The POD Project - Shape

Rhombi-what?

Most buildings today are variations on the cube. Walls rise straight up from the foundation to the roof, and stand at right angles to each other. It is not a natural shape and therefore stands out from nature, intruding on the landscape as a permanent and imposing structure.

For the POD, I wanted to use a shape that was different from everything done before, and which had a softer impact on the surroundings. It had to support a modular framework, be easy to make, and yet stand up to the harsh conditions of the planet. The best I could find is a sort of faceted sphere called a rhombicuboctahedron (pronounced ROM-buh-kyoo-BOCK-ta-HEED-ron, sounds like "rob the queue back to Hebron").

A the picture above shows, this polyhedron looks like a cube that has eaten too much. It is more like a sphere than a cube, having a larger ratio of volume to surface area. And unlike a cube, it is composed of square and triangular faces. Slice it in half, and the cross section is an octahedron (polygon with 8 sides).

Here are some more facts about the rhombicubactahedron:

• It has 26 sides (faces), with 18 squares and 8 equilateral triangles.

• There are 24 corners (vertices), each having the same set of angles: three at 90 degrees, and one at 60 degrees.

• All 48 of its face borders (edges) are the same length.

• It has rotational symmetry around 3 orthogonal axes, which means no matter which way you turn it, roll it, flip it, it looks the same.

• With the addition of a small cube, it "tiles" in three dimensions. This is a fancy way of saying that rhombicuboctahedrons can be stacked vertically and horizontally with little cubical shapes in between.

• Its volume to surface area ratio is 1.4. Compare this to a cube which is 1.0. Less surface area means it is roomier and easier to insulate.

Here is another illustration of a rhombicuboctahedron, rendered in a ray tracing program as a metallic frame (click for larger view):

And another view:

I have made a model with 6-inch edges out of aluminum tubes.

Vocabulary

This diagram shows the basic parts of the rhombicuboctahedron shape:

In math-speak, the lines are called edges. The points where they intersect are called vertices. The two-dimensional shapes they create are called faces. In this project, the edges will be called beams. They are connected together at the vertices with parts called nodes. The faces will be filled in with windows, doors, and walls called panels. The beams and nodes are load-bearing, meaning that they support the weight of the whole structure. Unlike a cubic house, the load is not transferred perpendicular to gravity, or straight up and down, but in a more complex distribution pattern.

Math geeks might be interested to know that the symmetry of edges and vertices puts this shape in the class of polyhedrons called archimedean solids. This symmetry is beneficial because it makes design and construction much simpler than if there were irregular features. Specifically, it means we only need to mass-produce one type of beam and one type of node. (In reality, there may be some departures from this rule to accomodate external connections such as feet for the pod, but this will be rare.)

The diagram below shows a cross-section of a rhombicuboctahedron (an octagon) with labels for various parts. Groups of vertices at each level are called tiers with the first tier at the bottom, and fourth tier at the top. The first and fourth tiers each have 4 vertices while the middle two have 8 vertices. The spaces between tiers, starting from the bottom will be called the bottom slice, middle slice, and top slice.

In cases where we need to discuss the shape from an overhead view, we will describe the directions using points on a compass: north, northeast, east, and so on.

Math

How big will the POD be? The formulae to calculate the size parameters are given in the following table. The variable x is the length for one edge, or beam. In the last column, I supply the final result using 6 ft. as the length of one beam.

Inside

How does one live inside a rhombicuboctahedron? Imagining the internal space might be challenging to cube-centric people, so I have made a diagram below to help.

The picture shows a cross-section with a person inside. The first tier is at ground level. The floor is at the second tier, 4.2 feet above the ground. We can call the region below this the sub-floor. Being so short and sloped on the bottom, it is not suitable for habitation, so we will likely use it for storage. The top two slices are for habitation, so we will call this region the living space. In the top slice, the ceiling approximates a dome, 6 feet high at the edge and 10.2 feet in the center.

One drawback to the rhombicuboctahedron shape is that it is unstable on the ground without some kind of support. It is not hard to imagine this structure rolling over at the slightest breeze, or an occupant jumping on the floor near the wall. There are several ways to overcome this problem. The easiest way to stabilize it is to add external legs. Four of these spaced around the POD, each connected to the lower portion with three struts, would keep the structure from falling over. The second method is to connect cables to the top and anchor them to the ground some distance away. Third, the POD can be suspended from trees by connecting cables to any of the vertices on the top half.

Viewed from above the POD is an octagon with the same dimensions as the side view. Each of the 8 faces is 6 feet by 6 feet. At least one of these will be a doorway, with the rest covered in some paneling material for walls. Windows could be placed in the ceiling or walls, depending on preference. Walls inside the octagonal space of a POD are possible with a variety of configurations, though rectangular rooms will be the exception rather than the rule.

References

Stephen Wolfram, Small Rhombicuboctahedron, Mathworld. http://mathworld.wolfram.com/SmallRhombicuboctahedron.html