan {\frac {1}{\sqrt {2}}}+\cdots +\arctan {\frac {1}{\sqrt {16}}}\approx 6.128731{\text{ rad}}\approx 351.1504^{\circ }}
and the subsequent addition of one more term arctan 1 17 {\displaystyle \arctan {\tfrac {1}{\sqrt {17}}}} gives a sum of ≈ 364.783 ∘ {\displaystyle \approx 364.783^{\circ }} , which exceeds a full turn, so that the triangles begin to overlap. The sum of the angles of the first k {\displaystyle k} triangles is written as φ ( k ) {\displaystyle \varphi (k)} . It grows proportionally to the square root of k {\displaystyle k} , with a small bounded correction term c 1 ( k ) {\displaystyle c_{1}(k)}
φ ( k ) = ∑ n = 1 k φ n = 2 k + c 1 ( k ) + 1 6 k − 1 120 k 3 / 2 + O ( k − 5 / 2 ) {\displaystyle \varphi \left(k\right)=\sum _{n=1}^{k}\varphi _{n}=2{\sqrt {k}}+c_{1}(k)+{\frac {1}{6{\sqrt {k}}}}-{\frac {1}{120k^{3/2}}}+O(k^{-5/2})}
where
lim k → ∞ c 1 ( k ) = − 2.157782996659 … = K . {\displaystyle \lim _{k\to \infty }c_{1}(k)=-2.157782996659\ldots =K.}
where K {\displaystyle K} denotes Hlawka's snail constant (OEIS: A105459). The aforementioned constant can be computed by the rapidly convergent series
K = ∑ m = 0 ∞ ( − 1 ) m ζ ( m + 1 2 ) 2 m + 1 {\displaystyle K=\sum _{m=0}^{\infty }{\frac {(-1)^{m}\zeta \left(m+{\frac {1}{2}}\right)}{2m+1}}}
where ζ ( x ) {\displaystyle \zeta (x)} is the Riemann zeta function.
Radius The stepwise growth of the radius of the spiral between triangles n {\displaystyle n} and n + 1 {\displaystyle n+1} , is
Δ r = n + 1 − n = 1 n + 1 + n ∼ 1 2 n {\displaystyle \Delta r={\sqrt {n+1}}-{\sqrt {n}}={\frac {1}{{\sqrt {n+1}}+{\sqrt {n}}}}\sim {\frac {1}{2{\sqrt {n}}}}}
Archimedean spiral The Spiral of Theodorus approximates the Archimedean spiral. Just as the distance between two windings of the Archimedean spiral equals mathematical constant π {\displaystyle \pi } , as the number of spins of the spiral of Theodorus approaches infinity, the distance between two consecutive windings quickly approaches π {\displaystyle \pi } . The following table shows successive windings of the spiral approaching π:
As shown, after only the fifth winding, the distance is a 99.97% accurate approximation to π {\displaystyle \pi } .
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
T ( x ) = ∏ k = 1 ∞ 1 + i / k 1 + i / x + k ( − 1 < x < ∞ ) {\displaystyle T(x)=\prod _{k=1}^{\infty }{\frac {1+i/{\sqrt {k}}}{1+i/{\sqrt {x+k}}}}\qquad (-1<x<\infty )}
which was further studied by his student Leader and by Iserles. This function can be characterized axiomatically as the unique function that satisfies the functional equation
f ( x + 1 ) = ( 1 + i x + 1 ) ⋅ f ( x ) , {\displaystyle f(x+1)=\left(1+{\frac {i}{\sqrt {x+1}}}\right)\cdot f(x),}
the initial condition f ( 0 ) = 1 , {\displaystyle f(0)=1,} and monotonicity in both argument and modulus. An analytic continuation of Davis' continuous form of the Spiral of Theodorus extends in the opposite direction from the origin. 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 n {\displaystyle n} with the integer value of the polar radius r n = ± | n | {\textstyle r_{n}=\pm {\sqrt {|n|}}} are numbered in the figure. The dashed circle in the coordinate origin O {\displaystyle O} is the circle of curvature at O {\displaystyle O} .
See also Fermat's spiral List of spirals
References
Further reading 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)