Salinon

The Salinon (salmon-colored) and the circle (blue) have the same area (purple: the common area of ​​Salinon and the circle).

The salinon (Greek presumably for "salt cellar") is one of four semicircles formed, mirror symmetrical geometric figure . It was probably first described by Archimedes in his book of lemmas .

construction

${\displaystyle O}$be the origin of a Cartesian coordinate system . On the outside of the axis are the two points and (each with the same distance from ) and inside the points and (also with the same distance from ); so is . Build a semicircle over and two smaller, equally large semicircles over and . Finally, draw a fourth semicircle underneath . The salinon is the figure delimited by these four semicircles (salmon-colored in the drawing). It intersects the -axis at points and . ${\displaystyle x}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle O}$ ${\displaystyle D}$ ${\displaystyle E}$ ${\displaystyle O}$ ${\displaystyle {\overline {AD}}={\overline {EB}}}$ ${\displaystyle AB}$ ${\displaystyle AD}$ ${\displaystyle EB}$ ${\displaystyle DE}$ ${\displaystyle y}$ ${\displaystyle C}$ ${\displaystyle F}$

properties

Salinon

Arbelos

Archimedes described the properties of Salinon as the  14th sentence in his Book of Lemmas with reference to Euclid's Elements , Book 2, Proposition 10.

If you denote the radius of the large semicircle ( ) with and that of the small, middle semicircle ( ) , then applies to the area of the Salinon: ${\displaystyle {\overline {AB}}/2={\overline {AO}}}$ ${\displaystyle R}$ ${\displaystyle {\overline {DE}}/2={\overline {DO}}}$ ${\displaystyle r}$ ${\displaystyle A}$

${\displaystyle A={\frac {1}{4}}\pi \left(R+r\right)^{2}.}$

In addition:

• The points on the four semicircles with the greatest distance to the axis (below and ) form a square . ${\displaystyle x}$ ${\displaystyle C}$ ${\displaystyle F}$
• The perimeter of this square has the same area (purple in the illustration) as the Salinon.
• When the diameter of the semicircle goes below zero (the points and therefore coincide), the salinon turns into an arbelos , mirror-symmetrical to the axis , another figure of semicircles whose study is attributed to Archimedes. ${\displaystyle DE}$ ${\displaystyle D}$ ${\displaystyle E}$ ${\displaystyle O}$ ${\displaystyle y}$

See also

• Arbelos

Web links

Commons : Salinon  - collection of images, videos and audio files

• Eric W. Weisstein : Salinon . In: MathWorld (English).
• Alexander Bogomolny: Salinon: From Archimedes' Book of Lemmas . In: Cut The Knot (English)
• Jürgen Köller: Salinon . In: Mathematical handicrafts

Individual evidence

1. ↑ For the origin of the name cf. Archimedes' works. With modern designations ed. by Sir Thomas L. Heath. German from Dr. Fritz Kliem. Berlin: Häring, 1914, pp. 21–23, note 3.