Sets of Edge Colored Octagons

2021/10/05

Given regular octagons with colored edges you can try to arrange the pieces using an orthogonal grid so that touching edges are same colored. This is rather easy. But what about the square holes in such a construction? Jared McComb suggested that opposite sides of the square holes should also be same colored. This way it's much harder to get a valid solution. Rectangular figures have 3 matching conditions at the corners, 5 matching conditions at the border and 8 matching conditions for pieces inside. No wonder that rectangles with long borders are easier to construct than almost squares.

If n colors are available the number of different one-sided pieces is (n^8+4n+2n^2+n^4)/8. For n>2 these numbers are rather high and I decided to look at subsets where only two colors out of n are used for each piece. It's also possible to include or exclude same colored pieces. If three or four colors out of n are allowed you can demand a fixed ratio of frequencies for these colors.

The picture above shows a construction choosing one or two colors out of three for the pieces. Typically pieces with many sides of same color are making kind of clusters.

The table shows the number of pieces for a lot of sets and subsets. Click the numbers to see some constructions.

Colors	One or Two-sided	Number of Colors on Each Piece				Ratio of Color Frequencies
Colors	One or Two-sided	≥ 1	> 1	≤ 2	= 2	6:2	5:3	4:4	2:2:4	2:3:3	1:3:4	2:2:2:2	1:2:2:3
2	One-sided	36	34	-	-	8	14	10	-	-	-	-	-
2	Two-sided	30	28	-	-	8	10	8	-	-	-	-	-
3	One-sided	834	831	105	102	24	42	30	162	210	210	-	-
3	Two-sided	498	495	87	84	24	30	24	99	114	114	-	-
4	One-sided	8230	8226	208	204	48	84	60	648	840	840	318	2520
4	Two-sided	4435	4431	172	168	48	60	48	396	456	456	171	1296

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