
Space
filling with Platonic and Archimedean solids 
. 
Uniary space fillers 
Just as the Platonic and Archimedean solids can be sectioned and rearranged 
. 
into each
other, so also can their whole bodies be packed together in various 
. 
combinations to fill space with no voids between them.
In order for them to pack 
. 
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 360^{0}. 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 
. 
octahedron is the only uniary space filler of
the Archimedean solids. 
. 


Fig. 23  Filling
space with cubes 
Fig. 24  Filling
space with 
. 
truncated octahedra 

. 
. 
. 
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 

individual, discrete polyhedra. 




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Geometry rules!  Space filling polyhedra 