Category Archives: Sacred Geometry

Quasicrystals, Shape Shifting Human Invisibility??

The universe brings me information to my great surprise. I was prompted to discover quasicrystals. My first thought on reading below was the technology is too advanced, the humanity underdeveloped and the population kept in a space of ignorance. This is not unlike silent weapons for silent wars philosophy, making haste slowly, or the boiling frog whereby these technologies are created quietly in the background and then suddenly humanity is exposed to them with no ability to defend themselves as they have no idea what they are dealing with.  How can they deal with this type of technology coming out of military applications and this silent war that regards any form of opposition as an enemy?  This is why democracy must be reclaimed.

I am becoming deeply aware of this mindset and I will sit with it in this now moment.  I will relay what I feel… crystals are intelligent firstly, quasi crystals are not.  The intent behind the creation of this by scientists is typically exploratory those funding it are not seeking to expand the vistas of human ingenuity but are focused on control.  Why?  They seek to ensure a pattern is non changing and predictable yet use a technology where the patterns change thus changing its shape as its atomic number changes. See https://en.wikipedia.org/wiki/Atomic_number I feel strongly the connection with genetic modification and the commercial interests that have driven the expansion of this biotechnology.  My inner feeling shows me that those inventing this have no awareness of the whole, the symmetry of the natural geometry.  How the sum of the parts is the whole. So when the atomic numbers change, the parts change and the whole changes as all are interconnected, not separate.  When those in power do not have highly evolved abilities to understand and feel a harmonic resonance with the whole they de-construct what is stable.  I get the image of a nuclear bomb, thus the atom split.  Massive release of energy, then I see fusion, forcing atoms together through an unnatural application to produce endless energy outcomes.  This is unstable. 

So back to homeostasis, this arises when the scientist is in harmony (balance) with his or her true nature.  The subject and object are not separate, one creates the other in synergy and symmetry. Thus those funding the exercise are in harmony with their true nature produces harmony if not, then the opposite. I felt the wailing wall here, releasing the pain, not atomic energy, at the wall of their own making.  There is no them or us, there is only us as we are made of the same stuff and what you do to another returns to the self.   The law of cause and effect, is a universal natural law.

So shapeshifting robots, humans etc. This is to gain advantage, the element of surprise rather than the element of celebration or indeed the element of natural science.  The purpose is to be ahead, to be free, not stopped and still fighting the perceived ‘other’.  Shapeshifting robots adapt to the environment, nature does this through adaptation as genes in harmony with nature, are naturally selected.  Because technocracy disconnects from humanity it is unable to see the whole, this is the weakness in technocracy.  AI cannot feel humanity, it is based on iterations, bio-feedback algorithms, changing neural wired networks to adapt to change, but unable to feel the god particle, as it is not a biological organism created by higher intelligence. This is not to say higher intelligences are not involved in shape-shifting, it is likely they are.  The malevolent intent is about power not love, this is why in the end it fails.  Intent is either life producing or destructive, the latter is not naturally selected as the resiliency of a species arises from harmony.  There are many forces at work here, on many levels is my feeling.  Again, the idea of the camel as the thread passing through the needle is the metaphor for the impossible becoming possible or indeed a portal to higher awareness.  This portal cannot open unless frequency changes, as one cannot see beyond their frequency no matter the technology.  The operative word appears to be ‘matter’.

Inspiration is an amazing journey.

 

Quasicrystals Are Nature’s Impossible Matter

 

“There can be no such creature.”

 
by Daniel Oberhaus
May 3 2015, 9:02pm

 

Al-Pd-Re, a lab made quasicrystal. Image: via

What do a frying pan, an LED light, and the most cutting edge camouflage in the world have in common? Well, that largely depends on who you ask. Most people would struggle to find the link, but for University of Michigan chemical engineers Sharon Glotzer and Michael Engel, there is a substantial connection, indeed one that has flipped the world of materials science on its head since its discovery over 30 years ago.

 

The magic ingredient common to all three items is the quasiperiodic crystal, the “impossible” atomic arrangement discovered by Dan Shechtman in 1982. Basically, a quasicrystal is a crystalline structure that breaks the periodicity (meaning it has translational symmetry, or the ability to shift the crystal one unit cell without changing the pattern) of a normal crystal for an ordered, yet aperiodic arrangement. This means that quasicrystalline patterns will fill all available space, but in such a way that the pattern of its atomic arrangement never repeats. Glotzer and Engel recently managed to simulate the most complex quasicrystal ever, a discovery which may revolutionize the field of crystallography by blowing open the door for a whole host of applications that were previously inconceivable outside of science-fiction, like making yourself invisible or shape-shifting robots.

While most of the current applications of quasicrystals are rather mundane, such as the coating for frying pans or surgical utensils, Glotzer and Engel’s simulation of a self-assembling icosahedral quasicrystal opens up exciting new avenues for research and development, such as improved camouflage.

“Camouflage is all about redirecting light to change the appearance of something,” said Glotzer. “Making camouflage materials or any kind of transformation optics materials is all about controlling the structure of the material, controlling the spacing of the building blocks to control the way light is absorbed and reflected.”

 

Icosahedral quasicrystals (IQCs) are one of the several unique structures which have something called a photonic band gap, which dictates the range of photon frequencies which are permitted to pass through the material. Photonic band gaps are determined by the spatial arrangement of an atomic lattice. In other words, whether or not a photon becomes “trapped” in the lattice depends on the photonic frequency (measured as a wavelength) in relation to the space between atoms and the way these atoms are arranged (periodically, aperiodically, etc). If the wavelength falls within the range of the photonic band gap for the specific material, then the photons will not be able to propagate through the structure.

Thus, being able to manipulate photonic band gaps means that one can manipulate atomic structures in such a way that the material will only be visible within determined photonic frequencies, a critical advancement for those concerned with making people invisible, which probably at least partly accounts for why the US Department of Defense and the US Army both helped fund Glotzer and Engel’s study.

While the existence of photonic bandgaps is nothing new, being able to manipulate solid-state matter in such a way that allows one to fully exploit these bandgaps has remained elusive. In this sense, Glotzer and Engel’s simulated quasicrystal represents a return to the fundamentals of crystallography, rather than something entirely novel.

 

According to the team, before their simulation, scientists knew that mixing certain metals in the right thermodynamic conditions (pressure, temperature) would result in the formation of a quasicrystal. They also knew that given the correct environmental conditions, it was possible for quasicrystals to form in nature (two natural quasicrystals have been discovered to date: the first in 2009 and the second was reported on March 13, coming from a 4.5-billion year old meteorite in Russia).

What scientists didn’t understand, said Engel, was what was happening in the reaction to make these quasicrystals form. There was an input and output, but what went on inside the blackbox remained a mystery. Glotzer and Engel’s experiment was a first step in solving this a-list conundrum in materials science.

“For a long time people have looked for methods to actually model [how icosahedral quasicrystals form],” said Engel. “This is more of a fundamental importance, it doesn’t necessarily make [IQCs] have better properties or applications, but it allows us to study how these crystals form.”

Understanding how these quasicrystals form is the first step in manipulating them toward desired ends. While this ability to manipulate quasicrystals is still in a very young phase, increasing technical sophistication could conceivably lead to some pretty wild developments in the future, like Terminator-style shape-shifting robots.

 

Part of the reason robots modeled after T-1000 don’t roam the Earth already is because our understanding of matter and our ability to find useful applications for the staggering variety of metals found in nature is still relatively rudimentary. Understanding how quasicrystals form will fill in a huge gap in our knowledge of solid-state physics and chemistry. Increasing this knowledge in all of its forms is essential to future physical manipulation, whether or not this manipulation is directly linked to quasicrystals.

“It’s not that the icosahedral quasicrystal itself would necessarily be the structure you would shoot for [in shapeshifting materials], but it represents the kind of complexity and control that one would like to have over the building blocks of matter,” said Glotzer. “If you understand what is required to get a certain structure, than you could imagine that we could change conditions and change the structure that we get. Everything about a material depends on its structure.”

T-1000 Shapeshifter in Terminator

The quasicrystalline structure was discovered by Dan Shechtman, a professor of materials science at the Technion-Israel Institute of Technology, in 1982 while he was observing an alloy of rapidly cooled aluminum and manganese with an electron microscope.

What he saw defied the laws of nature as they were understood at the time.

Rather than finding a random collection of atoms as expected, Shechtman observed a diffraction pattern with ten-fold rotational symmetry, something which was thought to be impossible (subsequent experiments would demonstrate that what Shechtman had discovered was actually five-fold symmetry).

 

Shechtman’s five-fold symmetry defied the basic definition of a crystal which had stood unchallenged since crystallography’s inauguration as a science some 70 years prior. According to the received wisdom at the time, a crystal was something which by definition was both ordered and periodic, meaning that it exhibited a certain pattern at regular intervals. On this definition crystals were only capable of exhibiting a two, three, four, or six-fold rotational symmetry (the ability to retain symmetry after being rotated so many times along an axis—in other words, after rotating the crystal along an axis so many times, it will look the same as when you started).

Upon his discovery of a diffraction pattern with five-fold symmetry, Shechtman allegedly exclaimed that “there can be no such creature.” His colleagues agreed with him.

Electron diffraction pattern showing five-fold symmetry from an aluminum-copper-iron quasicrystal. Image via

“Since 1912 all crystals that had been studied were periodic—hundreds of thousands of different crystals were studied. People did not believe that there was anything different because so many thousands of excellent scientists developed the field and found only crystals which were periodic,” Shechtman told me over Skype.

Thus, when Shechtman revealed his discovery, which would earn him a Nobel Prize in chemistry in 2011, he was met with not only incredulity, by outright hostility. Upon hearing of Shechtman’s discovery, the head of his laboratory allegedly told him to revisit a textbook covering the basics of x-ray diffraction, so that he might understand why his “discovery” was impossible. When Shechtman informed him that he had no need of the book since his discovery was not included in the material, he was told that he was a disgrace to the team.

 

He would be discredited by scientists around the world, including heavyweights such as Linus Pauling, the two time Nobel Prize winning chemist who dismissed Shechtman’s results as the product of “twinning,” the fusion of two normal crystals at an angle.

“When I came out with my results, people found it difficult to accept. It was easier to say ‘Don’t they know anything about crystallography at Technion? Don’t they read the books?’ I had to defend it awhile,” said Shechtman. It took two years from the initial discovery for Shechtman to publish his results. After their publication, according to Shechtman, “all hell broke loose.

“Shortly after the first publication, there was a growing community of avant-garde young scientists from around the world who all supported me and joined the fight, so I was not alone anymore,” he said. “But in the first two years I was alone.”

Dan Shechtman displays a model in his lab in Israel. Image via

Shechtman’s discovery prompted the International Union of Crystallography to redefine just what was meant by a crystal in 1992. The current definition now reads that a crystal is defined by “discrete diffraction patterns,” which accounts for both the periodic structures which traditionally defined a crystal, as well as the aperiodic quasicrystalline structures discovered by Shechtman.

“Quasiperiodic crystals are still crystals—they have nothing to do with amorphous materials,” said Shechtman. “Amorphous materials are non-ordered (like glass), quasicrystals are crystals, but the atomic relation within them is different than periodic crystals. It is perfectly ordered, but not periodic.”

 

The math underlying Shechtman’s design has a long history, dating back to Leonardo Fibonacci who in 1202 sought to discover how fast rabbits could breed in ideal circumstances (the sequence ‘discovered’ thereby actually long pre-dated Fibonacci in Indian mathematics—Fibonacci is most accurately credited with introducing it to the West).

Fibonacci began his thought experiment by assuming that two rabbits are placed in a field and produce a new pair of rabbits at the end of a month. It takes each new pair one month before they are able to breed another pair. The question Fibonacci sought to answer was how many pairs there would be at the end of one year. The sequence inaugurated by this pattern (1,1,2,3,5,8,13,21,34,55,89,144) is known as the Fibonacci sequence wherein the next number can always be derived by adding the two numbers which precede it in the sequence.

The Fibonacci sequence can be seen as a 1-D analog to Shechtman’s quasicrystal, in which there is order without repetition. The 2-D analog was discovered in 1974 by the famous English mathematician and physicist Roger Penrose.

That’s what order means: there is correlation between how it looks in one place versus another.

In addition to proving that black holes could result from the gravitational collapse of stars, Penrose discovered a method of tiling a plane aperiodically, which became the first demonstration of five-fold rotational symmetry. (Or the ability to rotate 72 degrees without changing the pattern.) In Penrose’s initial iteration, he used four different shapes all related to a pentagon. He would eventually narrow this down to an aperiodic tiling which used only two rhombuses, a “fat” rhombus and a “skinny” rhombus.

 

“Locally [Penrose tiles] are a very simple structure: there are only two building blocks and the way that they are put together means that it is not perfectly repeating,” said Engel. “But still there’s always a discrete number, a finite number of ways that you can arrange them. The fact that you only have these finite ways of arranging them makes it such that even if you are infinitely far away from where you started building the structure, it’s still in a way predictable. So there is correlation, meaning they are not independent of one another and that’s what order means: there is correlation between how it looks in one place versus another.”

Translate Penrose Tiling to a three-dimensional atomic lattice and you have the essence of a quasicrystal. The important takeaway here, according to Shechtman, is that “there is not a motif of any size that repeats itself. So there is order, and yet there is no periodicity.” The order is derived from the fact that anyone could reconstruct the Fibonacci sequence or Penrose tiles, yet despite this order, if the sequence or tiling is shifted in anyway it is impossible to derive an exact repetition.

Roger Penrose stands on Penrose Tiling, the first instance of Penrose inception in recorded history

In the 30 some years since Shechtman’s discovery, hundreds of quasicrystals have been discovered, many of which are aluminum-based alloys. The first naturally occurring quasicrystal, icosahedrite, was found in Russia in 2009. Quasicrystals, both natural and artificial, are divided into two primary types: polygonal and icosahedral quasicrystals. The former category exhibits periodicity in one direction (perpendicular to the quasiperiodic layers); the latter exhibits no periodicity whatsoever, which is precisely what makes Glotzer and Engel’s simulation such a big deal.

“The icosahedral quasicrystal is the most exotic,” said Engel. “It’s the most spatially or geometrically complex.”

 

In their experiment, Glotzer and Engel set out to answer one of the fundamental questions dogging the field of crystallography: How can long range order be generated from local interactions which exhibit no periodicity? While most real quasicystals are made of two or more elements, the University of Michigan team ran simulations using only one type of particle, another first in the field.

In essence, the team was attempting to determine what thermodynamic conditions favored the formation of icosahedral quasicrystals given certain initial parameters which determined the force field, or the way the particles would interact with one another. These parameters were designed so that they could be recreated in a laboratory setting. For instance, one parameter dictated that the particles were only allowed to interact with other particles which were within three particle distances of themselves.

10 fold???”: A page from Shechtman’s notebook the day he discovered the quasicrystal. Image: via

“Basically, we were solving Newton’s equation of motion,” said Glotzer. “What you have is a bunch of particles and they interact according to a certain force field. So that means at a given time, every atom in the system has a force exerted on it by every other atom in the system. You add up all those forces on every atom, and then you solve F=ma. By adding up all those forces you can solve for the acceleration which tells you how to move the particles. Then you do this for all the particles in the system.”

The end results of these calculations tells the team where the particles “want be” under varying thermodynamic conditions, such as pressure and temperature. Given these initial conditions, the crystal “self-assembles” in the simulation.

 

“All we know are the force fields between the particles and Newton’s equation,” said Glotzer. “We don’t know what will come out when we start—it’s very different from building [icosahedral quasicrystal] by hand.”

As the team discovered, the interaction of their particles in such a way that a quasicrystal was formed was favored by interactions governed by the golden ratio. The golden ratio is an irrational number which starts as 1.61803, and is derived from the ratio of two numbers whose ratio to one another is the same as that between their sum and the larger integer. It is related to the Fibonacci sequence insofar as each digit you progress in the sequence, the ratio between the current digit and the one before it approaches the golden ratio—it is an infinite approximation.

In a talk given by Penrose at the Royal Institution in 2014, the renowned scientist speculated that icosahedral quasicrystals might be governed by quantum mechanical interactions, given that a complex aperiodic structure demonstrated long range order solely from local potentials, or interactions. Glotzer and Engel’s findings suggest this might not be the case.

“Our simulations suggest that maybe quantum mechanics are not even necessary,” said Engel. “Maybe you can get it from classical, non-quantum interactions. How that works, exactly, is a wide open question. Right now we hope to address this question with our model.”

 

While shape-shifting robots derived from quasicrystalline principles may be a long way off, quasicrystals are already beginning to play a major role in everyday life. They are most commonly found as a reinforced coating (such as on a frying pan or surgical tool) but are increasingly being added in small quantities to normal metal alloys to reinforce them while retaining lightness. They are also becoming very popular in additive manufacturing, otherwise known as 3D printing, due to their low friction and resistance to wear.

Where the future of quasicrystals will take us is relatively uncertain at the moment. What is, known however is that quasicrystals, nature’s “impossible matter,” provide us with a very important missing link in the study of matter, and may very well hold the key to the total manipulation of the solid universe in the future.

The Golden Ratio is The Equation of Expanding Balance

Version:1.0 StartHTML:000000205 EndHTML:000281169 StartFragment:000057786 EndFragment:000281131 StartSelection:000057786 EndSelection:000281109 SourceURL:https://en.wikipedia.org/wiki/Golden_ratio Golden ratio – Wikipedia

Golden ratio

From Wikipedia, the free encyclopediaJump to navigationJump to searchThis article is about the number. For the Ace of Base album, see The Golden Ratio (album). For the calendar dates, see Golden number (time).

Line segments in the golden ratio

A golden rectangle with longer side a and shorter side b, when placed adjacent to a square with sides of length a, will produce a similar golden rectangle with longer side a + b and shorter side a. This illustrates the relationship a + b a = a b ≡ φ {\displaystyle {\frac {a+b}{a}}={\frac {a}{b}}\equiv \varphi }

{\frac {a+b}{a}}={\frac {a}{b}}\equiv \varphi

.

In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. The figure on the right illustrates the geometric relationship. Expressed algebraically, for quantities a and b with a > b > 0, a + b a = a b   = def   φ , {\displaystyle {\frac {a+b}{a}}={\frac {a}{b}}\ {\stackrel {\text{def}}{=}}\ \varphi ,}

{\displaystyle {\frac {a+b}{a}}={\frac {a}{b}}\ {\stackrel {\text{def}}{=}}\ \varphi ,}
\varphi
\phi
x^{2}-x-1=0

where the Greek letter phi (φ {\displaystyle \varphi } or ϕ {\displaystyle \phi } ) represents the golden ratio.[a] It is an irrational number that is a solution to the quadratic equation x 2 − x − 1 = 0 {\displaystyle x^{2}-x-1=0} , with a value of: φ = 1 + 5 2 = 1.6180339887 … . {\displaystyle \varphi ={\frac {1+{\sqrt {5}}}{2}}=1.6180339887\ldots .}

\varphi ={\frac {1+{\sqrt {5}}}{2}}=1.6180339887\ldots .

[1]

The golden ratio is also called the golden mean or golden section (Latin: sectio aurea).[2][3] Other names include extreme and mean ratio,[4] medial section, divine proportion, divine section (Latin: sectio divina), golden proportion, golden cut,[5] and golden number.[6][7][8]

Mathematicians since Euclid have studied the properties of the golden ratio, including its appearance in the dimensions of a regular pentagon and in a golden rectangle, which may be cut into a square and a smaller rectangle with the same aspect ratio. The golden ratio has also been used to analyze the proportions of natural objects as well as man-made systems such as financial markets, in some cases based on dubious fits to data.[9] The golden ratio appears in some patterns in nature, including the spiral arrangement of leaves and other plant parts.

Some twentieth-century artists and architects, including Le Corbusier and Salvador Dalí, have proportioned their works to approximate the golden ratio—especially in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratio—believing this proportion to be aesthetically pleasing.

Contents

Calculation

List of numbersIrrational numbersζ(3)√2√3√5φeπ
Binary 1.1001111000110111011…
Decimal 1.6180339887498948482…[1]
Hexadecimal 1.9E3779B97F4A7C15F39…
Continued fraction 1 + 1 1 + 1 1 + 1 1 + 1 1 + ⋱ {\displaystyle 1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+\ddots }}}}}}}}}
Algebraic form 1 + 5 2 {\displaystyle {\frac {1+{\sqrt {5}}}{2}}}

The Greek letter phi symbolizes the golden ratio. Usually, the lowercase form (φ or φ) is used. Sometimes the uppercase form (Φ {\displaystyle \Phi }

\Phi

) is used for the reciprocal of the golden ratio, 1/φ.[10]

Two quantities a and b are said to be in the golden ratio φ if a + b a = a b = φ . {\displaystyle {\frac {a+b}{a}}={\frac {a}{b}}=\varphi .}

{\frac {a+b}{a}}={\frac {a}{b}}=\varphi .

One method for finding the value of φ is to start with the left fraction. Through simplifying the fraction and substituting in b/a = 1/φ, a + b a = a a + b a = 1 + b a = 1 + 1 φ . {\displaystyle {\frac {a+b}{a}}={\frac {a}{a}}+{\frac {b}{a}}=1+{\frac {b}{a}}=1+{\frac {1}{\varphi }}.}

{\displaystyle {\frac {a+b}{a}}={\frac {a}{a}}+{\frac {b}{a}}=1+{\frac {b}{a}}=1+{\frac {1}{\varphi }}.}

Therefore, 1 + 1 φ = φ . {\displaystyle 1+{\frac {1}{\varphi }}=\varphi .}

1+{\frac {1}{\varphi }}=\varphi .

Multiplying by φ gives φ + 1 = φ 2 {\displaystyle \varphi +1=\varphi ^{2}}

\varphi +1=\varphi ^{2}

which can be rearranged to φ 2 − φ − 1 = 0. {\displaystyle {\varphi }^{2}-\varphi -1=0.}

{\varphi }^{2}-\varphi -1=0.

Using the quadratic formula, two solutions are obtained: 1 + 5 2 = 1.618 033 988 7 … {\displaystyle {\frac {1+{\sqrt {5}}}{2}}=1.618\,033\,988\,7\dots }

{\displaystyle {\frac {1+{\sqrt {5}}}{2}}=1.618\,033\,988\,7\dots }

and 1 − 5 2 = − 0.618 033 988 7 … {\displaystyle {\frac {1-{\sqrt {5}}}{2}}=-0.618\,033\,988\,7\dots }

{\displaystyle {\frac {1-{\sqrt {5}}}{2}}=-0.618\,033\,988\,7\dots }

Because φ is the ratio between positive quantities, φ is necessarily positive: φ = 1 + 5 2 = 1.61803 39887 … {\displaystyle \varphi ={\frac {1+{\sqrt {5}}}{2}}=1.61803\,39887\dots }

\varphi ={\frac {1+{\sqrt {5}}}{2}}=1.61803\,39887\dots

History

Further information: Mathematics and artSee also: Fibonacci number § History

According to Mario Livio:

Some of the greatest mathematical minds of all ages, from Pythagoras and Euclid in ancient Greece, through the medieval Italian mathematician Leonardo of Pisa and the Renaissance astronomer Johannes Kepler, to present-day scientific figures such as Oxford physicist Roger Penrose, have spent endless hours over this simple ratio and its properties. … Biologists, artists, musicians, historians, architects, psychologists, and even mystics have pondered and debated the basis of its ubiquity and appeal. In fact, it is probably fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics.[11]

Ancient Greek mathematicians first studied what we now call the golden ratio because of its frequent appearance in geometry;[12] the division of a line into “extreme and mean ratio” (the golden section) is important in the geometry of regular pentagrams and pentagons.[13] According to one story, 5th-century BC mathematician Hippasus discovered that the golden ratio was neither a whole number nor a fraction (an irrational number), surprising Pythagoreans.[14] Euclid‘s Elements (c. 300 BC) provides several propositions and their proofs employing the golden ratio[15][b] and contains the first known definition:[16]

A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser.[17][c]

Michael Maestlin, the first to write a decimal approximation of the ratio

The golden ratio was studied peripherally over the next millennium. Abu Kamil (c. 850–930) employed it in his geometric calculations of pentagons and decagons; his writings influenced that of Fibonacci (Leonardo of Pisa) (c. 1170–1250), who used the ratio in related geometry problems, though never connected it to the series of numbers named after him.[19] Luca Pacioli named his book Divina proportione (1509) after the ratio and explored its properties including its appearance in some of the Platonic solids.[8][20] Leonardo da Vinci, who illustrated the aforementioned book, called the ratio the sectio aurea (‘golden section’).[21] 16th-century mathematicians such as Rafael Bombelli solved geometric problems using the ratio.[22]

German mathematician Simon Jacob (d. 1564) noted that consecutive Fibonacci numbers converge to the golden ratio;[23] this was rediscovered by Johannes Kepler in 1608.[24] The first known decimal approximation of the (inverse) golden ratio was stated as “about 0.6180340” in 1597 by Michael Maestlin of the University of Tübingen in a letter to Kepler, his former student.[25] The same year, Kepler wrote to Maestlin of the Kepler triangle, which combines the golden ratio with the Pythagorean theorem. Kepler said of these:

Geometry has two great treasures: one is the Theorem of Pythagoras, and the other the division of a line into extreme and mean ratio; the first we may compare to a measure of gold, the second we may name a precious jewel.

18th-century mathematicians Abraham de Moivre, Daniel Bernoulli, and Leonhard Euler used a golden ratio-based formula which finds the value of a Fibonacci number based on its placement in the sequence; in 1843 this was rediscovered by Jacques Philippe Marie Binet, for whom it was named “Binet’s formula”.[26] Martin Ohm first used the German term goldener Schnitt (‘golden section’) to describe the ratio in 1835.[27] James Sully used the equivalent English term in 1875.[28]

By 1910, mathematician Mark Barr began using the Greek letter Phi (φ) as a symbol for the golden ratio.[29][d] It has also been represented by tau (τ), the first letter of the ancient Greek τομή (‘cut’ or ‘section’).[32][33]

Between 1973 and 1974, Roger Penrose developed Penrose tiling, a pattern related to the golden ratio both in the ratio of areas of its two rhombic tiles and in their relative frequency within the pattern.[34] This led to Dan Shechtman‘s early 1980s discovery of quasicrystals,[35][36] some of which exhibit icosahedral symmetry.[37][38]

Applications and observations

Architecture

Further information: Mathematics and architecture

A 2004 geometrical analysis of earlier research into the Great Mosque of Kairouan (670) reveals a consistent application of the golden ratio throughout the design.[39] They found ratios close to the golden ratio in the overall layout and in the dimensions of the prayer space, the court, and the minaret. However, the areas with ratios close to the golden ratio were not part of the original plan, and were likely added in a reconstruction.[39]

It has been speculated that the golden ratio was used by the designers of the Naqsh-e Jahan Square (1629) and the adjacent Lotfollah Mosque.[40]

The Swiss architect Le Corbusier, famous for his contributions to the modern international style, centered his design philosophy on systems of harmony and proportion. Le Corbusier’s faith in the mathematical order of the universe was closely bound to the golden ratio and the Fibonacci series, which he described as “rhythms apparent to the eye and clear in their relations with one another. And these rhythms are at the very root of human activities. They resound in man by an organic inevitability, the same fine inevitability which causes the tracing out of the Golden Section by children, old men, savages and the learned.”[41][42]

Le Corbusier explicitly used the golden ratio in his Modulor system for the scale of architectural proportion. He saw this system as a continuation of the long tradition of Vitruvius, Leonardo da Vinci’s “Vitruvian Man“, the work of Leon Battista Alberti, and others who used the proportions of the human body to improve the appearance and function of architecture. In addition to the golden ratio, Le Corbusier based the system on human measurements, Fibonacci numbers, and the double unit. He took suggestion of the golden ratio in human proportions to an extreme: he sectioned his model human body’s height at the navel with the two sections in golden ratio, then subdivided those sections in golden ratio at the knees and throat; he used these golden ratio proportions in the Modulor system. Le Corbusier’s 1927 Villa Stein in Garches exemplified the Modulor system’s application. The villa’s rectangular ground plan, elevation, and inner structure closely approximate golden rectangles.[43]

Another Swiss architect, Mario Botta, bases many of his designs on geometric figures. Several private houses he designed in Switzerland are composed of squares and circles, cubes and cylinders. In a house he designed in Origlio, the golden ratio is the proportion between the central section and the side sections of the house.[44]

Art

See also: Mathematics and art and History of aesthetics before the 20th century

Leonardo‘s illustration of a dodecahedron from Pacioli‘s Divina proportione (1509)

Divina proportione (Divine proportion), a three-volume work by Luca Pacioli, was published in 1509. Pacioli, a Franciscan friar, was known mostly as a mathematician, but he was also trained and keenly interested in art. Divina proportione explored the mathematics of the golden ratio. Though it is often said that Pacioli advocated the golden ratio’s application to yield pleasing, harmonious proportions, Livio points out that the interpretation has been traced to an error in 1799, and that Pacioli actually advocated the Vitruvian system of rational proportions.[45] Pacioli also saw Catholic religious significance in the ratio, which led to his work’s title.

Leonardo da Vinci‘s illustrations of polyhedra in Divina proportione[46] have led some to speculate that he incorporated the golden ratio in his paintings. But the suggestion that his Mona Lisa, for example, employs golden ratio proportions, is not supported by Leonardo’s own writings.[47] Similarly, although the Vitruvian Man is often shown in connection with the golden ratio, the proportions of the figure do not actually match it, and the text only mentions whole number ratios.[48][49]

Salvador Dalí, influenced by the works of Matila Ghyka,[50] explicitly used the golden ratio in his masterpiece, The Sacrament of the Last Supper. The dimensions of the canvas are a golden rectangle. A huge dodecahedron, in perspective so that edges appear in golden ratio to one another, is suspended above and behind Jesus and dominates the composition.[47][51]

A statistical study on 565 works of art of different great painters, performed in 1999, found that these artists had not used the golden ratio in the size of their canvases. The study concluded that the average ratio of the two sides of the paintings studied is 1.34, with averages for individual artists ranging from 1.04 (Goya) to 1.46 (Bellini).[52] On the other hand, Pablo Tosto listed over 350 works by well-known artists, including more than 100 which have canvasses with golden rectangle and root-5 proportions, and others with proportions like root-2, 3, 4, and 6.[53]

Depiction of the proportions in a medieval manuscript. According to Jan Tschichold: “Page proportion 2:3. Margin proportions 1:1:2:3. Text area proportioned in the Golden Section.”[54]

Books and design

Main article: Canons of page construction

According to Jan Tschichold,

There was a time when deviations from the truly beautiful page proportions 2:3, 1:√3, and the Golden Section were rare. Many books produced between 1550 and 1770 show these proportions exactly, to within half a millimeter.[55]

According to some sources, the golden ratio is used in everyday design, for example in the proportions of playing cards, postcards, posters, light switch plates, and widescreen televisions.[56][57][58][59]

Music

Ernő Lendvai analyzes Béla Bartók‘s works as being based on two opposing systems, that of the golden ratio and the acoustic scale,[60] though other music scholars reject that analysis.[61] French composer Erik Satie used the golden ratio in several of his pieces, including Sonneries de la Rose+Croix. The golden ratio is also apparent in the organization of the sections in the music of Debussy‘s Reflets dans l’eau (Reflections in Water), from Images (1st series, 1905), in which “the sequence of keys is marked out by the intervals 34, 21, 13 and 8, and the main climax sits at the phi position”.[62]

The musicologist Roy Howat has observed that the formal boundaries of Debussy’s La Mer correspond exactly to the golden section.[63] Trezise finds the intrinsic evidence “remarkable”, but cautions that no written or reported evidence suggests that Debussy consciously sought such proportions.[64]

Pearl Drums positions the air vents on its Masters Premium models based on the golden ratio. The company claims that this arrangement improves bass response and has applied for a patent on this innovation.[65]

About this sound

Though Heinz Bohlen proposed the non-octave-repeating 833 cents scale based on combination tones, the tuning features relations based on the golden ratio. As a musical interval the ratio 1.618… is 833.090… cents (Play (help·info)).[66]

Nature

Detail of Aeonium tabuliforme showing the multiple spiral arrangement (parastichy)Main article: Patterns in natureSee also: Fibonacci number § Nature

Johannes Kepler wrote that “the image of man and woman stems from the divine proportion. In my opinion, the propagation of plants and the progenitive acts of animals are in the same ratio”.[67]

The psychologist Adolf Zeising noted that the golden ratio appeared in phyllotaxis and argued from these patterns in nature that the golden ratio was a universal law.[68][69] Zeising wrote in 1854 of a universal orthogenetic law of “striving for beauty and completeness in the realms of both nature and art”.[70]

In 2010, the journal Science reported that the golden ratio is present at the atomic scale in the magnetic resonance of spins in cobalt niobate crystals.[71]

However, some have argued that many apparent manifestations of the golden ratio in nature, especially in regard to animal dimensions, are fictitious.[72]

Optimization

The golden ratio is key to the golden-section search.

Mathematics

Irrationality

The golden ratio is an irrational number. Below are two short proofs of irrationality:

Contradiction from an expression in lowest terms

If φ were rational, then it would be the ratio of sides of a rectangle with integer sides (the rectangle comprising the entire diagram). But it would also be a ratio of integer sides of the smaller rectangle (the rightmost portion of the diagram) obtained by deleting a square. The sequence of decreasing integer side lengths formed by deleting squares cannot be continued indefinitely because the integers have a lower bound, so φ cannot be rational.

Recall that: the whole is the longer part plus the shorter part;the whole is to the longer part as the longer part is to the shorter part.

If we call the whole n and the longer part m, then the second statement above becomes n is to m as m is to n − m,

or, algebraically n m = m n − m . ( ∗ ) {\displaystyle {\frac {n}{m}}={\frac {m}{n-m}}.\qquad (*)}

{\frac {n}{m}}={\frac {m}{n-m}}.\qquad (*)

To say that the golden ratio φ is rational means that φ is a fraction n/m where n and m are integers. We may take n/m to be in lowest terms and n and m to be positive. But if n/m is in lowest terms, then the identity labeled (*) above says m/(n − m) is in still lower terms. That is a contradiction that follows from the assumption that φ is rational.

By irrationality of √5

\textstyle {\frac {1+{\sqrt {5}}}{2}}
\textstyle 2\left({\frac {1+{\sqrt {5}}}{2}}\right)-1={\sqrt {5}}

Another short proof—perhaps more commonly known—of the irrationality of the golden ratio makes use of the closure of rational numbers under addition and multiplication. If 1 + 5 2 {\displaystyle \textstyle {\frac {1+{\sqrt {5}}}{2}}} is rational, then 2 ( 1 + 5 2 ) − 1 = 5 {\displaystyle \textstyle 2\left({\frac {1+{\sqrt {5}}}{2}}\right)-1={\sqrt {5}}} is also rational, which is a contradiction if it is already known that the square root of a non-square natural number is irrational.

Minimal polynomial

The golden ratio is also an algebraic number and even an algebraic integer. It has minimal polynomial x 2 − x − 1. {\displaystyle x^{2}-x-1.}

{\displaystyle x^{2}-x-1.}

Having degree 2, this polynomial actually has two roots, the other being the golden ratio conjugate.

Golden ratio conjugate

The conjugate root to the minimal polynomial x2 − x − 1 is − 1 φ = 1 − φ = 1 − 5 2 = − 0.61803 39887 … . {\displaystyle -{\frac {1}{\varphi }}=1-\varphi ={\frac {1-{\sqrt {5}}}{2}}=-0.61803\,39887\dots .}

-{\frac {1}{\varphi }}=1-\varphi ={\frac {1-{\sqrt {5}}}{2}}=-0.61803\,39887\dots .
\Phi

The absolute value of this quantity (≈ 0.618) corresponds to the length ratio taken in reverse order (shorter segment length over longer segment length, b/a), and is sometimes referred to as the golden ratio conjugate.[10] It is denoted here by the capital Phi (Φ {\displaystyle \Phi } ): Φ = 1 φ = φ − 1 = 0.61803 39887 … . {\displaystyle \Phi ={1 \over \varphi }=\varphi ^{-1}=0.61803\,39887\ldots .}

\Phi ={1 \over \varphi }=\varphi ^{-1}=0.61803\,39887\ldots .
\Phi

Alternatively, Φ {\displaystyle \Phi } can be expressed as Φ = φ − 1 = 1.61803 39887 … − 1 = 0.61803 39887 … . {\displaystyle \Phi =\varphi -1=1.61803\,39887\ldots -1=0.61803\,39887\ldots .}

\Phi =\varphi -1=1.61803\,39887\ldots -1=0.61803\,39887\ldots .

This illustrates the unique property of the golden ratio among positive numbers, that 1 φ = φ − 1 , {\displaystyle {1 \over \varphi }=\varphi -1,}

{1 \over \varphi }=\varphi -1,

or its inverse: 1 Φ = Φ + 1. {\displaystyle {1 \over \Phi }=\Phi +1.}

{1 \over \Phi }=\Phi +1.

This means 0.61803…:1 = 1:1.61803….

Alternative forms

Approximations to the reciprocal golden ratio by finite continued fractions, or ratios of Fibonacci numbers

The formula φ = 1 + 1/φ can be expanded recursively to obtain a continued fraction for the golden ratio:[73] φ = [ 1 ; 1 , 1 , 1 , … ] = 1 + 1 1 + 1 1 + 1 1 + ⋱ {\displaystyle \varphi =[1;1,1,1,\dots ]=1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+\ddots }}}}}}}

\varphi =[1;1,1,1,\dots ]=1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+\ddots }}}}}}

and its reciprocal: φ − 1 = [ 0 ; 1 , 1 , 1 , … ] = 0 + 1 1 + 1 1 + 1 1 + ⋱ {\displaystyle \varphi ^{-1}=[0;1,1,1,\dots ]=0+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+\ddots }}}}}}}

\varphi ^{-1}=[0;1,1,1,\dots ]=0+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+\ddots }}}}}}

The convergents of these continued fractions (1/1, 2/1, 3/2, 5/3, 8/5, 13/8, …, or 1/1, 1/2, 2/3, 3/5, 5/8, 8/13, …) are ratios of successive Fibonacci numbers.

The equation φ2 = 1 + φ likewise produces the continued square root, or infinite surd, form: φ = 1 + 1 + 1 + 1 + ⋯ . {\displaystyle \varphi ={\sqrt {1+{\sqrt {1+{\sqrt {1+{\sqrt {1+\cdots }}}}}}}}.}

\varphi ={\sqrt {1+{\sqrt {1+{\sqrt {1+{\sqrt {1+\cdots }}}}}}}}.

An infinite series can be derived to express φ:[74] φ = 13 8 + ∑ n = 0 ∞ ( − 1 ) n + 1 ( 2 n + 1 ) ! 4 2 n + 3 n ! ( n + 2 ) ! . {\displaystyle \varphi ={\frac {13}{8}}+\sum _{n=0}^{\infty }{\frac {(-1)^{n+1}(2n+1)!}{4^{2n+3}n!(n+2)!}}.}

{\displaystyle \varphi ={\frac {13}{8}}+\sum _{n=0}^{\infty }{\frac {(-1)^{n+1}(2n+1)!}{4^{2n+3}n!(n+2)!}}.}

Also: φ = 1 + 2 sin ⁡ ( π / 10 ) = 1 + 2 sin ⁡ 18 ∘ {\displaystyle \varphi =1+2\sin(\pi /10)=1+2\sin 18^{\circ }}

\varphi =1+2\sin(\pi /10)=1+2\sin 18^{\circ }

φ = 1 2 csc ⁡ ( π / 10 ) = 1 2 csc ⁡ 18 ∘ {\displaystyle \varphi ={1 \over 2}\csc(\pi /10)={1 \over 2}\csc 18^{\circ }}

\varphi ={1 \over 2}\csc(\pi /10)={1 \over 2}\csc 18^{\circ }

φ = 2 cos ⁡ ( π / 5 ) = 2 cos ⁡ 36 ∘ {\displaystyle \varphi =2\cos(\pi /5)=2\cos 36^{\circ }}

\varphi =2\cos(\pi /5)=2\cos 36^{\circ }

φ = 2 sin ⁡ ( 3 π / 10 ) = 2 sin ⁡ 54 ∘ . {\displaystyle \varphi =2\sin(3\pi /10)=2\sin 54^{\circ }.}

\varphi =2\sin(3\pi /10)=2\sin 54^{\circ }.

These correspond to the fact that the length of the diagonal of a regular pentagon is φ times the length of its side, and similar relations in a pentagram.

Geometry

Approximate and true golden spirals. The green spiral is made from quarter-circles tangent to the interior of each square, while the red spiral is a Golden Spiral, a special type of logarithmic spiral. Overlapping portions appear yellow. The length of the side of one square divided by that of the next smaller square is the golden ratio.

The number φ turns up frequently in geometry, particularly in figures with pentagonal symmetry. The length of a regular pentagon‘s diagonal is φ times its side. The vertices of a regular icosahedron are those of three mutually orthogonal golden rectangles.

There is no known general algorithm to arrange a given number of nodes evenly on a sphere, for any of several definitions of even distribution (see, for example, Thomson problem). However, a useful approximation results from dividing the sphere into parallel bands of equal surface area and placing one node in each band at longitudes spaced by a golden section of the circle, i.e. 360°/φ ≅ 222.5°. This method was used to arrange the 1500 mirrors of the student-participatory satellite Starshine-3.[75]

Dividing a line segment by interior division

Dividing a line segment by interior division according to the golden ratio

  1. Having a line segment AB, construct a perpendicular BC at point B, with BC half the length of AB. Draw the hypotenuse AC.
  2. Draw an arc with center C and radius BC. This arc intersects the hypotenuse AC at point D.
  3. Draw an arc with center A and radius AD. This arc intersects the original line segment AB at point S. Point S divides the original line segment AB into line segments AS and SB with lengths in the golden ratio.

Dividing a line segment by exterior division

Dividing a line segment by exterior division according to the golden ratio

  1. Draw a line segment AS and construct off the point S a segment SC perpendicular to AS and with the same length as AS.
  2. Do bisect the line segment AS with M.
  3. A circular arc around M with radius MC intersects in point B the straight line through points A and S (also known as the extension of AS). The ratio of AS to the constructed segment SB is the golden ratio.

Application examples you can see in the articles Pentagon with a given side length, Decagon with given circumcircle and Decagon with a given side length.

Both of the above displayed different algorithms produce geometric constructions that determine two aligned line segments where the ratio of the longer one to the shorter one is the golden ratio.

Golden triangle, pentagon and pentagram

Golden triangle. The double-red-arched angle is 36 degrees, or π 5 {\displaystyle {\frac {\pi }{5}}}

{\displaystyle {\frac {\pi }{5}}}

radians.

Golden triangle

The golden triangle can be characterized as an isosceles triangle ABC with the property that bisecting the angle C produces a new triangle CXB which is a similar triangle to the original.

If angle BCX = α, then XCA = α because of the bisection, and CAB = α because of the similar triangles; ABC = 2α from the original isosceles symmetry, and BXC = 2α by similarity. The angles in a triangle add up to 180°, so 5α = 180, giving α = 36°. So the angles of the golden triangle are thus 36°-72°-72°. The angles of the remaining obtuse isosceles triangle AXC (sometimes called the golden gnomon) are 36°-36°-108°.

Suppose XB has length 1, and we call BC length φ. Because of the isosceles triangles XC=XA and BC=XC, so these are also length φ. Length AC = AB, therefore equals φ + 1. But triangle ABC is similar to triangle CXB, so AC/BC = BC/BX, AC/φ = φ/1, and so AC also equals φ2. Thus φ2 = φ + 1, confirming that φ is indeed the golden ratio.

Similarly, the ratio of the area of the larger triangle AXC to the smaller CXB is equal to φ, while the inverse ratio is φ − 1.

Pentagon

In a regular pentagon the ratio of a diagonal to a side is the golden ratio, while intersecting diagonals section each other in the golden ratio.[8]

Odom’s construction

Let A and B be midpoints of the sides EF and ED of an equilateral triangle DEF. Extend AB to meet the circumcircle of DEF at C.
| A B | | B C | = | A C | | A B | = ϕ {\displaystyle {\tfrac {|AB|}{|BC|}}={\tfrac {|AC|}{|AB|}}=\phi }

{\tfrac {|AB|}{|BC|}}={\tfrac {|AC|}{|AB|}}=\phi

George Odom has given a remarkably simple construction for φ involving an equilateral triangle: if an equilateral triangle is inscribed in a circle and the line segment joining the midpoints of two sides is produced to intersect the circle in either of two points, then these three points are in golden proportion. This result is a straightforward consequence of the intersecting chords theorem and can be used to construct a regular pentagon, a construction that attracted the attention of the noted Canadian geometer H. S. M. Coxeter who published it in Odom’s name as a diagram in the American Mathematical Monthly accompanied by the single word “Behold!” [76]

Pentagram

A pentagram colored to distinguish its line segments of different lengths. The four lengths are in golden ratio to one another.

The golden ratio plays an important role in the geometry of pentagrams. Each intersection of edges sections other edges in the golden ratio. Also, the ratio of the length of the shorter segment to the segment bounded by the two intersecting edges (a side of the pentagon in the pentagram’s center) is φ, as the four-color illustration shows.

The pentagram includes ten isosceles triangles: five acute and five obtuse isosceles triangles. In all of them, the ratio of the longer side to the shorter side is φ. The acute triangles are golden triangles. The obtuse isosceles triangles are golden gnomons.

Ptolemy’s theorem

The golden ratio in a regular pentagon can be computed using Ptolemy’s theorem.

The golden ratio properties of a regular pentagon can be confirmed by applying Ptolemy’s theorem to the quadrilateral formed by removing one of its vertices. If the quadrilateral’s long edge and diagonals are b, and short edges are a, then Ptolemy’s theorem gives b2 = a2 + ab which yields b a = 1 + 5 2 . {\displaystyle {b \over a}={{1+{\sqrt {5}}} \over 2}.}

{b \over a}={{1+{\sqrt {5}}} \over 2}.

Scalenity of triangles

Consider a triangle with sides of lengths a, b, and c in decreasing order. Define the “scalenity” of the triangle to be the smaller of the two ratios a/b and b/c. The scalenity is always less than φ and can be made as close as desired to φ.[77]

Triangle whose sides form a geometric progression

If the side lengths of a triangle form a geometric progression and are in the ratio 1 : r : r2, where r is the common ratio, then r must lie in the range φ−1 < r < φ, which is a consequence of the triangle inequality (the sum of any two sides of a triangle must be strictly bigger than the length of the third side). If r = φ then the shorter two sides are 1 and φ but their sum is φ2, thus r < φ. A similar calculation shows that r > φ−1. A triangle whose sides are in the ratio 1 : √φ : φ is a right triangle (because 1 + φ = φ2) known as a Kepler triangle.[78]

Golden triangle, rhombus, and rhombic triacontahedron

One of the rhombic triacontahedron’s rhombi

All of the faces of the rhombic triacontahedron are golden rhombi

A golden rhombus is a rhombus whose diagonals are in the golden ratio. The rhombic triacontahedron is a convex polytope that has a very special property: all of its faces are golden rhombi. In the rhombic triacontahedron the dihedral angle between any two adjacent rhombi is 144°, which is twice the isosceles angle of a golden triangle and four times its most acute angle.[79]

Relationship to Fibonacci sequence

The mathematics of the golden ratio and of the Fibonacci sequence are intimately interconnected. The Fibonacci sequence is: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, …

A closed-form expression for the Fibonacci sequence involves the golden ratio: F ( n ) = φ n − ( 1 − φ ) n 5 = φ n − ( − φ ) − n 5 . {\displaystyle F\left(n\right)={{\varphi ^{n}-(1-\varphi )^{n}} \over {\sqrt {5}}}={{\varphi ^{n}-(-\varphi )^{-n}} \over {\sqrt {5}}}.}

{\displaystyle F\left(n\right)={{\varphi ^{n}-(1-\varphi )^{n}} \over {\sqrt {5}}}={{\varphi ^{n}-(-\varphi )^{-n}} \over {\sqrt {5}}}.}

A Fibonacci spiral which approximates the golden spiral, using Fibonacci sequence square sizes up to 34. The spiral is drawn starting from the inner 1×1 square and continues outwards to successively larger squares.

Golden squares with T-branching

Golden square fractal

The golden ratio is the limit of the ratios of successive terms of the Fibonacci sequence (or any Fibonacci-like sequence), as shown by Kepler:[80] lim n → ∞ F n + 1 F n = φ . {\displaystyle \lim _{n\to \infty }{\frac {F_{n+1}}{F_{n}}}=\varphi .}

{\displaystyle \lim _{n\to \infty }{\frac {F_{n+1}}{F_{n}}}=\varphi .}

In other words, if a Fibonacci number is divided by its immediate predecessor in the sequence, the quotient approximates φ; e.g., 987/610 ≈ 1.6180327868852. These approximations are alternately lower and higher than φ, and converge to φ as the Fibonacci numbers increase, and: ∑ n = 1 ∞ | F n φ − F n + 1 | = φ . {\displaystyle \sum _{n=1}^{\infty }|F_{n}\varphi -F_{n+1}|=\varphi .}

{\displaystyle \sum _{n=1}^{\infty }|F_{n}\varphi -F_{n+1}|=\varphi .}

More generally: lim n → ∞ F n + a F n = φ a , {\displaystyle \lim _{n\to \infty }{\frac {F_{n+a}}{F_{n}}}=\varphi ^{a},}

{\displaystyle \lim _{n\to \infty }{\frac {F_{n+a}}{F_{n}}}=\varphi ^{a},}
{\displaystyle a=1.}

where above, the ratios of consecutive terms of the Fibonacci sequence, is a case when a = 1. {\displaystyle a=1.}

Furthermore, the successive powers of φ obey the Fibonacci recurrence: φ n + 1 = φ n + φ n − 1 . {\displaystyle \varphi ^{n+1}=\varphi ^{n}+\varphi ^{n-1}.}

{\displaystyle \varphi ^{n+1}=\varphi ^{n}+\varphi ^{n-1}.}

This identity allows any polynomial in φ to be reduced to a linear expression. For example: 3 φ 3 − 5 φ 2 + 4 = 3 ( φ 2 + φ ) − 5 φ 2 + 4 = 3 [ ( φ + 1 ) + φ ] − 5 ( φ + 1 ) + 4 = φ + 2 ≈ 3.618. {\displaystyle {\begin{aligned}3\varphi ^{3}-5\varphi ^{2}+4&=3(\varphi ^{2}+\varphi )-5\varphi ^{2}+4\\&=3[(\varphi +1)+\varphi ]-5(\varphi +1)+4\\&=\varphi +2\approx 3.618.\end{aligned}}}

{\begin{aligned}3\varphi ^{3}-5\varphi ^{2}+4&=3(\varphi ^{2}+\varphi )-5\varphi ^{2}+4\\&=3[(\varphi +1)+\varphi ]-5(\varphi +1)+4\\&=\varphi +2\approx 3.618.\end{aligned}}

The reduction to a linear expression can be accomplished in one step by using the relationship φ k = F k φ + F k − 1 , {\displaystyle \varphi ^{k}=F_{k}\varphi +F_{k-1},}

\varphi ^{k}=F_{k}\varphi +F_{k-1},
F_{k}

where F k {\displaystyle F_{k}} is the kth Fibonacci number.

However, this is no special property of φ, because polynomials in any solution x to a quadratic equation can be reduced in an analogous manner, by applying: x 2 = a x + b {\displaystyle x^{2}=ax+b}

x^{2}=ax+b
\mathbb {Q} (\alpha )
\mathbb {Q}
{\displaystyle \{1,\alpha ,\dots ,\alpha ^{n-1}\}.}

for given coefficients a, b such that x satisfies the equation. Even more generally, any rational function (with rational coefficients) of the root of an irreducible nth-degree polynomial over the rationals can be reduced to a polynomial of degree n ‒ 1. Phrased in terms of field theory, if α is a root of an irreducible nth-degree polynomial, then Q ( α ) {\displaystyle \mathbb {Q} (\alpha )} has degree n over Q {\displaystyle \mathbb {Q} } , with basis { 1 , α , … , α n − 1 } . {\displaystyle \{1,\alpha ,\dots ,\alpha ^{n-1}\}.}

Symmetries

\varphi _{\pm }=(1\pm {\sqrt {5}})/2
x,1/(1-x),(x-1)/x,
1/x,1-x,x/(x-1)
1/2

The golden ratio and inverse golden ratio φ ± = ( 1 ± 5 ) / 2 {\displaystyle \varphi _{\pm }=(1\pm {\sqrt {5}})/2} have a set of symmetries that preserve and interrelate them. They are both preserved by the fractional linear transformations x , 1 / ( 1 − x ) , ( x − 1 ) / x , {\displaystyle x,1/(1-x),(x-1)/x,} – this fact corresponds to the identity and the definition quadratic equation. Further, they are interchanged by the three maps 1 / x , 1 − x , x / ( x − 1 ) {\displaystyle 1/x,1-x,x/(x-1)} – they are reciprocals, symmetric about 1 / 2 {\displaystyle 1/2} , and (projectively) symmetric about 2.

\operatorname {PSL} (2,\mathbf {Z} )
S_{3},
\{0,1,\infty \}
S_{3}\to S_{2}
C_{3}<S_{3}
()(01\infty )(0\infty 1)

More deeply, these maps form a subgroup of the modular group PSL ⁡ ( 2 , Z ) {\displaystyle \operatorname {PSL} (2,\mathbf {Z} )} isomorphic to the symmetric group on 3 letters, S 3 , {\displaystyle S_{3},} corresponding to the stabilizer of the set { 0 , 1 , ∞ } {\displaystyle \{0,1,\infty \}} of 3 standard points on the projective line, and the symmetries correspond to the quotient map S 3 → S 2 {\displaystyle S_{3}\to S_{2}} – the subgroup C 3 < S 3 {\displaystyle C_{3}<S_{3}} consisting of the 3-cycles and the identity ( ) ( 01 ∞ ) ( 0 ∞ 1 ) {\displaystyle ()(01\infty )(0\infty 1)} fixes the two numbers, while the 2-cycles interchange these, thus realizing the map.

Other properties

The golden ratio has the simplest expression (and slowest convergence) as a continued fraction expansion of any irrational number (see Alternate forms above). It is, for that reason, one of the worst cases of Lagrange’s approximation theorem and it is an extremal case of the Hurwitz inequality for Diophantine approximations. This may be why angles close to the golden ratio often show up in phyllotaxis (the growth of plants).[81]

The defining quadratic polynomial and the conjugate relationship lead to decimal values that have their fractional part in common with φ: φ 2 = φ + 1 = 2.618 … {\displaystyle \varphi ^{2}=\varphi +1=2.618\dots }

\varphi ^{2}=\varphi +1=2.618\dots

1 φ = φ − 1 = 0.618 … . {\displaystyle {1 \over \varphi }=\varphi -1=0.618\dots .}

{1 \over \varphi }=\varphi -1=0.618\dots .

The sequence of powers of φ contains these values 0.618…, 1.0, 1.618…, 2.618…; more generally, any power of φ is equal to the sum of the two immediately preceding powers: φ n = φ n − 1 + φ n − 2 = φ ⋅ F n + F n − 1 . {\displaystyle \varphi ^{n}=\varphi ^{n-1}+\varphi ^{n-2}=\varphi \cdot \operatorname {F} _{n}+\operatorname {F} _{n-1}.}

\varphi ^{n}=\varphi ^{n-1}+\varphi ^{n-2}=\varphi \cdot \operatorname {F} _{n}+\operatorname {F} _{n-1}.

As a result, one can easily decompose any power of φ into a multiple of φ and a constant. The multiple and the constant are always adjacent Fibonacci numbers. This leads to another property of the positive powers of φ:

\lfloor n/2-1\rfloor =m

If ⌊ n / 2 − 1 ⌋ = m {\displaystyle \lfloor n/2-1\rfloor =m} , then:   φ n = φ n − 1 + φ n − 3 + ⋯ + φ n − 1 − 2 m + φ n − 2 − 2 m {\displaystyle \!\ \varphi ^{n}=\varphi ^{n-1}+\varphi ^{n-3}+\cdots +\varphi ^{n-1-2m}+\varphi ^{n-2-2m}}

\!\ \varphi ^{n}=\varphi ^{n-1}+\varphi ^{n-3}+\cdots +\varphi ^{n-1-2m}+\varphi ^{n-2-2m}

  φ n − φ n − 1 = φ n − 2 . {\displaystyle \!\ \varphi ^{n}-\varphi ^{n-1}=\varphi ^{n-2}.}

\!\ \varphi ^{n}-\varphi ^{n-1}=\varphi ^{n-2}.

When the golden ratio is used as the base of a numeral system (see Golden ratio base, sometimes dubbed phinary or φ-nary), every integer has a terminating representation, despite φ being irrational, but every fraction has a non-terminating representation.

\mathbb {Q} ({\sqrt {5}})
\mathbb {Q} ({\sqrt {5}})
\varphi ^{n}={{L_{n}+F_{n}{\sqrt {5}}} \over 2}
L_{n}
n

The golden ratio is a fundamental unit of the algebraic number field Q ( 5 ) {\displaystyle \mathbb {Q} ({\sqrt {5}})} and is a Pisot–Vijayaraghavan number.[82] In the field Q ( 5 ) {\displaystyle \mathbb {Q} ({\sqrt {5}})} we have φ n = L n + F n 5 2 {\displaystyle \varphi ^{n}={{L_{n}+F_{n}{\sqrt {5}}} \over 2}} , where L n {\displaystyle L_{n}} is the n {\displaystyle n} -th Lucas number.

4\log(\varphi )

The golden ratio also appears in hyperbolic geometry, as the maximum distance from a point on one side of an ideal triangle to the closer of the other two sides: this distance, the side length of the equilateral triangle formed by the points of tangency of a circle inscribed within the ideal triangle, is 4 log ⁡ ( φ ) {\displaystyle 4\log(\varphi )} .[83]

Decimal expansion

The golden ratio’s decimal expansion can be calculated directly from the expression φ = 1 + 5 2 {\displaystyle \varphi ={1+{\sqrt {5}} \over 2}}

\varphi ={1+{\sqrt {5}} \over 2}

with √5 ≈ 2.2360679774997896964 OEISA002163. The square root of 5 can be calculated with the Babylonian method, starting with an initial estimate such as xφ = 2 and iterating x n + 1 = ( x n + 5 / x n ) 2 {\displaystyle x_{n+1}={\frac {(x_{n}+5/x_{n})}{2}}}

x_{n+1}={\frac {(x_{n}+5/x_{n})}{2}}

for n = 1, 2, 3, …, until the difference between xn and xn−1 becomes zero, to the desired number of digits.

The Babylonian algorithm for √5 is equivalent to Newton’s method for solving the equation x2 − 5 = 0. In its more general form, Newton’s method can be applied directly to any algebraic equation, including the equation x2 − x − 1 = 0 that defines the golden ratio. This gives an iteration that converges to the golden ratio itself, x n + 1 = x n 2 + 1 2 x n − 1 , {\displaystyle x_{n+1}={\frac {x_{n}^{2}+1}{2x_{n}-1}},}

x_{n+1}={\frac {x_{n}^{2}+1}{2x_{n}-1}},

for an appropriate initial estimate xφ such as xφ = 1. A slightly faster method is to rewrite the equation as x − 1 − 1/x = 0, in which case the Newton iteration becomes x n + 1 = x n 2 + 2 x n x n 2 + 1 . {\displaystyle x_{n+1}={\frac {x_{n}^{2}+2x_{n}}{x_{n}^{2}+1}}.}

x_{n+1}={\frac {x_{n}^{2}+2x_{n}}{x_{n}^{2}+1}}.

These iterations all converge quadratically; that is, each step roughly doubles the number of correct digits. The golden ratio is therefore relatively easy to compute with arbitrary precision. The time needed to compute n digits of the golden ratio is proportional to the time needed to divide two n-digit numbers. This is considerably faster than known algorithms for the transcendental numbers π and e.

An easily programmed alternative using only integer arithmetic is to calculate two large consecutive Fibonacci numbers and divide them. The ratio of Fibonacci numbers F 25001 and F 25000, each over 5000 digits, yields over 10,000 significant digits of the golden ratio.

The decimal expansion of the golden ratio φ[1] has been calculated to an accuracy of two trillion (2×1012 = 2,000,000,000,000) digits.[84]

Pyramids

Further information: mathematics and art

A regular square pyramid is determined by its medial right triangle, whose edges are the pyramid’s apothem (a), semi-base (b), and height (h); the face inclination angle is also marked. Mathematical proportions b:h:a of 1 : φ : φ {\displaystyle 1:{\sqrt {\varphi }}:\varphi }

1:{\sqrt {\varphi }}:\varphi

and 3 : 4 : 5 {\displaystyle 3:4:5}

{\displaystyle 3:4:5}

and 1 : 4 / π : 1.61899 {\displaystyle 1:4/\pi :1.61899}

{\displaystyle 1:4/\pi :1.61899}

are of particular interest in relation to Egyptian pyramids.

Both Egyptian pyramids and the regular square pyramids that resemble them can be analyzed with respect to the golden ratio and other ratios.

Mathematical pyramids

{\sqrt {\varphi }}
{\sqrt {\varphi }}

A pyramid in which the apothem (slant height along the bisector of a face) is equal to φ times the semi-base (half the base width) is sometimes called a golden pyramid. The isosceles triangle that is the face of such a pyramid can be constructed from the two halves of a diagonally split golden rectangle (of size semi-base by apothem), joining the medium-length edges to make the apothem. The height of this pyramid is φ {\displaystyle {\sqrt {\varphi }}} times the semi-base (that is, the slope of the face is φ {\displaystyle {\sqrt {\varphi }}} ); the square of the height is equal to the area of a face, φ times the square of the semi-base.

1:{\sqrt {\varphi }}:\varphi
{\sqrt {\varphi }}={\sqrt {\varphi ^{2}-1}}
\varphi ={\sqrt {1+\varphi }}
{\sqrt {\varphi }}

The medial right triangle of this “golden” pyramid (see diagram), with sides 1 : φ : φ {\displaystyle 1:{\sqrt {\varphi }}:\varphi } is interesting in its own right, demonstrating via the Pythagorean theorem the relationship φ = φ 2 − 1 {\displaystyle {\sqrt {\varphi }}={\sqrt {\varphi ^{2}-1}}} or φ = 1 + φ {\displaystyle \varphi ={\sqrt {1+\varphi }}} . This Kepler triangle[85] is the only right triangle proportion with edge lengths in geometric progression,[86][78] just as the 3–4–5 triangle is the only right triangle proportion with edge lengths in arithmetic progression. The angle with tangent φ {\displaystyle {\sqrt {\varphi }}} corresponds to the angle that the side of the pyramid makes with respect to the ground, 51.827… degrees (51° 49′ 38″).[87]

A nearly similar pyramid shape, but with rational proportions, is described in the Rhind Mathematical Papyrus (the source of a large part of modern knowledge of ancient Egyptian mathematics), based on the 3:4:5 triangle;[88] the face slope corresponding to the angle with tangent 4/3 is, to two decimal places, 53.13 degrees (53 degrees and 8 minutes). The slant height or apothem is 5/3 or 1.666… times the semi-base. The Rhind papyrus has another pyramid problem as well, again with rational slope (expressed as run over rise). Egyptian mathematics did not include the notion of irrational numbers,[89] and the rational inverse slope (run/rise, multiplied by a factor of 7 to convert to their conventional units of palms per cubit) was used in the building of pyramids.[88]

{\sqrt {\varphi }}\approx 4/\pi

Another mathematical pyramid with proportions almost identical to the “golden” one is the one with perimeter equal to 2π times the height, or h:b = 4:π. This triangle has a face angle of 51.854° (51°51′), very close to the 51.827° of the Kepler triangle. This pyramid relationship corresponds to the coincidental relationship φ ≈ 4 / π {\displaystyle {\sqrt {\varphi }}\approx 4/\pi } .

Egyptian pyramids very close in proportion to these mathematical pyramids are known.[90][78]

Egyptian pyramids

The Great Pyramid of Giza

One Egyptian pyramid that is close to a “golden pyramid” is the Great Pyramid of Giza (also known as the Pyramid of Cheops or Khufu). Its slope of 51° 52′ is close to the “golden” pyramid inclination of 51° 50′ – and even closer to the π-based pyramid inclination of 51° 51′. However, several other mathematical theories of the shape of the great pyramid, based on rational slopes, have been found to be both more accurate and more plausible explanations for the 51° 52′ slope.[78]

In the mid-nineteenth century, Friedrich Röber studied various Egyptian pyramids including those of Khafre, Menkaure, and some of the Giza, Saqqara, and Abusir groups. He did not apply the golden ratio to the Great Pyramid of Giza, but instead agreed with John Shae Perring that its side-to-height ratio is 8:5. For all the other pyramids he applied measurements related to the Kepler triangle, and claimed that either their whole or half-side lengths are related to their heights by the golden ratio.[91]

In 1859, the pyramidologist John Taylor misinterpreted Herodotus (c. 440 BC) as indicating that the Great Pyramid’s height squared equals the area of one of its face triangles.[e] This led Taylor to claim that, in the Great Pyramid, the golden ratio is represented by the ratio of the length of the face (the slope height, inclined at an angle θ to the ground) to half the length of the side of the square base (equivalent to the secant of the angle θ).[93] The above two lengths are about 186.4 metres (612 ft) and 115.2 metres (378 ft), respectively.[92] The ratio of these lengths is the golden ratio, accurate to more digits than either of the original measurements. Similarly, Howard Vyse reported the great pyramid height 148.2 metres (486 ft), and half-base 116.4 metres (382 ft), yielding 1.6189 for the ratio of slant height to half-base, again more accurate than the data variability.[86]

Eric Temple Bell, mathematician and historian, claimed in 1950 that Egyptian mathematics would not have supported the ability to calculate the slant height of the pyramids, or the ratio to the height, except in the case of the 3:4:5 pyramid, since the 3:4:5 triangle was the only right triangle known to the Egyptians and they did not know the Pythagorean theorem, nor any way to reason about irrationals such as π or φ.[94] Example geometric problems of pyramid design in the Rhind papyrus correspond to various rational slopes.[78]

Michael Rice[95] asserts that principal authorities on the history of Egyptian architecture have argued that the Egyptians were well acquainted with the golden ratio and that it is part of the mathematics of the pyramids, citing Giedon (1957).[96] Historians of science have long debated whether the Egyptians had any such knowledge, contending that its appearance in the Great Pyramid is the result of chance.[97]

Disputed observations

Examples of disputed observations of the golden ratio include the following:

Nautilus shells are often erroneously claimed to be golden-proportioned.

  • Some specific proportions in the bodies of many animals (including humans)[98][99] and parts of the shells of mollusks[3] are often claimed to be in the golden ratio. There is a large variation in the real measures of these elements in specific individuals, however, and the proportion in question is often significantly different from the golden ratio.[98] The ratio of successive phalangeal bones of the digits and the metacarpal bone has been said to approximate the golden ratio.[99] The nautilus shell, the construction of which proceeds in a logarithmic spiral, is often cited, usually with the idea that any logarithmic spiral is related to the golden ratio, but sometimes with the claim that each new chamber is golden-proportioned relative to the previous one.[100] However, measurements of nautilus shells do not support this claim.[101]
  • Historian John Man states that both the pages and text area of the Gutenberg Bible were “based on the golden section shape”. However, according to his own measurements, the ratio of height to width of the pages is 1.45.[102]
  • Studies by psychologists, starting with Gustav Fechner c. 1876,[103] have been devised to test the idea that the golden ratio plays a role in human perception of beauty. While Fechner found a preference for rectangle ratios centered on the golden ratio, later attempts to carefully test such a hypothesis have been, at best, inconclusive.[104][47]
  • In investing, some practitioners of technical analysis use the golden ratio to indicate support of a price level, or resistance to price increases, of a stock or commodity; after significant price changes up or down, new support and resistance levels are supposedly found at or near prices related to the starting price via the golden ratio.[105] The use of the golden ratio in investing is also related to more complicated patterns described by Fibonacci numbers (e.g. Elliott wave principle and Fibonacci retracement). However, other market analysts have published analyses suggesting that these percentages and patterns are not supported by the data.[106]

The Parthenon

Many of the proportions of the Parthenon are alleged to exhibit the golden ratio, but this has largely been discredited.[107]

The Parthenon‘s façade (c. 432 BC) as well as elements of its façade and elsewhere are said by some to be circumscribed by golden rectangles.[108] Other scholars deny that the Greeks had any aesthetic association with golden ratio. For example, Keith Devlin says, “Certainly, the oft repeated assertion that the Parthenon in Athens is based on the golden ratio is not supported by actual measurements. In fact, the entire story about the Greeks and golden ratio seems to be without foundation.”[109] Midhat J. Gazalé affirms that “It was not until Euclid … that the golden ratio’s mathematical properties were studied.”[110]

From measurements of 15 temples, 18 monumental tombs, 8 sarcophagi, and 58 grave stelae from the fifth century BC to the second century AD, one researcher concluded that the golden ratio was totally absent from Greek architecture of the classical fifth century BC, and almost absent during the following six centuries.[111] Later sources like Vitruvius (first century BC) exclusively discuss proportions that can be expressed in whole numbers, i.e. commensurate as opposed to irrational proportions.

Modern art

Albert Gleizes, Les Baigneuses (1912)

The Section d’Or (‘Golden Section’) was a collective of painters, sculptors, poets and critics associated with Cubism and Orphism.[112] Active from 1911 to around 1914, they adopted the name both to highlight that Cubism represented the continuation of a grand tradition, rather than being an isolated movement, and in homage to the mathematical harmony associated with Georges Seurat.[113] The Cubists observed in its harmonies, geometric structuring of motion and form, the primacy of idea over nature, an absolute scientific clarity of conception.[114] However, despite this general interest in mathematical harmony, whether the paintings featured in the celebrated 1912 Salon de la Section d’Or exhibition used the golden ratio in any compositions is more difficult to determine. Livio, for example, claims that they did not,[115] and Marcel Duchamp said as much in an interview.[116] On the other hand, an analysis suggests that Juan Gris made use of the golden ratio in composing works that were likely, but not definitively, shown at the exhibition.[116][117][118] Art historian Daniel Robbins has argued that in addition to referencing the mathematical term, the exhibition’s name also refers to the earlier Bandeaux d’Or group, with which Albert Gleizes and other former members of the Abbaye de Créteil had been involved.[119]

Piet Mondrian has been said to have used the golden section extensively in his geometrical paintings,[120] though other experts (including critic Yve-Alain Bois) have discredited these claims.[47][121]

Sacred Geometry is the Symmetry of Balance

Sacred Geometry is considered the basic geometry of the universe. When I was travelling the world I went to sacred sites. Some of the symbols that I found interesting were the spirals at NewGrange in Ireland, the Ankh was prolific on the Egyptian sarcophagus, the serpent at the Mayan Pyramids and stone monolithic circles at various henges I’ve visited. It seemed to me to be something like a code or my feeling was by looking at the image something impressed upon you, perhaps on some level we understand what it is. In my own poetry I’ve mentioned harmonics and have learned this is the vibration. It leads me to look at harmony, how to harmonise with nature and within ourselves. Harmony is peace it is symmetry, hence balance. I am interested to learn more, as I am curious about the universal blueprints. A blueprint is defined as a reproduction and my interest is the very nature of the universe from which we have all come. I thought to investigate sacred geometry tonight.

Courtesy of wikipedia.

Sacred Geometry

Sacred geometry involves sacred universal patterns used in the design of everything in our reality, most often seen in sacred architecture and sacred art. The basic belief is that geometry and mathematical ratios, harmonics and proportion are also found in music, light, cosmology. This value system is seen as widespread even in prehistory, a cultural universal of the human condition.

It is considered foundational to building sacred structures such as temples, mosques, megaliths, monuments and churches; sacred spaces such as altars, temenoi and tabernacles; meeting places such as sacred groves, village greens and holy wells and the creation of religious art, iconography and using “divine” proportions. Alternatively, sacred geometry based arts may be ephemeral, such as visualization, sand painting and medicine wheels.

Sacred geometry may be understood as a worldview of pattern recognition, a complex system of religious symbols and structures involving space, time and form. According to this view the basic patterns of existence are perceived as sacred. By connecting with these, a believer contemplates the Great Mysteries, and the Great Design. By studying the nature of these patterns, forms and relationships and their connections, insight may be gained into the mysteries – the laws and lore of the Universe.

Music

The discovery of the relationship of geometry and mathematics to music within the Classical Period is attributed to Pythagoras, who found that a string stopped halfway along its length produced an octave, while a ratio of 3/2 produced a fifth interval and 4/3 produced a fourth. Pythagoreans believed that this gave music powers of healing, as it could “harmonize” the out-of-balance body, and this belief has been revived in modern times. Hans Jenny, a physician who pioneered the study of geometric figures formed by wave interactions and named that study cymatics, is often cited in this context. However, Dr. Jenny did not make healing claims for his work.

Even though Hans Jenny did pioneer cymatics in modern times, the study of geometric relationships to wave interaction (sound) obviously has much older roots (Pythagoras). A work that shows ancient peoples understanding of sacred geometry can be found in Scotland. In the Rosslyn Chapel, Thomas J. Mitchell, and his son, my friend Stuart Mitchell, have has found what he calls “frozen music”. Apparently, there are 213 cubes with different symbols that are believed to have musical significance. After 27 years of study and research, Mitchell has found the correct pitches and tonality that matches each symbol on each cube, revealing harmonic and melodic progressions. He has fully discovered the “frozen music”, which he has named the Rosslyn Motet, and is set to have it performed in the chapel on May 18, 2007, and June 1, 2007.

Cosmology

At least as late as Johannes Kepler (1571-1630), a belief in the geometric underpinnings of the cosmos persisted among scientists. Kepler explored the ratios of the planetary orbits, at first in two dimensions (having spotted that the ratio of the orbits of Jupiter and Saturn approximate to the in-circle and out-circle of an equilateral triangle). When this did not give him a neat enough outcome, he tried using the Platonic solids. In fact, planetary orbits can be related using two-dimensional geometric figures, but the figures do not occur in a particularly neat order. Even in his own lifetime (with less accurate data than we now possess) Kepler could see that the fit of the Platonic solids was imperfect. However, other geometric configurations are possible.

Natural Forms

Many forms observed in nature can be related to geometry (for sound reasons of resource optimization). 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. But some scientists see such phenomena as the logical outcome of natural principles.

Art and Architecture

The golden ratio, 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. For examples of sacred geometry in art and architecture refer:

# Labyrinth (an Eulerian path, as distinct from a maze)
# Mandala
# Parthenon
# Taijitu (Yin-Yang)

Labyrinths

A labyrinth is an ancient symbol that relates to wholeness. It combines the imagery of the circle and the spiral into a meandering but purposeful path. It represents a journey to our own center and back again out into the world. Labyrinths have long been used as meditation and prayer tools. A labyrinth is an archetype with which we can have a direct experience. Walking the labyrinth can be considered an initiation in which one awakens the knowledge encoded within their DNA.

A labyrinth contains non-verbal, implicate geometric and numerological prompts that create a multi-dimensional holographic field. These unseen patterns are referred to as sacred geometry. They allegeldy reveal the presence of a cosmic order as they interface the world of material form and the subtler realms of higher consciousness.The contemporary resurgence of labyrinths in the west is stemmimg from our deeply rooted urge to honor again the Sacredness of All Life. A labyrinth can be experienced as the birthing womb of the Great Goddess. Thus, the labyrinth experience is a potent practice of Self-Integration as it encapsulates the spiraling journey in and out of incarnation. On the journey in, towards the center, one cleanses the dirt from the road. On the journey out, one is born anew to consciously dwell in a human body, made holy by having got a taste of the Infinite Center.

Mandals

Mandala derives from the Hindu language meaning ‘concentric energy circle.’ A circle represents protection, good luck, or completion. Mandalas link with the spiraling movement of consciousness sacred geometry, psychology and healing.

Psychoanalyst Carl Jung saw the mandala as “a representation of the unconscious self,” and believed his paintings of mandalas enabled him to identify emotional disorders and work towards wholeness in personality.

Theoretical Foundation for Jung’s ‘Mandala Symbolism’

The mandala as psychological phenomena appear spontaneously in dreams, in certain states of conflict, and in cases of schizophrenia. The main goal of this presentation is to give a theoretical explanation and a mathematical model for computer simulation of mandalas, based on the mechanisms of biochemical reactions in a human brain and on discrete chaotic dynamics.

The discrete dynamics of physicochemical reactions is a new theory based on the analogy between the p -Theorem of the theory of dimensionality, the principle of maximum entropy and the stoichiometry of complex chemical reactions.

Application of this theory to the spatiotemporal behavior of complex biochemical reactions has revealed symmetric patterns similar to the mandalas (link to mandals pictures) presented by C.G.Jung in his book “Mandala Symbolism”. This theory has also been shown to possess the ability to generate complex oscillations, that may be used for mathematical modeling of EEG and ECG and of living systems dynamics in general .

According to the results obtained, when the human brain is generating mandalas, it can be regarded as a complex’biochemical reactor’ that creates different images reflecting its internal state (or the distribution of chemicals and their biochemical interactions) and all these processes based on the laws of nature.

When I taught Special Ed students in High School – those with those who were emotionally challenges – I used sand and string with them to create mandalas. The project was very therapeutic, having a balancing effect on the students. Without realizing it, they were able to work with both sides of the brain in a creative positive way.

The use of right/left brain functioning also goes to walking the labyrinth. It is all about creation balance and activating the consciousness to higher thought forms.

In practice, the term ‘Mandala’ has become a generic term for any plan, chart, or geometric pattern which represents the cosmos metaphysically or symbolically, a microcosm of the universe from the human perspective. A mandala, especially its center, can be used during meditation as an object for focusing attention. The symmetrical geometric shapes which mandalas tend to have, draw the attention of the eyes towards their center.

In Hindu cosmology the surface of the Earth is represented as a square, the most fundamental of all Hindu forms. The earth is represented as four cornered with reference to the horizon’s relationship with sunrise and sunset, the north and south direction.

The Earth is thus called ‘Caturbhrsti (four-cornered)’ and is represented in the symbolic form of the Prithvi Mandala. The astrological charts or horoscopes (Rasi, Navamsa, etc) also represent in a square plan the ecliptic – the positions of the sun, moon, planets, and zodiacal constellations with reference to the native’s place and time of birth. The Vaastu Purusha Mandala is the metaphysical plan of a building, temple, or site that incorporates the course of the heavenly bodies and supernatural forces.

The square is the box. We experience ‘inside the box. 4 goes to 4th dimension or time. We experience through the spiraling patterns of sacred geometry in the alchemy of time m moving away from the central source, and then returning.

A mandala in tantric Buddhism usually depicts a landscape of the Buddha land or the enlightened vision of a Buddha. Mandalas are commonly used by tantric Buddhists as an aid to meditation. This pattern is painstakingly created on the temple floor by several monks who use small tubes to create a tiny flow of grains. The various aspects of the traditionally fixed design represent symbolically the objects of worship and contemplation of the Tibetan Buddhist cosmology.

To symbolize impermanence (a central teaching of Buddhism), after days or weeks of creating the intricate pattern, the sand is brushed together and is usually placed in a body of running water to spread the blessings of the Mandala.

The visualization and concretization of the mandala concept is one of the most significant contributions of Buddhism to religious psychology. Mandalas are seen as sacred places which, by their very presence in the world, remind a viewer of the immanence of sanctity in the universe and its potential in himself. In the context of the Buddhist path the purpose of a mandala is to put an end to human suffering, to attain enlightenment and to attain a correct view of Reality. It is a means to discover divinity by the realization that it resides within one’s own self.

The mandala is usually a symbolic representation which depicts the qualities of the enlightened mind in harmonious relationship with one another. A mandala may also be used to represent the path of spiritual development. On another level a mandala can be a symbolic representation of the universe, as in one of the four foundation practices of the Vajrayana, in which a mandala representing the universe is offered to the Buddha.

Yin and Yang

Yang is the sunny place, south slope (hill), north bank (river); sunshine”) is the brighter element; it is active, light, masculine, upward-seeking and corresponds to the day.

The two concepts yin and yang or the single concept yin-yang originate in ancient Chinese philosophy and metaphysics, which describe two primal opposing but complementary principles said to be found in all objects and processes in the universe.

Yin literally “shady place, north slope (hill), south bank (river); cloudy, overcast”) is the darker element; it is passive, dark, feminine, downward-seeking, and corresponds to the night. Yin is often symbolized by water or earth, while yang is symbolized by fire, or wind. Yin (receptive, feminine, dark, passive force) and yang (creative, masculine, bright, active force) are descriptions of complementary opposites rather than absolutes.

Any yin/yang dichotomy can be seen as its opposite when viewed from another perspective. The categorization is seen as one of convenience. Most forces in nature can be seen as having yin and yang states, and the two are usually in movement rather than held in absolute stasis. In Western culture, the dichotomy of good and evil is often taken as a paradigm for other dichotomies. In Hegelian dialectics, dichotomies are linked to progress. In Chinese philosophy, the paradigmatic dichotomy of yin and yang does not generally give preference or moral superiority to one side of the dichotomy, and dichotomies are linked to cyclical processes rather than progress. However, taoism often values yin above yang, and Confucianism often values yang above yin.

Yin and yang do not exclude each other. Yin and yang are interdependent. Yin and yang consume and support each other. Yin and yang can transform into one another. Part of yin is in yang and part of yang is in yin.