Acrylic, 17 cm x 15 cm x 17 cm, 2007.

The regular tiling of the dodecahedron requires a different approach than the
other Platonic solids, as five-fold symmetry is not possible in the 17 plane
pattern classes. As a consequence of the duality expressed by the Platonic solids,
the icosahedron may be inscribed into the dodecahedron, so that the vertices
of the icosahedron correspond with the centre of the faces of the dodecahedron.
A pattern may then be projected outwards from the icosahedron onto the faces
of the dodecahedron with only minor distortion. This projection method can be
applied to other polyhedra and is dependent on the inter-relationships between
solids. Hall of the Ambassadors 3, results from the projection of a p6mm pattern
from the surface of the icosahedron.

Briony Thomas, Lecturer in Design Theory, School of Design, University of Leeds,
UK

"An understanding of the symmetry characteristics of patterns and polyhedra
can help provide a means by which patterns may be applied to polyhedra in a
systematic and complete way, avoiding gaps or overlaps. Research has recently
been undertaken to discover which of the 17 pattern classes can regularly repeat
around the Platonic solids, applying only the restriction that the unit cell
must repeat across the solid in exactly the same way that it does in the plane
pattern. The study focused on the application of areas of the unit cell to act
as a tile when applied to the faces of the polyhedra. Emphasis was placed on
the pattern's underlying lattice structure and the inherent symmetry operations.
The results of this enquiry have given rise to the creation of a series of remarkable
mathematical solids. Tiling designs were inspired by geometric patterns at the
Alhambra Palace in Granada, Spain and created through the use of routing and
laser-cutting techniques."

**Other works by the artist**

Repeating patterns applicable to regularly tiling the octahedron must be constructed on a hexagonal lattice, where the unit cell comprises two equilateral triangles. When applied to the octahedron, pattern class p6 maintains the rotational symmetries of the octahedron due to the higher symmetry characteristics of the plane pattern. Four-fold rotation is present at each vertex; two-fold rotation is present at the mid-point of each edge and three-fold rotation at the centre of each face. Hall of the Ambassadors 1, results from the application of a p6 pattern where 36 unit cells repeat across the faces of the octahedron.

All pattern classes constructed on a square lattice are considered suitable to repeat across the surface of the cube, tiling the faces without gap or overlap. Connecting points of four-fold rotation in plane pattern class p4 produces a square grid, which will lend itself readily to the tiling of the cube. The application of this pattern class to the cube maintains cubic rotational symmetry; three-fold rotation is preserved at the vertices, two-fold rotation at the mid-point of each edge and four-fold rotation at the centre of each face. 24 unit cells repeat across the cube with four cells present on each face.