The Pentagonal Hexecontahedron

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The pentagonal hexecontahedron is a 3D Catalan solid bounded by 60 scalene pentagons, 150 edges (90 short, 60 long), and 92 vertices. It is the dual of the snub dodecahedron.

The 60 pentagonal faces are transitive with bilateral symmetry, but non-regular, with 3 consecutive short edges joined to two long edges. The ratio of edge lengths is approximately 0.571 : 1 (see the Coordinates section for precise value).

The 92 vertices are of three kinds: 12 vertices of the 1st kind where 5 long edges meet, corresponding to an inscribed icosahedron, 20 vertices of the 2nd kind where 3 short edges meet, corresponding to an inscribed dodecahedron, and 60 vertices of the 3rd kind where 3 edges (2 short, 1 long) meet.

Like the snub dodecahedron, the pentagonal hexecontahedron is chiral, and comes in two enantiomers, or mirror-images.

Projections

The following are images of the pentagonal hexecontahedron from various viewpoints:

Projection	Description
	Centered on order-5 vertex.
	Centered on vertex of the 2nd kind.
	Centered on edge between two vertices of the 3rd kind.
	Centered on face.

Animation

Here's an animation of the pentagonal hexecontahedron rotating around one of the axes with 5-fold symmetry:

Coordinates

The approximate Cartesian coordinates for the pentagonal hexecontahedron are the even permutations of coordinate and all changes of sign of:

• (0, 0.55655387422852, 0.90052308507219)
• (0, 0.94315125924388, 0.36025172449890)
• (0.58289953474498, 0.58289953474498, 0.58289953474498)

and even permutations of coordinates and even changes of sign of:

• (0.09198319476103, 1, −0.10418586612062)
• (0.17873726072917, 0.83268555705720, −0.54222076455363)
• (0.27072045549020, 0.88953429781332, 0.39338882903648)
• (0.34731353325969, 0.72849969093657, 0.60661117096352)
• (0.43929672802072, 0.78534843169269, −0.45777923544637)

Algebraically, these coordinates are the even permutations of coordinate and all changes of sign of:

• (0, A, φA)
• (0, Dφ², D)
• (Dφ, Dφ, Dφ)

and even permutations of coordinates and even changes of sign of:

• (C, 1, −(2τ²−3τ−φ))
• (Cτ, C/(2τ−τ²), −C(τ²/φ+τ+φ))
• (C(τ+1), (τ−1)², (2τ−1)/τ³)
• (Cτ², 1/φ+2τ−τ², 2φ+1/τ−φτ)
• (C(τ²+1), C(τ−1)/(2τ−τ²), −(τ+1/τ−2))

where:

• φ = (1+√5)/2 is the Golden Ratio;
• τ ≈ 1.94315125924388 is the unique root of the polynomial τ³ − τ² − τ − φ = 0;
• u = √(τ+1) ≈ 1.71556149969737;
• A = (6u¹¹ − 8u¹⁰ − 47u⁹ + 55u⁸ + 135u⁷ − 128u⁶ − 186u⁵ + 126u⁴ + 145u³ − 67u² − 55u + 8) / (−9u⁴ − 3u³ + 35u² + 4u − 16) ≈ 0.55655387422852;
• C = (2−τ)φ ≈ 0.09198319476103;
• D = (τ−1)/(φ²) ≈ 0.36025172449890.

The dual snub dodecahedron corresponding to these coordinates has edge length 2√(2−τ), or approximately 0.47685947932748.

The short to long edge ratio of the pentagonal hexecontahedron is √E : √F, or approximately 0.57147671695861 : 1, where:

• E = 600u¹¹ − 3672u¹⁰ − 5364u⁹ + 27714u⁸ + 17856u⁷ − 73416u⁶ − 26712u⁵ + 80336u⁴ + 16412u³ − 32832u² − 2188u + 3896
• F = 196u¹¹ − 57u¹⁰ − 2255u⁹ + 307u⁸ + 9099u⁷ + 20u⁶ − 15685u⁵ − 1821u⁴ + 10300u³ + 1849u² − 952u + 145