TESSELLATIONS

 

In 1900, David Hilbert (1862 - 1943) proposed a total of 23 problems about different areas of mathematics. 18th of these problems was about building spaces with congruent polyhedra. Later, this topic has been widely studied and is still being studied.

 

            The regular tiling of polygons or polyhedra (or polytopes in general) is called a tessellation. Tessellations in two dimensions are abundantly studied, but tessellations in higher dimensions still need to be extensively explored. A recent thesis published about space-filling polyhedra belongs to Kara Joy Duckett. [Duckett, K. J., Close-Packing Polyhedra: Three Dimensional Tessellations, Department of Mathematics and Computer Science, Meredith College, 2003 – the thesis is available online at page http://www.angelfire.com/nc3/karaduckettthesis/] Some three dimensional tessellation examples of Ducket are given below:

 

 

 

 

            Another mathematician working on space-filling polyhedra is Guy Inchbald. Three of his original space-filling polyhedra (the bisymmetric hendecahedra, The sphenoid hendecahedra, The rhombic dodecahemioctahedron) are illustrated below:

 

Fig 4Fig 10  Fig 18

 

Three of Inchbald’s tessellations [http://www.queenhill.demon.co.uk/polyhedra/five_sf/five.htm]