## Schafli is Alive and Well

and Living in a Hollywood Studio

By Tom Wilson, Literary Editor SuperAM. Circa 1996

I have been wondering for some time about when I was going to transcend this life as I know it. Wondering what impact reading books like Bart Kosko’s Fuzzy Thinking and The Secret Life of Salvador Dali would have on me. I longed to transcend the clarity of linear black and white thinking and rise above into the fog of not only circular but into spherical thinking. This is about getting a glimpse of transcendence. I’m talking about the “Schlafli Double 6”.

Ludwig Schlafli was a mathematician. He lived and worked throughout the eighteen hundreds when information was transferred by the written word and delivered by horses and boats. He started out as a translator in the field of mathematics. When the French, Italians, Germans or Americans wanted to share information about mathematics, they took Ludwig along to translate for them. As a result he had the opportunity to learn from the leading mathematicians of his time. Imagine transferring such information from one genius to another. Imagine the skill he must have obtained in understanding new concepts. As a result of this, Schlafli developed some of math’s most far reaching theories. Theories with names like The Theory of Manifold Continuity and Euclidean space Rn of n dimensions. “Transcending Math”, as it were. Schlafli was said by the experts in his field, “to have had the sad misfortune of those who are ahead of their time.” Sadly even in the field of mathematics his main theories were “rejected (until after his death) by the academies of Vienna and Berlin because of their great length” . Although he laid the groundwork for the scientific explosions that followed his death.

I had just met a fellow at a party and he was telling me about a sculpture called the Schlafli Double 6. “Man if you look at this thing from one direction you will see five dimensions to it”.

He went on, “I can show this to you from five different sides and it looks like a completely different object from each side.” Then he told me about Ludwig Schlafli and I was hooked. The “ever popular” Schlafli Double 6 sculpture is the coming to life of the work of a man who was ahead of his time and might have been onto an explanation of that which lies beyond the third dimension.

### A Man and His Accomplishments

**LUDWIG SCHLAFLIBern, Switzerland1814-1895**

1863 – Full professor University of Bern

1868 – Corresponding member of the Istitudo Lombardo di Scienze e Lettere in Milan

1870 – Won the Jacob Steiner prize for his geometric works

1871 – Corresponding member of the Akademie der Wissenschafter in Gottingen

1883 – Corresponding member of the Accademia dei Lincei

While at Bern he was concerned with two major problems. The first in elimination theory and the other in “n” dimensional geometry. He brought his results together in two extensive works. Drawing on the works of Hesse, Jocobi and Cayley. Schlafli found the thirty-six “double sixes”. Straight lines on a third-degree surface, the division of the cubicsurface into twenty-two species according to the nature of singularities. His posthumous manuscripts include ninety notebooks of Sanskrit and commentary on the Rig-Veda.

The ever popular, Schlafli Double Six is a visual representation of just one of the contributions made by Schlafli – One of the preeminent minds of the ages.

**Acceptors**

There are always those who will see when others can’t. Whether what they see truly “exists” is solely for them to decide. And when scientists use terms such as “randomness”, “unexpected”, and “violates rules” when referring to something that they have themselves discovered… the door of interpretation opens to all.

What does it mean when science is capable of placing a “phenomenon” down on paper in a mathematics formula but is incapable of applying the formula in the real world as we know it…does this mean that the phenomenon doesn’t exist or does it mean that it exists somewhere else. And if accepted that it may exist somewhere else how are we to relate to it here in our world? Here are a few spiritual men who find “this world” significance in the Schlafli Double Six.

Lama Wangchen Rinpoche, who is from Tibet and living in Darjeeling India, has the sculpture. He is the head Lama of the Buddhist Meditation Institute. When he saw it he felt that it “..represents the original order of constellations of the cosmos.”

Abbot John H Drais who has a Schlafli Double Six is founder of the Pracelsian order. An intercultural religious monastic teaching and healing order.

Yogi John also possesses one of the sculptures. He has been a Yogi for thirty years. Old friends of Dennis Hopper and Jack Nicholson and a well known figure from the sixties. Yogi John – “It’s an interdimensional meditational piece, that one can rest their eye on and fathom the unfathomable,”

The Boditree bookstore in Hollywood California has it on display.

If it touches one of us, it touches all of us.

**A Legacy**

In 1984, researchers at the National Institute of Standards and Technology dropped a molten alloy of aluminum and manganese onto the surface of a rapidly spinning wheel. The purpose of the spinning wheel was to solidify the molten metal as rapidly as possible so as to observe the effects that rapid cooling would have on the sample. The crystalline structure of the resulting solid was quite unexpected.(source: Scientific American, April 1991)

The crystalline structure of the resulting solid revealed a lattice structure having five-fold symmetry. When viewed in 3-D, the sample exhibited Icosahedral symmetry. An Icosahedron, one of the five platonic solids, has five-fold and three-fold symmetry. This five-fold symmetry violates the rules of classical crystallography because crystals are formed by a process in which adjacent polygons become attached. In the case of five-fold symmetry, adjacent pentagons cannot be joined without gaps. The lattice formed in the experiment was described as being “quasipriodic”, meaning that there was some randomness in its structure that allowed the five-fold symmetry to exist. Because this new substance had some characteristics of crystals, and had a quasiperiodic structure, they were termed quasicrystals. The quasiperiodicity presented a problem – how could it have formed?

Mathematicians observed that quasiperiodicity can be produced in a given dimensionality by taking a cut through a regular lattice of the next higher dimensionality. For the actual quasicrystals, it was found that their five-fold symmetry could be replicated mathematically by taking a cut through a regular six-dimensional lattice. The laws of classical crystallography are satisfied by viewing five-fold (icosahedral) crystals as 3-D hypersurfaces in 6 dimensional space.

This new shape, this five fold (icosahedral configuration), is described by Canadian geometer H.S.M. Coexeter as consisting of 27 points that are evenly distributed over the surface of a 5D sphere in six-dimensional space. The 3D projection of this 27 line figure is called a “Schlafli Double-Six.

These observations beg a very important question. In what way can a dimension greater than three exist? Did some other dimension manifest itself in our three-dimensional space during the test done at the National Institute of Standards and Technology in 1984? Is the beauty of the Schlafli sculpture a result of us getting a glimpse into another dimension?

F.Y.I – Commercially, quasicrystals are being used as heat conductors on food preparation and thawing utensils. Look into Zero-point energy.

**Pursuers**

The sculptors of the Schlafli Double Six have some interesting comments and questions they welcome any and encourage any response to their views.

Mike – “What did Ludwig accomplish with his concept known as the Schafli Double Six and what was its impact on the Mathematical world as it was utilized?

Mike – “Can someone show us the ‘formula’ for the Schafli Double Six?”

Laslo – “…and can you describe the formula in everyday terms?”

Mike – “It seems to show that mathematically at least that there is basically a virtually unlimited energy source out there for free if we could tap into it…I think?”

Laslo – “The Schlafli double six as far as I am concerned is making a connection between a triangular shape and a square shape in a golden section manner…a golden mean cutting through by squaring the triangle or triangulating the square. Finding symmetrical reference points that put these two bodies or shapes in a symmetrical relation to one another. And so the solution of this problem is three dimensional and I think the Schlafli Double Six underlines the marking of the symmetrical reference points.”

Laslo – “My impression…about the subject is that the (Quasicrystal) experiments by the National Institute of Standards and Technology, tried achieve some hints about what’s happening to matter before it takes on certain shapes, or crystallizes. Because before that it is in a liquid state and all it’s particles are chaotic, moving manner…and so the first hardened spatial arrangement that it can take up is as quasicrystals. (resulting in the Schlafli Double Six shape).”

Laslo – (Right after chaos stops and elements harden.) “This is the first possible constellation of elements that can be measured.”

Mike – “The hard definition of chaos is that if you break things down small enough you can’t exactly duplicate it.”

Laslo – “You can have reference to what kind of an order would be prevailing in a higher dimension and from a mathematical point of view it is just a matter of symbols. And it is easier to see than it is to understand.”

Laslo – “The Schlafli Double Six lines (yellow) best describes the relationships between those two possible constellations. The tetrahedron(red) and the cube(green).”

Mike – “For whatever reason we thought let’s make the biggest cube you could put inside the tetrahedron that doesn’t violate its boundaries or planes. This resulted in the magic points and views when you see the sculpture, boom you see this…boom you see that…I mean there is a definite visual magic to it that can get you.”

Laslo – “The closest thing I could find is like the Sri Lantra which is like a creation symbol of the universe it demonstrated the passage from formlessness where form takes place, and in the Indian view they look at the cosmos as a continuum not as abrupt elements. To me that refers to Schlafli’s work he calls the theory of continuity.”

Mike – “I think when Schlafli came up to his final double.. when his light bulb went on and there’s the Schlafli Double Six the beauty and perfection that he saw in the formula or the relationship … for the common guy could only, I think, be sort of semi-appreciated by seeing it.. you know…in a physical representation.”

Laslo – “…definitely it is a representation of a really high order of symmetry with references to higher dimensions as well, so maybe you can get glimpses of those by contemplating the essence of this.” (the sculpture of the Schlafli Double Six)”

**The Schafli Double Six**

Einstein, “Emptiness is form-form is emptiness.”

T. P. Wilson, “Where do the abstract and mathematics rendezvous.”

S. F. Johnson, “Time is in all things visual.”