distances in a dodecahedron
- Start from a regelar pentagon with center and radius .
- Create a dodecahedron out of it.
- Out of the properties of a dodecahedron you can define the coordinates of the points V and W:
V is the center of gravity of a pentagon adjacent to the base of the dodecahedron.
W is the center of gravity of a pentagon, connected to the base by one pentagon.
- You can calculate |OV| en |OW| out of the coordinates of V and W.
- The proportion
- The points O, V and W define a rectangle.
With edges [OW] en [OV] you get .
- This rectangle connects 4 midmidpoints of faces of the dodecahedron.
In a dodecahedron you can create 3 such quartets, defining 3 rectangles, perpendicular to each other.
In other words: "In a dodecahedron one can create 3 perpendicular golden ractangles" follows from the property:
"The proportion of the distance between the centers of gravity of a face and the center of gravity of a face connected to the first face by just one face and the distance between the centers of gravity of two adjacent faces equals ."