Cluster 1: Space Filling (Irregular) Tetrahedra – Sommerville

| Daniela Kröhnert

Alongside Hill’s tetrahedra there are also other irregular tetrahedral cell types to fill space, of which the Sommerville 1 tetrahedron is a very promising one regarding our intentions.

Fig. 1: Chart from Michael Goldberg, “Three Infinite Families of Tetrahedral Space-Fillers,” Journal of Combinatorial Theory A, 16, pp. 348–354, 1974.
Fig. 2: Construction of a Sommervile No. 1 tetrahdron

The particularity here is, that the No. 1 tetrahedron consist of two “brackets” of each 2 triangles with two edges the length of the square root of 3 and one edge length of 2, connected at an right angle at their edges of the length 2.

Fig. 3: Aggregation logic of Sommerville No. 1 tetrahedra

As with the Hill Type 1, three Sommerville No. 1 tetrahedra can be joined to form a triangular prism. However, unlike Hill, these prism can be aggregated with all neighbouring vertices, edges and faces aligning by rotating each neighbouring triangular beam (or the individual cells) by 180 degrees.

Fig .4: left: Comparison of a regular tetrahedral cell to Somemrville No. 1 tetrahedron | right: even though these tetrahedra could be aggregated space filling (in six different orientations in total), there is also the possibility to look at them as building blocks of individual beams running in different directions but which can always meet, connect and match at a full face.
Fig. 5: When looked at systematically (by translating each combination of two faces into aggregation rules), four distinct directions emerge, an two sequences close in on themselves.

It is also possible to stretch (= scale 1-dimensional) the whole aggregation in one of the four directions. This for example might fit bike-frame clusters made with a larger range of frame geometries (e.g. 2 average trekking bike frames and one BMX bike frame), as it generates flatter and more directional cells. It could also be useful to fit other types of re-use artifacts. But it needs to be mentiones that a scale in one dimension results in a slightly more complex aggregation made up of two different tetrahedral cells, but which are mirror images of each other.

Fig. 6: Stretching/scale1D of the whole aggregation results in 2 different cell types, which are mirror images. This might allow to accompany a larger range of bike-frames to be connected into clusters fitting the tetrahedral boundary.
Fig. 7: 1-dimensionally scaled space filled aggregation with the sequence of the two different (mirrored) cell types.

References:
Michael Goldberg, “The Space-Filling Pentahedra,” Journal of Combinatorial Theory A, 13, pp. 437–443, 1972.
Michael Goldberg, “Three Infinite Families of Tetrahedral Space-Fillers,” Journal of Combinatorial Theory A, 16, pp. 348–354, 1974.
Michael Goldberg, “The Space-Filling Pentahedra,” Journal of Combinatorial Theory A, 17, pp. 375–378, 1974.