Synergetic Discovery Between a Pyramid and an Octagon. The featured picture uses octagons. Their synergy parallels the synergetic behavior of the hexagon and tetrahedron. First, to define the terms: Synergy, Pyramid and Octagon. For my reference I quote Buckminster Fuller. His book is Synergetics: Explorations in the Geometry of Thinking.
Synergy: 101.01 is the “Behavior of whole systems unpredicted by their parts taken separately. “
Syneretics 108.00- Four Triangles out of Two. Synergy acts on the transformation of the hexagon into a 6-sided tetrahedron. In nature the six sided hexagon (two open ended triangles) synergetically transform into the six-edged tetrahedron.
Parallel between the square based pyramid and the octagon. A square based pyramid (like the Great Pyramid) has 8 edges. The octagon has 8 edges. The pyramid holds a total of 1080°. The square base has 360 °. Its 4 triangles total 4 x 180° = 720° . The octagon, be it regular or irregular, also has 8 vertices with a total of 1080°.
Here’s the parallel between my synergetic discovery in #3 with Fuller’s in #2.
As the six edges of 2 open ended triangles become a tetrahedron; Two open-ended squares can become a squared based pyramid. What practical application can this have? Many books have been written about the beneficial effects of the pyramid. These include:
What ever beneficial effects the 3 dimensional pyramid may have, the two dimensional octagon should also have the same effects. As two open ended triangles can converge to form a tetrahedron, two open ended squares can converge to form a pyramid. I have a number of blogs about Buckminster Fuller. Below are a couple of internal links. Geometry can be applied to benefit our daily lives. I think the octagon pyramid parallel is original with myself- Though I am not positive. Feel free to share.
Mirrored Pyramid Makes an Octahedron. Look at a square based pyramid. An upside down pyramid can form an octahedron with it. See picture below. Now do this with the Great Pyramid of Egypt. The featured picture shows such a blueprint. See the featured picture. It is an excerpt from page 16 of my Staff of God, Volume II.
First, what is the Great Pyramid of Egypt? The Great Pyramid of Giza (also known as the Pyramid of Khufu or the Pyramid of Cheops) is the oldest and largest of the three pyramids. They are in the Giza pyramid complex bordering what is now El Giza, Egypt. It is the oldest of the Seven Wonders of the Ancient World. Also, it the only ancient wonder that remains largely intact. Feel free to read these two internal links. As will can see, I have written on subjects for quite a number of links.
The basis of a lost Golden Age was the 3 x 3 number square. It has countless hidden number codes. They were once understood and activated. An example of its use is in the featured picture.
The radius height is 275 shorter Egyptian cubits. That radius has a circumference of 1728 cubits. Note the 3 x 3 number square gnomon. As pictured, 8 x 3 x 4 x 9 x 2 = 1728.
The core number of the 3 x 3 number square is 5. The 5 cubit tip is missing from the tip. This blog shows its number square source.
Circle’s diameter of 550 cubits is also numbered by the 3 x 3 number square. DSOworks.com gives ample instructions for discovering lost number codes. 55 is the average of any two opposite numbers as 49 + 61 = 110. They average 55. Similarly, you can find 110 in 16 different ways on this square. Zero as in 550, was not a recognized as an actual number. On the 3 x 3 number square it is a composite. It is the sum of any two sets of opposite numbers.
Dozen – Why Does This Number Have its Own Special Name? Most likely you’ve looked at the featured picture. Most likely you have said:”Where’s 12? What does this have to do with 12″? I see 9 boxes and 9 numbers.” Take another look. In the framework of the tick tack toe board, two vertical lines that cross two horizontal lines. This creates: 6 horizontal line segments and 6 vertical line segments. Consider the following:
This 12 line system can be bent around to form a cube. The same 12 lines can be bent around to form an octahedron. Both of these solids consist of 12 lines as seen in the picture. These regular polyhedrons are 2 of the 5 Platonic solids.
FINDING THE DOZEN IN MUSIC
First of all, the cube and octahedron are harmonic figures. Consider the following:
The cube has 12 edges, 6 faces and 8 corners.
The octahedron had 12 edges, 8 faces and 6 corners.
Now I will explain why they are harmonic. Greek philosophers believed figures were harmonic if their extreme numbered features related to the middle one by the same fraction. In the case of 6, 8 and 12; Number 8 is one third greater than 6. Number 8 is one third less than 12. That explains why many temples of the ancients were conceived as cubes or square based pyramids. Of course, a square based pyramid is the top half of an octahedron. Their object was to spread harmony through musical ratios throughout the country side. Yes, life was difficult. We had no modern medicine. But civilizations still manged to last for a long time. Egypt has a span of 10,000 years. John Michell and Christine Rhone discuss this in their book: Twelve-Tribe Nations and the Science of Enchanting the Landscape.
Besides Harmony, How Does A dozen Relate to Music?
Within the scope of an octave are 12 half tones. Any octave is divided into 12 equal semitones. However, a dozen relates to more than music. It relates to sacred building. Consider that the Holy Muslim shrine, the Ka’aba is cubic. The back third of Solomon’s Temple, where the Holy of Holies was kept, was cubic. Again, the cube is defined by its 12 edges. Also consider the following:
In the packing of spheres, 12 spheres totally encompass a central sphere. Atoms pack in this manner. Twelve around one make a complete system. There’s your dozen.
In Explorations in the Geometry of Thinking, Synergetics, Buckminister Fuller describes how there are 12 vectors of restraint. These are based on the tetrahedron. They define a minimum system. They have 6 positive and 6 negative vectors of restraint that stabilize a system. See 401.00 in his book.
Now, take a second look at our featured picture. It has 6 vertical lines. They are the two trisected vertical lines into 3 parts. These are positive. The board also has 2 trisected horizontal lines. These become the 6 negative lines.
Conclusion: Somehow and somewhere, someone in Neolithic times knew and worked with the 3 x 3 number square. It unifies the sciences and arts. It gives all mankind a common vision. That is why I been blogging so much about this square. It believe it truly can help to initiate a new era of peace and plenty. Music of the ancients totally comes from the mathematics and hidden number codes of this simple number square. Read my other blogs on the subject.
A Cube and 26-The Deity Does Enjoy the Shape of Dice. A concordance can be defined as an “harmonious agreement in ideas, feelings, etc” . This is Webster’s definition. I will present a concordance of #26. An ancient Egyptian tradition believed that numbers had personalities. Most of us today think: “Numbers are numbers”. “Words are words.” There is no connection. Therefore, an emotional response to numbers seems ridiculous. It’s possible to call that attitude: “schizophrenia: ” The brain has two sides. They are verbal and mathematical. Why not use them together?
The cube is also the overall shape of the Ka’aba, the Islamic sacred shrine
The cube is one of the 5 regular polyhedrons. It has 26 features
The cube is dual* to the octahedron. The octahedron also has 26 features: Mark a point in the center of each face of the cube. Connect these points and you have an octahedron.
When bisected, the octahedron becomes two square-based pyramids. The Square based pyramid is a sacred temple design all over the world.
The sacred, unutterable 4 letter name called the Tetragrammaton, has a numerical equivalent of 26 Strong’s Hebrew:. יְהֹוָה (Yhvh) In Hebrew yud =10; hei =5; vav = 6; and the final hei =5. This totals 26.
WHAT ARE WE TO CONCLUDE?
By common number, being 26, “the Blessed Name” is associated with the cube. He seems quite comfortable in that residence be it in Islam or Judaism. Also, the octahedron with its 26 features, when bisected into a square base pyramid, makes another comfortable dwelling. A pyramid built at ground level, implies an inverted pyramid underground. This completes the octahedron.