Near-miss Johnson Solid - Possible Vertex Figures

Possible Vertex Figures

The near-misses, like all convex polyhedra made of regular polygons, have a countably infinite set of vertex figures that they can use, defined by a positive angle defect. A secondary constraint for the triples requires the angle sum of the two smaller polygons to exceed the angle of the larger one.

The set of polygons that can create convex vertex figures include:

• Triples p.q.r:
  • 3.3.(3-5), 3.4.(4-11), 3.5.(5-29), 3.6.(6+), 3.7.(7-41), 3.8.(8-23), 3.9.(9-17), 3.10.(10-14), 3.11.(11-13), 4.4.(4+), 4.5.(5-19), 4.6.(6-11), 4.7.(7-9), 5.5.(5-9), 5.6.(6-7).
• Quadruples p.q.r.s:
  • 3.3.3.(3+), 3.3.4.(4-11), 3.3.5.(5-7), 3.4.4.(4-5)
• Quintuples p.q.r.s.t:
  • 3.3.3.3.(3-5)

NOTE:

• (a-b) means any polygon for which the number of sides is between a and b inclusive.
• (n+) means any polygon with n or more sides.

Permutations of these polygon lists further extend possible vertex figures.

Each vertex figure has an angle defect, and a convex polyhedron will have a combined angle defect of 720 degrees.

These vertex figures and angle defect sums constrain the possible existence of convex polyhedra of regular or near regular polygon faces.

See Vertex configuration for the convex vertex figures used in the regular and semiregular solids.