haz un resumen de lo que tenga que ver con el ratio dorado de manera esquematica y punto por punto: "URL: https://www.a-higher-view.com/golden-ratio-sacred-geometry/
Título: The Golden Ratio: Divine Proportion and Cosmic Harmony - A Higher View
Fecha de publicación: None

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The Golden Ratio: Divine Proportion and Cosmic Harmony
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Shilo Kherington
June 17, 2024
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Often referred to as the divine proportion or nature’s design secret, the Golden Ratio carries a wealth of significance within sacred geometry and beyond.
What exactly is the Golden Ratio? In the simplest terms, it’s a mathematical principle that involves a specific ratio (phi) —approximately 1.61803398875. But it’s far more than just a number. When translated into a spatial relationship, it often creates proportions that are naturally pleasing to the human eye.
Our journey here is not just a mathematical or artistic one but a spiritual one as well. The Golden Ratio has long been associated with notions of divine balance and cosmic harmony. It appears in religious symbols and structures, and many believe it provides a mathematical window into deeper spiritual truths.
In the sections to come, we’ll dive deeper into the world of the Golden Ratio, exploring its historical roots, mathematical underpinnings, occurrence in nature and art, role in sacred geometry, and spiritual implications.
Table of ContentsHistory and OriginsEgyptians and the Great Pyramid of GizaGreeks and their Fascination with MathematicsThe Fibonacci SequenceThe Golden Ratio and Sacred GeometryThe PentagramVesica PiscisMetatron’s CubeThe TorusThe Golden Ratio in SpiritualityThe Golden Ratio in Nature and ArtWrapping UpFrequently Asked Questions
History and Origins
Our fascinating journey into the past reveals that knowledge of the Golden Ratio can be traced back to some of the earliest civilizations in human history. These ancient societies recognized its existence and harnessed its power to create awe-inspiring structures and timeless works of art.
Egyptians and the Great Pyramid of Giza
One of the most iconic examples of the Golden Ratio in ancient architecture is the Great Pyramid of Giza, built around 2580-2560 BCE.
This remarkable structure has long been believed to incorporate divine proportion within its design, as evidenced by the proportionate relationship between its base length, height, and slope angles.
While it remains uncertain whether this was an intentional design choice or a mere coincidence, it is clear that these proportions imbue the pyramid with a sense of harmony and balance that continues to captivate visitors today.
Greeks and their Fascination with Mathematics
The Greeks held a deep appreciation for mathematics as they believed it to be a key to understanding the universe’s underlying orderliness. Euclid (circa 300 BCE), one of Greece’s most famous mathematicians, defined what would later become known as the Golden Ratio in his groundbreaking work “Elements.”
He referred to it as “the extreme and mean ratio,” which he demonstrated through a line divided into two segments so that their relationship reflected this unique proportion.
The Fibonacci Sequence
In addition to these historical observations, another essential aspect linking our understanding of phi comes from Leonardo Fibonacci’s sequence, discovered around 1202 CE.
This numerical series (0, 1, 1, 2, 3, 5, 8, 13…) is formed by adding the two preceding numbers to get the next one. Interestingly enough, as we progress in this sequence and divide each number by its predecessor, the resulting quotient converges toward the Golden Ratio. This unsuspecting connection between Fibonacci numbers and the divine proportion further solidifies its significance across various fields of study.
As we delve deeper into the intricate world of sacred geometry and the Golden Ratio’s role within it, it becomes apparent that this enigmatic number has been subtly shaping our understanding of beauty and harmony since time immemorial.
The Golden Ratio and Sacred Geometry
The Golden Ratio is central in sacred geometry, representing a fundamental proportion found throughout nature and human creations.
Phi (Φ), approximately 1.618, holds this unique ratio that conveys balance and harmony when incorporated into various geometrical forms.
The Pentagram
The pentagram, or five-pointed star is one of the most well-known symbols embodying the Golden Ratio within sacred geometry.
Each segment within this shape adheres to phi proportions, making it an excellent example of how this enigmatic number can manifest in simple yet powerful geometrical patterns.
Vesica Piscis
The Ves
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URL: https://www.goldennumber.net/golden-ratio-history/
Título: History of the Golden Ratio - The Golden Ratio: Phi, 1.618
Fecha de publicación: None

ts While the proportion known as the Golden Mean has always existed in mathematics and in the physical universe, it is unknown exactly when it was first discovered and applied by mankind. It is reasonable to assume that it has perhaps been discovered and rediscovered throughout history, which explains why it goes under several names. Uses in architecture potentially date to the ancient Egyptians and Greeks It appears that the Egyptians may have used both pi and phi in the design of the Great Pyramids. The Greeks are thought by some to have based the design of the Parthenon on this proportion, but this is subject to some conjecture. Phidias (500 BC – 432 BC), a Greek sculptor and mathematician, studied phi and applied it to the design of sculptures for the Parthenon. Plato (circa 428 BC – 347 BC), in his views on natural science and cosmology presented in his “Timaeus,” considered the golden ratio to be the most binding of all mathematical relationships and the key to the physics of the cosmos. It was not known as the golden ratio in his time, but he describes it with his first reference to proportion: “Now it is not possible for two things to be combined well on their own without a third, for some bond is required between the two to draw them together. The very best bond is that which, as much as possible, makes itself and the conjoined entities, one; and it is proportion that by nature best accomplishes this. So  whenever the middle item of three numbers or volumes or powers is such that the first is to the middle as the middle is to the last, and again, that the last is to the middle as the middle is to the first, then the middle becomes first and last, and the last and first for their part both become middles. Accordingly it follows, of necessity, that they all turn out to be the same, and since they have all become the same as one another, they will all be one.” Translation © 2021 by David Horan This is analogous to what Euclid later wrote. Euclid (365 BC – 300 BC), in “Elements,” referred to dividing a line at the 0.6180399… point as “dividing a line in the extreme and mean ratio.” This later gave rise to the use of the term mean in the golden mean.  He also linked this number to the construction of a pentagram. The Fibonacci Sequence was written of in India in about 200-300 BC and brought to the Western world around 1200 AD What we now as the Fibonacci sequence is named after Leonardo Pisano Bonacci (aka Bigollo) of Pisa, an Italian born in 1175 AD, who later became known as Leonardo Fibonacci. His book Liber Abaci, published in 1202, introduced this sequence to Western European mathematics in the form of a math problem on the breeding of rabbits. He learned of it though while being educated in North Africa with an Arab master, where he was exposed to the much earlier knowledge of Indian mathematicians. Liber Abaci became a pivotal influence in adoption by the Europeans of the Arabic decimal system of counting over Roman numerals. (3) The sequence itself though had been described as early as the 2nd or 3rd century BC in the works of Acharya Pingala, an Indian mathematician who enumerated the possible patterns of Sanskrit poetry that could be formed from syllables of two lengths. The contributions of Pingala and Fibonacci are important, but it’s not apparent that anyone even realized its connection to the Golden Ratio until the 1600’s by Johannes Kepler and others. It was first called the “Divine Proportion” in the 1500’s Leonardo Da Vinci provided illustrations for a dissertation published by Luca Pacioli in 1509 entitled “De Divina Proportione” (1), perhaps the earliest reference in literature to another of its names, the “Divine Proportion.”  This book contains drawings made by Leonardo da Vinci of the five Platonic solids. The Renaissance artists used the Golden Mean extensively in their paintings and sculptures to achieve balance and beauty. Leonardo Da Vinci, for instance, used it to define all the fundamental proportions of his painting of “The Last Supper,” from the dimensions of the table at which Christ and the disciples sat to the proportions of the walls and windows in the background. Johannes Kepler (1571-1630), discoverer of the elliptical nature of the orbits of the planets around the sun, also made mention of the “Divine Proportion,” saying this about it: “Geometry has two great treasures: one is the theorem of Pythagoras; 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.” The “Golden Ratio” was coined in the 1800’s It is believed that Martin Ohm (1792–1872) was the first person to use the term “golden” to describe the golden ratio. to use the term. In 1815, he published “Die reine Elementar-Mathematik” (The Pure Elementary Mathematics). This book is famed for containing the first known usage of the term “goldener schnitt” (golden section). The term “Phi” was not used until the 1900’s It
--------------------------------------------------------------------------------

URL: https://www.ancient-origins.net/unexplained-phenomena/golden-ratio-sacred-number-links-past-present-001091
Título: The Golden Ratio - A Sacred Number Linking the Past to the Present
Fecha de publicación: None

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The Golden Ratio – A Sacred Number Linking the Past to the Present
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There is one thing that ancient Greeks, Renaissance artists, a 17th century astronomer and 21st century architects all have in common – they all used the Golden Mean, otherwise known as the Golden Ratio, Divine Proportion, or Golden Section.
Precisely, this is the number 1.61803399, represented by the Greek letter Phi, and considered truly unique in its mathematical properties, prevalence throughout nature, and its ability to achieve a perfect aesthetic composition.
According to astrophysicist Mario Livio in his book  ‘The Golden Ratio: The Story of PHI, the World's Most Astonishing Number’:
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. But the fascination with the Golden Ratio is not confined just to mathematicians. 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.
What is the Golden Ratio?
In mathematics and the arts, 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. When the Golden Mean is conceptualized in two dimensions it is typically presented as a regular spiral that is defined by a series of squares and arcs, each forming “Golden Rectangles”.
This symbolic potential arises because of the way the mean’s spiral shape resembles growth patterns observed in nature and its proportions are reminiscent of those in human bodies. Thus, these simple spirals and rectangles, which served to suggest the presence of a universal order underlying the world, were thereby dubbed “golden” or “divine”.
The Golden Mean’s spiral shape resembles growth patterns observed in nature. (Dean Marston /Adobe Stock)
The Golden Ratio in History
The golden ratio has fascinated Western intellectuals of diverse interests for at least 2,400 years. The earliest known monuments believed to have been built according to this alluring number are the statues of the Parthenon in Greece, dating back between 490 and 430 BC. However, there are many who have argued that it goes back much further than this and that the Egyptians were well versed in the properties of this unique number.
According to some historians, the Egyptians thought that the golden ratio was sacred. Therefore, it was very important in their religion. They used the golden ratio when building temples and places for the dead.
Ancient Egyptians are said to have used the Golden Ratio in their buildings. (John Smith /Adobe Stock)
In addition, the Egyptians found the golden ratio to be pleasing to the eye. They used it in their system of writing and in the arrangement of their temples. The Egyptians were aware that they were using the golden ratio, but they called it the "sacred ratio."
The first recorded definition of the golden ratio dates back to the period when Greek mathematician Euclid (c. 325–c. 265 BC) described what he called the "extreme and mean ratio". However, the ratio
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URL: https://posterhouse.org/blog/follow-the-golden-ratio-from-africa-to-the-bauhaus-for-a-cross-cultural-aesthetic-for-images/
Título: Follow the Golden Ratio from Africa to the Bauhaus for a Cross-Cultural ...
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February 24, 2020.Follow the Golden Ratio from Africa to the Bauhaus for a Cross-Cultural Aesthetic for Images.Rating: G.
Audrey Bennett is a University Diversity and Social Transformation Professor at the University of Michigan and a Graduate Program Director and Professor of Art and Design in the Penny W. Stamps School of Art and Design. She is also a former Andrew W. Mellon Distinguished Scholar of the University of Pretoria, South Africa, and a former College Art Association Professional Development Fellow. Her publications include: “Towards an Autochthonic Black Aesthetic for Graphic Design Pedagogy”; “How Design Education Can Use Generative Play to Innovate for Social Change”; Engendering Interaction with Images; “The Rise of Research in Graphic Design”; “Interactive Aesthetics”; and “Good Design is Good Social Change.” Audrey Bennett is the co-editor of the Icograda Design Education Manifesto 2011 and a member of the Editorial Boards of the journals Image and Text (South Africa) and New Design Ideas (Azerbaijan). She sits on the Board of Directors of the College Art Association.
Bennett recently presented “Radical Design Pedagogy: Towards an Autochthonic Black Aesthetic for Graphic Design Pedagogy” at BIPOC Design History in an educational series called “Black Design in America: African Americans and the African Diaspora in Graphic Design.”
This essay was originally published in Volume 6, Issue 1 of Critical Interventions in 2012. PDF version available here.
Follow the Golden Ratio from Africa to the Bauhaus for a Cross-Cultural Aesthetic for Images
The golden ratio, a mathematical relation that often arises in fractals and other scaling geometries, is known for its ability to affect visual beauty (Livio 2002, 179). As a result, communication designers have used it throughout history to compute a plethora of visual compositions. For instance, many designers use the golden rectangle, a popular compositional grid derived from the golden ratio, to organize verbal and visual information into eye-catching images. However, within the discipline’s literature our European predecessors, primarily ancient Greece, receive most of the recognition for being the sole contributors of the golden ratio to interdisciplinary discourse. Very few references in the discipline’s literature acknowledge Africa as the one of the primary contributors of knowledge about the golden ratio. In this paper, I challenge the assumption that golden ratio structures originated solely in ancient Europe. Scaling designs reflecting growth patterns are actually more common in Africa, and thus it is no surprise that geometric relationships for growing forms have an independent origin there. There is even some evidence that this African tradition of dynamic geometry may have made its way to Europe via sources such as Fibonacci and others.
Herein I respond to Robert Bringhurst’s The Elements of Typographic Style, Kimberly Elam’s Geometry of Design, and Tim Samara’s Making and Breaking the Grid: A Graphic Design Layout Workshop by noting that the African contributions are conspicuously absent in their accounts. My aim is to weave into the discourse other perspectives from anthropology, architecture, and mathematics that suggest that Africa’s golden ratio exemplars predate the instances that create the current origin story.
Specifically, I introduce perspectives that suggest that the structures derived from the golden ratio that organize typographic compositions and even structure living spaces also has presence in traditional African architecture, and that this presence extends further back than sites and artifacts in Greece. Two important examples derive from the analysis in Eglash (1999). The first is the chief’s palace in Logone-Birni, Cameroon. This historical architectural site has a golden ratio scaling pattern embedded in its spatial design. The second is a similar scaling pattern in the Temple of Karnak from ancient Egypt.
These additions of African design in the story of the golden ratio need not be limited to merely correcting an incomplete history. We can use it to broaden perspectives on how the golden ratio can be used; to encourage its incorporation in the design of images that resonate cross-culturally. From existing cultural artifacts and spatial designs that reflect the golden ratio and other patterns, they can extract cultural grids—a phrase I introduce to explain the phenomenon of grids found in cultural artifacts and man-made and natural spaces that can apply to communication design practice, 
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URL: https://philosophy.stackexchange.com/questions/38239/what-are-some-arguments-for-the-golden-ratio-making-things-more-aesthetically-pl
Título: What are some arguments for the golden ratio making things more ...
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What are some arguments for the golden ratio making things more aesthetically pleasing?
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What are some (not necessarily good) arguments that painters, architects, designers, musicians, etc. basing their work on the golden ratio φ makes their work more aesthetically pleasing?
I think these lines of reasoning are heard the most:
(a) The golden ratio can easily be found in nature [I include mathematics as part of nature, thus including all instances when the constant crops up, but this can be relaxed] in a lot of unrelated and unexpected places. Therefore, the constant is part of nature.
Anything made relating the thing to nature in some way (e.g. using a material found in nature: a wooden chair, a golden ring, etc.) is always more aesthetically pleasing because nature itself is the epitome of aesthetic beauty, therefore always adding aesthetic value to the thing made.
(b) The simplest visual representation of φ, the golden rectangle, is aesthetically pleasing; same goes for other simple geometric figures based on φ.
(c) All classic art is aesthetically pleasing. There is classic art incorporating φ, therefore there is aesthetically pleasing art involving the golden ratio.
Note: I am not talking about art exclusively.
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Michael SmithMichael Smith
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The golden ratio has an important political interpretation as well; consider one way to state the geometric definition: the greater is to the lesser as the whole is to the greater...
– Joseph Weissman
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I'm not sure of the philosophy behind it, but so of  Da Vinci's work related proportions in the (ideal?) human form to this ratio. en.wikipedia.org/wiki/De_divina_proportione
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DaVinci attributes these to en.wikipedia.org/wiki/Vitruvius#Proportions_of_Man
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More often than not artists do not give arguments about using the golden ratio, they are sufficiently motivated by the long tradition of singling it out as "golden", which accumulated since Pythagoreans and the ancient Greek sculptor/architect Phidias. The perceived presence of golden ratios in his Parthenon now appears to be spurious, but the letter φ often used to denote it comes from the first letter of his name. Plato helped too by promoting dodecahedron, "which the god used for embroidering the constellations on the whole heaven", and which is full of golden ratios just like the Pythagorean pentagram. Both Pythagoreanism and Platonism flourished during the Renaissance, when much of our modern artistic tradition was forged, and even Copernicus and Kepler were self-proclaimed Pythagoreans. This said, the extent of the ratio's use in arts is greatly exaggerated today. Here is Falbo:
"It is remarkable that prior to Fischler’s and Markowsky’s papers, there seemed to have been no set standards for obtaining measurements of artwork. Often, a proponent of the golden ratio will choose to frame some part of a work of art in an arbitrary way to create the appearance that the artist made use of an approximation of φ. Markowsky shows an example in which Bergamini arbitrarily circumscribes a golden rectangle about the figure of St. Jerome in a painting by Leonardo Da Vinci, cutting off the poor
fellow’s arm in order to make the picture fit."
Livio in The Golden Ratio and Aesthetics is similarly skeptical:
Many books claim that if you draw a rectangle around the face of Leonardo da Vinci's Mona Lisa, the ratio of the height to width of that rectangle is equal to the Golden Ratio. No documentation exists to indicate that Leonardo consciously used the Golden Ratio in the Mona Lisa's composition, nor to where precisely the rectangle should be drawn. Nevertheless, one has to acknowledge the fact that Leonardo was a close personal friend of Luca Pacioli, who published a three-volume treatise on the Golden Ratio in 150
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URL: https://ia801409.us.archive.org/22/items/PythagorasPlatoAndTheGoldenRatio/PythagoreanGoldenRatio.pdf
Título: PDF Pythagoras, Plato and the the Golden RatioGolden RatioGolden Ratio
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By: Ken L. WheelerBy: Ken L. Wheeler    
 Pythagoras, Plato and Pythagoras, Plato and Pythagoras, Plato and Pythagoras, Plato and 
the the the the Golden Ratio Golden RatioGolden RatioGolden Ratio    
The Golden Ratio and the Pentagram in the philosophy of the PythagoreansThe Golden Ratio and the Pentagram in the philosophy of the PythagoreansThe Golden Ratio and the Pentagram in the philosophy of the PythagoreansThe Golden Ratio and the Pentagram in the philosophy of the Pythagoreans    
 
Author: Ken L. Wheeler 
 
1st Edition 2005 
All Rights Reserved. Copyright © Oct. 2005 DARKSTAR PUBLICATIONS 
 Printed on acid free paper for archiaval purposes. 
 
No part of this book may be reproduced or transmitted in any form or by 
any means, graphic, electronic, or mechanical, including photocopying, 
recording, taping or by any information storage or retrieval system, 
without the permission in writing from Darkstar Publications 
 
 
For information, please contact: Darkstar Publications, 533 Plainview Rd., 
Lexington, Ky 40517 
 
Published by Darkstar PublicationsPublished by Darkstar PublicationsPublished by Darkstar PublicationsPublished by Darkstar Publications    
 
 
ISBN  0-9712541-5-X 
$15.00 U.S. Printed in the United States of America 
 Introduction  
 
     “The Pythagorean Philolaus says that the sun is a vitrescent body which receives the light reflected by the fire of the Co smos, and 
sends it back to us, after having filtered them, light and heat; so that you might say that there are two suns, the body of the  fire which 
is in the heaven, and the igneous light which emanates there from, and reflects itself in a kind of a mirror. Perhaps we might consider 
as a third light that which, from the mirror in which it reflects, and falls back on us in dispersed rays.” [FRAGMENTS OF 
PHILOLAUS From Boeckh 14. (Stob.Ecl.l:25:3:p.530)] 
 
      Pythagoras, Plato, and Plotinus after them were well aware, by their intense study of geometry and nature, the Golden secti on. The 
purpose of this very small but pithy book is to examine some key passages of Plato and the Phi incommensurability of the Pythagorean pentagram and unveil a heretofore unknown ‘secret’ rediscovered which links the Golden ratio, Pythagoras, the ontological preeminence of the pentagram, and the two most important passages in the Republic of Plato all together. I will sho w that 
the Emanationist philosophy/religion of the Platonists employed the Golden ratio into both a “Divided Line” analogy and the “Ca ve” 
symbolism to show the variant degrees of proportion and ratio between the visible (aisthetos) and intelligible (noetos) kosmos in a 
divine Logos (proportion) of which the Monad is Arche. Also I hope to demonstrate that the “Pythagorean oath”, the tetraktys wa s 
shown only inside a special ‘Pythagorean triangle’, the 1-1-Phi triangle which is not only the “triple intertwined triangle” of  the 
pentagram, but also the 1-1-Phi-1/Phi analogy of Plato’s Cave and Divided line symbolism for the totality of all things.       The Logos (proportion) which comprises the visible and intelligible, I hope to show, formed the foundational doctrine of t he 
Pythagoreans and Platonists who despised all who were “ignorant of geometry” to partake of their instruction, for “As above, so  
below”, the Platonists before and after understood the “third thing which unified” was a Logos of the Monad, geometrically and 
arithmetically represented by a power of Phi, which itself is the logos of the necessary emanation of the Monad.       Phi is the ratio and relationship of the Monad to its increasingly phenomenal self-image in emanation. “But employing the objects 
of which the shadows below are resemblances in their turn as images” [Rep. 509]; thru the many ages of nature study the Pythagoreans 
understood that all nature exhibits a Logos (proportion) to a Principle (1), and this very special ratio was Phi (1 to 1.618033 ), and so 
being the case, the 2
nd and 3rd Hypostases (the Nous and the Psyche tou pantos [world soul]), though noetic, MUST exhibit this same 
proportion. After who knows how long a time, the symbolism and analogies I mention below will spell out that which the Emanationists settled upon as the visible models for their knowledge of the ordered kosmos into powers and or proportions of Ph i.  
     One very special triangle, its significance uncovered for the first time here in this book, serves as the model for the te traktys, the 
pentagram, the Divided line, and the Cave analogies, and its geometers contain the seven sections of Platonic totality. The 1-1 -Phi 
‘Pythagorean triangle’, the 108-36-36 degrees isosceles triangle. At no time does this book affiliate its discussion with the 
bastardization of the pentagram in meaning and usage, or the theology of numbers with those groups as commonly denoted in the 
practice of the white-lighter new-age movement, but only to the root meaning and importance as taught by the Pythagoreans alone 
--------------------------------------------------------------------------------

URL: https://illustrarch.com/articles/25256-golden-ratio-in-architectural-design-from-ancient-times.html
Título: Unveiling the Golden Ratio: Its Impact on Architectural Design From ...
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Unveiling the Golden Ratio: Its Impact on Architectural Design From Antiquity to Present
byElif Ayse Sen25 April 2024
Explore the timeless impact of the golden ratio in architectural design, from ancient structures to modern marvels like the CN Tower. Discover how architects, including Le Corbusier, have used this mathematical principle to create aesthetic balance and harmony in their designs over different periods.
We’ve all marveled at the grandeur of ancient architectural wonders. But have you ever wondered what makes them so aesthetically pleasing? The secret lies in the golden ratio, a mathematical concept that’s been used in design since antiquity.The golden ratio, often symbolized by the Greek letter phi (Φ), is a special number approximately equal to 1.618. It’s found in nature, art, and of course, architecture. This ratio has been used by architects and designers for thousands of years, shaping the world’s most iconic structures.From the Great Pyramids of Egypt to the Parthenon in Greece, the golden ratio has been a key component in architectural design. It’s this ratio that gives these structures their balanced, harmonious appearance. In this article, we’ll delve deeper into the golden ratio’s role in architectural design from ancient times.
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Exploration of the Golden Ratio in Architectural DesignOrigin of the Golden Ratio in Ancient Architectural PracticesAncient Civilizations’ Use of the Golden RatioThe Influence of Greek Architecture on the Golden RatioApplication of the Golden Ratio in Architectural ElementsProportional Harmony in Building FacadesSymmetry and Balance in Architectural DesignModern Interpretations and CalculationsConclusion
Exploration of the Golden Ratio in Architectural DesignUpon dissecting the formula of aesthetics in historical structures, we stumble across the persistent and prevalent use of the golden ratio, or phi (Φ). This divine proportion, approximately 1.618, is not incidental. From the Parthenon in Greece to the Great Pyramids of Egypt, it’s a fundamental element in architectural design.The Parthenon in Greece serves as an emblematic specimen of the golden ratio. Its façade, columns, and interior structures all align with Φ, giving it an overwhelming sense of balance and harmony. These visually pleasing proportions immediately catch the eye, playing into our inherent appreciation for patterns and symmetry.The Great Pyramids of Egypt, another timeless creation, embody architectural design principles based on the golden ratio. Its near perfect proportional relationships have stirred speculation that the Egyptians understood and applied Φ. The Great Pyramid’s base to height ratio closely mirrors 1:1.6180339887, a precision that’s hardly coincidental.We can also spot instances of the golden ratio in the Notre-Dame Cathedral in Paris. Erected during an era when the golden ratio was consciously employed, the multiple instances of Φ found within its intricate design is testament to the period’s architectural mastery.Even in modern architecture, the golden ratio’s implementation continues. Our insatiable appetite for visual balance, harmony, and appealing proportions drives its continued use. This timeless mathematical concept is far from being a historical relic – it remains a vital, celebrated component in our quest to erect structures that resonate with our innate sense of beauty.
Origin of the Golden Ratio in Ancient Architectural PracticesDelving back into the annals of history, one discovers the origins of the golden ratio (Φ) in architectural practices are deeply rooted in the grandeur of ancient civilizations.Ancient Civilizations’ Use of the Golden RatioEons ago, our forebears had a keen understanding of the golden ratio (Φ), quite evident in their architectural marvels. From the majestic Parthenon in Greece to the impressive Great Pyramids of Egypt, the golden ratio
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URL: https://artistichive.com/golden-ratio-in-art/
Título: The Secret Math Behind Beauty: A Guide to Golden Ratio Art
Fecha de publicación: None

lance, order, and beauty. The golden ratio, also known as the golden mean or divine proportion, is a mathematical ratio of 1:1.618 that has been used by artists for thousands of years to create aesthetically pleasing compositions. Understanding and applying the golden ratio in art can help take your artwork to the next level.
The golden ratio is found throughout nature, architecture, and the human body. When used in art and design, the golden ratio helps create compositions that are balanced, harmonious, and naturally pleasing to the eye. In this article, we will explore what the golden ratio is, its origins, and how artists over the centuries have made use of the golden ratio principles in their paintings, drawings, and sculptures to achieve the perfect composition.
What is the Golden Ratio?
The golden ratio is a mathematical proportion defined by the ratio of 1 to 1.618, represented by the Greek letter phi. It is an irrational number, meaning its value cannot be expressed exactly as a simple fraction. The golden ratio is also called the golden section, golden mean, divine proportion, extreme and mean ratio, and the divine proportion.
The golden ratio is related to the Fibonacci sequence, a pattern of numbers where each number is the sum of the two preceding numbers (0, 1, 1, 2, 3, 5, 8, 13, 21, 34…). As the Fibonacci sequence progresses, the ratio between consecutive numbers approaches phi, the golden ratio.
When a line is divided into two parts and the whole length divided by the long part is equal to the long part divided by the short part, it is in the golden ratio proportion. Visually, this looks like a square next to a rectangle where the ratio of the rectangle’s length to width is phi.
The History and Origins of the Golden Ratio
The golden ratio has been mathematically pondered for millennia, at least since the time of Pythagoras around 500 BC. It was studied further by the ancient Greek mathematicians Euclid and Phidias. The golden ratio was first defined as the “extreme and mean ratio” in Euclid’s Elements around 300 BC.
The mathematical properties of the golden ratio were written about extensively by the ancient Greek mathematician Euclid in his book Elements around 300 BC. Euclid demonstrated geometrically that a line divided into two parts where the ratio of the whole to the larger part is the same as the ratio of the larger part to the smaller part produces the golden ratio.
In 1509, Luca Pacioli wrote a book on divine proportion called De divina proportions, exploring the mathematics behind the golden ratio. The illustrations in Pacioli’s book were drawn by Leonardo da Vinci and likely introduced the concept more widely in art composition. Pacioli called the golden ratio “the divine proportion” and this term persists today.
The golden ratio has been rediscovered and pondered throughout history by eminent mathematicians such as Leonardo Fibonacci, Johannes Kepler, and Martin Ohm. The ratio’s unique mathematical properties continue to intrigue mathematicians today.
Using the Golden Ratio in Art Composition
The golden ratio is a powerful compositional tool because its proportions are aesthetically pleasing to the human eye. When artworks are composed using the golden ratio, they have a natural beauty, harmony, and balance that viewers intuitively respond to.
The golden ratio provides the perfect formula for laying out the key elements of an artwork such as focal points, objects, negative space, and frames. Placing compositional elements according to the golden ratio creates compositions that are balanced and proportional. The ratio helps guide the placement of elements in an ideal way.
Artists employ a variety of techniques and tools to plan and execute golden ratio compositions. Some common methods include:
Drawing a golden rectangle as the basis for the composition
Using the golden spiral to map key elements
Dividing a composition according to major golden ratio points
Composing along the golden curve
The golden rectangle, where the ratio of the sides is the golden ratio, can provide the underlying framework. A square can be nested inside, with a golden rectangle extending outwards, creating a natural splitting point.
The golden spiral is another great composition tool. It geometrically divides spaces based on a spiraling golden ratio. The spiral can guide placement of focal objects along its curves and intersections.
Dividing up the picture plane into major horizontal and vertical lines based on golden ratio proportions provides strong guidance for element placement. Objects and areas of interest can be aligned on these division points.
Examples of the Golden Ratio in Famous Artworks
Many of history’s greatest artists intuitively used the golden ratio to compose aesthetically pleasing, balanced artworks that stand the test of time. Tracing the golden ratio overtop paintings reveals its timeless presence.
Leonardo da Vinci incorporated the golden ratio extensively in his artworks and was likel
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URL: https://www.architectureadrenaline.com/the-golden-ratio-in-architecture/
Título: Unveiling the Golden Ratio: Fibonacci Principles in Architecture and ...
Fecha de publicación: None

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Unveiling the Golden Ratio: Fibonacci Principles in Architecture and DesignSeptember 11th, 2024
Architecture Adrenaline > Recent > General > Unveiling the Golden Ratio: Fibonacci Principles in Architecture and DesignThe Golden Ratio is a magical mathematical concept that has captivated artists, architects, and mathematicians for centuries. This divine proportion, also known as Phi, can be found in nature, art, and even architecture. In this blog post, we will delve into the fascinating world of the Golden Ratio and explore how this mathematical principle shapes some of the most stunning architectural wonders around the globe. From the enchanting spirals of the nautilus shell to the majestic proportions of the Parthenon, the Golden Ratio is truly a key player in creating architectural masterpieces that stand the test of time. So join us as we unveil the secrets behind the Golden Ratio and discover the incredible impact it has on the world of architecture.
Architects of ancient times have been known to use the golden ratio in their design. The golden ratio is an important concept in architecture because it is believed to promote balance between form (aesthetics) and function. By using this concept, architects are able to create designs that are optimal for both aesthetics and function.
Balance between form (aesthetics) and function
Creating a balanced equation between form (aesthetics) and function (functionality) is no small feat. To do it well, you'll need to know what you're doing and a little bit of luck. The key to making the right call is to do your homework and get input from professionals who have a stake in your success.
The best way to do this is to hire an architect. Architects have a unique perspective on designing spaces that meet the needs of occupants. In addition, they are often tasked with designing something that is functional and aesthetically pleasing. A good architect knows how to bend the rules of architecture in order to achieve the desired outcome.
The best architecture is the result of careful planning and a hefty dose of foresight. The most important component of a well-executed design is function, but function alone isn't enough to make a good building. A good architect also knows how to maximize the aesthetics of a given space by making use of materials with aesthetic properties. A good example of this is the use of decorative columns.
The best example of a balanced equation between form (aesthetics) in function is the Empire State Building. This iconic structure, which stands atop Manhattan's Manhattan island, was completed during the Great Depression. Aside from its storied history, the building also demonstrates the most important architectural maxim: form and function should be in sync. This is the most practical way to approach designing a new building.
It's not hard to see why this building is considered to be one of the most iconic structures in the world. Aside from its functional aspects, it also proves that architecture is an art. Its many components can be reused in other incarnations. This is a good thing, because architecture isn't limited to new buildings. Many ancient buildings could be repurposed for new purposes. The most difficult challenge is finding the best architecture for your particular space and budget.
Hopefully, the next time you're designing a new building, you'll have a better grasp on how to achieve balance between form (aesthetics)and function in architecture.
Biomimetics research allows derivation of optimal designs
Architects, engineers, and designers can use biomimetics to derive optimal designs from nature. Biomimetics is a process that takes the designs of living organisms and applies them to engineered technologies. It can be used to solve existing problems and create new solutions.
One example is the application of biomimetics in space optics. The Lobster-Eye X-ray optical system is a prime example. This system has many unique features, including active optics. In addition, it is capable of capturing images at a great distance.
A second example is the development of an upper limb robotic exoskeleton. This design was inspired by bone morphology and muscle force transmission. It was created using CATIA software.
A third example is the application of biomimetics to designing legs for agile quadrupeds. This design uses key features of horse legs. This includes the horse's powerful locomotion, which is a well-known fact. The resulting leg is also a good example of biomimetics, as it demonstrates how to replicate organic appendages that actuate out of plane.
One example is the biomimetic design of a branched vascular network in solid organ tissue engineering. This netwo
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