Golygons

As I track my diurnal age, every day's number always holds some mathematical significance that Numbers Aplenty or the Online Encyclopaedia of Integer Sequences will usually uncover. Every day holds the promise that I'll be introduced to some mathematical concept that I've never encountered before. Today was one such day because I was introduced to the concept of a golygon.

The number that lead me there was 25603, a twin prime with 25601, and one of the OEIS sequences to which it belonged caught my attention:

A292618: the first prime of 8 consecutive primes a, b, c, d, e, f, g, h such that a + g = c + e and b + h = d + f. 

The sequence begins: 359, 389, 839, 853, 937, 1019, 2213, 2221, 2237, 2593, 3019, 3821, 3823, 4111, 4231, 4801, 5407, 5839, 6997, 12241, 13499, 14741, 15473, 25603, ...

At first I was inclined to dismiss this as another of the many highly contrived sequences that litter the OEIS but then an ASCII diagram (see Figure 1) in the comments section caught my attention:

Figure 1

This was interesting. The relationship between the octuplet of primes now had a clear geometrical significance. It was a link in the OEIS entry that lead me to the golygon. So what is a golygon? Wikipedia states that:

A golygon is any polygon with all right angles (a rectilinear polygon) whose sides are consecutive integer lengths. Golygons were invented and named by Lee Sallows, and popularised by A.K. Dewdney in a 1990 Scientific American column (Smith). Variations on the definition of golygons involve allowing edges to cross, using sequences of edge lengths other than the consecutive integers, and considering turn angles other than 90°.

The least possible number of sides that a golygon can have is eight and there is only one of this type (see Figure 2):

Figure 2 (from Wikipedia)

The sequence of numbers shown 1, 2, 3, 4, 5, 6, 7, 8 can be modified to 2, 3, 4, 5, 6, 7, 8, 9 or 3, 4, 5, 6, 7, 8, 9, 10 etc. However, the 8-sided golygon with its sides formed from eight consecutive prime numbers is an instance of the variations alluded to in the definition. The number of sides that a golygon can have must be a multiple of eight and so the next largest golygon will have 16 sides. There are 112 possibilities. Figure 3 shows a 16-sided golygon with sides whose lengths form the first 16 prime numbers:

Figure 3 (from blog)

Figure 4 shows the tessellation of the plane using the unique 8-sided golygon:

Figure 4: from blog

There is a three dimensional analogue of the two dimensional golygon. It's called a golyhedron but I'll save that for a possibly later post. Finally, Figure 5 shows the 8-sided golygon that I created using geoboard (by trial and error because I didn't refer to the original diagram in creating it).

Figure 5: golygon created using geoboard

This would be a good exercise for primary students who could also easily count the number of squares (52) that comprise it and work out its perimeter (36). They could then investigate the area of a square and rectangles with the same perimeter as the golygon. 

Figure 6 shows another variation on the classical 8-sided golygon. This one has sides that are not in sequence but alternate from odd to even. In the classic 8-sided polygon, the horizontal sides have opposite parity to the vertical sides.

Figure 6: from blog

Figure 8 shows a 6-sided "golygon" (although strictly speaking a golygon must have eight sides) in which the sides are formed from the Fibonacci numbers 1, 2, 3, 5, 8, 13:

Figure 8: from blog