Self-Intersecting Polygons


A polygram is like a polygon but the sides are allowed to cross. A polygram of n-sides or vertices is called an n-gram and for particular values of n we can give them individual names as for polygons. However there is no such thing as a 'trigram', since three successive line segments cannot cross anywhere; the first case is a tetragram, followed by pentagram, hexagram, heptagram, octagram, and so on.

A regular polygram has all sides and angles equal. A tetragram cannot be regular, it can however be quasiregular, that is have the same vertices as a regular polygon; the first case of a regular polygram is the pentagram; there are also two quasiregular pentagrams. From six points we can derive eleven quasiregular hexagrams; none regular, one is asymmetric. The suffixes in the dagram indicate the numbers of different orientations in which each polygram can be seen if the points are fixed in place (thus 1 indicates regular). [The problem of enumerating the 6-point regular and quasiregular polygons and polygrams appears as problem 51 in K. Fujimura, The Tokyo Puzzles, where it is attributed to S. Kobayashi (c.1976).]

The condition for a regular polygram with n vertices to exist is that there be a number m > 1 and < n/2 such that m and n have no common factor other than 1; the polygram is then formed by connecting each vertex to the mth vertex round the circle successively. The number of points of intersection within and on the circle is m.n, since they form m circles of n. There are two regular polygrams of 7 vertices (m = 2, 3); one of 8 vertices (m = 3); two of 9 vertices (m = 2, 4); one of 10 vertices (m = 3); four of 11 vertices (m = 2, 3, 4, 5); one of 12 vertices (m = 5).

Stars, Asterisks and Atoms

A related question to that of the regular polygrams is that of the other regular patterns that can be formed by joining each of n points, regularly arranged round a circle, to the points m steps away in each direction, where m < n/2. When m = 1 we have the ordinary polygons. When hcf (m,n) = 1 we have the regular polygrams. For other cases of m we have stars. All the stars for n from 4 to 12 are illustrated. The number n must be composite and hcf (m, n) > 1.

When m = n/2 the star consists of m superimposed diametral lines, intersecting at the centre. I call these stars asterisks. When m = n/k the star consists of m superimposed regular k-gons or k-grams.

If the digons in the asterisks are drawn as long narrow ellipses we get diagrams often used as illustrations of atoms, based on the old (1911) Rutherford orbital model, so we use this name for them. How many intersections are there in these atoms? The number of intersections in and on the circle is again m.n when the digons are drawn as two arcs, i.e. when m is k and n is 2k the intersection number is 2k² which takes the values 2, 8, 18, 32, 50, 72, ... It is interesting that these numbers 2, 8, 18, 32 are the numbers of chemical elements in the successive periods of the atomic number sequence ending at the inert gases: H-He (2), Li-Ne (8), Na-Ar (8), K-Kr (18), Rb-Xe (18), Cs-Rn (32). So perhaps calling these patterns 'atoms' is not so fanciful after all.