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.

Hill’s tetrahedra seemed very interesting as aggregation modules because they can be scaled 1-dimensional and still be put together to form “triangular beams”. The adjustability of the shape of the individual cells accomodates a wide range of bike-frame clusters.

These triangular beams can be arranged space-fillingly on a 2-dimensional plane in various orientations toward each other. However it is nearly impossible to fit in bike-frame cluster modules whose connection planes/points meet with the various neighboring tetrahedra. The vertical shift prevents the “matching” of the cluster modules. Faces, edges and vertices of neighbouring cells ever only partly align or not at all.

Bike-frame cluster inside Hill type 1 tetrahadron. Connections to neighboring cells at the inscribed tetrahedron’s vertex points.

Stacked Hill type 1 tetrahedra showing that the connections of neighboring cells do not perfectly meet up. Translations along the vertical axis result in only some connections meeting at various “degrees of almost”.

Another interesting type for fitting the bike-frame clusters and spatial aggregation typology – where the neighbouring tetrahedra faces, edges, and vertices are all aligned – is the Sommerville 1 Tetrahedron.