Hendecagon

A Elfeck (also Hendekagon of Greek ἕνδεκα, hendeka, eleven) is a polygon with eleven sides and eleven corners. Often thereby a planar, regular Elfeck is meant, in which all sides are equal and all vertices lie on a common circle.

Properties

For the Inkreisradius, the radius radius and the area of ​​a regular Elfecks with side length applies:

The regular Elfeck is not constructible with compass and straightedge, because is a prime number that is not a Fermat prime, see konstruierbares polygon. It can not even construct with the aid of an auxiliary for trisection of an angle and it is the regular polygon with the smallest number of vertices with this property.

History

Area calculation to Heron

Heron of Alexandria constructed in his book Metrika in the 1st century. BC faces regular polygons with 3, 5, 6, 8, 10 and 12 pages and gave approximate solutions for the heptagon, the Neuneck and Elfeck to. For the Neuneck and Elfeck he relied on this angle approximations of the work On the tendons ( Περὶ τῶν ἐν κὐκλῳ εὐθειῶν, probably the Chordentafel of Hipparchus of Nicaea ). The approximate formula for the area of a regular Elfecks is therefore

The side length of the Elfecks is.

Construction after Dürer

Albrecht Dürer describes in his work Underweysung the measurement with the zirckel and richtscheyt in lines pave unnd gantzen corporen ( 1525) the construction of a circle inscribed in a regular Elfecks:

"As soon jch a eylf eck in a zirckel reyssen wants nym jch a vierteyl of the zirckels diameter VND Erleng jn an eight teyl Auss jm perso VND far with images this leng herumb in a zirckel the tryt beileuoftig So it is Mechanice but nit demonstratiue place "

Thus, it takes one quarter of the circle diameter, it is decomposed into eight equal parts, and it extends around a part. This route is then placed eleven times on the circle. Dürer explicitly points out that this is an approximate ( " mechanical " ) and not an exact concerns ( " demonstrative " ) construction. The resulting approximation to the side length of the Elfecks

Wherein the circle diameter.

Use

The front and back of the Susan B. Anthony dollar, a U.S. one-dollar coin that was minted from 1979 to 1981 and 1999, shows the figure of a Elfecks. The introduced in 1987 Canadian one-dollar coins are in the shape of a rounded Elfecks. The Indian two -rupee coin is elfeckig.