Ken Wheeler – The Pythagorean Tetraktys

“Some called the Tetraktys the great oath of the Pythagoreans, because they considered it the perfect number, or even because it is the principle of wholeness; among them is Philolaus.

“The number 10 is complete at 4” “To him that gave to our generation the Tetraktys, which contains the fount and root of all eternal nature”
“Arithmetic, geometric, and harmonics were the three principles by which the Divine Artifice proportioned out the world soul”

Number is the first principle, a thing which is undefined, incomprehensible, having in itself all numbers which could reach infinity in amount. And the first principle of numbers is in substance the first Monad, which is a male monad, begetting as a father all other numbers. Secondly, the Dyad is a female number, and the same is called by the arithmeticians even.

Thirdly, the Triad is a male number; this the arithmeticians have been wont to call odd. Finally, the Tetrad is a female number, and the same is called even because it is female. ….

Pythagoras said this sacred Tektractys is: `the spring having the roots of ever-flowing nature.’…. the four parts of the Decad, this perfect number, are called number, monad, power and cube. And the interweavings and minglings of these in the origin of growth are what naturally completes nascent number.

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Golden Ratio in Life and Science

So called empty space is most likely structured. Buckminster Fuller already considered the possibility that the geometric structure of space was given by his Isotropic Vector Matrix (IVM): a network of interconnected tetrahedra and octahedra with a Vector Equilibrium in its center. This is what I have called the inner structure of Metatron’s Cube, a structure that scales inwards and outwards, and whose cartesian coordinates can be derived exclusively from integer and rational numbers, in fact from powers of 2 and 3 as in Aristoxenus musical scale. For Fuller, the IVM was a conceptual framework describing the symmetry of space, with which energy events could interact through its jitterbug property, producing a radiating wave of activity [3, p.192]. So the hypothesis is that, depending on the frequency of the sound source, a different geometric energy propagation pattern takes place in empty (structured) space. This geometric pattern may not captured by microphones, but it may interact with the subtle bodies of human beings, and it may be the source of the inherent qualities of sound that we are able to perceive but not yet to quantify.

Render Heads


For more than 7 years RenderHeads has specialized in developing beautiful, bespoke interactive software. Our work can be seen in permanent museum installations as well as trade shows, storefronts and product launches all over the world. Recently, we have grown to have offices both in the United Kingdom and South Africa.

Shane Marks

Director (SA) With a background in animation and visual effects, Shane started out as a technical artist with a passion for all things that cross the divide between the creative and technical. This passion led him to head up the South African Branch of RenderHeads. In his spare time he likes to play around with up-and-coming technology, like webGL.

Nathan Lance

Co-Founder & Director (UK) With a degree in Computer Science and programming experience in the games industry, Nathan has a sharp eye for detail and a keen instinct for what will look and feel right. It is his personal mission to make all systems, procedures and workflows as simple and efficient as possible. Despite being a self-confessed geek, due to Nathan’s easy going and sociable nature you wouldn’t know it if you met him on the street.

Andrew Griffiths

Co-Founder & Director (UK) Andrew started his passion for real-time graphics programming way back on the Commodore 64. He continued through to the PC where he spent many years as a professional games programmer, specialising in gameplay, tools, special effects, animation and 3D engine development. Andrew has a love of digital arts and is a proud and active member of the demo scene.