Cluster 3 – Curved Space Diamond Structure Analysis

In an effort to advance discrete element aggregation using the Grasshopper3D plugin WASP (and Andrea building new features into it along the way), we are exploring a ‘sequence-based design’ approach and identified Peter Pearce’s Curved Space Diamond Structure as an ideal foundation.

Digital model of Peter Pearce’s Curved Space Diamond Structure

In our prior research, we encountered Pearce’s publication “Structure in Nature is a Strategy for Design” and his educational toy designs, sparking our interest in the geometric logic behind his playground structures. With guidance from his online documentation on the system’s top-down geometric principles, and aided by ‘a few’ key numbers* from the book (a bottom-up approach), we managed to reconstruct his once-patented design—though not without some detours along the way.

Sketches of possible geometric construction methods relating to angle and edge length correlations

Considering that computers were only beginning to emerge in the mid-1970s, it is remarkable that Pearce was able to develop this construction system based on saddle polyhedra and planar hexagons. Although the design appears straightforward once the correct angles and measurements are established, it is, in fact, a highly specific and geometrically complex achievement.

The system used for the playground structures primarily comprises two elements: a ‘plane hexagon’ and a ‘saddle pentagon.’ When the edge length of the plane hexagon is set to 1 unit, the resulting saddle pentagon has three edges of 1 unit in length and two edges measuring √2 units. Four of such saddle pentagons can be assembled into one large “saddle octagon”, having alternating edge lengths of 1 or 2 unit(s).

The primary building blocks of the curved space diamond structure (plane hexagon (left), and saddle pentagon (middle))
one plane hexagon connected to one saddle octagon (4 saddle pentagons) in space
Aggregation only using the strict rules of the curved space diamond grid structure, top view (left) and perspective view (right)

When the angles and edge lengths are precisely configured, all elements fit together seamlessly, forming a continuous surface that loops back on itself along the connections between individual ‘atoms’ within a diamond cubic crystal grid—essentially, an arrangement of stacked regular tetrahedra.

Interestingly, our research continually leads us back to this tetrahedral dimension! The beauty of this system lies in the fact that the plane hexagons consistently align with one of the four tetrahedral planes. Architecturally, this alignment provides regular ‘flat’ planes for spatial orientation and opens possibilities for utilizing or “inhabiting” such a spatially curved shell structure.

Plane hexagons colored according to their correlating tetrahedral plane of the diamond cubic grid.