Polypons

Only recently, on November 25th 2020, I posted about polyiamonds, a particular type of polyform, constructed from equilateral triangles. On the other hand, polypons are constructed from isosceles triangles with angles of 30°, 30° and 120°. The building block, shown in Figure 1, is the monopon or 1-pon.

Figure 1: monopon

What prompts my current investigation into polypons is the fact that today I turned 26230 days old and one of the properties of this number is that is a member of OEIS A151531:

A151531

Number of 1-sided polypons with \(n\) cells.

The sequence, from \(n\)=1 to 17, runs 1, 2, 3, 5, 7, 15, 24, 49, 91, 183, 356, 734, 1465, 3017, 6153, 12721, 26230. What is meant by 1-sided is that the polypons cannot be flipped over as they could be if they were 2-sided. Wolfram Player comes to our aid here and we can see, in Figures 2 and 3, the difference in the case of \(n\)=3 for 1-sided and 2-sided tripons.

Figure 2: the three possible types of 1-sided tripons

Figure 3: the two possible types of 2-sided tripons

In Figure 2, the two shapes on the right are mirror images of each other but different because they cannot be flipped over into their mirror counterpart. In Figure 3, the two merge into one because they can be flipped. It is important to realise that the shapes need to be formed from a triangular grid as shown in Figure 4:

Figure 4: source

Without this restriction we could get far more tripons and tetrapons as shown in Figure 5:

Figure 5: source

So getting back to the triangular grid and the more restricted arrangements of monopons, Figure 6 shows the possible shapes for the case of \(n\)=6, the hexapons:

Figure 6: the 15 possible types of 1-sided hexapons

A word should be said about the origin of the word pon in polypon. Figure 7 is a snapshot taken from the Collins Dictionary:

Figure 7: source

So getting back to 26230, it can be seen that 17 1-sided monopons can be arranged to form 26230 different 17-pons. The Wolfram Player only stretches to \(n\)=9 and the 91 possible nonapons or 9-pons are shown in Figure 8:

Figure 8: the 91 different types of nonapons

As with the polyiamonds, the polypons can be arranged into interesting shapes. Figure 9 shows a star shape made from ten hexapons:

Figure 9: source

Here is a link to a post of mine on pentominoes from March 1st 2020. Like polypons and polyiamonds, the polyominoes are particular types of polyforms.