Archimedean Solid Study: Truncated Icosahedron

Understanding the geometry of truncated icosahedron.  A truncated icosahedron is formed by 12 identical faces of regular pentagons and 20 identical faces of hexagons.

To be able to construct this geometry in Revit, I have to understand the relationship between the dimensions of the pentagon and the hexagon.  The pentagon and the hexagon share the same side.  I can determine the relationship of the width of the pentagon (measured from vertex to vertex) to its side.  The relationship of the width of the hexagon to its side can be determined too.  Considering joining the geometry’s vertice at the same horizontal plane, I can see the geometry can be formed by blending 8 polygons of different sizes.

The lowest (1st) and the top (8th) ones are the basic pentagons.  The 2nd and the 7th are pentagons with side equal to the width of the basic hexagon.  The 3rd and the 6th are decagons with 5 sides equal to the side of the basic pentagon and the other 5 sides equal to the width of the basic pentagon.  The 4th and the 5th are decagons with 5 sides equal to the side of the basic pentagon and the other 5 sides equal to the width of the basic hexagon.  With proper placement of the 8 polygons at different elevations (requires some math to determine each elevation), I can blend them (two at a time) to make the form.

Finally join all the geometries together.

Parameters and Math

TRUNCATE-ICOSAHEDRON5

This study led to the construction of the soccer ball.

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