Building with Geometry

:: From the design journals of architect Gregg Fleishman

 

Date: 12.15.2025

One of the many goals of an architect is to design a chair, I’ve done that many times over.

Another is to solve the problem of building, ultimately prefabricating houses. This is what we are working on now. 

For this we have started with 3D geometry and the presumption that it will be a more humanistic and inspirational solution. By this I mean that we can take the building process into our own hands, the parts are smaller, and with simple connections we can avoid the complexity of conventional construction. We can build more effectively, more efficient, faster, more sustainably, versatile and easy to reconfigure. And, additionally, we will be elevating the aesthetic to a universally recognized higher plane.

The most basic forms of 3D geometry are the Platonic Solids. There are five of these “polyhedra”. They have basic regular polygons for faces (equal angles, equal edges), all faces and vertices are the same. The three simplest are cubic, (tetrahedron, cube, octahedron) the other two are pentagonal (pentagonal dodecahedron, icosahedron).

The next class of polyhedra are the Archimedean Solids. There are 13. They can have different regular polygons for faces, but all their vertices are identical. Again there are some cubic based and some pentagonally based.

My original experiments included pentagonal variations as well as cubic.

The pentagonally based polyhedra build primarily domes, generally single isolated structures, while the cubic based polyhedra have the ability to build networks of spaces that attach together. Sometimes called space filling, or tessellation, it has the ability to be infinitely expandable. So I’ve gravitated to the cubic world because of that, not that we need the infinite part, but larger groups of geometrical assemblies have the potential to be functionally useful and architecturally inspiring

And there are only three of these somewhat regular polyhedra that fill space called parallelohedra. That makes it pretty easy. The CUBE, the Rhombic Dodecahedron, and the truncated Octahedron !! Well, to make it even more simple, the Cube and the Rhombic Dodecahedron are pretty closely related.

I have spent almost 30 years studying/working with the rhombic dodecahedron, it is actually two cubes by volume, and can be seen as a central cube with pyramidal extensions into the centers of adjacent cubes in a 3D checkerboard. The faces are inclined at 45 degrees to the faces of the cube so adjacent faces are coplanar. The resulting diamond/rhombic faces have a width to length aspect ratio of one to the square root of two. Which also defines most of the dihedral angles of “cubic” polyhedra.

While searching for ways to build more easily with the cube and the rhombic dodecahedron we can find related “cubic” polyhedra in the interstices between the neighboring rhombic dodecahedra if we truncate, that is cut off, their points.

Depending on the degree of truncation, the tetrahedron and a cube form in the interstices of the 3v and 4v points or at a higher degree, a truncated tetrahedron and a truncated cube will result.

Alternately the long axis of the diamond defines an octahedron. With a tetrahedron on each face it is the most efficient of structural frames.

The arrangements of truncated rhombic dodecahedra are the framework of the pods of the Otic: Wonderfruit structures while the associated octahedral frame provides support.