Cluster 1: Space Filling (Irregular) Tetrahedra – Sommerville

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.

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.

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Cluster 1: Space Filling (Irregular) Tetrahedra – Hill

While it is not possible to fill space with regular tetrahedra, there are – according to WOLFRAM – five known irregular space-filling tetrahedral cells, when mirror cells are excluded.

At first we looked at Hill’s tetrahedra and Izidor Hafner “Definitions of Hill’s Tetrahedra” on the Wolfram Demonstrations Project was very helpful in setting up our Grasshopper definitions.

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Cluster 1: Regular Tetrahedral Cells vs. Cluster Modules

After aggregating regular tetrahedra following different set of rules these tetra-units are replaced by by various bike-frame cluster modules.
Each of these cluster modules is made up of 2-3 bike frames from the scanned set and fitted into a regular tetrahedral container. For this study only copies of one single cluster type are populated throughout the aggregated frameworks replacing the regular tetra-units. This is done assuming that other bike-frames within a certain tolerance range can be used to form the same combinations (possibly with a slightly different clipping at the joint-plate areas). The bigger the inventory of bike frames, the more likely it is to find very similar bike-frames.

Fig.1: Bike Cluster TYPE 12 is used for the first study. Dependening on the bike-frames orientation within one tetrahedral cell informed by the respective connection logic (from left to right: face to face, edge to edge, vertex to vertex) the amount of “empty space” locally within one unit varies immensely.

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Cluster 1: Regular Tetrahedral Cells – Packing and Stacking

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.

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