In geometry, a generalized polygon can be called a polygram, and named specifically by its number of sides. All polygons are polygrams, but can also include disconnected sets of edges, called a compound polygon. For example, a regular pentagram, {5/2}, has 5 sides, and the regular hexagram, {6/2} or 2{3}, has 6 sides divided into two triangles.
A regular polygram {p/q} can either be in a set of regular star polygons (for gcd(p,q) = 1, q > 1) or in a set of regular polygon compounds (if gcd(p,q) > 1). [1]
The polygram names combine a numeral prefix, such as penta-, with the Greek suffix -gram (in this case generating the word pentagram). The prefix is normally a Greek cardinal, but synonyms using other prefixes exist. The -gram suffix derives from γραμμῆς (grammos) meaning a line. [2]
A regular polygram, as a general regular polygon, is denoted by its Schläfli symbol {p/q}, where p and q are relatively prime (they share no factors) and q ≥ 2. For integers p and q, it can be considered as being constructed by connecting every qth point out of p points regularly spaced in a circular placement. [3] [1]
{5/2} |
{7/2} |
{7/3} |
{8/3} |
{9/2} |
{9/4} |
{10/3}... |
In other cases where n and m have a common factor, a polygram is interpreted as a lower polygon, {n/k, m/k}, with k = gcd(n,m), and rotated copies are combined as a compound polygon. These figures are called regular compound polygons.
Triangles... | Squares... | Pentagons... | Pentagrams... | ||||
---|---|---|---|---|---|---|---|
{6/2}= 2{3} |
{9/3}= 3{3} |
{12/4}= 4{3} |
{8/2}= 2{4} |
{12/3}= 3{4} |
{10/2}= 2{5} |
{10/4}= 2{5/2} |
{15/6}= 3{5/2} |
In geometry, a generalized polygon can be called a polygram, and named specifically by its number of sides. All polygons are polygrams, but can also include disconnected sets of edges, called a compound polygon. For example, a regular pentagram, {5/2}, has 5 sides, and the regular hexagram, {6/2} or 2{3}, has 6 sides divided into two triangles.
A regular polygram {p/q} can either be in a set of regular star polygons (for gcd(p,q) = 1, q > 1) or in a set of regular polygon compounds (if gcd(p,q) > 1). [1]
The polygram names combine a numeral prefix, such as penta-, with the Greek suffix -gram (in this case generating the word pentagram). The prefix is normally a Greek cardinal, but synonyms using other prefixes exist. The -gram suffix derives from γραμμῆς (grammos) meaning a line. [2]
A regular polygram, as a general regular polygon, is denoted by its Schläfli symbol {p/q}, where p and q are relatively prime (they share no factors) and q ≥ 2. For integers p and q, it can be considered as being constructed by connecting every qth point out of p points regularly spaced in a circular placement. [3] [1]
{5/2} |
{7/2} |
{7/3} |
{8/3} |
{9/2} |
{9/4} |
{10/3}... |
In other cases where n and m have a common factor, a polygram is interpreted as a lower polygon, {n/k, m/k}, with k = gcd(n,m), and rotated copies are combined as a compound polygon. These figures are called regular compound polygons.
Triangles... | Squares... | Pentagons... | Pentagrams... | ||||
---|---|---|---|---|---|---|---|
{6/2}= 2{3} |
{9/3}= 3{3} |
{12/4}= 4{3} |
{8/2}= 2{4} |
{12/3}= 3{4} |
{10/2}= 2{5} |
{10/4}= 2{5/2} |
{15/6}= 3{5/2} |