“Philosophy is written in this grand book – I mean the universe — which stands continually open to our gaze. But it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures, without which it is humanly impossible to understand a single word of it.” — Galileo Galilei, Il Saggiatore (1623)
In the ancient world certain numbers had symbolic meaning, aside from their ordinary use for counting or calculating …
plane figures, the polygons, triangles, squares, hexagons, and so forth,
were related to the numbers (e.g. three and the triangle), were thought of in a similar way, and in fact, carried even more emotional value than the numbers themselves, because they were visual.
Sacred Geometry as worldview and cosmology
The belief that God created the universe according to a geometric plan has ancient origins. Plutarch attributed the belief to Plato, writing “Plato said God geometrizes continually” (Convivialium disputationum, liber 8,2). In modern times the mathematician Carl Friedrich Gauss adapted this quote, saying “God arithmetizes.”
At least as late as Johannes Kepler (1571–1630), a belief in the geometric underpinnings of the cosmos persisted among scientists.
According to Stephen Skinner, the study of sacred geometry has its roots in the study of nature, and the mathematical principles at work therein. Many forms observed in nature can be related to geometry, for example, the chambered nautilus grows at a constant rate and so its shell forms a logarithmic spiral to accommodate that growth without changing shape. Also, honeybees construct hexagonal cells to hold their honey.
These and other correspondences are seen by believers in sacred geometry to be further proof of the cosmic significance of geometric forms.
Art and architecture
Geometric ratios, and geometric figures were often employed in the design of Egyptian, ancient Indian, Greek and Roman architecture. Medieval European cathedrals also incorporated symbolic geometry. Indian and Himalayan spiritual communities often constructed temples and fortifications on design plans of mandala and yantra.
Many of the sacred geometry principles of the human body and of ancient architecture have been compiled into the Vitruvian Man drawing by Leonardo Da Vinci, itself based on the much older writings of the roman architect Vitruvius.
Stephen Skinner suggests that it is possible to place a geometric diagram over virtually any image of a natural object or human created structure, and find some lines intersecting the image. If the geometric diagram does not intersect major physical points in the image, the result is what Skinner calls “unanchored geometry.”
Pythagoras is often credited for discovering that an oscillating string stopped halfway along its length produces an octave relative to the string’s fundamental, while a ratio of 2:3 produces a perfect fifth and 3:4 produces a perfect fourth. However the Chinese culture already featured the same mathematical positions on the Guqin and the tone holes in flutes, so Pythagoras was not the first. Pythagoreans believed that these harmonic ratios gave music powers of healing which could “harmonize” an out-of-balance body.
The Golden Ratio Phi
In mathematics and the arts, two quantities are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller.
The golden ratio is often denoted by the Greek letter “phi”. The figure of a golden section illustrates the geometric relationship that defines this constant.
The golden ratio is an irrational mathematical constant, approximately 1.6180339887.
How to draw a line equal Phi?
Click to Enlarge
The number Pi is a mathematical constant that is the ratio of a circle’s circumference to its diameter, and is approximately equal to 3.14159. It has been represented by the Greek letter “pi” since the mid-18th century, though it is also sometimes written as pi. Pi is an irrational number, which means that it cannot be expressed exactly as a ratio of two integers (such as 22/7 or other fractions that are commonly used to approximate Pi); consequently, its decimal representation never ends and never settles into a permanent repeating pattern. The digits appear to be randomly distributed, although no proof of this has yet been discovered. Pi is a transcendental number – a number that is not the root of any nonzero polynomial having rational coefficients. The transcendence of Pi implies that it is impossible to find exact solution to the ancient challenge of squaring the circle with a compass and straight-edge.
For thousands of years, mathematicians have attempted to extend their understanding of Pi, sometimes by computing its value to a high degree of accuracy. Before the 15th century, mathematicians such as Archimedes and Liu Hui used geometrical techniques, based on polygons, to estimate the value of Pi. Starting around the 15th century, new algorithms based on infinite series revolutionized the computation of Pi, and were used by mathematicians including Madhava of Sangamagrama, Isaac Newton, Leonhard Euler, Carl Friedrich Gauss, and Srinivasa Ramanujan.
In the 20th and 21st centuries, mathematicians and computer scientists discovered new approaches that – when combined with increasing computational power – extended the decimal representation of Pi to, as of late 2011, over 10 trillion (1013) digits. Scientific applications generally require no more than 40 digits of Pi, so the primary motivation for these computations is the human desire to break records, but the extensive calculations involved have been used to test supercomputers and high-precision multiplication algorithms.
How to draw a line equal Pi (close approximation)?
Click to Enlarge
“God’s Fingerprint” equation
Phi and Pi relate to each other:
Pi = (6/5) * phi2
How to draw Regular (equilateral) triangle, square, pentagon and hexagon using only rope and peg
Here is one of the simplest and most elegant methods – only string/rope and peg is needed to mark the key points of this construction method:
Note: There are many other construction methods to draw a regular pentagon, for example
- Approximate Construction of Regular Pentagon by A. Durer
- Construction of Regular Pentagon by H. W. Richmond
- Inscribing a regular pentagon in a circle – and proving it
- Regular Pentagon Construction by Y. Hirano
- Regular Pentagon Inscribed in Circle by Paper
- Mascheroni Construction of a Regular Pentagon
- Regular Pentagon Construction by K. Knop
Tetractys and Sacred Numbers
The Pythagoreans adored numbers. Aristotle, in his Metaphysica, sums up the Pythagorean’s attitude towards numbers:
“The (Pythagoreans were) … the first to take up mathematics … (and) thought its principles were the principles of all things. Since, of these principles, numbers … are the first, … in numbers they seemed to see many resemblances to things that exist … more than [just] air, fire and earth and water, (but things such as) justice, soul, reason, opportunity …”
One of fascinating ancient discoveries is Tetractys. It is a symbol composed of ten dots in an upward-pointing triangular formation. It was a sacred pattern for the school of philosophers who followed the teachings of the Greek sage Pythagoras.
Tetractys itself can be interpreted as the symbolic blueprint of creation.
Its image is an equilateral triangle based on the essential numbers 1 (top), 2, 3 and 4 (base), whose sum is the “perfect” number 10 ( 1 + 2 + 3 + 4 = 10).
These numbers were considered by the Pythagoreans to be holy and at the origins of the universe. They believed that a four-fold pattern permeated the natural world, examples of which are the point, line, surface and solid and the four elements Earth, Water, Air and Fire.
Musically they represent the perfect consonants: the unison, the octave, the fifth and the fourth.
The importance of the tetractys to the Pythagoreans is illustrated by their oath of fellowship:
I swear by the discoverer of the Tetractys, Which is the spring of all our wisdom, The perennial fount and root of Nature.
One particular triangular number that they especially liked was the number ten. It was called a Tetractys, meaning a set of four things, a word attributed to the Greek Mathematician and astronomer Theon (c. 100 CE). The Pythagoreans identified ten such sets.
In Plato’s Timaeus, we find that God created the Cosmic Soul using two mathematical strips of 1, 2, 4, 8 and 1, 3, 9, 27. These two strips have the shape of an inverted “V” or the “Platonic Lambda” since it resembles the shape of the 11th letter of the Greek alphabet “Lambda” (?).
The Platonic Lambda interestingly describes 3 dimensional space:
- Top – 1 — single point
- 1st row: 2 – 3 — linear dimension
- 2nd row: 4 – 9 — surface (2-D: 2×2 and 3×3)
- 3rd row: 8 – 27 — cubic volume (3-D: 2x2x2 and 3x3x3)
It starts with a single point (a dot).
Two points define a line – and they also define a radius of a circle.
Three points define a single plane – and also produce a line, curve and/or triangle…
Four dots can be used to define 3 dimensional space (unless they are located on the same plane) – they can produce variety of lines, flat shapes and solids…
The Platonic Solids
Below we present simple method of creating 2-D projection of Platonic solids
Sacred Geometry: “Eye Candies”
Dance of the Planets
Orbits of Venus and the Earth are yet another example of sacred geometry.
This image shows the proportions of the orbit of the Earth and Venus (if the diameter the Earth’s orbit is equal to diagonal of a square, the orbit of Venus would fit inside such square as shown on the drawing.
The image below has been made by hand with help of computer software for drafting. First, let’s draw a square and two circles so one fits inside a square and the other outside the square. Strangely this represents accurately proportions of the orbits of Venus and Earth.
Second, let’s draw equally spaced points on each circle representing number of days per orbit (365 points for the orbit of the Earth and 225 points for the orbit of Venus). Next, let’s connected lines between orbital positions of each planet for each day – picking for the starting point a day when both planets are closest to each other. After repeating this process for 8 orbits of the Earth the lines form this amazing image:
This image (to scale) shows dance of Venus and Earth – each of the fine lines connects both planets over time
required to complete 8 orbits by Earth (and 13 orbits by Venus). Orbit of Earth is cropped out.
Copyright by World-Mysteries.com
Computer generated image. Fractal Type: Mandelbrot.
Magnification: 36,028,797,018,963,968 times (Number of zooms: 55)
Software used: Fractal eXtreme by Cygnus Software and Photoshop.
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