Tukey hexagons

  

Figure 61: Since each of the six edges of the first level flessagon spawns four new vertices, the full second level flessagon has 30 vertices since (4 * 6 + 6 = 30).

+	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+
1	3	3	5	5	27	27	29	29	1	21	23	23	25	25	17	17	19	19	21	11	13	13	15	15	7	7	9	9	11	1
2	2	4	4	6	26	28	28	30	30	22	22	24	24	26	16	18	18	20	20	12	12	14	14	16	6	8	8	10	10	2

If the strips in Figures 63 through 71 had been displayed as six pages with strings of five hexagons each, there would have been an artistic parity problem, but the structure of the second order flexagon would have been that much clearer. Anyway, once the strips have been cut out, pasted, and made ready for folding, the second order periodicity is evident enough.

The reason for this is that any n-gon in a regular flexagon can be replaced by a fanfolded strip of n-1 n-gons turned upside down while still making the same connections as before. When all the original n-gons have been so replaced, the next higher order of flexagon results. Treating these subpats as units, everything remains as before; but each of them can be opened up via the mountain-valley transition (pinching), to get cycles based just on the subpat.