Golden Ratio in Art and Architecture (2024)

According to NCTM's Principles and Standard for School Mathematics(2000),rich problems contexts involve connections to other disciplines,(e.g. science, social studies, art) as well as to the real worldand to the daily life experience of middle-grades students(NCTM,2000, p.374).
An exploration with the golden ratio offers opportunities to connectan understanding the conceptions of ratio and proportion to geometry.The mathematical connections between geometry and algebra canbe highlighted by connecting Phi to the Fibonacci numbers andsome golden figures.
Also, the golden ratio is a good topic to introduce historic andaesthetic elements to a mathematical concept, because we can findthat not a few artists and architects were connected with thegolden ratio of their works through much of the art history.

The one of purposes of this project is to overview the goldenratio briefly. The other is to introduce the occurrences of thegolden ratio in art and architecture.

The content includes the following :

I.A discovery of the Golden Ratio
A. A brief history of the Golden Ratio
B. Definitions of the Golden Ratio related to Fibonacci sequencenumber
II. Some Golden Geometry
III. The Golden Ratio in Art and Architecture
IV. Resources
I. A discove ry of Golden Ratio
A. A brief history of Golden Ratio

There are many different names for the golden ratio; The GoldenMean, Phi, the Divine Section, The Golden Cut, The Golden Proportion,The Divine Proportion, and tau(t).

The Great Pyramid of Giza built around 2560 BC is one of theearliest examples of the use of the golden ratio. The length ofeach side of the base is 756 feet, and the height is 481 feet.So, we can find that the ratio of the vase to height is 756/481=1.5717..The Rhind Papyrus of about 1650 BC includes the solution to someproblems about pyramids, but it does not mention anything aboutthe golden ratio Phi.

Euclid (365BC - 300BC) in his "Elements" calls dividinga line at the 0.6180399.. point dividing a line in the extremeand mean ratio. This later gave rise to the name Golden Mean.He used this phrase to mean the ratio of the smaller part of thisline, GB to the larger part AG (GB/AG) is the same as the ratioof the larger part, AG, to the whole line AB (AG/AB).Then thedefinition means that GB/AG = AG/AB.
­ proposition 30 in book VI

Plato, a Greek philosopher theorised about the Golden Ratio.He believed that if a line was divided into two unequal segmentsso that the smaller segment was related to the larger in the sameway that the larger segment was related to the whole, what wouldresult would be a special proportional relationship.

Luca Pacioli wrote a book called De Divina Proportione (TheDivine Proportion) in 1509. It contains drawings made by Leonardoda Vinci of the 5 Platonic solids. Leonardo Da Vinci first calledit the sectio aurea (Latin for the golden section).

Today, mathematicians also use the initial letter of the GreekPhidias who used the golden ratio in his sculptures.

B. Definitions of Golden Ratio

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ..

If we take the ratio of two successive numbers in this seriesand divide each by the number before it, we will find the followingseries of numbers.

1/1 = 1
2/1 = 2
3/2 = 1.5
5/3 = 1.6666...
8/5 = 1.6
13/8 = 1.625
21/13 = 1.61538...
34/21 = 1.61904...

The ratio seems to be settling down to a particular value,which we call the golden ratio(Phi=1.618..).

2) Geometric definition

We can notice if we have a 1 by 1 square and add a square withside lengths equal to the length longer rectangle side, then whatremains is another golden rectangle. This could go on forever.We can get bigger and bigger golden rectangles, adding off thesebig squares.

Step 1 Start with a square 1 by 1
Step 2 Find the longer side
Step 3 Add another square of that side to whole thing

Here is the list we can get adding the square;
1 x 1, 2 x 1, 3 x 2, 5 x 3, 8 x 5, 13 x 8, 21 x 13, 34 x 21.
with each addition coming ever closer to multiplying by Phi.

start 1 by 1, add 1 by 1 => Now, it is 2 by 1, add 2 by2

Now, it is 3 by 2, add 3 by 3 => Now, it is 5 by 3, add5 by 5

Now, it is 8 by 5.

3) Algebraic and Geometric definition

We can realize that Phi + 1 = Phi * Phi.

Start with a golden rectangle with a short side one unit long.
Since the long side of a golden rectangle equals the short sidemultiplied by Phi, the long side of the new rectangle is 1*Phi= Phi.

If we swing the long side to make a new golden rectangle, theshort side of the new rectangle is Phi and the long side is Phi* Phi.

We also know from simple geometry that the new long side equalsthe sum of the two sides of the original rectangle, or Phi + 1.(figure in page4)

Since these two expressions describe the same thing, they areequivalent, and so
Phi + 1 = Phi * Phi.

II. Some Golden Geometry

1) The Golden Rectangle

A Golden Rectangle is a rectangle with proportions that aretwo consecutive numbers from the Fibonacci sequence.

The Golden Rectangle has been said to be one of the most visuallysatisfying of all
geometric forms. We can find many examples in art masterpiecessuch as in edifices of ancient Greece.

2) The Golden Triangle

If we rotate the shorter side through the base angle untilit touches one of the legs, and then, from the endpoint, we drawa segment down to the opposite base vertex, the original isoscelestriangle is split into two golden triangles. Aslo, we can findthat the ratio of the area of the taller triangle to that of thesmaller triangle is also 1.618. (=Phi)

The Golden Spiral above is created by making adjacent squaresof Fibonacci dimensions and is based on the pattern of squaresthat can be constructed with the golden rectangle.
If you take one point, and then a second point one-quarter ofa turn away from it, the second point is Phi times farther fromthe center than the first point. The spiral increases by a factorof Phi.

This shape is found in many shells, particularly the nautilus.

4) Penrose Tilings

The British physicist and mathematician, Roger Penrose,has developed an aperiodic tiling which incorporates the goldensection. The tiling is comprised of two rhombi, one with anglesof 36 and 144 degrees (figure A, which is two Golden Triangles,base to base) and one with angles of 72 and 108 degrees (figureB).

When a plane is tiled according to Penrose's directions, theratio of tile A to tile B is the Golden Ratio.

In addition to the unusual symmetry, Penrose tilings reveala pattern of overlapping decagons. Each tile within the patternis contained within one of two types of decagons, and the ratioof the decagon populations is, of course, the ratio of the GoldenMean.

5) Pentagon and Pentagram

We can see there are lots of lines divided in the goldenratio. Such lines appear in the pentagon and the relationshipbetween its sides and the diagonals.

We can get an approximate pentagon and pentagram using theFibonacci numbers as lengths of lines. In above figure, thereare the Fibonacci numbers; 2, 3, 5, 8. The ratio of these threepairs of consecutive Fibonacci numbers is roughly equal to thegolden ratio.
III. Golden Ratio in Art and Architecture

A. Golden Ratio in Art

1) An Old man by Leonardo Da Vinci

Leonardo Da Vinci explored the human body involvingin the ratios of the lengths of various body parts. He calledthis ratio the "divine proportion" and featured it inmany of his paintings.

Leonardo da Vinci's drawing of an old man can be overlaid witha square subdivided into rectangles, some of which approximateGolden Rectangles.

2) The Vetruvian Man"(The Man in Action)"by Leonardo Da Vinci

We can draw many lines of the rectangles into this figure.
Then, there are three distinct sets of Golden Rectangles: Eachone set for the head area, the torso, and the legs.

3) Mona-Risa by Leonardo Da Vinci

This picture includes lots of Golden Rectangles. In above figure,we can draw a rectangle whose base extends from the woman's rightwrist to her left elbow and extend the rectangle vertically untilit reaches the very top of her head. Then we will have a goldenrectangle.
Also, if we draw squares inside this Golden Rectangle, we willdiscover that the edges of these new squares come to all the importantfocal points of the woman: her chin, her eye, her nose, and theupturned corner of her mysterious mouth.
It is believed that Leonardo, as a mathematician tried to incorporateof mathematics into art. This painting seems to be made purposefullyline up with golden rectangle.

4) Holy Family by Micahelangelo

We can notice that this picture is positioned to the principalfigures in alignment with a Pentagram or Golden star.

5) Crucifixion by Raphael

his picture is a well-known example, in which we can find aGolden Triangle and also Pentagram. In this picture, a goldentriangle can be used to locate one of its underlying pentagrams.

6) self-portrait by Rembrandt

We can draw three straight lines into this figure. Then, theimage of the feature is included into a triangle. Moreover, ifa perpendicular line would be dropped from the apex of the triangleto the base, the triangle would cut the base in Golden Section.

7) The sacrament of the Last Supperby Salvador Dali(1904-1989)

This picture is painted inside a Golden Rectangle. Also, wecan find part of an enormous dodecahedron above the table. Sincethe polyhedron consists of 12 regular Pentagons, it is closelyconnected to the golden section.

The title of this work itself includes the Golden Section.It simply means that it is cut into sections of Golden Proportion.

9) Bathers by Seurat

Seurat attached most of canvas by the Golden Section. Thispicture has several golden subdivisions.
10) Composition with Gray and Light Brownby Piet Mondrian 1918

Mondrian believed that mathematics and art were closely connected.He used the simplest geometrical shapes and primary colours (blue,red, yellow).
His point of view lies in the fact that any shape is possibleto create with basic geometric shapes as well as any color canbe created with different combinations of red, blue, and yellow.The golden rectangle is one of the basic shapes appear in Mondrian'sart.

Composition in Red, Yellow, and Blue(1926)

We can find that the ratio of length to width for some rectanglesis Phi.
B. Golden Ratio in Architecture

1) The Great Pyramid

The Ahmes papyrus of Egypt gives an account of the buildingof the Great Pyramid of Giaz in 4700 B.C. with proportions accordingto a "sacred ratio."

2) Parthenon

The Greek sculptor Phidias sculpted many things includingthe bands of sculpture that run above the columns of the Parthenon.

Even from the time of the Greeks, a rectangle whose sides arein the "golden proportion" has been known since it occursnaturally in some of the proportions of the Five Platonic. Thisrectangle is supposed to appear in many of the proportions ofthat famous ancient Greek temple in the Acropolis in Athens, Greece.

3) Porch of Maidens, Acropolis, Athens

4) Chartres Cathedral

The Medieval builders of churches and cathedrals approachedthe design of their buildings in much the same way as the Greeks.They tried to connect geometry and art.
Inside and out, their building were intricate construction basedon the golden section.

5) Le Corbussier

In 1950, the architect Le Corbussier published a book entitled"Le modulator. Essai
sur une mesure harmonique a l'echelle humaine applicable universalementa l'architecture et a la mecanique ". He invented the word"modulator" by combining "modul" (ratio) and"or" (gold); another expression for the well-known goldenratio.

III. Resoureces

Internet

Michael's Crazy Enterprises, Inc., The Golden Mean
(http://www.vashti.net/mceinc/)

The Golden Ratio
(http://www.math.csusb.edu/course/m128/golden/)

Ron Knott, The Golden section ratio : Phi
(http://www.ee.surrey.ac.uk/Personal/R.Knott/)

The Golden Ratio
(http://library.thinkquest.org/C005c449/)

Ron Knott, Fibonacci Numbers and Nature-part 2, Why is theGolden section the "best" arrangement?
(http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/)

Ron Knott, The Golden Section in Art, Architecture and Music
(http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/)

Steve Blacker, Jeantte Polanski, and Marc Schwach, The GoldenRatio
(http://www.geom.umn.edu/~demo5337/s97b/)

Ethan, The relations of the Golden ratio and the FibonacciSeries
(http://mathforum.org/dr.math/problems)

Golden Section in Art and Architecture
(http://www.camosun.bc.ca/~jbritton/goldslide/)

Sheri Davis and Danny Rhee, Mathematical Aspects of Arichitecture
(http://www.ma.uyexas.edu/~lefcourt/SP97/M302/projects/lefc023/)

Mathematics and Art
(http://www.q-net.au/~lolita/)

Leonardo da Vinci
(http://libray.thinkquest.org/27890/)

Math & Art : The golden Rectangle
(http://educ.queensu.ca/~fmc/october2001/)

Sue Meredith, Some Explorations with the Golden Ratio
( http://jwilson.coe.uga.edu/EMT668/)

What is a Fractal?
(http://ecsd2.re50j.k12.co.us/ECSD/)

Ron Knott, Phi's Fascinating Figures
(http://www.euler.slu.edu/teachmaterial/)

Cynthia Lanius, Golden ratio Algebra
(http://math.rice.edu/~lanius/)

Newsletter, Mathematical Beauty
(http://www.exploremath.com/news/

Some Golden Geometry
(http://galaxy.cau.edu/tsmith/KW/)

Book
Robert L. (1989). Scared goemetry: philosophy and practice, NewYork: Thames
and Hudson.

Article

Donald, T. S. (1986). The Geometric Figure Relating the GoldenRatio and Phi,
Mathematics Teacher 79, 340-341.

Edwin, M. D. (1993). The Golden Ratio: A good opportunity toinvestigate multiple
representations of a problem, Mathematics Teacher 86, 554-557.

Susan, M. P. (1982). The Golden Ratio in Geometry, E. M. Maletsky,C. Hirsch, & D.
Yates(Eds.), Mathematics Teacher 75, 672-676.

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