Spiral of Theodorus
In geometry, the spiral of Theodorus (also called "square root spiral", "Einstein 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 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 formula_1th triangle in the sequence is a right triangle with the side lengths formula_2 and 1, and with hypotenuse formula_3. For example, the 16th triangle has sides measuring formula_4, 1 and hypotenuse of formula_5. 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. 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. Hypotenuse. Each of the triangles' hypotenuses formula_6 gives the square root of the corresponding natural number, with formula_7. Plato, tutored by Theodorus, questioned why Theodorus stopped at formula_5. The reason is commonly believed to be that the formula_5 hypotenuse belongs to the last triangle that does not overlap the figure. Overlapping. 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. Extension. Theodorus stopped his spiral at the triangle with a hypotenuse of formula_5. If the spiral is continued to infinitely many triangles, many more interesting characteristics are found. Growth rate. Angle. If formula_11 is the angle of the formula_1th triangle (or spiral segment), then: formula_13 Therefore, the growth of the angle formula_11 of the next triangle formula_1 is: formula_16 The sum of the angles of the first formula_17 triangles is called the  total angle formula_18 for the formula_17th triangle. It grows proportionally to the square root of formula_17, with a bounded correction term formula_21: formula_22 where formula_23 Radius. The growth of the radius of the spiral at a certain triangle formula_1 is formula_25 Archimedean spiral. The Spiral of Theodorus approximates the Archimedean spiral. Just as the distance between two windings of the Archimedean spiral equals mathematical constant formula_26, as the number of spins of the spiral of Theodorus approaches infinity, the distance between two consecutive windings quickly approaches formula_26. The following is a table showing of two windings of the spiral approaching  pi: As shown, after only the fifth winding, the distance is a 99.97% accurate approximation to formula_26. Continuous curve. The question of how to interpolate the discrete points of the spiral of Theodorus by a smooth curve was proposed and answered in by analogy with Euler's formula for the gamma function as an interpolant for the factorial function. Davis found the function formula_29 which was further studied by his student Leader and by Iserles (in an appendix to ). An axiomatic characterization of this function is given in as the unique function that satisfies the functional equation formula_30 the initial  condition formula_31 and monotonicity in both argument and modulus; alternative conditions and weakenings are also studied therein. An alternative derivation is given in . An analytic continuation of Davis' continuous form of the Spiral of Theodorus which extends in the opposite direction from the origin is given in . 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 formula_1 with the integer value of the polar radius formula_33 are numbered in the figure. The dashed circle in the coordinate  origin formula_34 is the circle of curvature at formula_34.