Regular starry polygon – Wikipedia

Example of regular starry polygon .
Its twenty-four peaks are also spread over a circle.

In geometry, a regular starry polygon (not to be confused with a starry part) is a regular non -convex polygon. Non -regular star polygons are not formally defined.

Branko Grünbaum identifies two primary concepts used by Kepler, one being the regular starry polygon with secante edges that do not generate new peaks, and the other being simple polygons concave ^{[ first ]} .

When the star polygon has summits or sides in low numbers, its name can combine a numeral prefix, such as penta- For a number five of the summits or sides, with the Greek suffix -gone or -gramme (the name of the polygon is then Pentagon , or pentagram for a star pentagon). The most common prefix comes from the Greek, but have- For example comes from Latin, in the name ” nonagramme “a polygon with nine peaks, also called” enneagram “.

A regular starry polygon is a team and equilateral polygon that self-intersecte, created by connecting a summit of a regular polygon to p sides at another non -adjacent summit and continuing the process until returning to the first summit ^{[ 2 ]} . Alternatively, for integers p And q , we can consider it as a construction connecting all q -Iemes heights of a set of p regularly spaced peaks and circularly placed ^{[ 3 ]} . For example, in a regular pentagon, a five -pointed star can be obtained by drawing a line from the first point to the third, then from the third to the fifth, then from the fifth to the second, then from the second to the fourth, and finally from the fourth to first. In short, a regular starry polygon can be obtained by stirring a regular convex polygon.

A regular star polynomial is denoted by its symbol of Schläfli { p / q }, Or p And q are first between them and q ≥ 2.

The symmetry group of { n / k } is the diédral group D _n of order 2 n independent of k .

Star regular polygons have been studied for the first time systematically by Thomas Bradwardine, then later Kepler ^{[ 4 ]} .

Degenerate star polygon [ modifier | Modifier and code ]

And p And q are not primary between them, a generated polygon results from the heights and edges coinciding. For example, {6/2} will appear as a triangle, but can be designated by two sets of summits 1-6. You have to see this phenomenon not as two superimposed triangles but as a single polygon rebuke on itself ^{[ 5 ] , [ 6 ]} .

When the senior lines are removed, the star polygons are no longer regular, but can be seen as 2n -Intestoxal concaves. Branko Grünbaum represents these stars by | n/d |. They have the same geometry as polygons { n/d } with a notation {n _a } more general which represents a star at n Branches each with an internal angle of α <180 (1 - 2/ n ) degrees. For | n/d |, the internal vertices have an β external angle of 360 ( d -first)/ n

Examples of simple isotoxal star polygons

|n/d| {n a }	{3 30 ° }	{6 30 ° }	| 5/2 | {5 36 ° }	{4 45 ° }	| 8/3 | {8 45 ° }	| 6/2 | {6 60 ° }	{5 72 ° }
a	30 °		36 °	45 °		60 °	72 °
b	150 °	90 °	72 °	135 °	90 °	120 °	144 °
Isotoxal star
Associated polygon {n/d}	{12/5}		{5/2}	{8/3}		{6/2}	{10/3}

These polygons are often observed in pavement models. The α parametric angle can be chosen to correspond to the internal angles of neighboring polygons in such models.

Example of pavages with isotoxal star polygons ^{[ 7 ]}

Star triangles	Star squares	Star hexagons		Star octagons
(3.3* A.3.3 ** a)	(8.4* P/4.8.4* II/4)	(6.6* P/3.6.6* II/3)	(6.6* P/3.6.6* II/3)	Not edge-to-edge

The interior of a star polygon can be interpreted in several ways. Three of these interpretations are illustrated for a pentagram. Branko Grünbaum and Geoffrey Shephard consider two of them as regular starry and starred and 2n -Sogonal concave ^{[ 8 ]} .

These include:

• At the place on one side, one side is treated like the outside and the other as the interior: it is the interpretation of the left (in red) on the illustration;
• The number of times a polygon covers a given region determines its density. A zero density is given outside, and any region with non -zero density is interior; This is the interpretation of the environment (in yellow);
• where a line can be drawn between two sides, the region where it is located is considered interior in the figure. This is right interpretation (in blue).

Each of the above approaches gives the polygon a different area.