Root snail

The spiral up ${\displaystyle {\sqrt {17}}}$

The first three turns

The root snail , root spiral or spiral of Theodorus (after Theodoros of Cyrene (5th century BC)) is a spiral that is created by right triangles with sides 1, and . ${\displaystyle {\sqrt {n}}}$ ${\displaystyle {\sqrt {n+1}}}$

Construction and properties

The first triangle has the side lengths 1, and . On the hypotenuse of this triangle, the right-angled triangle with the side lengths 1, and etc. is erected, etc. The adjacent cathets then form a spiral. ${\displaystyle {\sqrt {1}}}$ ${\displaystyle {\sqrt {2}}}$ ${\displaystyle {\sqrt {2}}}$ ${\displaystyle {\sqrt {3}}}$

In contrast to the Archimedean or logarithmic spiral , the root snail consists of straight lines. It can not be differentiated as a curve , but it can be described exactly by the countable number of corner points.

In 1958 Erich Teuffel proved that two of the hypotenuses will never coincide, no matter how far you draw the spiral.

The smallest number of triangles that complete the kth turn of the spiral can be found in the On-Line Encyclopedia of Integer Sequences . The first members are 17, 54, 110, 186, ...

use

With the help of the root snail, the square roots of positive integers can be constructed geometrically.

It is believed that Theodoros, with the help of the root snail, proved that the roots of the non-square integers from 3 to 17 are irrational numbers . (That the root of 2 is irrational was known long before Theodoros.)

Connection with the Archimedes' spiral

As the number of turns increases, the root snail asymptotically approaches an Archimedean spiral.

The spiral spacing thus approaches the number as the number of turns increases . ${\displaystyle \pi }$

Winding number:	Calculated average winding distance	Accuracy of the average coil spacing compared to ${\displaystyle \pi }$
2	3.1592037	99.44255%
3	3.1443455	99.91245%
4th	3.14428	99.91453%
5	3.142395	99.97447%
→ ∞	→ ${\displaystyle \pi }$	→ 100%

literature

• Detlef Gronau: The Spiral of Theodorus . The American Mathematical Monthly, Vol. 111, No. 3 (March, 2004), pp. 230-237 ( JSTOR 4145130 )
• James Tanton: Mathematics Galore! MAA, 2012, ISBN 978-0-88385-776-2 , pp. 8-9
• Julian Havil: The Irrationals . Princeton University Press, 2012, ISBN 978-0-691-14342-2 , pp. 7, 272-274
• Paul J. Nahin: An Imaginary Tale: The Story of √-1 . Princeton University Press, 2012, ISBN 978-1-4008-3389-4 , pp. 33-34

Web links

Commons : root snail  - collection of images, videos and audio files

• Root spiral

Individual evidence

1. ↑ Erich Teuffel: A property of the square root snail. In: Math.-Phys. Semesterber. 6, pp. 148-152 (1958).
2. ↑ http://oeis.org/A072895
3. ^ Paul J. Nahin: An Imaginary Tale: The Story of √-1 . Princeton University Press, 2012, ISBN 978-1-4008-3389-4 , pp. 33-34
4. ↑ http://kociemba.org/themen/spirale/spirale.htm