Sacred Geometry

The Golden Ratio in Life and Science
3D Flower of Life and Sacred Solids
and much more...

ON THE CONCERT PITCH A=432Hz

Scales define the intervals between frequencies of all the notes in an octave. Modern instruments are tuned by fixing the sixth note in the scale to a frequency or concert pitch A=440Hz. Maria Renold, who has spent many years of her life to the study and comparison of musical scales and intervals, reiteratively observed that the intervals and tones become antisocial, and indeed cause people to provoke one another when the concert pitch A=440Hz is used. On the other hand, intervals and tones have a beautiful, pleasant and harmonic effect on the human being when instruments are tuned to A=432Hz. She made comparisons over many years, with many people, in many places, and with many instruments, and the same phenomena always took place. These findings show that tones of certain frequencies hide characteristic qualities which can have major effects on human beings.

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Sacred Geometry - Proportion in Musical Scales

PROPORTION IN MUSICAL SCALES

Did you know that the so-called "Pythagorean musical scale" was actually introduced by Aristoxenus? The earliest written record on Greek music on which most scholars base their studies is known as the Harmonic Elements of Aristoxenus. These books contain an attack and critique of those whom Aristoxenus calls the harmonists of the Pythagorean school. Pythagorean music was most likely based on the ancient Greek Aulos Modes , rediscovered and carefully documented by Kathleen Schlesinger. This article introduces the basic mathematical foundation behind musical scales so that anyone without musical knowledge as myself can understand their structure. It then shows how the structure of both actual and ancient musical scales can be easily derived from the arithmetic, geometric and harmonic means; and how the great diversity of musical intervals available in the ancient Greek Aulos Modes has been gradually lost from common musical practice.

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SACRED SOLIDS IN THE ATOMIC NUCLEUS

This article describes a tiny portion of the research conducted by the eminent nuclear physicist Dr. Robert J. Moon (1911-1989): He proposed a nice, coherent, geometrical model of the arrangement of protons in the atomic nucleus, which involves the Platonic Solids. At present, there is no theoretical model capable of describing into detail the structure of the nucleus. Each of the available nuclear models describes some of the known experimental observations, but there is no definite model that explains them all. Dr. Moon's nuclear model accounts for some of the periodicities found in many properties of the atomic elements, and it also explains why some elements -like uranium- may participate in nuclear fission.

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Sacred Geometry - The Golden Ratio in Sacred Solids

PHI IN SACRED SOLIDS

Most discussions on Metatron's Cube hold that this structure contains all the Platonic solids inside. In this article on Metatron's Cube it is shown that this is not true, because the inner grid of the cube is a tesselation of space composed of alternating tetrahedrons and octahedrons. The solids that can be found in Metatron's Cube do not involve the Golden Ratio: its inner grid generates the tetrahedron, the octahedron, the star tetrahedron, and the cuboctahedron, but neither the icosahedron nor the dodecahedron. In this article it is shown that the inner interconnection of their vertices and the outer prolongation of their edges lead to a set of interconnected Stellations, such as the one known elsewhere as the Double Pentadodecahedron.

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THE GOLDEN RATIO

Simply stated, the Golden Ratio establishes that the small is to the large as the large is to the whole. This is usually applied to proportions between segments. This ratio has been venerated by every culture in the planet. We can find it in art, music composition, even in the proportions of our own body, and elsewhere in Nature "hidden" behind the Fibonacci sequence. In this site we also provide some examples of disciplines where the presence of the Golden Ratio was unsuspected until recently. Such is the case, for example, of Atomic Physics or DNA codon populations of the whole human genome.

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Sacred Geometry - The Flower of Life

THE FLOWER OF LIFE

The Flower of Life is an ancient symbol full wisdom. In this article we describe its essential process of construction and several derived figures. Some of them can be considered as their constituent parts (like the Seed of Life, and the Egg of Life) whereas some others are derived from it (like the Fruit of Life or Metatron's Cube). We also provide links to the articles where we derive our proposed Three-Dimensional views of some of all those figures.

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METATRON'S CUBE

Here we have a non conventional presentation of Metatron's Cube. This does not mean it is better or worse than others, it's just different. The aim is to deduce de three-dimensional structure of the Cube. The article shows how one can arrive to a 64-spheres three dimensional representation of Metatron's Cube. It contains a study of the inner 3D grid that supports those spheres, and shows that this grid supports the three dimensional construction of some solids at different scales (tetrahedron, octahedron, star tetrahedron, ...) but not of others (such as the icosahedron or the dodecahedron).

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Sacred Geometry - Metatrons Cube Three Dimensional 3D
Sacred Geometry - Platonic Solids Dodecahedron

PLATONIC SOLIDS

There are only five polyhedra that contain the same regular polygon in all of their faces, and which have the same number of these polygons meeting at each vertex. They are known as the five Platonic Solids, namely the tetrahedron, the cube, the octahedron, the icosahedron and the dodecahedron.

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ARCHIMEDEAN SOLIDS

The Archimedean solids are the only 13 polyhedra which are convex, have identical vertices, and their faces are regular polygons, although not all equal as in the Platonic solids. Since all the vertices are identical to one another, these solids can be described by indicating which regular polygons meet at a vertex and in what order. Some Archimedean solids are derived from the Platonic solids by truncating a percentage less than 1/2. The percentage of truncation varies in each solid and it depends on the shape of the original face.

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Sacred Geometry - Archimedean Solids Snub Dodecahedron
Sacred Geometry - Catalan Solids Dysdiakis Triacontahedron

ARCHIMEDEAN SOLIDS

The Catalan Solids are the duals of the Archimedean Solids. The vertices of the dual of a solid are the midpoints of the faces of the original solid. Catalan Solids have nothing to do with Calalunya, they are named after their first description by belgian mathematician Eugène Catalan in 1865. The faces of these solids are not regular polygons, but they are all equal.

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3D CONSTRUCTIVE COEFFICIENT

The three-dimensional constructive coefficient is a general quantity that can be computed for any polyhedron. Adrià Garcia defines it as the sum of the number of faces times the number of edges per face (which represents the constructive geometry of the polygon) over the fixed numer 12. This normalization is possible because the coefficient, at least in the Platonic, Catalan and Archimedean solids, always happens to be a multiple of 12. The three-dimensional constructive coefficient is the same for a solid and its dual. Therefore, it provides a means to know which solids can be dual of each other.

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Sacred Geometry - Pyramids and Prisms Heptagrammic Pyramid

PYRAMIDS AND PRISMS

A pyramid is a polyhedron formed by connecting each vertex of a polygonal base to a point called the apex. It is a conic solid with polygonal base. When unspecified, the base is usually assumed to be square. A closely related polyhedron is the dipyramid (or bipyramid), which is formed by joining two pyramids base to base.. The dipyramids are the dual polyhedra of the uniform prisms. Antiprisms are similar to prisms except the bases are twisted relative to each other, and the side faces are triangles rather than quadrilaterals.

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STELLATIONS

The stellation is the process of constructing a new polyhedron from an existing one by extending some elements such as edges or faces, usually in a symmetrical way, until they meet to form a new polygon or polyhedron. The new figure is a stellation of the original. Back in 1619 Kepler already stellated the dodecahedron to obtain two of the regular star polyhedra, also known as the Kepler-Poinsot polyhedra. He also stellated the octahedron obtaining the stellated octahedron.

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Sacred Geometry - Stellations Great Dodecahedron
Sacred Geometry - Golden_Ratio in the Great Pyramid Squaring the Circle

GOLDEN RATIO IN THE GREAT PYRAMID

The Great Pyramid of Gizeh fascinates many of us. It is an astonishing construction built with an incredible accuracy, far beyond the one achievable by our current technology. In this article we discuss some important properties of its design which are closely related to Sacred Geometry. We will show that the ratio of the apotem to half the base obeys the Golden Ratio, and that the perimeter of its base equals that of a circle with a radius equal to its height. We will also see that the angles of the Great Pyramid hide the Euler number. As a preparing we need to review the properties of the Kepler triangle and the problem of the Squaring of the Circle.

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GOLDEN RATIO AND MUSIC IN THE DNA

Jean-Claude Perez has been studying for years the populations of codons from the whole human genome -not just the 2% coding DNA. By cumulating the ratios of these populations in different ways, he obtained the Golden Ratio related attractor (3-φ)/2. While trying to reproduce his results, Jordi Solà-Soler found that the relative populations of several groups of eight codons in the whole human DNA correspond to the ratios of musical notes. He also found that the relative populations of some specific groups of eight codons in the whole human DNA is exactly the Golden Ratio.

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Sacred Geometry - Golden Ratio and Music in the DNA
Sacred Geometry - Golden Ratio in Atoms and Fine Structure constant

GOLDEN RATIO IN ATOMIC STRUCTURE

Dr. Rajalakshmi Heyrovska has recently shown that the Golden Ratio provides a quantitative link between various known quantities in atomic physics. While searching for the exact values of ionic radii and for the significance of the ionization potential of hydrogen, Dr. Heyrovska found that the Bohr radius can be divided into two Golden sections pertaining to the electron and the proton. More generally, she found that the Golden Ratio is also the ratio of anionic to cationic radii of any atom, their sum being the covalent bond length. In addition, she discovered a new interpretation of the Fine Structure constant in terms of the Golden Angle.

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GOLDEN RATIO IN DNA STRUCTURE

Almost 20 years ago, Mark E. Curtis began a research into the structure of DNA with the intention of producing a series of drawings and paintings of the double helix. During his work, it became clear to him that the Crick and Watson structure for the DNA molecule did not conform to geometric principles. Without compromising the essence of their structure, Mark E. Curtis proposed a resolution of the geometrical inconsistencies by means of a simple change in the position of alignment between the purines and pyrimidines. This realignment is founded entirely upon geometric principles. Furthermore, a set of simple mathematical relationships involving the Golden Ratio describe a three-dimensional geometric helix that conforms to the known ratios of DNA.

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Sacred Geometry - Golden Ratio in Three Dimensional DNA Structure
Sacred Geometry - Golden Ratio in Particle Physics

GOLDEN RATIO IN PARTICLE PHYSICS

Engineer M.S. El Nachie has developed a new theory of elemetary particle physics. This theory provides a fractal model of quantum space-time which allows the precise determination of the mass-energy of most elementary particles in close agreement with their experimental values. The Golden Ratio emerges naturally in this theory, and turns out to be the central piece that connects the fractal dimension of quantum space-time with the mass-energy of every fundamental particle, and also with several fundamental physical quantities such as the Fine Structure constant.

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GOLDEN RATIO IN THE HUMAN BODY

The Roman architect Marcus Vitruvius Pollio (c. 25 B.C.) inscribed the human body into a circle and a square, the two figures considered images of perfection. It is widely accepted that the proportions in the human body follow the Golden Ratio. In this article we will review some studies on the subject. We will show the nineteenth century findings of the Golden Ratio in the human body by Adolf Seizing, actually approximated by a Fibonacci sequence of measures. Then we will examine the Golden proportions of the human body proposed by architects Erns Neufert and Le Corbusier in the twentieth century. Finally we will show how a common study with a german and an indian population samples confirmed the presence of the Golden Ratio in some proportions of the human body.

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Sacred Geometry - Golden Ratio based human body proportions