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 midpoint-stretching polygon of a

cyclic polygon In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...

is another cyclic polygon inscribed in the same circle, the polygon whose vertices are the

midpoint In geometry, the midpoint is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment. Formula The midpoint of a segment in ''n''-dimens ...

s of the

circular arc Circular may refer to: * The shape of a circle * ''Circular'' (album), a 2006 album by Spanish singer Vega * Circular letter (disambiguation) ** Flyer (pamphlet), a form of advertisement * Circular reasoning, a type of logical fallacy * Circular ...

s between the vertices of .. It may be derived from the

midpoint polygon In geometry, the midpoint polygon of a polygon is the polygon whose vertices are the midpoints of the edges of . It is sometimes called the Kasner polygon after Edward Kasner, who termed it the ''inscribed polygon'' "for brevity". Examples Tr ...

of (the polygon whose vertices are the edge midpoints) by placing the polygon in such a way that the circle's center coincides with the

origin Origin(s) or The Origin may refer to: Arts, entertainment, and media Comics and manga * Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002 * The Origin (Buffy comic), ''The Origin'' (Bu ...

, and stretching or normalizing the vector representing each vertex of the midpoint polygon to make it have

unit length Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * ''Unit'' (alb ...

Musical application

The midpoint-stretching polygon is also called the shadow of ; when the circle is used to describe a repetitive

time sequence In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Exa ...

and the polygon vertices on it represent the onsets of a

drum beat A drum beat or drum pattern is a rhythmic pattern, or repeated rhythm establishing the meter and groove through the pulse and subdivision, played on drum kits and other percussion instruments. As such a "beat" consists of multiple drum strokes o ...

, the shadow represents the set of times when the drummer's hands are highest, and has greater rhythmic evenness than the original rhythm.

Convergence to regularity

The midpoint-stretching polygon of a

regular polygon In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either convex p ...

is itself regular, and iterating the midpoint-stretching operation on an arbitrary initial polygon results in a sequence of polygons whose shape converges to that of a regular polygon.

References

{{Reflist Polygons