Space filling with Platonic and Archimedean solids

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Uniary space fillers

     Just as the Platonic and Archimedean solids can be sectioned and rearranged

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into each other, so also can their whole bodies be packed together in various

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combinations to fill space with no voids between them.  In order for them to pack 

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together the faces they share with each other must be parallel and congruent.  And
the sum of the dihedral angles of faces surrounding a commonly shared edge must
equal 3600.  Of the Platonic solids, only the cube can fill space by simply repeating

itself in three dimensions.  Hence it is called a uniary space filler.  The truncated

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octahedron is the only uniary space filler of the Archimedean solids.

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spacecube140x150transgif256.gif (10028 bytes)

spacetruncocta167x125res100transgif256.gif (12467 bytes)

Fig. 23 - Filling space with cubes

Fig. 24 - Filling space with
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truncated octahedra
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Note:  In all of the Polymorf models of space filling polyhedra shown here, each

constituent polyhedron is constructed in its entirety with all its faces.  This is to

demonstrate unambiguously that space is being is filled from the packing together of

page 18 of lesson

 individual, discrete polyhedra.

page 20 of lesson
   

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