geometry

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".