Estándar

Eleonora Dorrego and Julian Gil, his partners,  with deep sorrow announce the death of Kuky Valle, creator of this blog, on August 11, 2015 in Buenos Aires. Formidable architect and exquisite artist, as his numerous works on the blog show, his greatest personal virtue was his huge and generous heart.

In honor of its creator, we have decided to establish a time interval in the blog post.

Anuncios

# The Platonic Solids

Estándar

The 5 Platonic solids (Tetrahedron, Cube or (Hexahedron), Octahedron, Dodecahedron and Icosahedron) are ideal, primal models of crystal patterns that occur throughout the world of minerals in countless variations.

These are the only five regular polyhedra, that is, the only five solids made from the same equilateral, equiangular polygons. To the Greeks, these solids symbolized fire, earth, air, spirit (or ether) and water respectively. The cube and octahedron are duals, meaning that one can be created by connecting the midpoints of the faces of the other. The icosahedron and dodecahedron are also duals of each other, and three mutually perpendicular, mutually bisecting golden rectangles can be drawn connecting their vertices and midpoints, respectively. The tetrahedron is a dual to itself.

There are 13 Archimedean solids, each of which are composed of two or more different regular polygons. Interestingly, 5 (Platonic) and 13 (Archimedean) are both Fibonacci numbers, and 5, 12 and 13 form a perfect right triangle.

Metatron’s Cube

Metatron’s Cube contains 2-dimensional images of the Platonic Solids (as shown above) and many other primal forms.

The Flower of Life

Indelibly etched on the walls of temple of the Osirion at Abydos, Egypt, the Flower of Life contains a vast Akashic system of information, including templates for the five Platonic Solids.

Life itself as we know it is inextricably interwoven with geometric forms, from the angles of atomic bonds in the molecules of the amino acids, to the helical spirals of DNA, to the spherical prototype of the cell, to the first few cells of an organism which assume vesical, tetrahedral, and star (double) tetrahedral forms prior to the diversification of tissues for different physiological functions. Our human bodies on this planet all developed with a common geometric progression from one to two to four to eight primal cells and beyond.

Estándar

TOROIDS

Rotating a circle about a line tangent to it creates a torus, where the center exactly touches all the “rotated circles.” The surface of the torus can be covered with 7 distinct areas, all of which touch each other.

Fractals and Recursive Geometries

fractal is a natural phenomenon or a mathematical set that exhibits a repeating pattern that displays at every scale. If the replication is exactly the same at every scale, it is called a self-similar pattern.

Perfect Right Triangles

The 3/4/5, 5/12/13 and 7/24/25 triangles are examples of right triangles whose sides are whole numbers. The graphic above contains several of each of these triangles. The 3/4/5 triangle is contained within the so-called “King’s Chamber” of the Great Pyramid.

# The Golden Ratio

Estándar

The golden ratio is another fundamental measure.   (The golden ratio is about 1.618033988749894848204586834365638117720309180…) The golden ratio is the unique ratio such that the ratio of the whole to the larger portion is the same as the ratio of the larger portion to the smaller portion.

The golden ratio (phi) has some unique properties and makes some interesting appearances:

• phi = the ratio of segments in a 5-pointed star (pentagram) considered sacred to Plato and Pythagoras in their mystery schools. Note that each larger (or smaller) section is related by the phi ratio

Spirals

This spiral generated by a succession of Golden Triangles (triangles with relative side lengths of 1, phi and phi) is the classic shape of the Chambered Nautilus shell. The creature building this shell uses the same proportions for each expanded chamber that is added; growth follows a law which is everywhere the same. The outer triangle is the same as one of the five “arms” of the pentagonal graphic above.

# BASIC ELEMENTS OF ANCIENT GEOMETRY (extracted from “Introduction to Sacred Geometry” by B. Rawles

Estándar

In nature, we find patterns, designs and structures from the most minuscule particles, to expressions of life discernible by human eyes, to the greater cosmos. These follow geometrical archetypes, which reveal to us the nature of each form.

We will analyze some of the main elements and figures

The Sphere

This is one of the simplest and most perfect of forms. All points on the surface are equally accessible and regarded by the center from which all originate.

The Circle

The circle is a two-dimensional shadow of the sphere, and shares with it the perfection of its simplicity.

The ratio of the circumference of a circle to its diameter, Pi, is the original transcendental and irrational number. (Pi equals about 3.14159265358979323846264338327950288419716939937511…)

The essence of the circle exists in a dimension that transcends the linear rationality that it contains.

The circle is a two-dimensional shadow of the sphere, and shares with it the perfection of its simplicity.

The ratio of the circumference of a circle to its diameter, Pi, is the original transcendental and irrational number. (Pi equals about 3.14159265358979323846264338327950288419716939937511…)

The Point

At the center of a circle or a sphere is always an infinitesimal point. The point needs no dimension, yet embraces all dimension.

.

# ANCIENT GEOMETRY, 4. RENAISSANCE

Estándar

The Renaissance saw a rebirth of Classical Greek and Roman culture and ideas, among them the study of mathematics as a relevant subject needed to understand nature and the arts. Two major reasons drove Renaissance artists towards the pursuit of mathematics. First, painters needed to figure out how to depict three-dimensional scenes on a two-dimensional canvas.

Second, philosophers and artists alike were convinced that mathematics was the true essence of the physical world and that the entire universe, including the arts, could be explained in geometric terms. In light of these factors, Renaissance artists became some of the best applied mathematicians of their times.

The first printed illustration of a rhombicuboctahedron, byLeonardo da Vinci, published in De divina proportione.

Written by Luca Pacioli in Milan from 1496–98, published in Venice in 1509, De Divina Proportione was about mathematical and artistic proportionLeonardo da Vinci drew illustrations of regular solids in De divina proportione while he lived with and took mathematics lessons from Pacioli. Leonardo’s drawings are probably the first illustrations of skeletonic solids, which allowed an easy distinction between front and back. Skeletonic solids, such as the rhombicuboctahedron, were one of the first solids drawn to demonstrate perspective by being overlaid on top of each other. Additionally, the work also discusses the use of perspective by painters such as Piero della FrancescaMelozzo da Forlì, and Marco Palmezzano.

It is in De Divina Proportione that the golden ratio is defined as the divine proportion. Pacioli also details the use of the golden ratio as the mathematical definition of beauty when applied to the human face.

“The Ancients, having taken into consideration the rigorous construction of the human body, elaborated all their works, as especially their holy temples, according to these proportions; for they found here the two principal figures without which no project is possible: the perfection of the circle, the principle of all regular bodies, and the equilateral square.” from De Divina Proportione (1509)

# ANCIENT GEOMETRY, 3. Great Mosque of Kairouan

Estándar

The oldest mosque in North Africa is the Great Mosque of Kairouan (Tunisia), built by Uqba ibn Nafi in 670 A.D. Boussora and Mazouz’s study of the mosque dimensions reveals a very consistent application of the golden ratio in its design.

Floor plan of the Great Mosque of Kairouan.

The geometric technique of construction of the golden section seems to have determined the major decisions of the spatial organization. The golden section appears repeatedly in some part of the building measurements. It is found in the overall proportion of the plan and in the dimensioning of the prayer space, the court and the minaret. The existence of the golden section in some parts of Kairouan mosque indicates that the elements designed and generated with this principle may have been realised at the same period.

Because of urban constraints, the mosque floor plan is not a perfect rectangle. Even so, for example, the division of the courtyard and prayer hall is almost a perfect golden ratio.