4.3 the s rectangle

258 Geometric Gems

Some Scenarios Involving Chance Th at Confound Our IntuitionFinding Aesthetics in Life, Art, and Math Th rough the Golden Rectangle

Geometry has two great trea-sures: one is the theorem of

Pythagoras; the other, the division of a line into extreme and mean

ratio. Th e fi rst we may compare to a measure of gold; the second we

may name a precious jewel.

JOHANNES KEPLER

Are you into it?

On our journeys through various mathematical landscapes we have become conscious of the issue of aesthetics—in particular, the intrinsic beauty of mathematical truths. We’re discovering that mathematics is not just a collection of formulas tied together by algebra but is instead a wealth of creative ideas that allows us to investigate, explore, and dis-cover new realms. Now, however, we wonder if mathematics can be used to discover structure behind the aesthetics of art and nature.

4.3 THE SEXIEST RECTANGLE

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4.3 / Th e Sexiest Rectangle 259

Rectangular AppealIn our discussion of Fibonacci numbers we asked the following geometri-cal question that begs to be asked again: What are the dimensions of the most attractive rectangle—the rectangle we might imagine when we close our eyes on a dark starry night and dream of the ideal rectangle? When someone says rectangle, we think of a shape. What shape is it? From the rectangles given here, choose the one you fi nd most appealing:

Given these choices, a high percentage of people think that the second rectangle from the left is the most aesthetically pleasing—the one that captures the true spirit of rectangleness. That rectangle is referred to as the Golden Rectangle. It is the length of the base relative to the length of the height that makes it a Golden Rectangle.

What precisely is the ratio of base to height that produces the Golden Rectangle? Recall that, in our conversations about numbers, we found a ratio that was an especially attractive number. The ratio arose in our dis-cussions of the Fibonacci numbers, and we denoted it by the Greek letter phi, ϕ. It was called the Golden Ratio because it satisfi ed the symmetrical equation of ratios:

ϕϕ1

11

��

.

Specifi cally, we found that the Golden Ratio, ϕ, is the number ( ) /1 5 2+ = 1.618 . . .

You may want to glance back at the Fibonacci discussion in Section 2.2 and revisit the relationship ϕ/1 � 1/(ϕ � 1). (The Greek letter ϕ used to denote the Golden Ratio was introduced in the past century to honor the famous ancient Greek sculptor Phidias, much of whose work appears to involve the Golden Ratio.)

The Golden Ratio gives us the satisfying relationship of height to width for those rectangles that many deem extremely pleasing to the eye. The precise mathematical defi nition of a Golden Rectangle is any rectangle having base b and height h such that

b

h= =

+ϕ

1 5

2.

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260 Geometric Gems

We have already discovered how the Fibonacci numbers and the Golden Ratio appear in nature’s spirals. Do the proportions of the Golden Ratio make the Golden Rectangle especially attractive and, if so, why? These questions have given rise to heated debate and much controversy. In 1876, Gustav Fechner, a German psychologist and physi-cist, conducted a study of people’s taste in rectangles—a taste test—and found that 35% of the people surveyed selected the Golden Rectangle. So, although the Golden Rectangle seems likely to win an election, we would not expect the outcome to be a landslide.

The Golden Rectangle in GreeceThe Greeks appear to have been captivated by the proportions of the Golden Rectangle as evidenced by its frequent occurrence in their architecture and art. As a classic illustration, consider the magnifi cent Parthenon in Athens, built in the 5th century bce.

The Parthenon today is pretty run-down—in fact, it’s in ruins. How-ever, perhaps you’re a step ahead of us, guessing that the big rect-angle contained in the Parthenon is a Golden Rectangle. Actually, if we measure the sides and do the division, we will see that the rect-angle is not a Golden Rectangle! So what’s the point? Well, when the Parthenon was built, it was much fancier—in particular, it had a roof. Imagine now that the roof is in place. If we form the rectangle from the tip of the rooftop to the steps, we will see a nearly perfect Golden Rectangle.

Another example of the Golden Rectangle in Greek sculpture is the Grecian eye cup. The one pictured is inscribed inside a perfect Golden Rectangle.

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1

1 � 5_______2

1_2

�1

1_2

It remains an unanswered question whether Greek artists and design-ers intentionally used the Golden Rectangle in their work or chose those dimensions solely based on aesthetic tastes. In fact, we are not even certain that such artists were consciously aware of the Golden Rectangle. Although we will likely never know the truth, it is romantic to hypoth-esize that the Greeks were not conscious of the Golden Rectangle, because this then shows how aesthetically appealing its dimensions are and that we are naturally attracted to such shapes. Some people, however, believe that the occurrence of Golden Rectangle proportions is simply coincidental and random. While some believe that ancient Greek works defi nitely contain Golden Rectangles, others believe that it is nearly impossible to measure such works or ruins accurately; thus, there is plenty of room for error. In the preceding pictures, all the superimposed rectangles are perfect Golden Rectangles. Was their presence random or deliberate? Are Golden Rectangles really there? What do you think?

The Golden Rectangle in the RenaissanceIt appears that mathematicians in the Middle Ages and the Renaissance were fascinated by the Golden Rectangle, but there is much question as to whether this enthusiasm was shared by artists of the time. Leonardo da Vinci was a math enthusiast, but did he know about the Golden Rect-angle? Did he deliberately use it in his work? While historians debate such issues, let’s take a look at Leonardo’s unfi nished portrait of St. Jerome from 1483. In the reproduction on page 262, we have superim-posed a perfect Golden Rectangle around the great scholar’s body.

Intentional or otherwise, Leonardo selected proportions that were aes-thetically appealing, and such dimensions resemble those of the Golden Rectangle. Although we are not certain whether Leonardo intentionally used the Golden Rectangle, we do know that 26 years later he was aware of its existence. In 1509, Leonardo was the illustrator for Luca Pacioli’s text on the Golden Ratio titled De Divina Proportione. It was famous mainly for the reproductions of 60 geometrical drawings illustrating the Golden Ratio.

(Bohams, London, UK/The Bridgeman Art Library International)

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262 Geometric Gems

St. Jerome by Leonardo da Vinci (1480, Pinacoteca, Vatican Museums, Vatican State, Scala/Art Resource, NY)

Leonardo da Vinci’s illustration for Luca Pacioli’s De Divina Proportione (The Vitruvian Man, 1492, Accademia, Venice, Italy. Scala/Art Resource, NY)

The Divine Proportion is a synonym for the Golden Ratio. In fact, many people, including Johannes Kepler, referred to the Golden Ratio as the Divine Proportion, or as the Mean and Extreme Ratio. Sometimes imaginations ran a bit too wild. Pacioli claimed that one’s belly button divides one’s body into the Divine Proportion. If you’re not ticklish, you can easily check that this is not necessarily true.

Note the Fibonacci-like pattern in Le Corbusier’s 1946 Modulor Proportional System: 6 � 9 � 15, 9 � 15 � 24, and so on. [Le Corbusier Modular Man. © 2004 Artists Rights Society (ARS), New York.]

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The Golden Rectangle and ImpressionismLet’s now leap ahead about 300 years to the creative age of French Impressionism. Painter Georges Seurat was captivated by the aesthetic appeal of the Golden Ratio and the Golden Rectangle. In his painting La Parade from 1888, he carefully planted numerous occurrences of the Golden Ratio through the positions of the people and the delineation of the colors. The use of the Golden Ratio in works of art is now known as the technique of dynamic symmetry.

Seurat’s La Parade (1888) (The Metropolitan Museum of Art)

ABCD, FGHJ, EBIK are all golden rectangles; we also

note that GE

EA

EA

FE� � ϕ.

G H

J

K

I

B

E

A

F

C

D

The Golden Rectangle in the 20th CenturyIn the 20th century, artists were still fascinated with the beautiful propor-tions of the Golden Rectangle. French architect Le Corbusier believed that people are comforted by mathematics. In this spirit, he deliberately designed this villa (below right) to conform with the Golden Rectangle.

Le Corbusier was one of the architects involved in the design of the United Nations Headquarters in New York City. Here we again see the infl uence of the Golden Rect-angle in this monolithic structure (right).

Finally, we note that the Golden Rect-angle appears often in other art forms, including musical works. As an illustration,

Le Corbusier, Villa (© 2009 Artists Rights Society, New York)

United Nations

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264 Geometric Gems

consider the work of French composer Claude Debussy. In his 1894 work “Prelude to the Afternoon of a Faun,” he deliberately placed numer-ous ratios of musical pulses (called quaver units) that approximate the Golden Ratio.

Quaver units for “Prelude to the Afternoon of a Faun.” Note: 817

5151 5864� . . . . ≈ ϕ

Why the Appeal?Why do we see proportions conforming to the Golden Ratio in so many works of art? To answer this question, let’s return to Le Corbusier’s villa and notice that the living area creates a large square, whereas the open patio on the left has a rectangular shape. Look what happens when we compare the proportions of the whole villa to the small rectangular patio:

Le Corbusier, Villa.

From Roy Howait, Debussy in proportion: A Musical Analysis, Cambridge University Press.

Patio turned on its side and enlarged. (Courtesy of Corbusier Foundation and the Mathematics of the Ideal Villa and other Essays by Rowe. ARS.)

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Both are Golden Rectangles!

This rectangular similarity is actually a fundamental and beautiful mathematical property of the Golden Rectangle. This property might explain why the Golden Rectangle is so aesthetically pleasing.

To examine this property in general, let’s picture a Golden Rect-angle with base equal to ( )/1 5 2� and height equal to 1, so that b h/ ( )/ .� � �ϕ 1 5 2

We now divide this Golden Rectangle aefd into a square (abcd) and a smaller rectangle befc. The smaller rectangle is formed by removing that largest square from the Golden Rectangle. We will soon prove that it was the Golden Ratio pro-portions of the Golden Rectangle that automati-cally made the smaller rectangle, befc, golden!

An Unexpected RectangleThe fact that a Golden Rectangle comprises a square and a smaller Golden Rectangle may well explain its aesthetic appeal. This “self-proliferation” feature represents an attractive regenerating property: If we look at the smaller Golden Rectangle and now remove the largest possible square inside it, we are left with an even smaller Golden Rectangle. Can you visualize continuing this process of removing the square and getting another even smaller Golden Rectangle forever? There is, in some sense, a self-similarity property at work here: At any stage in this process, no matter how small the Golden Rectangle is, when we chop off the biggest square possible, we have created an even smaller Golden Rectangle. We will observe a similar situation when we consider fractals.

Why is this surprising mathematical fact true? It comes from the pleas-ing algebraic relationship that the Golden Ratio satisfi es:

ϕϕ1

11

��

.

d c f

eba

d c f

eba

Th e Golden Rectangle Within a Golden Rectangle.If a Golden Rectangle is divided into a square and a smaller rectan-gle, then the small rectangle is another Golden Rectangle.

ProofLet’s begin with our picture of a Golden Rectangle.

As before, we might as well declare that the length ad is 1 unit, and ae has length ϕ. To show that rectangle befc is a Golden Rectangle, we

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266 Geometric Gems

must show that the ratio of its longer side to its shorter side, that is, ef/be, is ϕ. So we will need the lengths of the sides of the smaller rectangle. Well, ef is easy to fi gure out: It equals ad. So ef � 1. What is be?

We note that be is just ae minus ab. So,

be ae ab� � .

But ae � ϕ, and ab � 1. So,

be � �ϕ 1.

So, the ratio

efbe

��

11ϕ

..

But recall our pleasing identity:

ϕϕ1

11

��

.

Therefore, ef/be equals ϕ, and the small rectangle befc is indeed a Golden Rectangle. This observation completes our proof.

Constructing Your Own Golden RectanglePerhaps you are now convinced that the Golden Rectangle is aestheti-cally intriguing and downright cool. You want one for yourself. Sure, you can call 1-800-COOL-REC and order one (operators are standing by), but why waste your money? We can make a perfect Golden Rectangle ourselves for free. It may appear that such a perfectly propor-tioned rectangle would be complicated to create. Not so. In fact, it’s easy to construct a perfect Golden Rectangle. Here’s how: First we build a square.

Next, we connect the midpoint of the base of the square to the northeast corner of the square with a straight line segment. We then extend the base of the square with a straight line segment off to the east, like a landing strip. We now have a picture that looks like this:

Now we draw part of a circle whose center is the midpoint of the base and whose radius extends to the northeastern corner of the square. We note where the circle portion hits

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the landing strip. The line segment drawn inside the square from the mid-point to the northeastern corner is actually a radius of the circle arc drawn. We now have the picture to the right.

Next, we construct a line perpendicular to the landing strip and passing through the point where the circle hit the landing strip. We then extend the top edge of the square to the right with a straight line until it hits the perpendicu-lar line just drawn. Finally, we erase the excess

landing strip to the right of the arc, giving us the diagram shown here.

That was pretty easy. Now take a look at that big rectangle we just constructed (we made ours a bit darker). Do you fi nd yourself drawn to that tall, dark, and handsome rectangle? If so, it’s all right, because that rectangle is a perfectly precise Golden Rectangle.

Why This Procedure Produces a Golden RectangleWe begin by recalling the fi nal picture of our construction and labeling all of the vertices.

To prove that the rectangle aefd is really a Golden Rectangle, we must show that the length of ae divided by the length of ad is equal to the Golden Ratio ( )/1 5 2� . So, we want to prove that

aead

��1 5

2.

The size of the rectangle is not important. What matters is the ratio of the two sides. We can call the length of ad 1 unit and note that this now completely determines the length of everything else in the rectangle. Given this agreement, our goal is to fi gure out what the length of ae is. Notice that ae is just am plus me. If we can fi nd am and then me, then we will have ae, since ae � am � me. Remember that we started with a square, and m bisected the bottom side. So am � mb � 1/2. Great—all we need to do is fi nd me.

The truth is that the length of me is mysterious. Let’s see if we can fi nd another line segment having the exact same length as me. Examine the preceding picture and fi nd another line that has the same length as me. Try this before reading on.

Did you guess mc? If so, great. Note that both mc and me are radii for the same circle, so the segments must have the same length. Instead of fi nd-ing the length of me, let’s fi nd the length of mc. Why is this quest easier? The answer is that mc is part of a right triangle. In fact, it is the hypotenuse

m

d c f

eba

Often in life when faced with a diffi culty, it is valuable to look for something

else that is comparable, but easier to resolve.

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268 Geometric Gems

of the triangle mbc. Notice that we already saw that bc is equal to 1 and mb is equal to 1/2. Thus, using the Pythagorean Theorem, we can fi gure out the length of mc. Why not try to fi gure it out on your own before reading on?

Here we go:

( ) ( ) .112

2

2

2� �⎛⎝⎜

⎞⎠⎟

mc

That is,

114

54

2 2� � �( ) ( ) .mc mcor

Notice the 5 making its debut in this discussion. This development is great news since we want a 5 at some point. In fact, note that to solve for mcwe need to take �/� the square root of both sides, but, because the length mc is positive, we have

mc � �5

22( ).because 4

Remember that mc has the same length as me, so,

me �5

2.

Therefore,

ae � � ��1

25

21 5

2.

Now for the big fi nish:

aead

�

�

��

�

1 52 1 5

2

⎛⎝⎜

⎞⎠⎟

1ϕ.

So, we have a Golden Ratio, which proves we’ve constructed a perfect Golden Rectangle.

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Golden SpiralsWe close with one last aesthetically pleasing construction. Let’s take a

Golden Rectangle and start drawing successive squares. Within each square, we will draw a quar-ter of a circle having a radius equal to the side of the square. If we do this, we get a spiral.

This spiral closely approximates the logarith-mic spiral, and it occurs in nature in various forms, such as the nautilus sea shell. The natural and aes-

thetic beauty of this spiral may be described mathematically. We fi rst consider the center of the spiral. By the center we mean that point at which the spiral spins around infi -nitely often—the point that the spi-ral is heading toward. How can we locate the very center of the spiral?

Locating the center is surprisingly simple. We need only draw a diago-nal in the largest Golden Rectangle from the northwest corner down to the southeast corner and then draw the diagonal in the next largest Golden Rectangle from its north-east corner to its southwest corner.

These two diagonals intersect at the precise center of the spiral. You may also have observed another

unexpected fact: All analogous diagonals on all subsequent pairs of Golden Rectangles lie on the fi rst two diagonals. This follows from the fact that each rectangle has exactly the same proportions. Thus, we see structure and beauty in the construction of the Golden Rectangle and the associated spiral.

What makes the curve of the spiral so appealing? Here is a mathemati-cal observation that may account for its appeal. Select any point on the spiral and connect that point with the center of the spiral. Now draw

the line that passes through that chosen point on the spiral but just grazes the curve of the spi-ral (such a line is called a tangent line). Notice the angle made by these two lines (the tangent at the point and the line connecting the point to the center). These angles are nearly the same, no matter which point on the spiral you selected.Angles are nearly equal.

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270 Geometric Gems

Finally, we note that this beautiful spiral inspired Henri Matisse’s 1953 work L’Escargot. On the Heart of Mathematics Web site, you can fi nd a program to generate these spirals and thus create your own works of art.

We’ll now close our discussion of the Golden Rectangle, but not forever. Several other examples of Golden Rectangles occur in surpris-ing places; but for them we will have to wait until we talk about the Pla-tonic solids.

A Look BackA rectangle is a Golden Rectangle if the ratio of its base to its height equals the Golden Ratio. If we remove the largest square from a Golden Rectangle, the small remaining rectangle is itself another Golden Rectangle. Thus, we can create a sequence of smaller and smaller Golden Rectangles. This sequence of Golden Rectangles leads to spirals that occur in nature.

We can build a Golden Rectangle by starting with a square and elongating it by using a simple geometric procedure. We can verify that the ratio of base to height is the Golden Ratio by applying the Pythagorean Theorem.

Art, aesthetics, geometry, and numbers all meet in the Golden Rectangle. Its appealing proportions have appeared in art through-out history and we can also fi nd them in nature. Do the mathemati-cal properties of the Golden Ratio somehow create the beauty of the Golden Rectangle? Some ideas span the artifi cial boundaries of subjects—in this case from the algebra of numbers (the Golden Ratio) to the geometry of rectangles (the Golden Rectangle). Seek-ing connections across disciplines often leads to new insights and creative ways of understanding.

Take ideas from one domain and explore them in another.

� �

Matisse’s L’Escargot (Henri Matisse, L'Escargot, 1953. Tate Gallery, London, Great Britain/Art Resource.[© 2009 Succession H. Matisse/Artists Rights Society (ARS), NY.)

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4.3 the s rectangle

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