Near-miss Johnson solid

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In geometry, a near-miss Johnson solid is a convex polyhedron, where every face is a regular or nearly regular polygon, and excluding the 5 Platonic solids, the 13 Archimedean solids, the infinite set of prisms, the infinite set of antiprisms, and the 92 Johnson solids.

The set of near-misses is not exactly defined, but can be loosely defined as convex polyhedra that can be approximately constructed from rigid regular polygon faces as a physical model.

Contents 1 An example 2 Possible vertex figures 3 See also 4 External links

[edit] An example

A polyhedron can be built by starting with a hexagon and surrounding it with 6 pentagons. It is completed by a reflected copy across the lowest 5 vertices, and finally inserting triangles between the two copies along the equator. The triangles are nearly equilateral but are required to be 2.8% shorter to fit, a difference too small to be visibly noticeable. Alternately if exact equilateral triangles are used, it can still be closed in a physical model by the pentagons being slightly bent off from a plane.

This polyhedron has three types of vertex configurations: 5.5.6, 3.5.3.5, and 3.3.5.5:

• The 5.5.6 vertex does not exist in any polyhedron constructed of all regular polygon faces.
• The 3.5.3.5 vertex is very common, existing in an Archimedean solid: icosidodecahedron, and also in 14 Johnson solids: pentagonal rotunda (J₆) and its 11 augmentations (J₂₁, J₂₅, J₃₂, J₃₃, J₃₄, J₄₀, J₄₁, J₄₂, J₄₃, J₄₇, J₄₈), bilunabirotunda (J₉₁), triangular hebesphenorotunda (J₉₂).
• The 3.3.5.5 vertex exists in 6 Johnson solids: pentagonal orthobirotunda (J₃₄), the four augmented or polyaugmented dodecahedra (J₅₈, J₅₉, J₆₀, J₆₁), and augmented tridiminished icosahedron (J₆₄).

[edit] 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.(5-7), 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.
• (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 contrain the possible existence of convex polyhedra of regular or near regular polygon faces.

[edit] See also

• Platonic solid
• Semiregular polyhedron
  • Archimedean solid
  • prism
  • antiprism
• Johnson solids
• Geodesic dome
• Tetrated dodecahedron (a near-miss Johnson solid)

[edit] External links

• Near Misses
  • <PDF> Symmetrohedra: Polyhedra from Symmetric Placement of Regular Polygons
• 16 Johnson Solid Near Misses