Phi appears in many basic geometric constructions.

3 lines:

Take 3 equal lines.  Lay the 2nd line against the midpoint of the 1st.  Lay the 3rd line against the midpoint of the 2nd.  The ratio of AG to AB is Phi, the Golden Ratio. (Contributed by Jo Niemeyer)

3 sides: Triangle

Insert an equilateral triangle inside a circle, add a line at the midpoint of the two sides and extend that line to the circle.  The ratio of AG to AB is Phi.

4 sides: Square

Insert a square inside a semi-circle.  The ratio of AG to AB is Phi.

5 sides: Pentagon

Insert a pentagon inside a circle.  Connect three of the five points to cut one line into three sections. The ratio of AG to AB is Phi.

When the basic phi relationships are used to create a right triangle, it forms the dimensions of the great pyramids of Egypt, with the geometry shown below creating an angle of 51.83 degrees, the cosine of which is phi, or 0.618.

A ruler and compass can be used to construct the “golden rectangle,” as shown by the animations below, which was used by the Greeks in the Parthenon.   (See also the Orthogons page.)

Phi also defines other dimensions of a pentagon.

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If you sum the squares of any series of Fibonacci numbers, they will equal the last Fibonacci number used in the series times the next Fibonacci number.  This property results in the Fibonacci spiral, based on the following progression and properties of the Fibonacci series:

1² + 1² + 2² + 3² + 5² = 5 x 8

 1² + 1² + . . . + F(n)² = F(n) x F(n+1)

A Golden spiral is very similar to the Fibonacci spiral but is based on a series of identically proportioned golden rectangles, each having a golden ratio of 1.618 of the length of the long side to that of the short side of the rectangle:

The Fibonacci spiral gets closer and closer to a Golden Spiral as it increases in size because of the ratio of each number in the Fibonacci series to the one before it converges on Phi, 1.618, as the series progresses (e.g., 1, 1, 2, 3, 5, 8 and 13 produce ratios of 1, 2, 1.5, 1.67, 1.6 and 1.625, respectively)

Fibonacci spirals and Golden spirals appear in nature, but not every spiral in nature is related to Fibonacci numbers or Phi.  Most spirals in nature are equiangular spirals, and Fibonacci and Golden spirals are special cases of the broader class of Equiangular spirals.  An Equiangular spiral itself is a special type of spiral with unique mathematical properties in which the size of the spiral increases but its shape remains the same with each successive rotation of its curve.  The curve of an equiangular spiral has a constant angle between a line from origin to any point on the curve and the tangent at that point, hence its name.  In nature, equiangular spirals occur simply because the forces that create the spiral are in equilibrium, and are often seen in non-living examples such as spiral arms of galaxies and the spirals of hurricanes.  Fibonacci spirals, Golden spirals and golden ratio-based spirals often appear in living organisms.

Alternate spirals in plants occur in Fibonacci numbers.

The most common appearances of a Fibonacci numbers in nature are in plants, in the numbers of leaves, the arrangement of leaves around the stem and in the positioning of leaves, sections and seeds.

Here a sunflower seed illustrates this principal as the number of clockwise spirals is 55 (marked in red, with every tenth one in white) and the number of counterclockwise spirals is 89 (marked in green, with every tenth one in white.)

Pinecones and pineapples illustrate similar spirals of successive Fibonacci numbers, with the example below showing the alternating pattern of 8 and 13 spirals in a pine cone.

Click on an image below to see the full size versions of each image above:

Golden spirals in sea shells

Golden ratios are also sometimes found in the proportions of successive spirals of a sea shell, as shown below.

The Nautilus shell spiral is not a Golden spiral but often still has Golden Ratio proportions.

The nautilus shell is often shown as an illustration of the golden ratio in nature, but the spiral of a nautilus shell is NOT a golden spiral, as illustrated below.  The golden spiral overlay is provided by PhiMatrix golden ratio software:

Nautilus shell spiral compared to a Golden Spiral

The Nautilus spiral, however, while not a Golden spiral, often displays proportions its dimensions that are close to a golden ratio, appearing in successive rotations of the shell as the Nautilus grows.  As with all living organisms, there is variation in the dimensions of individuals, so the appearance of the golden ratio is not universal.

See also other examples and explanations of the golden ratio in the nautilus spiral.

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A mandala is a geometric design, often symbolic of the universe and used in Eastern religions as an aid to meditation.

Phi-based geometric shapes can be used to create some interesting phi mandalas, embodying the phi proportion found throughout creation and iterating into the infinitely large and the infinitely small, much as the universe itself.

Phi Corbett contributed the mandala design below (inspired by Bruce Rawles’ Sacred Geometry Page) which uses the pentagram, also based on phi, to collapse into an infinite series of smaller pentagrams. (Download PDF version)

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Areas can be filled completely and symmetrically with tiles of 3, 4 and 6 sides, but it was long believed that it was impossible to fill an area with 5-fold symmetry, as shown below:

The solution was found in Phi, the Golden Ratio

In the early 1970’s, however, Roger Penrose discovered that a surface can be completely tiled in an asymmetrical, non-repeating manner in five-fold symmetry with just two shapes based on phi, now known as “Penrose tiles.”

This is accomplished by creating a set of two symmetrical tiles, each of which is the combination of the two triangles found in the geometry of the pentagon.

Phi plays a pivotal role in these constructions.  The relationship of the sides of the pentagon, and also the tiles, is Phi, 1 and 1/Phi.

The triangle shapes found within a pentagon are combined in pairs.	One creates a set of tiles, called “kites” and “darts” like this:	The other creates a set of diamond tiles like this:

 

The ratio of the two types of tiles in the resulting patterns is always phi!

As you expand the tiling to cover greater areas, the ratio of the quantity of the one type of tile to the other always approaches phi, or 1.6180339…, the Golden Ratio.

Within this tiling there can be small areas of five-fold symmetry. Decagons can also occur, which when grouped together can look like pentagons from a distance.

Explore further with these resources:

You can download a free program called “Bob” to generate Penrose tiling like the above at the site of Stephen Collins.

You can buy acrylic puzzles of Penrose darts, kites and tiles and other pentagon-based shapes at Kadon Enterprises.

Create Penrose tiling with an L-System fractal generator. (See more on L-System) The input variables shown on the left create the pattern on the right. Note that the 36 degree angle is based on 360 degrees divided by 5 and then by 2, which relates it to the five-sided symmetry of Penrose tiling.

Quasi-Crystals

Phi also gives 5-fold symmetry in 3D with a single shape, known as a quasi-crystal.

Phi is intrinsically related to the number 5

The appearance of the golden ratio in examples of five-fold symmetry occurs because phi itself is intrinsically related to the number 5, mathematically and trigonometrically.

• A 360 degree circle divided into five equal sections produces a 72 degree angle, and the cosine of 72 degrees is 0.3090169944, which is exactly one half of phi, the reciprocal of Phi, or 0.6180339887.

• Phi itself is computed using the square root of five, as follows:

5 ^ .5 * .5 + .5 = Phi

In this mathematical construction, “5 ^ .5” means “5 raised to the 1/2 power,” which is the square root of 5, which is then multiplied by .5 and to which .5 is then added.  See more on the Five and Phi page.

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Even before the foundations of the Great Pyramids were laid men have tried to “square the circle.” That is, in a finite number of steps, construct a square and a circle that are precisely equal in area using only the most primitive instruments; namely, an unmarked compass & straightedge. Some of the greatest men in all of history have attempted to solve this ancient riddle. They have included mathematicians, architects, politicians, artists, musicians, philosophers, astronomers and theologians.

The task was finally “proven impossible” in 1882 when Lindemann showed that pi was a transcendental number. In other words, it cannot be calculated as the root of a polynomial equation with rational coeffecients. Hence, the decimal values of pi are infinite, and since it is not possible to construct the square root of an infinite number, it is therefore “impossible” to square the circle with exact precision. One can only hope to come close.

Christopher Ricci has recently discovered an elegant method which comes about as close as it gets. The technique is extraordinary in that it employs a royal parade of three successive Phi constructions that ultimately converge on the same ratio attained by the well known equation: Phi Squared/5 = Pi/6. The procedure is outlined below.

Download the Squaring the Circle with the Golden Ratio pdf file or visit The Circle is Squared to explore the steps at your own leisure.

If we consider the Red Square as a unit square (Side = 1; Area = 1), the following calculations will result:

Golden Square: Side = Phi (1.618033988); Area = Phi Squared (2.618033986).

Golden Circle: Radius = (0.91287093); Radius Squared = (.833333334). Area = (2.61799388).

With respect to the area, there is virtually no difference between these two shapes. Measured in inches the difference is literally microscopic. And even if we convert them into square feet, the difference would remain undetectable by the naked eye. The area of the Golden Circle subtracted from the area of the Golden Square would be a miniscule .0057751 square inches. Converted to metric = a little over 144 sq. microns. This would enclose an area of 12.111 X 12.111 microns; which is roughly the size of two red blood cells.

As far as linear measurement is concerned, this construction yields a very tight approximation for pi as well; (3.141640784). [Note: The math for this is located on Figure #13 in the pdf file]. This is 99.85% accurate for true pi.  To illustrate just how significant this is we would need to enlarge the shapes astronomically. Imagine, for example, you have a planet with a diameter of a thousand miles. According to pi it would take a car racing along at 60 mph more than 52 hours & 21 minutes to circumnavigate the globe at its equator. If we were to extrapolate our travel time using Phi instead, the difference between the two times would be less than three seconds! Now that’s impressive no matter how you slice it.

The fact that we can attain such a high degree of precision without the aid of modern tools and in so few steps sets this construction apart from some of even the most ingenious techniques. If you have any comments or would like to discuss this further with Chris, you may contact him at Ricci1.1107207@yahoo.com

Thanks go to for Chris Ricci for his passion and dedication in developing this innovative response to a classic geometric challenge, finding another way to relate phi to pi and for sharing it first with GoldenNumber.net.

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Here’s a challenge to “all the real mathematicians in the back row,” as my college professor often said:

1. Picture the classic solids of geometry, each sitting inside a cube that encloses it on all sides.  What is the ratio of surface area of this tangent cube to the surface area of the solid, and which solid results in a ratio that is within 1% of phi?
2. Which solid has a silhouette projection from the x, y and z axes of a cube, a sphere and a convex parallelogram?

Give up?  Enter the DOR (the Direct Opposite Reverse), at TheDOR.net, a geometric solid discovered by David P. Sterner that is the answer to both of these questions.  Sterner sees the DOR as the missing geometric link, a new shape in geometry’s basic set of solids (cubes, cones and cylinders, etc.) that haven’t had a new member since the time of Euclid before 200 B.C.

The patented refractor lens of the DOR creates images that when printed directly to photographic paper create two opposite images, the normal inverse image created by any convex lens but also a positive image of the original subject matter in its true orientation.

The images photographed through the DOR also have an appearance of depth:

You can construct your own 3D model of the DOR using the template below, which is available in a DOR template PDF download.

 

When constructed, the model has these views:

Circle / Sphere view	Square / Cube view
Third / Convex view	3D view of all three

Now picture the various geometric solids sitting inside a tangent cube, that is a cube to which all sides of the solid are touching.  The area of the solid to the area of its tangent cube is below, and only the DOR is close to phi, 0.618.  Can anyone create a solid as simple in construction but with a tangent cube that is as close to phi, or closer yet?  If so, contact David Sterner with your discovery.

Solid	Solid Image	Surface Area Formula of Solid	Ratio of Surface Area of Solid to its Tangent Cube
DOR			0.6086903
Cone		pr² + p( r x s )wheres = √( r² + h² )	0.4236003…
Dodecahedron		3a² Ö(25+10Ö5)where a = length of an edge, with a width of two times phi and edge of 2/phi.	0.50202854
Pyramid		(½ x P x s ) + A where A = area of the base shape P = perimeter of base shape s = height of face triangles	0.5393446…
Sphere		4pr²	0.5235987…
Cylinder		2pr² + 2prh	0.7853981…
Prism		2A + Pdwhere A = area of the base shape P = perimeter of base shape d = height of prism	1.0000000

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√5 + 1
2

This mathematical expression can be expressed geometrically as shown below:

Three circle construction:

Put three circles with a diameter of 1 (AB and DE) side by side and construct a triangle that connects the bottoms of the outside circles (AC) and the top and bottom of the outside circles (BC).  The dimensions are as follows:

AB = 1

AC = 2

BC = √5

DE = 1

The line BC thus expresses the following embedded phi relationships:

BE = DC = (√5-1)/2+1  = (√5+1)/2 = 1.618 … = Phi

BD = EC = (√5-1)/2 = 0.618… = phi

This simple and elegant way of expressing the most standard mathematical expression of Phi was discovered and contributed by Bengt Erik Erlandsen on 1/11/2006.

Construction by compass and ruler:

Scott Beach developed a way to represent this calculation of phi in a geometric construction:

As Scott shares on his web site:

Triangle ABC is a right triangle, where the measure of angle BAC is 90 degrees. The length of side AB is 1 and the length of side AC is 2. The Pythagorean theorem can be used to determine that the length of side BC is the square root of 5. Side BC can be extended by 1 unit of length to establish point D. Line segment DC can then be bisected (divided by 2) to establish point E.

The length of line segment EC is equal to Phi (1.618 …).

The length of line segment DB is 1 and the length of line segment BE is the reciprocal of Phi (0.6180…).

Taken together, DB and BE constitute a graphic representation of the Golden Ratio.

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Most crystals in nature, such as those in sugar, salt or diamonds, are symmetrical and all have the same orientation throughout the entire crystal.  Quasicrystals represent a new state of matter that was not expected to be found, with some properties of crystals and others of non-crystalline matter, such as glass.

With five-fold symmetry, once thought to be impossible, they were first observed in 1982 in an aluminiun-manganese alloy (Al₆Mn).  Since then, quasicrystals have been found in other substances.

Quasi-crystals fill space with five-fold symmetry based on phi.

Penrose tiles allow a two-dimensional area to be filled in five-fold symmetry, using two shapes based on phi.  It was thought that filling a three-dimensional space in five-fold symmetry was impossible, but the answer was again found in phi.

Where the solution in 2D required two shapes, this can be accomplished in 3D with just one shape.  The shape has six sides, each one a diamond whose diagonals are in the ratio of phi:

The resulting solid looks like this:

Download and print this to make your own!

A Nobel Prize for Quasi-Crystal Discovery was awarded in 2011.

As reported in “2011 Nobel Prize in Chemistry: ‘Quasicrystals’ once thought impossible have changed understanding of solid matter” by ScienceDaily on October 14, 2011 athttp://www.sciencedaily.com/releases/2011/10/111005080232.htm:

The Royal Swedish Academy of Sciences has decided to award the Nobel Prize in Chemistry for 2011 to Daniel Shechtman of the Technion — Israel Institute of Technology in Haifa, Israel, for the discovery of quasicrystals: non-repeating regular patterns of atoms that were once thought to be impossible.

In quasicrystals, we find the fascinating mosaics of the Arabic world reproduced at the level of atoms: regular patterns that never repeat themselves. However, the configuration found in quasicrystals was considered impossible, and Daniel Shechtman had to fight a fierce battle against established science. The Nobel Prize in Chemistry 2011 recognizes a breakthrough that has fundamentally altered how chemists conceive of solid matter.

On the morning of April 8, 1982, an image counter to the laws of nature appeared in Daniel Shechtman’s electron microscope. In all solid matter, atoms were believed to be packed inside crystals in symmetrical patterns that were repeated periodically over and over again. For scientists, this repetition was required in order to obtain a crystal.

Shechtman’s image, however, showed that the atoms in his crystal were packed in a pattern that could not be repeated. Such a pattern was considered just as impossible as creating a football using only six-cornered polygons, when a sphere needs both five- and six-cornered polygons. His discovery was extremely controversial. In the course of defending his findings, he was asked to leave his research group. However, his battle eventually forced scientists to reconsider their conception of the very nature of matter.
Aperiodic mosaics, such as those found in the medieval Islamic mosaics of the Alhambra Palace in Spain and the Darb-i Imam Shrine in Iran, have helped scientists understand what quasicrystals look like at the atomic level. In those mosaics, as in quasicrystals, the patterns are regular — they follow mathematical rules — but they never repeat themselves.

When scientists describe Shechtman’s quasicrystals, they use a concept that comes from mathematics and art: the golden ratio. This number had already caught the interest of mathematicians in Ancient Greece, as it often appeared in geometry. In quasicrystals, for instance, the ratio of various distances between atoms is related to the golden mean.

Following Shechtman’s discovery, scientists have produced other kinds of quasicrystals in the lab and discovered naturally occurring quasicrystals in mineral samples from a Russian river. A Swedish company has also found quasicrystals in a certain form of steel, where the crystals reinforce the material like armor. Scientists are currently experimenting with using quasicrystals in different products such as frying pans and diesel engines.

History of the findings

These findings came about as follows:

• In the mid-1970s the mathematician Roger Penrose created an aperiodic mosaic, with a pattern that never repeats itself, using only two different tiles: one fat rhomboid and one thin rhomboid. He called these kites and darts and named this finding Penrose tiles.
• In 1982, Dan Shechtman captured a picture with an electronic microscope that seemed counter to all logic. The ten bright dots in each circle revealed that he was seeing tenfold symmetry. Conventional wisdom said that this was against the laws of nature.
• In 1982, Alan Mackay experimented with a model in which circles representing atoms were placed at intersections in Penrose’s mosaic. He illuminated the model and found a tenfold diffraction pattern.
• In 1984, Paul Steinhardt and Dov Levine connected Mackay’s model with Shechtman’s actual diffraction pattern. They realized that aperiodic mosaics helped to explain Shechtman’s unusual crystals.

Implications for other geometries

One of the most amazing implications of this property is that it’s not just 5-fold symmetry that is made possible. The unexpected find is that with quasiperiodicity, a whole new class of solids is possible! Any symmetry in any number of dimensions becomes attainable! Here are some examples of other symmetries from a presentation by P.J. Steinhardt titled “What are quasicrystals?”

References:

https://www.nobelprize.org/nobel_prizes/chemistry/laureates/2011/popular-chemistryprize2011.pdf

http://www.i-sis.org.uk/Golden_Mean_Wins_Chemistry_Nobe_Prize.php

http://www.physics.princeton.edu/~steinh/QuasiIntro.ppt

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The golden section can be constructed from a square with a compass and ruler:

This is the most commonly known of twelve orthogons which can be constructed using this technique.  Among orthogons, the golden section is known as the auron, coming from the root “aur,” meaning gold.

Orthogons provide a system of design that for centuries has allowed artists and artisans to create consistent, harmonious themes without the need for complicated calculations or measuring devices.  Examples of orthogons include:

Auron (Golden Section)	Diagon	Quadriagon	Hemidiagon
1/2 + √5/2 =1.618 … = Phi	√2 =1.414…	1/2 + √2/2 =1.207…	√5/2 =1.118…
Height to width ratio

Insights on orthogons provided by Valrie Jensen.  See additional information on the application of orthogons to the principles of design at her site,  Timeless by Design.

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Pythagoras discovered that a right triangle with sides of length a and b and a hypotenuse of length c has the following relationship:

a² + b² = c²

A foundational equality of phi has a similar structure:

1 + Phi = Phi²

^{( 1+ 1.618… = 2.618… )}

By taking the square root of each term in this equality, we have the dimensions of a triangle, known as a Kepler Triangle, a right triangle based on this phi equality, where:

This triangle is illustrated below.  It has an angle of 51.83° (or 51°50′), which has a cosine of 0.618 or phi.

The Pythagorean 3-4-5 triangle is the only right-angle triangle whose sides are in an arithmetic progression. 3 + 1 = 4, and 4 plus 1 = 5. The Kepler triangle is the only right-angle triangle whose side are in a geometric progression: The square root of phi times Φ = 1 and 1 times Φ = Φ.

Although difficult to prove with certainty due to deterioration through the ages, this angle is believed by some to have been used by the ancient Egyptians in the construction of the Great Pyramid of Cheops.

Other triangles with Golden Ratio proportions can be created with a Phi (1.618 0339 …) to 1 relationship of the base and sides of triangles:

The isosceles triangle above on the right with a base of 1 two equal sides of Phi is known as a Golden Triangle.  These familiar triangles are found embodied in pentagrams and Penrose tiles.

No three successive numbers in the Fibonacci series can be used to create a right triangle.  Marty Stange, however, contributed the following relationship in January 2007:  Every successive series of four Fibonacci numbers can be used to create a right triangle, with the base and hypotenuse being determined by the second and third numbers, and the other side being the square root of the product of the first and fourth numbers.  The table below shows how this relationship works:

Thus for the illustration highlighted in gold, Stange’s Treatise on Fibonacci Triangles reveals that a triangle with sides of 5 and the square root of 39 (e.g., 3 x 13) will produce a right triangle with a hypotenuse of 8.

As greater numbers in the series are used, the triangle approaches the proportions of the phi-based Kepler Triangle above, with a ratio of the hypotenuse to the base of Phi, or 1.618…

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