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 compl ex 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 c orrect and leads to elegant coordinates for all the sporadic complex p olygons. Among the by-products of producing these coordinates we count most significant our new insights about 2{6}3 and our simple proof th at 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.