Canonical Polygons

Today I turned 25571 days old and one of the OEIS entries for this number was A052436: canonical polygons of n sides. This prompted the inevitable question: what is a canonical polygon? Well, according to MathWorld:

A canonical polygon is a closed polygon whose vertices lie on a point lattice and whose edges consist of vertical and horizontal steps of unit length or diagonal steps (at angles which are multiples of 45 degrees with respect to the lattice axes) of length \(\sqrt 2\). In addition, no two steps may be taken in the same direction, no edge intersections are allowed, and no point may be a vertex of two edges. 

On this same site, all the possible shapes for polygons with sides from 3 to 8 are shown (see Figure 1):

Figure 1: all possible canonical polygons
with sides ranging from 3 to 8

There are exactly eight distinct convex canonical polygons (see Figure 2). It is not a requirement that the canonical polygons are convex however, and a great many more shapes are possible when concave polygons are included.

Figure 2: the eight possible convex canonical polygons

For polygons with 13 sides, there are 25571 different possible configurations. Geoboard of course is a great way to experiment with the different possible shapes. Figure 4 shows a concave canonical polygon with 13 sides. It has an area of 7.5 square units (the distance between successive vertical and horizontal points on the lattice is one unit). The double area is 15 square units and this way of expressing the area is preferred because it gets rid of any 0.5 decimals. The maximum possible area of a 13-sided canonical polygon is 15.5 square units or a double area of 31 square units. There is only one such shape and that is shown in Figure 3.

Figure 3: the 13-sided canonical polygon with
maximum area of 15.5 square units

Figure 4: 13-sided canonical polygon created
using geoboard. Area is 7.5 square units.

This link to a PDF file is titled:

A052436 Examples

Lars Blomberg

February 18, 2019

Abstract

Examples of Canonical Polygons are given ordered by number of edges

and area.

The file shows examples of canonical polygons from 3 up to 18 sides. Figure 5 shows an extract from this file.

Figure 5: extract from Lars Blomberg's file showing
examples of canonical polygons with sides up to 18