Hosohedra as regular polyhedra Hosohedron

for regular polyhedron schläfli symbol {m, n}, number of polygonal faces may found by:

n

2


=



4
n


2
m
+
2
n
−
m
n





{\displaystyle n_{2}={\frac {4n}{2m+2n-mn}}}

the platonic solids known antiquity integer solutions m ≥ 3 , n ≥ 3. restriction m ≥ 3 enforces polygonal faces must have @ least 3 sides.

when considering polyhedra spherical tiling, restriction may relaxed, since digons (2-gons) can represented spherical lunes, having non-zero area. allowing m = 2 admits new infinite class of regular polyhedra, hosohedra. on spherical surface, polyhedron {2, n} represented n abutting lunes, interior angles of 2π/n. these lunes share 2 common vertices.