On BBC

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Tonight on BBC Two at 8pm, Dan Cruickshank explores some of Ulster’s most remarkable buildings. He begins with the region’s castles and forts, testament to a history of invasion and conflict, before moving on to chart the rise of the country house in the 18th century.
http://www.bbc.co.uk/programmes/b01r5t40

Ancient Faces

de bene esse: literally, of well-being, morally acceptable but subject to future validation or exception 

Beautiful little Francine Beirach, age 4, with her teddy bear taken in 1942. 

Francine was deported to Auschwitz with her family where she lost her life. Never forget. 

Original: http://www.ancientfaces.com/photo/1942-francine-beirach/1258784

Lapham's Quarterly

de bene esse: literally, of well-being, morally acceptable but subject to future validation or exception 

DEJA VU: Think twice before ordering cocktails in New Jersey, though at Ned Ward's 18th century tavern you should be safe: http://bit.ly/174ZV3p

Greek Mythology

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John William Waterhouse: Circe Offering the Cup to Ulysses - 1891

Greek Mythology: Kirke (Circe) is the daughter of Helios (the Sun) and Perseis, making her the grand-daughter of Okeanos (Ocean).
http://www.jwwaterhouse.com/view.cfm?recordid=62

Chinese Mountain Trek

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The tourist trek thousands of feet up a Chinese mountain along wooden boards that were nailed together 700 years ago http://dailym.ai/12P1750

The Bellamy Salute

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Posted on the Being Liberal fan page.

Bird Evolution

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A dinosaur from the Middle-Late Jurassic period, found in China, gives scientists new understandings of how birds evolved
http://on.cnn.com/Ztxp9g

Edith Kroupa

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Women in Science ... read about Edith Kroupa: http://bit.ly/ZtpGa

Teddy Roosevelt

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Little self-portrait that Teddy Roosevelt included in a 1907 letter to his youngest son Quentin: http://slate.me/112Q3yP

Phi and Golden Ratio Overview

de bene esse: literally, of well-being, morally acceptable but subject to future validation or exception

What is Phi?

(or Phi for “Neo-Phi-tes”)

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What is Phi?

Phi ( Φ = 1.618033988749895… ), most often pronounced fi like “fly,” is simply an irrational number like pi ( p = 3.14159265358979… ), but one with many unusual mathematical properties.  Unlike pi, which is a transcendental number, phi is the solution to a quadratic equation.

Phi is the basis for the Golden Ratio, Section or Mean

The ratio, or proportion, determined by Phi (1.618 …) was known to the Greeks as the “dividing a line in the extreme and mean ratio” and to Renaissance artists as the “Divine Proportion“  It is also called the Golden Section, Golden Ratio and the Golden Mean.

Phi, like Pi, is a ratio defined by a geometric construction

Just as pi (p) is the ratio of the circumference of a circle to its diameter, phi () is simply the ratio of the line segments that result when a line is divided in one very special and unique way.

Divide a line so that:

the ratio of the length of the entire line (A)
to the length of larger line segment (B)

is the same as

the ratio of the length of the larger line segment (B)
to the length of the smaller line segment (C).

This happens only at the point where:

A is 1.618 … times B and B is 1.618 … times C.

Alternatively, C is 0.618… of B and B is 0.618… of A.

Phi with an upper case “P” is 1.618 0339 887 …, while phi with a lower case “p” is 0.6180339887, the reciprocal of Phi and also Phi minus 1.

What makes phi even more unusual is that it can be derived in many ways and shows up in relationships throughout the universe.

Phi can be derived through:

• A numerical series discovered by Leonardo Fibonacci
• Mathematics
• Geometry

Phi appears in:

• The proportions of the human body
• The proportions of many other animals
• Plants
• DNA
• The solar system
• Art and architecture
• Music
• Population growth
• The stock market
• The Bible and in theology
  
  
  
  

  The Golden Ratio

  An overview of its properties, appearances and applications

  Based on the number 1.618, Phi, and also known as
  the Divine Proportion, Golden Proportion, Golden Mean and Golden Section

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  “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.”  Johannes Kepler, 1571-1630 (Other Quotations)
  
  
  

  Phi-based grid lines created with PhiMatrix golden ratio software

  
  

   What makes a single number so interesting that ancient Greeks, Renaissance artists, a 17th century astronomer and a 21st century novelist all would write about it?  This “golden” number, 1.61803399, represented by the Greek letter Phi, is known as the Golden Ratio, Golden Proportion, Golden Mean, Golden Section and Divine Proportion.  It was written about by Euclid in “Elements” around 300 B.C., by Luca Pacioli, a contemporary of Leonardo Da Vinci, in “De Divina Proportione” in 1509, by Johannes Kepler around 1600 and by Dan Brown in 2003 in his best selling novel, “The Da Vinci Code.” With the movie release of the “The Da Vinci Code”, the quest to know Phi was brought even more into the mainstream of pop culture. The allure of “The Da Vinci Code” was that it creatively integrated fiction with both fact and myth from art, history,theology and mathematics, leaving the reader never really knowing what was truth and what was not. This site studies this golden number Phi, and its mathematical cousin, the Fibonacci series, both of which have roles in the plot of this murder mystery, and distinguishes between the myth and the math.

  Mathematics of the Golden Ratio

  This Golden Ratio truly is unique in its mathematical properties and pervasive in its appearance throughout nature. The “mathematically challenged” may be more interested in the appearances of Phi in nature, its application to art, architecture and design, and its potential for insights into the spiritual realm, but let’s begin with the purest of facts about Phi, which are found in mathematics.
  Most everyone learned about the number Pi in school, but relatively few curriculums included Phi, perhaps for the very reason that grasping all its manifestations often takes one beyond the academic into the realm of the spiritual just by the simple fact that Phi unveils a constant of design that applies to so many aspects of life. Both Pi and Phi are irrational numbers with an infinite number of digits after the decimal point, as indicated by “…”, the ellipsis.
  Where Pi or p (3.14…) is the ratio of the circumference of a circle to its diameter, Phi or Φ (1.618 …) is the Golden Ratio that results when a line is divided in one very special and unique way. To illustrate, suppose you were asked to take a string and cut it.  There’s any number of places that you could cut it, and each place would result in different ratios for the length of the small piece to the large piece, and of the large piece to the entire string. There is one unique point, however, at which the ratio of the large piece to the small piece is exactly the same as the ratio of the whole string to the large piece, and at this point this Golden Ratio of both is 1.618 to 1, or Phi.
  
  
  What makes this so much more than an interesting exercise in mathematics is that this proportion appears throughout creation and extensively in the human face and body. It’s found in the proportions of many other animals, in plants, in the solar system and even in the price and timing movements of stock markets and foreign currency exchange. Its appeal thus ranges from mathematicians to doctors to naturalists to artists to investors to mystics.
  Part of the uniqueness of Phi is that it can be derived in many other ways than segmenting a line. Phi is the only number whose square is greater than itself by one, expressedmathematically as Φ² = Φ + 1 = 2.618. Phi is also the only number whose reciprocal is less than itself by one, expressed as 1/Φ = Φ - 1 = 0.618.  These can be stated as quadratic equations, the only positive solution of which is:
  
  
  Where 1.618 is represented in upper case as Phi or Φ, its near twin or reciprocal, 0.618, is often represented in lower case as phi or φ.  The Fibonacci series, also a plot element in “The Da Vinci Code,” provides yet another way to derive Phi mathematically. The series is quite simple. Start with 0 and add 1 to get 1. Then repeat the process of adding each two numbers in the series to determine the next one: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, and so on. The relationship to the Golden Ratio or Phi is found by dividing each number by the one before it. The further you go in the series, the closer the result gets to Phi. For example, while 5/3 = 1.666 and 13/8 = 1.625, go further into the series and you’ll find that 233/144 = 1.61805, a very close approximation of Phi, which to ten decimal places is 1.6180339887.
  

  Geometry of the Golden Ratio

  The Golden Ratio is also found in geometry, appearing in basic constructions of an equilateral triangle, square and pentagon placed inside a circle, as well as in more complex three-dimensional solids such as dodecahedrons, icosahedrons and “Bucky balls,” which were named for Buckminster Fuller and are the basis for the shapes of both Carbon 60 and soccer balls.
  
  

  Life and the Golden Ratio

  There are many other fascinating mathematical relationships and oddities in both Phi and the Fibonacci series that can be explored in more depth, but for now let’s now take a step away from the purely mathematical and venture into nature, where Phi and the Fibonacci series manifest themselves pervasively, but not universally. Fibonacci numbers frequently appear in the numbers of petals in a flower.  The positions and proportions of the key dimensions of many, if not most, animals are based on Phi. Examples include the body sections of ants and other insects, the wing dimensions and location of eye-like spots on moths, the spirals of sea shells and the position of the dorsal fins on porpoises. Even the spirals of human DNA embody phi proportions.  More intriguing yet is the extensive appearance of Phi throughout the human form, in the face, body, fingers, teeth and even our DNA, and the impact that this has on our perceptions of human beauty.
  Some would argue that beauty is in the eye of the beholder, but there is sound basis in scientific study and evidence to support that what we perceive as beauty in both men and women is based on how closely the proportions of facial and body dimensions come to Phi. It seems that Phi is hard-wired into our consciousness as a guide to beauty. For this reason, Phi is applied in both facial plastic surgery and cosmetic dentistry as a guide to achieving the most natural and beautiful results in facial features and appearance.
  

  Art, Architecture and Design

  With all the unique mathematical properties of Phi and its appearance throughout creation, it’s little wonder that mankind would not only take notice of this number and the Golden Ratio it creates, but also use it to capture the beauty and harmony of nature in our own creations in art, architecture and other areas of design.  In some cases, mankind’s application of Phi is undeniable. In other cases, it is still the subject of debate. The Great Pyramid of Egypt appears to embody the Golden Ratio in the ratios of its base, height and hypotenuse, but its state of ruin and the absence of the mention of Phi in ancient Egyptian writings make it difficult to prove conclusively that this was by design. Since the Greeks knew of Phi at the time of the building of the Parthenon, it seems quite clear to many that the Golden Ratio relationships found therein were by design, yet the evidence is not complete and there are those who contest this as well. It is recognized that Leonardo Da Vinci used Phi, known in the 1500’s as “The Divine Proportion,” in a number of his paintings. While this is undeniable, some see Phi relationships where others do not believe they were intended. The dimensions of the treasured Stradivarius violinsbuilt around 1700 show clear Phi relationships. It plays a role in music and acoustics.  More modern applications of the Golden Ratio in architecture can be seen in Notre Dame in Paris, the United Nations building in New York and the CN Tower in Toronto. It’s commonly used in the design of products and logos and by many major corporations.  It’s even being used in high fashion clothing design in the “Phi Collection” announced in 2004 and covered by Vogue, Elle and Vanity Fair.
  

  New Discoveries involving the Golden Ratio

  The Golden Ratio continues to open new doors in our understanding of life and the universe. It appeared in Roger Penrose’s discovery in the 1970′s of “Penrose Tiles,” which allowed surfaces to be tiled in five-fold symmetry, a task previously thought impossible. It appeared again in the 1980′s in the three-dimensional molecular arrangement of quasi-crystals, a newly discovered form of matter. As we enter the 21st century, Phi seems to be having a rebirth in integrating knowledge across a wide variety of fields of study, including time and quantum physics.
  

  Spiritual Aspects

  The description of this golden proportion as the Divine proportion is perhaps fitting because it is seen by many as a door to a deeper understanding of beauty and spirituality in life, unveiling a hidden harmony or connectedness in so much of what we see. That’s an incredible role for a single number to play, but then again this one number has played an incredible role in human history and in the foundations of life itself. The line between its mathematical and mystical aspects is thus not easily drawn. Phi does not appear explicitly in the Bible or other ancient scriptures, yet we find that the dimensions given by God to Noah for the Ark and to Moses for the Ark of the Covenant both reflect a 5 to 3 proportion, Fibonacci numbers with a ratio of 1.666, and a reasonably close approximation to Phi.  The Kaaba, the most sacred site of Islam in Mecca, is located very close to the golden ratio of the distance between the Earth’s north and south poles.  Curiously enough, even the symbol for Phi, a circle with a line drawn through it, can be thought to represent a zero, or void, divided by one, or Unity, to create beauty, analogous to God creating the universe from nothing.

  A Journey of Discovery

  In matters of reason, seeing is believing but in matters of faith, it is believing that first opens the door to seeing. Just as we need two eyes to add depth to our perception in vision, both faith and reason serve us in adding depth to our understanding of life and the universe in which we live. The best way to know for yourself where Phi is present and where it is imagined is to explore with an open mind, learn and reach your own conclusions. You can learn more about the Golden Ratio here at www.GoldenNumber.net and download PhiMatrix golden ratio design software at www.phimatrix.com to easily discover and apply Phi proportions for yourself. Enjoy the “phi”nomemon, whether to enhance your own understanding and appreciation of beauty and harmony in life or to apply it to your own artistic creations, like Leonardo Da Vinci and other masters before you.  You may contribute your own insights and findings to share with our online community.
  
  Gary Meisner
  The Phi Guy
  Author of www.goldennumber.net
  Developer of PhiMatrix software at www.phimatrix.com
  
  
  

Aston Martin, James Bond and the Golden Ratio

de bene esse: literally, of well-being, morally acceptable but subject to future validation or exception

What do James Bond, Aston Martin and the Golden Ratio have in common?

James Bond, also known as 007, drove an Aston Martin DB5 in the movies GoldFinger and GoldenEye, and Aston Martin is now boasting its application of the Golden Ratio in the design of its latest DB9 and Rapide S automobiles.

The Aston Martin Rapide S is described as:

“Breathtaking Proportions – The ‘Golden Ratio’ sits at the heart of every Aston Martin. Balanced from any angle, each exterior line of Rapide S works in concert and every proportion is precisely measured to create a lithe, pure form. Our engineering follows the same principle. A near perfect weight distribution ensures Rapide S is balanced in form and balanced in function.”

The Aston Martin DB9 is described as:

“Perfectly Proportioned – Every inch of DB9′s form is designed for elegance and balance. The simple beauty of nature guides the design of DB9, with the ‘golden ratio’ setting all proportions. The result is a profile where every line, dimension and proportion works in harmony. Combine this with the near perfect weight distribution, provided by a lower engine placement, and you have a DB9 balanced on sight and in experience.”

Note how so many of the design choices in the dimensions and proportions of these cars are based on golden ratios, as shown by the colored lines divided at their golden ratio points.  Even the wheels highlight five spokes, with the number 5 being intrinsically related to phi, 1.618, the golden ratio in both its mathematics and geometry.  Most people would agree that these cars embody excellence in design and styling.
The golden ratio has been used for centuries for design and composition in the arts and architecture, and is frequently used by professional designers in:

• graphic design
• product design
• logo design
• Photo cropping and photo composition
• website design
• fonts and type spacing

Aston Martin is getting some good press on this, with articles appearing in Forbes and the New York Times, but is not the only car manufacturer to use the golden ratio in its design applications.  A number of manufacturers have golden ratio proportions in their logos.

Perfect Body Proportions for Men – The Adonis Golden Ratio

de bene esse: literally, of well-being, morally acceptable but subject to future validation or exception 

The Golden Ratio and Perfect Proportions for the Male Body

Beauty in the the human form is closely correlated to, if not defined by, the golden ratio.  It should be no surprise then that the perfect proportions for the male body would also be related to the Golden Ratio.  I recently reviewed a new body-building and exercise program called “Adonis Golden Ratio.”  I was impressed with not only its legitimate application of the golden ratio in defining the ideal proportions of the male physique, but also the research and methods developed to help men achieve this ideal.

The system is based on the concept that ideal male proportions occur when the circumference of the shoulders to that of the waist is a golden ratio.  Men’s suits sizes are generally based on the circumference of the chest, which results in a smaller ratio.  An example of ideal build then would be a 48″ shoulder measurement with a 30″ waist measurement.

Do you now look more like the Pillsbury Dough Boy than Michelangelo’s David?  No problem. The Adonis Golden Ratio system is customized to meet the needs of any man no matter what his condition when he starts the program.  The program makes it clear that this is not your average body building program, and that the look that you want for optimal health and attractiveness is not what you often see in muscle-building magazines.  It’s more about achieving the way your body was designed to look.  As they put it, “think Muscle Beach 1940, not Wrestle Mania.”  Adonis includes a software application that considers your current height, weight, waist, and shoulder measurements, age and available exercise time as inputs and then creates a custom program to progress you towards your own ideal physique.  It provides instructions on diet, exercise and weight training that change as you progress through the program.

The concepts behind the program appeared in an article called The Perfect Body Formula in Men’s Health Magazine in 2008 but the program has apparently been released only recently in this bundle of software, books and DVD’s and is being offered at a very low price.  Click on the image below for more information:

Phi, Pi and the Great Pyramid of Egypt at Giza

de bene esse: literally, of well-being, morally acceptable but subject to future validation or exception 

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The Great Pyramid of Egypt is based on Golden Ratio proportions

There is still some debate as to whether the Great Pyramid of Giza in Egypt, built around 2560 BC, was constructed with dimensions based on phi, the golden ratio.  Its once flat, smooth outer shell is gone and all that remains is the roughly-shaped inner core, so it is difficult to know with certainty.
There is compelling evidence, however, that the design of the pyramid embodied these foundations of mathematics and geometry:

• Phi, the Golden Ratio that appears throughout nature.
• Pi, the circumference of a circle in relation to its diameter.
• The Pythagorean Theorem – Credited by tradition to mathematician Pythagoras (about 570 – 495 BC), which can be expressed as a² + b² = c².

First, phi is the only number which has the mathematical property of its square being one more than itself:

 Φ + 1 = Φ²

or

1.618… + 1 = 2.618…

 By applying the above Pythagorean equation to this, we can construct a right triangle, of sides a, b and c, or in this case a Golden Triangle of sides √Φ, 1 and Φ, which looks like this:

This creates a pyramid with a base width of 2 (i.e., two triangles above placed back-to-back) and a height of the square root of Phi, 1.272.  The ratio of the height to the base is 0.636.
According to Wikipedia, the Great Pyramid has a base of 230.4 meters (755.9 feet) and an estimated original height of 146.5 meters (480.6 feet).  This also creates a height to base ratio of 0.636, which indicates it is indeed a Golden Triangles, at least to within three significant decimal places of accuracy.  If the base is indeed exactly 230.4 meters then a perfect golden ratio would have a height of 146.53567, so the difference of only 0.3567 meters appears to be just a measurement or rounding difference.

The Great Pyramid has a surface ratio to base ratio of Phi, the Golden Ratio

A pyramid based on a golden triangle would have other interesting properties.  The surface area of the four sides would be a golden ratio of the surface area of the base.  The area of each trianglular side is the base x height / 2, or 2 x Φ/2 or Φ.  The surface area of the base is 2 x 2, or 4.  So four sides is 4 x Φ / 4, or Φ for the ratio of sides to base.

The Great Pyramid also has a relationship to Pi

There is another interesting aspect of this pyramid.  Construct a circle with a circumference is 8, the same as the perimeter of this pyramid with its base width of 2.  Then fold the arc of the semi-circle at a right angle, as illustrated below in “Revelation of the Pyramids”.  The height of the semi-circle will be the radius of the circle, which is 8/pi/2 or 1.273.

This is less than 1/10th of a percent different than the height of 1.272 computed above using the Golden Triangle.  Applying this to the 146.5 meter height of the pyramid would result in a difference in height between the two methods of only 0.14 meters (5.5 inches).

Its near perfect alignment to due north shows that little was left to chance

Some say that the relationships of the Great Pyramid’s dimensions to phi and pi either do not exist or happened by chance.  Would a civilization with the technological skill and knowledge to align the pyramid to within 1/15th of a degree to true north leave the dimensions of the pyramid to chance?  If they didn’t intend the precise 51.83 degree angle of a golden triangle, why would they have not used another simpler angle found in divisions of a circle such as 30, 45, 54 or 60 degrees?  If the dimensions of the pyramid were not based on both phi and pi, would it not be most reasonable to assume that phi was used since it is based on the visible base of the pyramid and not an invisible circle with the same circumference as that base?

Other possibilities for Phi and Pi relationships

Even if the Egyptians were using numbers that they understood to be the circumference of the circle to its diameter and the golden ratio that appeared in nature, it’s difficult to know if they truly understood the actual decimal representations of pi and phi as we understand them now. Since references to phi don’t appear in the historical record until the time of the Greeks hundreds of years later, some contend that the Egyptians did not have this knowledge and instead used integer approximations that achieved the same relationships and results in the design.
A rather amazing mathematical fact is that pi and the square root of phi can be approximated with a high degree of accuracy using simple integers. Pi can be approximated as 22/7, resulting in a repeating decimal number 3.142857142857… which is different from Pi by only 4/100′s of a percent. The square root of Phi can be approximatey by 14/11, resulting in a repeating decimal number 1.2727…, which is different from Phi by less than 6/100′s of a percent.  That means that Phi can be approximated as 256/121.
The Great Pyramid could thus have been based on 22/7 or 14/11 in the geometry shown about.  Even if the Egyptians only understood pi and/or phi through their integer approximations, the fact that the pyramid uses them shows that there was likely some understanding and intent of their mathematical importance in their application. It’s possible though that the pyramid dimensions could have been intended to represent only one of these numbers, either pi or phi, and the mathematics would have included the other automatically.  We really don’t know with certainty how the pyramid was designed as this knowledge could have existed and then been lost. The builders of such incredible architecture may have had far greater knowledge and sophistication than we may know, and it’s possible that both pi and phi as we understand them today could have been the driving factors in the design of the pyramid.

Construct your own pyramid to the same proportions as the Great Pyramid

Use the template below in gif or pdf format:

Thanks go to Jacques Grimault for these insights, and for other fascinating facts and speculations presented about ancient pyramids in the movie on “The Revelations of the Pyramids.”

The Parthenon and Phi, the Golden Ratio

de bene esse: literally, of well-being, morally acceptable but subject to future validation or exception

The Parthenon in Athens, built by the ancient Greeks from 447 to 438 BC, is regarded by many to illustrate the application of the Golden Ratio in design.  Others, however, debate this and say that the Golden Ratio was not used in its design.  This article will answer that question.
It was not until about 300 BC that the Greek’s knowledge of the Golden Ratio was first documented in the written historical record by Euclid in “Elements.”  It states, “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 less.”

There are several challenges in determining whether the Golden Ratio was used is in the design and construction of the Parthenon:

• The Parthenon was constructed using few straight or parallel lines to make it appear more visually pleasing, a brilliant feat of engineering.
• It is now in ruins, making its original features and height dimension subject to some conjecture.
• Even if the Golden Ratio wasn’t used intentionally in its design, Golden Ratio proportions may still be present as the appearance of the Golden Ratio in nature and the human body influences what humans perceive as aesthetically pleasing.
• Photos of the Parthenon used for the analysis often introduce an element of distortion due to the angle from which they are taken or the optics of the camera used.

To provide better insight into the answer to this question, the photos below selected for photographic analysis are of very high resolution and were taken from an angle that is almost exactly perpendicular to the face of the Parthenon.  The grids overlayed on each of the photos are from PhiMatrix golden ratio software, so each line of the grid is in perfect golden ratio proportion to other grid lines.

As background, there are many geometric constructions of the Golden Ratio, but two of the most familiar are based on a golden rectangle whose ratio of the longer side to the shorter side is 1.618.  Golden rectangles can be divided to create a square and another golden rectangle:

 

 

The photo below shows a Golden Rectangles with a Golden Spiral overlay to the entire face of the Parthenon.  This illustrates that the height and width of the Parthenon conform closely to Golden Ratio proportions.  This construction requires a assumption though, that the bottom of the golden rectangle should align with the bottom of the second step into the structure and that the top should align with a peak of the roof that is projected by the remaining sections.  Given that assumption, the top of the columns and base of the roof line are in a close golden ratio proportion to the height of the Parthenon.  This demonstrates that the Parthenon has golden ratio proportions, but because of the assumptions is probably not strong enough evidence to demonstrate that the ancient Greeks used it intentionally in its overall design, particularly given the exacting precision found in many aspects of its overall design.

 

In the next photo, however, applies golden ratio grid lines to elements of the Parthenon that remaining standing.  The grid lines appear to illustrate golden ratio proportions in these design elements:

• Height of the columns – The structural beam on top of the columns is in a golden ratio proportion to the height of the columns.  Note that each of the grid lines is a golden ratio proportion of the one below it, so the third golden ratio grid line from the bottom to the top at the base of the support beam represents a length that is phi cubed, 0.236, from the top of the beam to the base of the column.
• Dividing line of the root support beam - The structural beam on top of the columns has a horizontal dividing line that is in golden ratio proportion to the height of the support beam.
• Width of the columns – The width of the columns is in a golden ratio proportion formed by the distance from the center line of the columns to the outside of the columns.

A magnified of the above photo view reveals that each of these golden ratio proportions is very close to perfect, but perhaps not as exact as one might hope, particularly given the preciseness of the design and construction of the Parthenon.  Click on the thumbnail image below to see this in more detail.  Are the small variations from perfect golden ratio proportions just a result of angular distortions in the photo or evidence that the golden ratio wasn’t actually used?  A more precise means of measure is required.



The photo below illustrates the golden ratio proportions that appear in the height of the roof support beam and in the decorative rectangular sections that run horizontally across it.  The gold colored grids below are golden rectangles, with a width to height ratio of exactly 1.618 to 1.



The animated photo below provides a closer look yet at the quite precise golden ratio rectangle that appear in the design work above the columns.  This, probably more than any other single feature of the Parthenon, provides rather compelling evidence that the Greeks knew of, and applied, the golden ratio in the construction of the Parthenon.


The photo below illustrates how this section of the Parthenon would have been constructed if other common ratios of 2/3′s or 3/5′s had been intended to be represented by its designers rather than the golden ratio:

If you examine all dimensions of the Parthenon, you’ll find a variety of numbers and proportions.  A floor plan view shows eight columns across the front view and seventeen columns from the side view.  Six columns are the inside entry way, with five by ten columns enclosing the large interior temple room.  Several interior rooms are found, some with proportions that are close to a golden rectangle, but clearly not exactly a golden rectangle.

If the Greeks had intended the Parthenon to highlight the golden ratio in its design, they could have taken advantage of many more opportunities to do so, or done it with the level of exacting precision in the various places that it seems to appear that is found throughout its design and construction.  If, however, the golden ratio was intended to be included among the many numbers and proportions included, then one can find some rather compelling evidence that they applied it, whether through a simple geometry construction below or with the deeper knowledge recorded by Euclid some 150 years later.

The Phi in Human Beauty

de bene esse: literally, of well-being, morally acceptable but subject to future validation or exception

A template for human beauty is found in phi and the pentagon

Dr. Stephen Marquardt has studied human beauty for years in his practice of oral and maxillofacial surgery.   Dr. Marquardt performed cross-cultural surveys on beauty and found that all groups had the same perceptions of facial beauty.  He also analyzed the human face from ancient times to the modern day.  Through his research, he discovered that beauty is not only related to phi, but can be defined for both genders and for all races, cultures and eras with the beauty mask which he developed and patented. This mask uses the pentagon and decagon as its foundation, which embody phi in all their dimensions.

1350 B.C. Egypt	500 B.C. Greece

164 A.D. Rome	1794 A.D.

Sojourner Truth

de bene esse: literally, of well-being, morally acceptable but subject to future validation or exception

On this day 162 years ago, anti-slavery and woman's rights leader Sojourner Truth would deliver her "Ain't I a Woman?" speech, saying, in part:

" Well, children, where there is so much racket there must be something out of kilter. I think that 'twixt the negroes of the South and the women at the North, all talking about rights, the white men will be in a fix pretty soon. But what's all this here talking about?

"That man over there says that women need to be helped into carriages, and lifted over ditches, and to have the best place everywhere. Nobody ever helps me into carriages, or over mud-puddles, or gives me any best place!"

Portrait of JFK

de bene esse: literally, of well-being, morally acceptable but subject to future validation or exception

President John F. Kennedy was born on this date, May 29, in 1917. He would have been 96 years old today.

Icon of American prep, JFK. http://huff.to/117xWbS

(Photo credit: U.S. National Archives/Wikimedia Commons)

JFKhttp://huff.to/13YQxeK

Photo credit: Getty Images

Photos from his historic 1960 campaign: http://ti.me/MpoCcV

(Paul Schutzer—Time & Life Pictures/Getty Images)

This 1960 portrait is the work of Boris Chaliapin (1904-1979) from the collection of the National Portrait Gallery, Smithsonian Institution, a gift of TIME magazine. The work is a graphite on illustration board.

More about the image: http://bit.ly/10zCcVk
More artwork from the exhibition: http://bit.ly/ZdXDtS