The following table gives the interior‑angle measures of regular polygons with a given number of sides. Number of sides 3 4 5 6 7 8 9 10 11 12 Interior‑angle measure 60∘ 90∘ 108∘ 120∘ 128.6∘ 135∘ 140∘ 144∘ 147.3∘ 150∘

Exactly five vertex types are possible with each involving one polygon with more than 12 sides. None of these vertex types leads to a semiregular tiling. The five many‑sided polygons involved in these five vertex types have 15, 18, 20, 24, and 42 sides. Give the notation for each of these five vertex types. (For each vertex, list the number of sides of the included polygons, separated by periods, in either clockwise or counterclockwise order, starting from the smallest number of sides. For example, 3.5.8 represents a vertex with polygons of 3, 5, and 8 sides all meeting at a point.)

vertex involving a 15‑ sided polygon:

vertex involving an 18‑ sided polygon:

vertex involving a 20‑ sided polygon:

vertex involving a 24‑ sided polygon:

vertex involving a 42‑ sided polygon:

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