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.
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.
*page 230: “The saddle pentagon, when viewed in plan, has mirror symmetry. Its included angles are: 120°, 90°, 90°, 90°, 120°. The boundary of this pentagon is comprised of two edge lengths. The longer edge is √2 times as long as the shorter length. The longer edges are adjacent and each falls between 90° included angles. It is a 1/4 subdivision of the 120° saddle octagon. […] When viewd in plan it has 2-fold rotational as well as mirror symmetry. Its included angles are 120°, 90°, 120°, 120°, 90°, 120°, and it has one edge length equal to the shorter edges of the pentagon. The plane hexagon is regular with all included angles equal to 120°; its single edge length is equal to the short length of the pentagon. […]
Since the only long edge occurs on the pentagon, it always joins itself along such edges. In both the case of the saddle pentagon and saddle hexagon, the edges which fall between 90° and 120° included angles are compatible; and the edges which fall between 120° and 120° included angles are compatible. Also, these latter edges are the only ones that are compatible with the plane hexagon and the square.”
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).
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.
