Jesús de la Peña Hernández

l l

l

HF 0 , 16245991 0 , 8506508

2

(^322)
2
 − =







hence, the icosahedron radius will be:
()l l
VV
2 0 , 5257311 0 , 8506508 0 , 9510565
2
1
2
= × + =
18.6.3 RELATIONS 3 (stellate pentagon)
which leads to the new relation:
D=l+ 2 d (2)
From system (1) (2) we can obtain the values d,l as a function of D,L.
L D
D
d
2
2

• = ;
  L D
  DL
  l


• 
  

  2

  
  

  Recalling Point 18.6.1 we can also write:
  = = 1 , 618034
  L
  D
  l
  d
  Fig. 2 shows a regular dodecahedron; in it we can see that each face pentagon is homothetic to
  another one with side d. Both pentagons are outlined in Fig. 3. Let ́s look for the center of homothecy.
  First of all we observe in Fig. 1 that the three angles in B are equal since all of them see the same
  chord L (same arc of circumference) from the same point B.
  1
  A
  E
  B L C
  B ́
  l
  C ́
  Finally we shall study some relations associated to the stellate
  pentagon; they will be of interest in connection with the regular stel-
  late polyhedra.
  In Fig. 1 we see two convex regular pentagons (consequently,
  similar). One of them is interior with side l (and diagonal d); the other
  is exterior with side L (and diagonal D). Therefore we ́ll have:
  D
  d
  L
  l
  = (1)
  thinking in similar ∆ ABC, AB ́C ́, this implies that
  AC ́ = d = L
  L
  0 , 618034
  3
  108
  2 cos