In
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 ...
, the Woo circles, introduced by Peter Y. Woo, are a set of
infinitely many Archimedean circle
In geometry, an Archimedean circle is any circle constructed from an arbelos that has the same radius as each of Archimedes' twin circles. If the arbelos is normed such that the diameter of its outer (largest) half circle has a length of 1 and '' ...
s.
Construction
Form 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 ...
with the two inner
semicircle
In mathematics (and more specifically geometry), a semicircle is a one-dimensional locus of points that forms half of a circle. The full arc of a semicircle always measures 180° (equivalently, radians, or a half-turn). It has only one line o ...
s
tangent
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...
at point ''C''. Let ''m'' denote any
nonnegative
In mathematics, the sign of a real number is its property of being either positive, negative, or zero. Depending on local conventions, zero may be considered as being neither positive nor negative (having no sign or a unique third sign), or it ...
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
. Draw two circles, with
radii
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 ...
''m'' times the radii of the smaller two arbelos semicircles, centered on the arbelos ground line, also tangent to each other at point ''C'' and with radius ''m'' times the radius of the corresponding small arbelos arc. Any circle centered on 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 ...
and externally
tangent to the circles is a Woo circle.
[
]
See also
*
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 ...
References
Arbelos
Circles
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