Coordinates for the Regular Complex Polygons

Certain projections of the real polytopes {3, 3, 4}, {3, 4, 3}, {3, 3, 5} suggest highly symmetric coordinates for the self‐reciprocal complex polygons 3{3}3, 4{3}4, 3{4}3, 5{3}5 and 3{5}3. Although there are a number of interesting complications, this suggestion is essentially correct and leads to elegant coordinates for all the sporadic complex polygons. Among the by‐products of producing these coordinates we count most significant our new insights about 2{6}3 and our simple proof that the 600 vertices of the real polytope {5, 3, 3} are quite unrelated to the 600 vertices of either 5{6}2 or 5{4}3.