SuperCube

SuperCube

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.

986.515 (Modified from R. Buckminster Fuller Synergetics (1997)) - Unfolding of the T-Quanta Module

986.515 (Modified from R. Buckminster Fuller Synergetics (1997)) - Unfolding of the T-Quanta Module

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.

T-Quanta modules and their unfolded, mirrored version.

T-Quanta modules and their unfolded, mirrored version.

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.

Rotating a Cube around its diagonal axis.

Rotating a Cube around its diagonal axis.

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.

Thorsteinn´s model of a cubeFiveCompound strapped to a car.

Thorsteinn´s model of a cubeFiveCompound strapped to a car.

Exploring the construction method of the rotation we can also create multiple variations of the SuperCube having the same abilities of spatial arrangement.

SuperCube Variations

SuperCube Variations

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.

 
 

Further Reading

Cube and compounds:

http://www.georgehart.com/virtual-polyhedra/compound-cubes-info.html

http://mathworld.wolfram.com/Cube5-Compound.html

Einar Thorsteinn:

http://crystaldesign.kingdomes.de/einar_stalke.html

 

FANG

Subdividing space according to the FANG

Subdividing space according to the FANG

FANG - Fivefold Symmetry - All Space Filling - Non-Periodic - Geometry

Einar Thorsteinn (1942 - 2015) dedicated a lot of his works to the exploration of what he called the FANG.

Using Buckminster Fuller´s element of the T-Quanta Module, he researched ways of subdividing space into fivefold symmetric units. Everything is achieved with the use of only this one tetrahedron by applying mirror- and scaling transformations.

In his essay FANG Report I (2004) , Thorsteinn lists properties of these subdivisions and how their vertices, angles, surfaces and volumes relate to each other.

FANG Report I cover

FANG Report I cover

The two-dimensional equivalent of the FANG would be the Amman bars.

T Quanta Module

Another tetrahedral shape introduced by R. Buckminster Fuller in his work Synergetics (1997) is the T Quanta Module:

As the A & the B Quantum Module, the T Quantum can be derived from many differer known polyhedral geometries.

In this case the most obvious is the Rhombic Triacontahedron.

To begin with one of its faces is divided into four right-angled triangles by referencing opposite vertices.

Fig. 986.405 B - Division of a face of the Rhombic Triacontahedron into four triangels

Fig. 986.405 B - Division of a face of the Rhombic Triacontahedron into four triangels

Connecting the corners of this triangle to the center of the Rhombic Triacontahedron creates an irregular tetrahedron, which is declared to be the T Quanta Module.

Fig. 986.411C - T Quanta Module dereived from the Rhombic Triacontahedron

Fig. 986.411C - T Quanta Module dereived from the Rhombic Triacontahedron

As the Rhombic Triacontahedron consists of 30 congruent Faces it can be represented as a set of 120 T Quanta Modules. Half of them will be the mirrored version. (Buckminster Fuller calls them positive and negative.)

Fig. 986.419 A - T Qunata Modules within the Rhombic Triacontahedron

Fig. 986.419 A - T Qunata Modules within the Rhombic Triacontahedron

The vertices created by this method also follow the general division of the icosahedral symmetry group.

(Note that Point C is not on the great circle though. Flattening the spherical sections of the icosahedral great-circles would result in a Disdyakis Triacontahedron.)

Fig. 986.502D Thirty Great-circles of the icosahedral symmerty group in realtion to the divison of the Rhombic Triacontahedron

Fig. 986.502D Thirty Great-circles of the icosahedral symmerty group in realtion to the divison of the Rhombic Triacontahedron

The T Quanta module has outstanding properties in relation to many other regular polyhedra, polyhedral stellations, space fillers and other symmetric or (a-)periodic spatial arrangements to be discussed further.

In an unfolded view we can see the faces add up to a surface of a square. To make this more obvious we have rotate the smallest triangle. The edge length of this square is also the radius of the former Rhombic Triacontahedron.

986.515 (Modified) - Unfolding of the T-Quanta Module

986.515 (Modified) - Unfolding of the T-Quanta Module

986.411A -  Edge Lengths of the T Quanta Module in a plane

986.411A -  Edge Lengths of the T Quanta Module in a plane

Dissecting the T Quanta module, it turns out its fundamental relation to the Golden Ratio ( φ ). The number can be found in many relations of edges, angles, areas and volumes.

As this sooner or later transforms into a fractal (research) process, which leads up to the "Answer to the Ultimate Question of Life, the Universe, and Everything", this shall not be discussed any further.

 
All illustrations are taken from Synergetics (1997), by R. Buckminster Fuller, but may be used in a different context to serve the authors simplified explanation.

A Quanta Module & B Quanta Module

In Synergetics (1997),R. Buckminster Fuller introduces two tetrahedral shapes:

A Quanta Module (913.00) & B Quanta Module (916.00)

The A Quanta Module can be constructed in many different ways, but the most obvious may be to divide the first of the five platonic solids , the equilateral tetrahedron, in quarters. The result are four identical irregular pyramids meeting in the centre of the original tetrahedron.  A further subdivision of this quarter into six irregular tetrahedra can be achieved by division at the three perpendicular bisectors.  The resulting shape is a therefore a 1/24th (a 1/6th of a 1/4th) of the original volume and assigned as being the A Quanta Module.

 

913.01 The A-Quanta Module from Synergetics

913.01 The A-Quanta Module from Synergetics

As the six parts are equal in all properties, they are mirror symmetrical in pairs of three. Buckminster Fuller calls them positive and negative (913.10). The A Quanta Module has a set of unique properties according to fold-ability and therefore further interesting relations to two-dimensional space, not to be further discussed here.

 

The second shape described in the following is the B Quanta Module. Similar to the earlier procedure, we now divide a regular octahedron in the same manner first into eight equal parts and afterwards into six pyramids. By subtracting the A Module from this shape we create the so called B Quanta Module equal in in volume and therefore also a 24th of original tetrahedron.

916.01 The B-Quanta Module from Synergetics

916.01 The B-Quanta Module from Synergetics

While Buckminster Fuller briefly mentions their functionality for closest packed space he also outlines their behaviour of asymmetrical polyhedral units.n His further investigations on the modules mainly concentrate on their meaning as the representative for energy deployment (921.00).

In polyhedral geometries these modules can be applied to many known forms.  

In Synergetics, Buckminster Fuller also explores their meaning for space-filling and further refers to Coexters "Regular Polytopes".