Ingeometry, thespiral of Theodorus (also called thesquare root spiral,Pythagorean spiral, orPythagoras's snail)^[1] is aspiral composed ofright triangles, placed edge-to-edge. It was named afterTheodorus of Cyrene.

The spiral is started with anisosceles right triangle, with eachleg having unitlength, and ahypotenuse with length thesquare root of 2. Another right triangle (which is theonlyautomedian right triangle) is formed, with one leg being the hypotenuse of the prior right triangle and the other leg having length of 1; the length of the hypotenuse of this second right triangle is thesquare root of 3. The process then repeats; the${\displaystyle n}$th triangle in the sequence is a right triangle with the side lengths${\displaystyle \sqrt{n}}$ and 1, and with hypotenuse${\displaystyle \sqrt{n+1}}$. For example, the 16th triangle has sides measuring${\displaystyle 4=\sqrt{16}}$, 1 and hypotenuse of${\displaystyle \sqrt{17}}$.

Although all of Theodorus' work has been lost,Plato put Theodorus into his dialogueTheaetetus, 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 areirrational by means of the Spiral of Theodorus.^[2]

Plato does not attribute the irrationality of thesquare 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]

Each of the triangles' hypotenuses${\displaystyle h_n}$ gives thesquare root of the correspondingnatural number, with${\displaystyle h_1=\sqrt{2}}$.

Plato, tutored by Theodorus, questioned why Theodorus stopped at${\displaystyle \sqrt{17}}$. The reason is commonly believed to be that the${\displaystyle \sqrt{17}}$ hypotenuse belongs to the last triangle that does not overlap the figure.^[4]

In 1958, Kaleb Williams proved that two hypotenuses will never overlap, regardless of how far the spiral is continued. Also, if the sides of unit length are extended into aline, they will never pass through any of the other vertices of the total figure.^[4][5]

Theodorus stopped his spiral at the triangle with a hypotenuse of${\displaystyle \sqrt{17}}$. If the spiral is continued to infinitely many triangles, many more interesting characteristics are found.

If${\displaystyle {\phi }_n}$ is the angle of the${\displaystyle n}$th triangle (or spiral segment), then:$${\displaystyle \tan \left({\phi }_n\right)=\frac{1}{\sqrt{n}}.}$$Therefore, the growth of the angle${\displaystyle {\phi }_n}$ of the next triangle${\displaystyle n}$ is:^[1]$${\displaystyle {\phi }_n=\arctan \left(\frac{1}{\sqrt{n}}\right).}$$

The sum of the angles of the first${\displaystyle k}$ triangles is called the total angle${\displaystyle \phi (k)}$ for the${\displaystyle k}$th triangle. It grows proportionally to the square root of${\displaystyle k}$, with abounded correction term${\displaystyle c_2}$:^[1]$${\displaystyle \phi \left(k\right)=\sum _{n=1}^k{\phi }_n=2\sqrt{k}+c_2(k)}$$where$${\displaystyle \underset{k\to \mathrm{\infty }}{lim}{c}_2(k)=-2.157782996659\dots }$$(OEIS: A105459).

The growth of the radius of the spiral at a certain triangle${\displaystyle n}$ is$${\displaystyle \mathrm{\Delta }r=\sqrt{n+1}-\sqrt{n}.}$$

The Spiral of Theodorusapproximates theArchimedean spiral.^[1] Just as the distance between two windings of the Archimedean spiral equalsmathematical constant${\displaystyle \pi }$, as the number of spins of the spiral of Theodorus approachesinfinity, the distance between two consecutive windings quickly approaches${\displaystyle \pi }$.^[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%
${\displaystyle \to \mathrm{\infty }}$	${\displaystyle \to \pi }$	${\displaystyle \to 100\mathrm{\%}}$

As shown, after only the fifth winding, the distance is a 99.97% accurate approximation to${\displaystyle \pi }$.^[1]

The question of how tointerpolate the discrete points of the spiral of Theodorus by a smooth curve was proposed and answered byPhilip J. Davis in 2001 by analogy with Euler's formula for thegamma function as aninterpolant for thefactorial function. Davis found the function^[7]which was further studied by his studentLeader^[8] and byIserles.^[9] This function can be characterized axiomatically as the unique function that satisfies thefunctional equation$${\displaystyle f(x+1)=\left(1+\frac{i}{\sqrt{x+1}}\right)\cdot f(x),}$$the initial condition${\displaystyle f(0)=1,}$ andmonotonicity in bothargument andmodulus.^[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${\displaystyle n}$ with the integer value of the polar radius${\displaystyle r_n=\pm \sqrt{|n|}}$ are numbered in the figure.The dashed circle in the coordinate origin${\displaystyle O}$ is the circle of curvature at${\displaystyle O}$.

• Fermat's spiral
• List of spirals

• 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)

• Spirals
• Pythagorean theorem
• Pi
• Greek mathematics

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• Short description is different from Wikidata