In geometry, the **spiral of Theodorus** (also called *square root spiral*, *Einstein spiral*, or *Pythagorean spiral*)^{ [1] } is a spiral composed of right triangles, placed edge-to-edge. It was named after Theodorus of Cyrene.

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 √2) and the other leg having length of 1; the length of the hypotenuse of this second triangle is √3. The process then repeats; the *n*th triangle in the sequence is a right triangle with side lengths √*n* and 1, and with hypotenuse √*n* + 1. For example, the 16th triangle has sides measuring 4 (=√16), 1 and hypotenuse of √17.

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] }

Each of the triangles' hypotenuses *h _{n}* gives the square root of the corresponding natural number, with

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

In 1958, Erich Teuffel 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] }

Theodorus stopped his spiral at the triangle with a hypotenuse of √17. If the spiral is continued to infinitely many triangles, many more interesting characteristics are found.

If φ_{n} is the angle of the *n*th triangle (or spiral segment), then:

Therefore, the growth of the angle φ_{n} of the next triangle *n* is:^{ [1] }

The sum of the angles of the first *k* triangles is called the total angle φ(*k*) for the *k*th triangle. It grows proportionally to the square root of *k*, with a bounded correction term *c*_{2}:^{ [1] }

where

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

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

The following is a table showing of two 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% |

→ ∞ | → π | → 100% |

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

The question of how to interpolate the discrete points of the spiral of Theodorus by a smooth curve was proposed and answered in ( Davis 2001 , pp. 37–38) by analogy with Euler's formula for the gamma function as an interpolant for the factorial function. Davis found the function

which was further studied by his student Leader ^{ [7] } and by Iserles (in an appendix to ( Davis 2001 ) ). An axiomatic characterization of this function is given in ( Gronau 2004 ) as the unique function that satisfies the functional equation

the initial condition and monotonicity in both argument and modulus; alternative conditions and weakenings are also studied therein. An alternative derivation is given in ( Heuvers, Moak & Boursaw 2000 ).

An analytic continuation of Davis' continuous form of the Spiral of Theodorus which extends in the opposite direction from the origin is given in ( Waldvogel 2009 ).

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 .

In mathematics, a **complex number** is a number that can be expressed in the form *a* + *bi*, where a and b are real numbers, and i is a symbol called the imaginary unit, and satisfying the equation *i*^{2} = −1. Because no "real" number satisfies this equation, i was called an imaginary number by René Descartes. For the complex number *a* + *bi*, a is called the **real part** and b is called the **imaginary part**. The set of complex numbers is denoted by either of the symbols or **C**. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers and are fundamental in many aspects of the scientific description of the natural world.

In integral calculus, an **elliptic integral** is one of a number of related functions defined as the value of certain integrals. Originally, they arose in connection with the problem of finding the arc length of an ellipse and were first studied by Giulio Fagnano and Leonhard Euler. Modern mathematics defines an "elliptic integral" as any function *f* which can be expressed in the form

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In mathematics, a **square root** of a number *x* is a number *y* such that *y*^{2} = *x*; in other words, a number *y* whose *square* (the result of multiplying the number by itself, or *y* ⋅ *y*) is *x*. For example, 4 and −4 are square roots of 16, because 4^{2} = (−4)^{2} = 16. Every nonnegative real number *x* has a unique nonnegative square root, called the *principal square root*, which is denoted by where the symbol is called the *radical sign* or *radix*. For example, the principal square root of 9 is 3, which is denoted by because 3^{2} = 3 ⋅ 3 = 9 and 3 is nonnegative. The term (or number) whose square root is being considered is known as the *radicand*. The radicand is the number or expression underneath the radical sign, in this case 9.

A **right triangle** or **right-angled triangle** (British), or more formally an **orthogonal triangle**, is a triangle in which one angle is a right angle. The relation between the sides and angles of the right angled is the basis for trigonometry.

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A **Fermat's spiral** or **parabolic spiral** is a plane curve named after Pierre de Fermat. Its polar coordinate representation is given by

In geometry, a **hypotenuse** is the longest side of a right-angled triangle, the side opposite the right angle. The length of the hypotenuse can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse equals the sum of the squares of the lengths of the other two sides. For example, if one of the other sides has a length of 3 and the other has a length of 4, then their squares add up to 25. The length of the hypotenuse is the square root of 25, that is, 5.

**Theodorus of Cyrene** was an ancient Greek who lived during the 5th century BC. The only first-hand accounts of him that survive are in three of Plato's dialogues: the *Theaetetus*, the *Sophist*, and the *Statesman*. In the former dialogue, he posits a mathematical theorem now known as the Spiral of Theodorus.

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In mathematics, the **sine** is a trigonometric function of an angle. The sine of an acute angle is defined in the context of a right triangle: for the specified angle, it is the ratio of the length of the side that is opposite that angle, to the length of the longest side of the triangle. For an angle , the sine function is denoted simply as .

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The **square root of 3** is the positive real number that, when multiplied by itself, gives the number 3. It is denoted mathematically as **√3**. It is more precisely called the **principal square root of 3**, to distinguish it from the negative number with the same property. The square root of 3 is an irrational number. It is also known as **Theodorus' constant**, after Theodorus of Cyrene, who proved its irrationality.

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The **square root of 5** is the positive real number that, when multiplied by itself, gives the prime number 5. It is more precisely called the **principal square root of 5**, to distinguish it from the negative number with the same property. This number appears in the fractional expression for the golden ratio. It can be denoted in surd form as:

A **Kepler triangle** is a right triangle with edge lengths in geometric progression. The ratio of the progression is √𝜙, where 𝜙 is the golden ratio, and can be written: , or approximately **1 : 1.272 : 1.618**. The squares of the edges of this triangle are also in geometric progression according to the golden ratio itself.

- 1 2 3 4 5 Hahn, Harry K. "The Ordered Distribution of Natural Numbers on the Square Root Spiral". arXiv: 0712.2184 .
- ↑ Nahin, Paul J. (1998),
*An Imaginary Tale: The Story of [the Square Root of Minus One]*, Princeton University Press, p. 33, ISBN 0-691-02795-1 - ↑ Plato; Dyde, Samuel Walters (1899),
*The Theaetetus of Plato*, J. Maclehose, pp. 86–87. - 1 2 Long, Kate. "A Lesson on The Root Spiral". Archived from the original on 11 April 2013. Retrieved 30 April 2008.
- ↑ Erich Teuffel, Eine Eigenschaft der Quadratwurzelschnecke,
*Math.-Phys. Semesterber.*6 (1958), pp. 148-152. - ↑ 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 .
- ↑ Leader, J.J. The Generalized Theodorus Iteration (dissertation), 1990, Brown University

- 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*, Mathematical Association of America,**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)

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