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GeoGebra

distances in a dodecahedron

Auteur :chris cambré
  • 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 ."

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