From Pythagoras to Pizza

Drawing by Rafael Araujo, with vector coding overlay by Andy Thomson

Who didn’t love making their first circle with a compass? Although a love of building blocks and form in general may appeal to any kindergartener, it was this subtle abstraction of  the ‘geometry’ – the graphic description of the rules of form that took hold of my imagination as a kid. I imagined that for every footfall, the arc of every movement of every limb was the sweep of some hidden radius, and that the whole world was somehow moving along finite pathways, energy in matter travelling through an invisible, and somehow perfect wireframe that encompassed vast grids of interconnected circles, spheres really, touching everything visible and invisible. I wondered what energized, powered and moved nature, but as with most mere mortals, this amazing world behind the world remained an opaque mystery, invisible to me except for my imagination.  The art of geometry would soon give way to mathematics in grade 2, 3 and beyond, and as the numbers and rules, the axioms of mathematics overshadowed shapes and forms.  Rather than dealing in forms and natural analogs, the introduction of the axiom was the beginning of my aversion to mathematics.  Math even looked mathy – all boring numbers (abstractions) and symbols and no pretty shapes or solids.   I hated the rules, because even though they could be proven to be true, they seemed arbitrary to me because nobody had taken the time to explain to me why they were true.  The times table was simply to be memorized. Through an infantile act of mental protest, I could not accept it and refused to memorize it – not because I couldn’t – but because I wanted to actually understand it. Now, at age 47, (why don’t remember it from the times table? because it’s prime :/), I know there are wonderful patterns and mechanics in even these routines, but to love them is to know them. The more mathy a thing looked, the more polarized I was to it. Why couldn’t I re-marry my enthusiasm for geometry with a more mature science of number?

After a stint at a high-school in Vermont, I had discovered a bunch of abandoned hippy projects in the back woods. Wonderful artefacts from another age, shingle-clad timber geodesic domes with hexagonal stained glass windows that looked like alien structures on Earth, organic, ancient, yet also futuristic somehow. Our library had a copy of Lloyd Kahn’s Shelter book and at age 18 the geodesic dome seduced me completely.  I needed to know more about these structures, which quickly led me to Buckminster Fuller who had proposed a kind of geometric theory of everything for comprehending and designing within the Universe.   I devoured his tome Synergetics, and many of his other works and believed that if only I could understand a tiny fraction of what Fuller did – I could design and build strong, beautiful buildings with minimal materials. Fuller’s Synergetics in fact rekindled the analytic part of my brain that had run on empty through every class on math in high school.

The fact that we have inherited so many unquestioned fundamentals, such as the base ten number system, the x,y,z cubic Cartesian spatial grid, 360 degrees in a circle, etc. was not lost on Fuller. There are many more ways to count and divide space than that, and in fact, there is a much more elegant and rational way to do so, and by rational Fuller meant without resolving (or not, in the case of Pi) in a bunch of irrational/infinite decimals. Synergetics supposed a system of various densities or frequencies of energies within energies* , all describe-able in terms of vectors and their structures, or agglomerations, or constellations, and these energies often shunted or were re-directed through such structures. Resultant and complimentary forces such as precession were key in describing not only the consequences of mechanical forces, but also strangely, human relationships. *(such as waves are an expression of energy moving through water, which  itself a ‘form’ of energy)

Now from the practical point of view of designing buildings, one need not have a cosmology underlying the geometry of the design. The Cartesian X, Y, Z system is perfectly adequate for designing almost anything, and with the power of computing, one is not constrained to designing boxes on top of boxes, but virtually any kind of curvy swoopy thing is possible. But for anybody craving a meaningful engagement with design, and seeking to find our place in an intelligible Universe, the return of the cosmological investigation that Geometry has always been is one clear path to be explored, and humanity has not finished this project. Design, according to principles of Synergetics adds a number of other meaningful dimensions to the study.  Rather than allowing all manner of mushy, blobby irrationalism and unintellectual (however fun!) form to emerge from computerized design, would it not make more sense to develop an architecture that expressed the elegance and efficiency of material and structural and mathematical and computational economy? This is fundamental to the Design Science Revolution that Fuller predicted was required to bring humanity past the challenges of resource ‘scarcity’ and the human-driven impacts (ie. Climate Change) we currently face. No political system could effect the kind of change that technology could, Fuller argued, citing Edison’s invention of the light bulb as having more far reaching effects than any political or religious doctrine, and while it would be hard to say the geodesic dome or the space frame truss has similarly revolutionized architecture, it would also be fair to say that this geometry has hardly become mainstream. The coordinate system required for this Design Science Revolution is in fact the Quadray coordinate system developed by Darrel Jarmush, David Chako, Gerald de Jong, Kirby Urner, and later Tom Ace, as described here by Kirby Urner  who has referred to the contributors as “…individuals who had all read Synergetics and so were thinking along these lines.”

To design in a Quadray-defined spatial system is entirely possible, but as the spatial coordinate system in any computerized spatial framework is orthogonal (or whose fundamental terms of reference are based on right-angles), there are inherent challenges. Since we do not yet have a Quadray or Synergetics-based coordinate system as our starting point (there are some C++ options developed by Tom Ace, but none of it is yet deployed in 3D/4D design software), we need to find ways to approximate this system and design within the X, Y, Z system as a kind of lattice by using workarounds such as square roots, fractions, Phi (the golden section ration 1.618…) and the Trigonometric functions of Sin, Cos and Tan.  Incredibly, if we had a Quadray system as the starting point, we would not need any of these functions or fractions, which are necessitated by describing 4D phenomena in terms of 90 degree coordinates, or as Fuller puts it:

Geometers and “schooled” people speak of length, breadth, and height as constituting a hierarchy of three independent dimensional states — “one-dimensional,” “two-dimensional,” and “three-dimensional” — which can be conjoined like building blocks. But length, breadth, and height simply do not exist independently of one another nor independently of all the inherent characteristics of all systems and of all systems’ inherent complex of interrelationships with Scenario Universe…. All conceptual consideration is inherently four-dimensional. Thus the primitive is a priori four-dimensional, always based on the four planes of reference of the tetrahedron. There can never be less than four primitive dimensions. Any one of the stars or point-to-able “points” is a system-ultratunable, tunable, or infratunable but inherently four-dimensional. (527.702, 527.712)

In other words, we kindof got the whole dimensions thing fundamentally wrong, or, only partially right. The 4th dimension is not ‘time’, not in Synergetics anyways. But more on that in another post, see Julian Barbour, The Janus Point re. a New Theory of Time.

If we start this investigation by describing the 5 Platonic Series of ‘Solids’ or Polyhedra (etymology: ‘Many Faces’, compiled from Wikipedia in the chart below) –  also the formal units of Fuller’s Synergetics – plus a few more, we have a kind of 3D dimensional key to start designing in this vector-based system. Interestingly, all of these basic solids have unique relationships with one another. For example, the diagonals of a cube form a tetrahedron, or the intersection of a right-side-up and upside-down tetrahedron form an octahedron. To understand the full extent of these inter-relationships, one can build a series of physical models (which is way more fun – dowels and surgical tubing for joints!), or as we have done in the computer, build a range of virtual models.  The full suite of objects has been described by Kirby Urner and others as canonical, and Kirby’s uses a 26 vector notation that conveniently correlates to our alphabet, from A to Z. These objects also have unique features such as 3 spheres per object that can be inscribed within the boundary faces (the insphere) , at the midpoint of the vectors or edges (the midsphere) and one touching all vertices equidistant and radial from the centre (the circumsphere)

1. Tetrahedron


  • 4 faces
  • 6 edges
  • 4 vertices

2. Hexahedron


  • 6 faces
  • 12 edges
  • 8 vertices

3. Octahedron


  • 8 faces
  • 8 edges
  • 6 vertices

4. Dodecahedron


  • 12 faces
  • 30 edges
  • 20 vertices
  • (illustration of its Euclidean construction by Rafael Araujo at the top of the post)

5. Icosahedron


  • 20 faces
  • 30 edges
  • 12 vertices

6. Cuboctahedron


  • 14 faces
  • 24 edges
  • 12 vertices

7. Rhombic Dodecahedron

  • 12 faces
  • 24 edges
  • 14 vertices

As an architect, I design in a 3D BIM (Building Information Modelling) software called ArchiCad, which uses a parametric sub-language called GDL (Geometric Descriptive Language), whose spatial system is of course also X, Y, Z based, and so it is in this software that I have worked to find, modify, and re-code these polyhedra in order to more easily manipulate them (namely to rotate and scale them) to build all possible relationships between them, and networks of these forms into structures such as octet or space frame trusses, domes, shells, and so on.  This is not something I have had the math or programming chops to pull off on my own, and I have been completely supported in this endeavour by the brilliant minds of Erich KarpKirby Urner and Olivier Dentan.  With these various polyhedra available to us, we can then rotate, scale, interlock and/or nest these solids together to understand the relationships among the forms, and find ways to design spaces and structures that are defined by their boundaries which, if we are sensitive to, can result in some unique, intelligent, rational and unexpected structures. These objects are available to download (just send me an email), free of charge, for any user of ArchiCad, but are copyright protected and cannot be offered for sale.

Without delving into the details of how these objects are coded (one can open them to study the scripts), suffice it to say that these should be the fundamental building blocks of form in any CAD/BIM system.  I have always imagined that the polyhedra, scaled however you like (you could use le Corbusier’s Modulor), could be the cursor used to draw in four dimensional space. Buckminster Fuller referred to this system of geometry as Nature’s Coordinate System, and most serious students of his thinking, from physicists to astronomers to programmers tend to be in agreement that Fuller had caught a glimpse of something that nobody else had.  It’s up to all of us to take it further. Eventually, we would hope to use this series of objects to form a kind of 3D cursor, that could ‘point’ to any direction in space, and cast vectors to them, building a framework of triangulated, strong and materially efficient architecture – like a kind of fractal geodesic spiderman.

Here is the link to the series of polyhedra developed for ArchiCad to date. Be sure to try them out with the ‘debugger’ script switched on, which will allow you to graphically rotate and scale the objects in the 3D window. Features to be added will include rotation around polygon normals (as axes).

Now the cool thing about Olivier Dentan’s objects is that it allows for a wide range of parametric options to be switched on or off, from wireframe mode to struts, nodes, faces, inspheres, outspheres, circumspheres, planar vectors, scaling handles and more.

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Some comments and corrections offered by Kirby Urner here:

“I’m looking over your handsomely constructed article, lots of good information, including about your own sequence of encounter, with different influences.  The correction I’d offer is Bucky Fuller didn’t publish about Quadrays as far as I know [corrected – ART].  Synergetics 1 & 2 (the two Macmillan volumes) don’t have Quadrays in the index, and that, combined with Synergetics Dictionary (both on-line) is pretty much the last word on Fuller’s own lexicon.

Quadrays arose in the context of listserv discussions, with one David Chako first sharing the idea of four rays from a common origin (0,0,0,0) and with coordinates (1,0,0,0) (0,1,0,0) (0,0,1,0) (0,0,0,1) in the configuration of four hydrogens around a carbon in the methane molecule.  Gerald de Jong, myself, and later Tom Ace committed time/energy towards their evolution.  A lot of our early work featured Waterman Polyhedrons, a series I rightly named for Steve Waterman.  A Waterman polyhedron is a maximal convex hull swept out within a fixed radius with all vertexes, and the origin, at IVM nodes.  They’re really pretty! (we were only concerned with the O1 column)

Given how we defined quadrays, all IVM vertexes have whole number non-negative coordinates.  Also, it turns out all Watermans have a whole number volume (proved by Dr. Bob Gray), expressing volume in tetravolumes.  Tom Ace has recently updated and cleaned up his web pages regarding quadrays (linked from Wikipedia).  He’s a professional C++ coder and produced some tight code.  He came up with the 4×4 rotation matrices that, applied to quadrays, rotate them.  Darrel Jarmusch also appears in the Wikipedia article, as the inventor of quadrays, with Chako further developing them.  From my angle, they were both pioneering the idea independently and then found out about each other through the web.  The Zeitgeist is like that.

Quadrays, or 4D IVM “caltrop” basis vectors, are a boon to Synergetics sharing because they loosen up our imaginations and get us thinking about vocabulary.  Are these “basis vectors”?  Why does XYZ have three basis vectors with the negatives not considered basis vectors?  What makes the negatives “secondary” (not basis vectors)?  With quadrays, all four vectors carry the same weight (none are secondary) so are they all “basis vectors”?  We’ve all seen Cartesian frames that are skew, so their non-90-degree “skewness” shouldn’t be a disqualifying attribute.

Linear combinations of them span all of space, just like linear combinations of XYZ basis vectors do (once we allow vector negation i.e. 180 degree rotation).  XYZ is a “jack” of six rays from the origin, three positive, three negative.   Quadrays are four positive, a “caltrop”.  Dividing space into four quadrants (quadrays) looks simpler than dividing it into eight octants (what XYZ does).

I’d say quadrays belong to “second generation” Synergetics, as does the rhombic triacontahedron of 7.5, not mentioned by Fuller, that shares vertexes with the RD of volume 6 (see below). The Wikipedia page on Synergetics includes it in the volumes table.

All 26 letters get used, but my graphic may leave some out to reduce clutter.  It’s a happy coincidence that exactly 26 letters give the points needed, but then the five-fold polyhedra get left out.  So don’t let that happy coincidence get in your way.  It’s more a mnemonic than anything.

The first tetrahedron (say orange) has corners A,B,C,D and its dual (say black) has E,F,G,H.  That’s your cube.  The octahedron edges cross the cube’s at 90 degrees, defining six more points: I, J, K, L, M, N.  The rhombic dodecahedron uses all of them.  Then finally, the 12 centers of the surrounding balls, even further out from the center: O, P, Q, R,   S, T, U, V,   W, X, Y, Z.  The fun thing is these are all vector sums of previously specified vectors, per that table in Wikipedia.  What distinguishes Synergetics from Neoplatonism is its relative emphasis on the rhombic dodecahedron over the pentagonal one.

The pentagonal dodecahedron (one of the Platonic Five) is certainly present, as the “wife” ([Fuller] calls it that) or “dual” of the canonical icosahedron (volume 18.51…), but it’s the rhombic dodecahedron (6) that embeds the cube (3) and the octahedron (4) as disjoint sets of face diagonals.

  • Rhombic Dodecahedron volume: 6
  • Cube of 12 short face diagonals volume: 3
  • Octahedron of 12 long face diagonals volume: 4

Then that Cube has the two tetrahedrons inside, the unit volumes.  A canonical tetrahedron is formed from four unit radius balls in closest packing, connecting their centers.  The best way to keep the IVM in the picture is to remember the ball packing that goes with it:  the CCP.  In the CCP, each sphere is encased by that same rhombic dodecahedron of volume 6, creating a space-filling arrangement known to Kepler.  The rods of the IVM are all perpendicular to those dodeca-diamond face-centers. Diamonds are forever.

Getting your 4D IVM cursor to be really Synergetics friendly may involve only XYZ coordinates as that’s what the existing software expects.  Quadrays are a conceptual overlay.  I have code for going back and forth: give me an (x,y,z) and I’ll give you an (a,b,c,d) and vice versa.  Getting the rhombic dodecahedron in the picture is probably the key ingredient, as it’s the space-filler that goes with the IVM.

Finally, if you haven’t seen it or don’t own it:  Popko’s Divided Spheres is a new classic primer in this area, I’d think a valuable addition to any architect’s collection.  I have it on my Kindle, but it has to be high end Kindle software that reads it, given all the colorful high resolution embedded graphics.

Notes: Cube: (±1, ±1, ±1)
Tetrahedron: (+1, +1, +1);
(−1, −1, +1);
(−1, +1, −1);
(+1, −1, −1).
Rhombic Dodec: (0, ±(1 + h), ±(1 − h2))
(±(1 + h), ±(1 − h2), 0)
(±(1 − h2), 0, ±(1 + h))


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