Excerpt from the book "Arqka Biological Architecture and Sacred Geometry" by Arturo Ponce de León, Ninón Fregoso and Michael Rice Acquire it here

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Article 41: The tetrahedron as a geometric minimum unit .

In the book Three Paths to Quantum Gravity we read how Lee Smolin provides very important bridges to link the cosmic micro and macro worlds. Space / time are described as an independent system, that is, the “empty” space / time can be a grid-like factory filled with interconnected energy grids. This would mean that the particles and waves that we experience are only emergent patterns of this fundamental factory.

In contrast, string theory seems like a waste of time to explain 3D reality. There is a beautiful mathematics in the vibrational resonances in their theories but it is not comprehensible three-dimensionally. An image of unification emerges when we see the fundamental patterns observed in Nature through biology, geology, and cosmology where the basic patterns are found in space / time.

The spiral galaxies, the tornadoes, the shell of the Nautilus lead us to conceive a flow created by a spiral of Phi proportion in different scales. Mapping the fundamental geometric patterns found in all fields of science, as well as in almost all the spiritual traditions of the planet, leads us to think of simplified primary patterns as three base geometries and their constitutive geometries. The pentagram (five-pointed star), the hexagram (six-pointed star), as well as the equilateral triangles that form it and the heptagram (seven-pointed star).

The pentagram consists of the proportion of Phi, which describes meteorological weather patterns such as tornadoes and hurricanes, the growth of plants (phyllotaxis), galactic spiral arcs, even the ratio of the wavelength width of DNA.

The hexagrams are formed by the simplest possible packing of circles and spheres are made of equilateral triangles in equilibrium with each one. This basically describes the most balanced state of energy possible in the universe, and can be understood as the ground state of any highly efficient energy lattice. We find it in the structures of the carbon molecules (essential for life) even in the hives, leaves, and flowers and at the poles of Saturn.

Heptagrams are most commonly seen in integrative symbolism through human societies and culture. Perhaps archetypally rooted in the seven planets visible to the human eye from Earth, the expression of the seven is found through the Christianity, Hinduism, Islam, Judaism, Norse Mythology, Greek, Alchemy, Kabbalah, Egyptian Magic, Astrology and many others. In nature, it is found in leaves, flowers and feathers. Also in the musical scales, the perception of color, the circles of the crops.

First, let’s look at the fundamental structure of space / time. If we consider the most basic possible structures in the Planck scale, a Planck length line is the most fundamental representation of distance, and a triangle made of three of these lines will be the most fundamental structure of the area (2D). Observe that the triangle 2D also forms the most stable array of energy in a highly energized matrix, which can be stacked in a bidirectional fractal array. Moving to the next level more fundamental structural, we obtain a tetrahedron in Planck scale (3D). From here we have a component that can form a reticular structure that has flat surfaces and depth, both of which are measured both in space and time.

Each unit of the Planck space is also a unit of time, since distance itself cannot exist in measurement without a measure of time. So we can say that the temporal dimension (4D) is embonado, entangled, or folded in each of these three space dimensions.

It is also important to look at the fundamental building block, the triangle. Through the laws of thermodynamics, we know that energy and matter are always trying to reach the state of balance. The heat will flow into the cold space until the temperature equalizes. If we throw spheres in a box they will be compacted with each other until they have reached their maximum state of spatial density, which at the same time is the lowest state of potential energy or mobility.

When we discuss an energy lattice, each node (or point of intersection) goes to try to reach the position where you have the highest balance with respect to its surrounding points. This is simply the way to describe why an equilateral triangle, and its 3D projection, the tetrahedron, is the structure of equalized form for space / time.

If the energy and therefore the temporary space factory are always trying to maintain and balance their state, then any condition from the outside will produce a force. An influence causes a body to accelerate, or to produce gravity. From here, we think that each of the permutations of this fundamental matrix would produce different levels of force, and these would cause different levels of gravity. Any polygon with more than three sides is unstable. Only the triangle is inherently stable.

On the surface of an equilateral triangular grid, if you remove one of the triangles, the entire surface will be bent to maintain equilibrium intact. It is the principle that keeps the secret of the work of B. Fuller and allows the curvature of the structures to be modulated by adjusting the distance between the areas where the triangles are removed. In these places, the hexagonal plate has been deformed in a pentagonal plate, and this change of geometry in a single position allows the curvature of the entire surface. In the pentagonal maximum curvature, these pentagonal plates are directly adjacent to each other and would form an icosahedron.

There are only three possible omnisimetric, omnitriangulated cases of structural systems of least effort in nature: the tetrahedron with three triangles at each vertex, the octahedron with four triangles at each vertex and the icosahedron with five triangles at each vertex. If there are six equilateral triangles around a vertex, we cannot define a three-dimensional structural system, just a plane, as B. Fuller would say.

Any time a triangle is removed from a plane of balance or a curvature cannot simply be added to another hexagonal network. If you try to insert an extra equilateral triangle into a hexagon, you will find that there is not enough angular adjustment so you can make it fit. Therefore two things happen: the equilateral triangles in the set are deformed and acute triangles are formed to accommodate the new triangular energy field, which would take a lot of energy since these structures want to remain in balance and this would change the balance of the factory temporary space; and second, the equilateral triangle that connects to one of the triangles of the set, and the new triangle now rotate to a position of connection with another of the equilibrium planes of the vector. The initial act of this rotation in the position would cause the torsional forces in the space / time factory.

Since the second option would require much less energy, it is a more likely scenario and generates notable effects. First, the rotation of a new triangle, or more specifically a single point of force (since it is really just a knot or intersection in the structure that has to be removed) of a surface in any other plane of the hexagonal equilibrium vector would produce a chain reaction, where that hexagonal plane will then transfer one of its points to the next adjacent plane and so on.

This would be seen as a wave effect, where an energy point in the space / time factory would be rotated through the grid until it finds a stable position to rest or nest. The energy transfer in this form is limited to the speed at which the each unit of Planck distance in the structure can be interchanged with the next, which would be a fundamental measurement of the speed of the energy exchange through the temporary space factory. This speed, of course, is the speed of light, which provides the fundamental unit of time of the space equilibrium vector, as well as provides the relative measurement of separation between any points in the grid with any other point.

The energy will always try to reach its equilibrium state, so the dynamics at this scale are in constant flux, working to return to the balance of the fundamental factory. In the book “Synergy: explorations in the geometry of thought”, B. Fuller makes an exhaustive study of the patterns and structures that are inherent in energy dynamics. The theory of Unified Physics and therefore of Biological Architecture are nourished by these concepts.

The triangle, as the basic unit of all structures. The tetrahedron, as the most basic system of energy dynamics. The isotropic, vector matrix as the cosmic energy grid that underlies as the source of all patterns and structures. The equilibrium vector as the conceptual zero phase of all energy systems. The packing of spheres based, on the flower of life that inherently build these energetic forms. The pulsation of the dynamics of the Equilibrium Vector that creates all the platonic solids and the spiral vortex flow dynamics, until reaching the toro tube.

The foundation of energetic dynamics is based on cosmic patterns that are independent of time and scale, therefore they are cosmic by nature, since there is an integrity (or continuity) of these patterns in all manifested forms. In B. Fuller’s own words “the integrity of patterns is a conceptual relation independent of its size”, that is, there is a principle of fractality.


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