## Friday, 12 April 2019

### Space Filling Spheres

A small study to see what happens when stacking spheres are converted in to space-filling polyhedra:

It is interesting to play with space-filling shapes in architectural design, just to see what can be created and examine their significance to architecture and design.  Tessellating polyhedra is a big subject.  Part of it are the forms which are created when stacking spheres are converted in to space filling polyhedra.  These are useful because they represent some of the most compact arrangements of forms.  Regularly stacked spheres occupy approximately 74% of space.

In architecture and design, it points to geometric shapes that offer a low surface area to volume ratio, with a potentially low energy loss through the envelope, or components which might prove easier to move and transport.

There are several ways to stack spheres in compact arrangements, but they seem to fall in to two basic arrangements.  Stacking spheres with a square base also creates an arrangement with a triangular base on the diagonal.  Working with a triangular base, the stacking arrangement offers the opportunity to rotate the second layer through 60ยบ.  The spheres still stack but create different arrangements and different space filling polyhedra.

Matching the arrangement of the layers gives a trapezo-rhombic dodecahedron as a space-filling polyhedra.

 Trapezo-rhombic dodecahedron

Rotating the arrangement of layers gives a rhombic dodecahedron as a space filling polyhedra.

 Rhombic dodecahedron

The trapezo-rhombic dodecahedron and the rhombic dodecahedron are similar.  If a rhombic dodecahedron is sliced across the central horizontal axis and one half mirrored, it creates the trapezo-rhombic dodecahedron.

 Trapezo-rhombic dodecahedron to rhombic dodecahedron slicing through the horizontal plane, making a mirror image and reapplying

It also replicates the arrangement of atoms and crystal structures in nature.

 Making the rotation about the horizontal centre of the arrangement creates a cuboctahedron. Keeping the same arrangement throughout creates a variation on the cuboctahedron. Because the layer arrangements can be changed, there is an infinate variety of ways space-filling spheres can be stacked. Reference: The Penguin Dictionary of Curious and Interesting Geometry, David Wells, 1991. The cuboctahedron is one of the key geometries of Buckminster Fuller's Jitterbug.

The economic stacking of these space filling polyhedra could inform the assembly of units in buildings.

 Polyhedra and Architecture
Space-filling polyhedra have been investigated in architecture, art and science fiction as a conceptual link between nature, technology, society and culture.