The number of and for first 6 terms of Moser's circle problem
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 c ...
, the problem of dividing a circle into areas by means of an inscribed
polygon
In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed '' polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two t ...
with ''n'' sides in such a way as to ''maximise'' the number of areas created by the edges and
diagonal
In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Gree ...
s, sometimes called
Moser
Moser may refer to:
* Moser (surname)
* An individual who commits the act of Mesirah in Judaism
Places
* Moser Glacier, a glacier on the west coast of Graham Land, Antarctica
* Moser River, Nova Scotia, Canada
* Moser Bay Seaplane Base, a p ...
's circle problem, has a solution by an inductive method. The greatest possible number of regions, , giving the sequence 1, 2, 4, 8, 16, 31, 57, 99,
163
Year 163 ( CLXIII) was a common year starting on Friday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Laelianus and Pastor (or, less frequently, year 916 ''Ab urbe cond ...
, 256, ... (). Though the first five terms match the
geometric progression
In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the ''common ratio''. For e ...
, it diverges at , showing the risk of generalising from only a few observations.
Lemma
If there are ''n'' points on the circle and one more point is added, ''n'' lines can be drawn from the new point to previously existing points. Two cases are possible. In the first case (a), the new line passes through a point where two or more old lines (between previously existing points) cross. In the second case (b), the new line crosses each of the old lines in a different point. It will be useful to know the following fact.
Lemma. The new point ''A'' can be chosen so that case ''b'' occurs for each
