The SuperCube is an geometric arrangement developed by Einar Thorsteinn (1942 - 2015) evolved by his research regarding the FANG (Fivefold Symmetry - All Space Filling - Non-Periodic -Geometry).
Inspired by the work with Buckminster Fuller Thorsteinn´s work deeply concentrated on polyhedral space fillers and spatial symmetries. In a simplified expression the FANG is the exploration of properties of Buckminster Fuller´s T-Quanta Module. Thorsteinn discussed the characteristics of the FANG-based fractal sub-divisions and their relation to Five Fold Symmetry Space in his essay Fang Report I (2004).
In order to develop the SuperCube out of, or in relation to, the T-Quanta Module various methods could be applied. As Thorsteinn started most of his work with two-dimensional drawings, it is believed he approached the SuperCube from one of its elevations.
Looking back at the unfolding of the T-Quanta Module, we see the faces of this non-regular tetrahedron could be brought into the arrangement of a square.
Considering this unrolled square being the face of a cube, we arrange six T-Quanta modules so they all meet three others at their vertices. Now we subtract these initial volumes from the cube. This process leaves us with a cube, where each corner is transformed into a positive and each of the former faces is transformed into a negative of the initial module.
For this to work, we need set three times the corner of the square, which connects to the longer edges of the triangle to meet in one corner of the cube. The two opposing corners are mirrored versions of each other.
Another way of a transformational approach could be to take a cube as the starting point. Around it´s diagonal axis the volume is rotated until it´s edges divide each other be the golden ratio. At first sight it seems the rotating angle would be exactly 45°, but it is slightly below.
This transformation subdivides the initial and the rotated cube into a series of T-Quanta modules.
Using this approach we subtract the initial from the rotated volume, mirror the left over modules towards the inside and subtract them again from the first cube.
As mentioned their would be several ways to reach the same result.
Operating on any two volumes of a regular compound of five cubes would get you there as well. By looking at this fact we already get an idea to the geometries dependency of tetrahedral and icosahedral symmetry.
Exploring the construction method of the rotation we can also create multiple variations of the SuperCube having the same abilities of spatial arrangement.
The SuperCube is an interesting geometry due to its dual integration of different symmetry groups and their space filling modules.
It describes the cube being the simplest periodic all space filling element of fourfold symmetry space and the T-Quanta representing the aperiodic space filler of fivefold symmetry space.
It allows for a large variety of configurations to be discussed further.
Cube and compounds: