Prisms & Antiprisms

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(Uniform #77) Heptagonal Antiprism
Vertices:  14  (14[4])
Faces:16  (14 equilateral triangles + 2 regular heptagons)
Symmetry:  7-fold Antiprismatic  (D7v)
Heptagon-Triangle Angle:  acos(−sqrt(root[189*(x^3)−315*(x^2)+63*x−1]))    
    = acos(−tan(π/14)*sqrt(3)/3)    
≈97.572257567 degrees
Triangle-Triangle Angle:  acos(-root[27*(x^3)+9*(x^2)−27*x−1])    
    = acos(−(4*cos(π/7)−1)/3)    
≈150.222262672 degrees
Dual Solid:  Heptagonal Trapezohedron
(values below based on edge length = 1)
Circumscribed Radius:  sqrt(root[64*(x^3)−144*(x^2)+80*x−13])    
    = sqrt(5+cot(π/14)^2)/4    
Midscribed Radius:  root[8*(x^3)−8*(x^2)−2*x+1] = 1/(4*sin(π/14))    ≈1.12348980185873353053
Heptagon Center Radius:  sqrt(root[448*(x^3)−112*(x^2)+1])    
    = sqrt(3−tan(π/14)^2)/4    
Triangle Center Radius:  sqrt(root[1728*(x^3)−2160*(x^2)+144*x+1])    
    = sqrt(3*(1−3*(cot(π/14)^2)))/12    
    = sqrt(7*(48*(cos(π/7)^2)+22*cos(π/7)−5))/6    

References:[1]Johannes Kepler, Harmonices Mundi (1619).
[2]Johannes Kepler with E. J. Aiton, A. M. Duncan, and J. V. Field, translators,
The Harmony of the World, American Philosophical Society (1997).