Spherical Cube or Hexahedron

The geodesic subdivision i.e. tessellation of the sphere derived from the spherical cube is little explored.

I wonder why?

Imagine a cube inscribed in a sphere. Now from the centre of the sphere project the edges of the cube on the circumsphere. The projected edges will follow great circle arcs i.e. geodesics. Each face of the cube can be divided into an orthogonal grid. Again projecting this grid from the centre of the sphere on the circumsphere will give a quadrilateral mesh with a geodesic pattern (all the lines follow great circle arcs). The number of divisions of the cube’s edge will give the frequency breakdown. The figures show a 6 frequency spherical cube.

This geodesic dome has a 4-fold symmetry, which makes it adaptable to plans with right angles, which is the norm.

Nowadays, when any imaginable configuration (sculpture) can be built, we should remind ourselves that the architectural language, for some 5000 years, is primarily based on the discipline of orthogonal ground plans! Also observe how the structures grow directly from the plans.

What is different and new?

Contemporary architects and engineers should ask: how much does their building weigh?

Of course, this design approach is not always appropriate.

Geodesic grid shell, 6 frequency spherical cube. 10 different strut lengths are required to construct this geodesic grid shell.

Geodesic grid shell, 6 frequency spherical cube. 10 different strut lengths are required to construct this geodesic grid shell.

Geodesic grid shell, 6 frequency spherical cube. 10 different strut lengths are required to construct this geodesic grid shell.

Geodesic dome plus orthogonal ground plan and elevation in concert! The grid shell is propped by a trabeated frame.

Truncated modular geodesic grid shells that can be tessellated along the X and Y axes.

Barrel vault capped by half domes

Barrel vaults also known as tunnel vaults (shells with single curvature) are the earliest kind of vaulting used. They are simple structures for roofing rectangular ground plans. How to enclose the arched openings of the barrel vault has always been a challenging structural and aesthetic problem.

The standard solutions are plane lattice structures which resist wind loads via bending. A much more efficient solution is a double curved space frame, which resist wind loads principally in an axial manner (via compression or tension). The figure show a barrel vault capped by half domes. Its geometry is a geodesic envelope!

It is not generally recognised that a series of circles (arches) parallel to one another, and a series of straight lines parallel to one another, intersecting the circles at right angles, form the simplest case of a geodesic pattern on the surface of a cylinder. Indeed, structures with geodesic patterns on single curved surfaces (Euclidean geometry) are nothing new! The geometry of the half dome is derived from the spherical cube.

Returning to lightweight roofs

The transept roof of the Crystal Palace, for the Great Exhibition of 1851, with its barrel vault construction was given lateral stability via diagonal bracing made from wrought iron rods. It was a highly efficient system. It is very instructive to note how Paxton used hierarchical systems for the transfer of loads — nature’s way. This seminal roof structure inspired engineers to develop ligtweight grid shells with rectangular meshes and diagonally pre-stressed cables. If the cladding is transparent the cables become almost invisible!

Geodesic grid shell with quadrilateral and rectangular meshes. It resembles a loaf of bread - organic.

Geodesic grid shell with quadrilateral and rectangular meshes. It resembles a loaf of bread - organic.