Poliedres i cossos de revolució

Característiques bàsiques

• Cares = 12
• Arestes = 30
• Vèrtexs = 20

Coordenades dels vèrtexs

Per un dodecaedre regular d'aresta unitat, les coordenades dels seus vèrtexs són:

• \(V_1=\left(0,a,c\right)\)
• \(V_2=\left(0,a,-c\right)\)
• \(V_3=\left(0,-a,c\right)\)
• \(V_4=\left(0,-a,-c\right)\)
• \(V_5=\left(c,0,a\right)\)
• \(V_6=\left(c,0,-a\right)\)
• \(V_7=\left(-c,0,a\right)\)
• \(V_8=\left(-c,0,-a\right)\)
• \(V_9=\left(a,c,0\right)\)
• \(V_{10}=\left(a,-c,0\right)\)
• \(V_{11}=\left(-a,c,0\right)\)
• \(V_{12}=\left(-a,-c,0\right)\)
• \(V_{13}=\left(b,b,b\right)\)
• \(V_{14}=\left(b,b,-b\right)\)
• \(V_{15}=\left(b,-b,b\right)\)
• \(V_{16}=\left(b,-b,-b\right)\)
• \(V_{17}=\left(-b,b,b\right)\)
• \(V_{18}=\left(-b,b,-b\right)\)
• \(V_{19}=\left(-b,-b,b\right)\)
• \(V_{20}=\left(-b,-b,-b\right)\)

on:

• \(a=\frac{1}{2}\)
• \(b=\frac{1+\sqrt{5}}{4}\)
• \(c=\frac{3+\sqrt{5}}{4}\)

Característiques avançades

Per un dodecaedre regular d'aresta unitat tenim:

• R_circumscrita = \(\frac{\sqrt{3}+\sqrt{15}}{4}\)
• R_{tangent arestes} = \(\frac{3+\sqrt{5}}{4}\)
• R_inscrita \(\frac{\sqrt{10\left(25+11\sqrt{5}\right)}}{20}\)
• Volum = \(\frac{15+7\sqrt{5}}{4}\)