106 Lattice polygons
Appendix: Regular solids
A regular solid is one whose faces of regular polygons of the same type,
say,n-gons, and each vertex belongs to the same number of faces, say,
mfaces. Note thatm≥ 3 andn≥ 3.
LetV,E, andF be the numbers of vertices, edges, and faces re-
spectively. ThennF =2E = mV, andV =^2 mE,F =^2 nE. Since
V−E+F=2,wehave^2 mE−E+^2 nE =2. From this,
E=
2 mn
2(m+n)−mn.
Sincem, n≥ 3 , and we require2(m+n)>mn, the only possibilities
are as follows.