geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, an Archimedean circle is any circle constructed from an

arbelos In geometry, an arbelos is a plane region bounded by three semicircles with three apexes such that each corner of each semicircle is shared with one of the others (connected), all on the same side of a straight line (the ''baseline'') that conta ...

that has the same

radius In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...

as each of

Archimedes' twin circles In geometry, the twin circles are two special circles associated with an arbelos. An arbelos is determined by three collinear points , , and , and is the curvilinear triangular region between the three semicircles that have , , and as their diame ...

. If the arbelos is normed such that the diameter of its outer (largest) half circle has a length of 1 and ''r'' denotes the radiius of any of the inner half circles, then the radius ''ρ'' of such an Archimedean circle is given by :

\rho=\fracr\left(1-r\right),

There are over fifty different known ways to construct Archimedean circles.

Origin

An Archimedean circle was first constructed by

Archimedes Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists ...

in his ''

Book of Lemmas The ''Book of Lemmas'' or ''Book of Assumptions'' (Arabic ''Maʾkhūdhāt Mansūba ilā Arshimīdis'') is a book attributed to Archimedes by Thābit ibn Qurra, though the authorship of the book is questionable. It consists of fifteen propositio ...

''. In his book, he constructed what is now known as

Radius

a

and

b

are the radii of the small semicircles of the arbelos, the radius of an Archimedean circle is equal to :

R = \frac

This radius is thus

\frac 1R = \frac 1a + \frac 1b

. The Archimedean circle with center

C

(as in the figure at right) is tangent to the tangents from the centers of the small semicircles to the other small semicircle.

Other Archimedean circles finders

Leon Bankoff

Leon Bankoff Leon Bankoff (December 13, 1908 – February 16, 1997), born in New York City, New York, was an American dentist. As an amateur mathematician he constructed the Bankoff circle. He was also an Esperantist. Life After a visit to the City ...

constructed other Archimedean circles called Bankoff's triplet circle and Bankoff's quadruplet circle.

Thomas Schoch

In 1978 Thomas Schoch found a dozen more Archimedean circles (the

Schoch circles In geometry, the Schoch circles are twelve Archimedean circles constructed by Thomas Schoch. History In 1979, Thomas Schoch discovered a dozen new Archimedean circles; he sent his discoveries to ''Scientific Americans "Mathematical Games" editor Ma ...

) that have been published in 1998. He also constructed what is known as the

Schoch line In geometry, the Schoch line is a line defined from an arbelos and named by Peter Woo after Thomas Schoch, who had studied it in conjunction with the Schoch circles. Construction An arbelos is a shape bounded by three mutually-tangent semicir ...

Peter Y. Woo

Peter Y. Woo considered the Schoch line, and with it, he was able to create a family of infinitely many Archimedean circles known as the

Woo circles In geometry, the Woo circles, introduced by Peter Y. Woo, are a set of infinitely many Archimedean circles. Construction Form an arbelos with the two inner semicircles tangent at point ''C''. Let ''m'' denote any nonnegative real number. Draw tw ...

Frank Power

In the summer of 1998, Frank Power introduced four more Archimedes circles known as

Archimedes' quadruplets In geometry, Archimedes' quadruplets are four congruent circles associated with an arbelos. Introduced by Frank Power in the summer of 1998, each have the same area as Archimedes' twin circles, making them Archimedean circles. Construction An ar ...

Archimedean circles in Wasan geometry (Japanese geometry)

In 1831, Nagata (永田岩三郎遵道) proposed a

sangaku Sangaku or San Gaku ( ja, 算額, lit=calculation tablet) are Japanese geometrical problems or theorems on wooden tablets which were placed as offerings at Shinto shrines or Buddhist temples during the Edo period by members of all social classes ...

problem involving two Archimedean circles, which are denoted by W6 and W7 in In 1853, Ootoba (大鳥羽源吉守敬) proposed a

problem involving an Archimedean circle.

References

{{reflist Arbelos Geometric shapes Circle packing