Cluster 1: Regular Tetrahedral Cells – Packing and Stacking

| Daniela Kröhnert

In parallel to the bike-frame 3d-scanning and finding of bike-frame cluster modules we are trying to find global systems of how to aggregate these regular tetrahedral cells into large frameworks that form closed loops at different scale levels.
Dependent on the aggregation system, such tetra-units containing the bike-frame clusters are connected at either at their 4 vertices, 4 faces, or 6 edges. Depending on the packing or stacking logic as well as on the bike-frame orientation within the cell, these large aggregations vary immensely in density due to different amounts of “negative space”: empty space globally not filled by tetrahedral cells, and locally – within one cell – not filled by bike frames.
This post covers the global aspects. An overview of – or rather zoom in on – these formations, with various types of bike-frame cluster modules replacing the tetra-units, can be found here.

a face to face aggregation is a classical L-system resulting exclusively in open ended growth, leaving gaps. Although these gaps could technically be bridged by specific bike-frame clusters, out of which bike frames would cantilever

One important property of identical regular tetrahedra is that they cannot be arranged in any way to fill 3-dimensional space. Currentlty the densest packing fraction is filling 85.63%, and consists of a double lattice of triangular bipyramids. There is a whole field within mathematics devoted to find even denser packings.
E.g. the following two papers are leaning more towards the geometric side of the spectrum, and are easier to start comprehending in order to dive deeper into that subject:

To be able to control the “gaps” (negative space) in between the cells, we tried various other stacking formations:

This centrosymmetric stacking pattern is the one we used in our previous research project “Conceptual Joining”. The “negative space” between the individual regular tetrahedral cells (grey) are regular truncated tetrahedra scaled by a factor of 3 (yellow and orange).
The resulting structural framework is a much less dense structure than the two before but as with “V2a-Vertex to Vertex” it forms closed loops within itself and is structurally stable.

Two spatial frameworks with closed loops of a medium density can be achieved through edge to edge aggregations:

eight aggregated regular tetrahedra (grey) enclose one octahedron (purple)
bottom row: two regular tetrahedra are firstly put together into a bipyramid. These regular bipyramids (grey and turquoise) are then stacked regularly to also enclose octahedra (pink).
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In a second step these tetrahedral cells/units are then replaced by various bike-frame cluster modules.
Some studies can be found here.