4.3 THE SEXIEST RECTANGLE
Some Scenarios
Involving
Finding
Aesthetics
in Life, Chance
Art, and Math
at Confound
Our Intuition
Through
the Golden
Rectangle
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
Are you into it?
O
n 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 discover new realms. Now, however, we wonder if mathematics can be used
to discover structure behind the aesthetics of art and nature.
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Rectangular Appeal
In our discussion of Fibonacci numbers we asked the following geometrical 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 find 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 discussions of the Fibonacci numbers, and we denoted it by the Greek letter
phi, ϕ. It was called the Golden Ratio because it satisfied the symmetrical
equation of ratios:
ϕ
1
.
1
ϕ 1
Specifically, 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 definition of a Golden Rectangle is any rectangle
having base b and height h such that
b
1+ 5
=ϕ=
.
h
2
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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 physicist, 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 Greece
The 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 magnificent
Parthenon in Athens, built in the 5th century bce.
The Parthenon today is pretty run-down—in fact, it’s in ruins. However, perhaps you’re a step ahead of us, guessing that the big rectangle contained in the Parthenon is a Golden Rectangle. Actually,
if we measure the sides and do the division, we will see that the rectangle 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
1
_
2
1 5
_______
2
1
_
2
(Bohams, London, UK/The Bridgeman Art Library International)
It remains an unanswered question whether Greek artists and designers 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 hypothesize 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
definitely 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 Renaissance
It 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 Rectangle? Did he deliberately use it in his work? While historians debate
such issues, let’s take a look at Leonardo’s unfinished portrait of
St. Jerome from 1483. In the reproduction on page 262, we have superimposed a perfect Golden Rectangle around the great scholar’s body.
Intentional or otherwise, Leonardo selected proportions that were aesthetically 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.
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Leonardo da Vinci’s
illustration for Luca Pacioli’s
De Divina Proportione
(The Vitruvian Man, 1492,
Accademia, Venice, Italy.
Scala/Art Resource, NY)
St. Jerome by Leonardo da
Vinci (1480, Pinacoteca,
Vatican Museums, Vatican
State, 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 Impressionism
Let’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.
G
H
I
B
C
J
F
K
E
D
A
Seurat’s La Parade (1888) (The Metropolitan
Museum of Art)
ABCD, FGHJ, EBIK are all golden rectangles; we also
note that
GE
EA
EA
ϕ.
FE
The Golden Rectangle in the 20th Century
In the 20th century, artists were still fascinated with the beautiful proportions 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, Villa (© 2009 Artists
Rights Society, New York)
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 influence of the Golden Rectangle in this monolithic structure (right).
United Nations
Finally, we note that the Golden Rectangle appears often in other art forms,
including musical works. As an illustration,
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consider the work of French composer Claude Debussy. In his 1894 work
“Prelude to the Afternoon of a Faun,” he deliberately placed numerous ratios of musical pulses (called quaver units) that approximate the
Golden Ratio.
Quaver units for “Prelude
to the Afternoon of a Faun.”
Note:
817
1.5864 . . . ≈ ϕ
515
From Roy Howait,
Debussy in proportion:
A Musical Analysis,
Cambridge University
Press.
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.
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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 Rectangle 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 proportions of the Golden Rectangle that automatically made the smaller rectangle, befc, golden!
d
c
f
a
b
e
An Unexpected Rectangle
The 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 pleasing algebraic relationship that the Golden Ratio satisfies:
ϕ
1
.
1
ϕ 1
The Golden Rectangle Within a Golden Rectangle.
If a Golden Rectangle is divided into a square and a smaller rectangle, then the small rectangle is another Golden Rectangle.
Proof
Let’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
d
c
f
a
b
e
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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 figure 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
ef
1
.
be ϕ 1 .
But recall our pleasing identity:
ϕ
1
.
1
ϕ 1
Therefore, ef/be equals ϕ, and the small rectangle befc is indeed a Golden
Rectangle. This observation completes our proof.
Constructing Your Own Golden Rectangle
Perhaps you are now convinced that the Golden Rectangle is aesthetically 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 proportioned 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 midpoint 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 perpendicular 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 find 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 Rectangle
We begin by recalling the final picture of our construction and labeling
all of the vertices.
d
a
m
c
f
b
e
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
ae
1 5
.
ad
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 figure out what the length of ae is.
Notice that ae is just am plus me. If we can find 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 find me.
The truth is that the length of me is mysterious. Let’s see if we can find
another line segment having the exact same length as me. Examine the
preceding picture and find 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 finding the length of me, let’s find 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
Often in life
when faced with
a difficulty, it is
valuable to look
for something
else that is
comparable, but
easier to resolve.
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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 figure
out the length of mc. Why not try to figure it out on your own before
reading on?
Here we go:
2
⎛ 1⎞
(1) ⎜ ⎟ ( mc)2 .
⎝ 2⎠
2
That is,
1
1
5
( mc)2 or
( mc)2 .
4
4
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 mc
we need to take / the square root of both sides, but, because the length
mc is positive, we have
mc 5
(because
2
4 2).
Remember that mc has the same length as me, so,
me 5
.
2
Therefore,
ae 1
5
1 5
.
2
2
2
Now for the big finish:
⎛1 5⎞
⎜⎝ 2 ⎟⎠
ae
1 5
ϕ.
ad
1
2
So, we have a Golden Ratio, which proves we’ve constructed a perfect
Golden Rectangle.
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Golden Spirals
We 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 quarter 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 logarithmic spiral, and it occurs in nature in various forms,
such as the nautilus sea shell. The natural and aesthetic beauty of this spiral may be
described mathematically. We first
consider the center of the spiral. By
the center we mean that point at
which the spiral spins around infinitely often—the point that the spiral is heading toward. How can we
locate the very center of the spiral?
Locating the center is surprisingly
simple. We need only draw a diagonal 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 northeast 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 first 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 mathematical 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 spiral (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
Angles are nearly equal.
matter which point on the spiral you selected.
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Finally, we note that this beautiful
spiral inspired Henri Matisse’s 1953
work L’Escargot. On the Heart of
Mathematics Web site, you can find
a program to generate these spirals
and thus create your own works of
art.
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.)
We’ll now close our discussion
of the Golden Rectangle, but not
forever. Several other examples of
Golden Rectangles occur in surprising places; but for them we will have
to wait until we talk about the Platonic solids.
A Look Back
A 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 throughout history and we can also find them in nature. Do the mathematical properties of the Golden Ratio somehow create the beauty of
the Golden Rectangle? Some ideas span the artificial boundaries
of subjects—in this case from the algebra of numbers (the Golden
Ratio) to the geometry of rectangles (the Golden Rectangle). Seeking connections across disciplines often leads to new insights and
creative ways of understanding.
c04.indd Sec3:270
270
Take ideas from one domain and
explore them in another.
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Mindscapes
Invitations to Further Thought
In this section, Mindscapes marked (H) have hints for solutions at the
back of the book. Mindscapes marked (ExH) have expanded hints at
the back of the book. Mindscapes marked (S) have solutions.
I. Developing Ideas
1. Defining gold. Explain what makes a rectangle a Golden Rectangle.
2. Approximating gold. Which of these numbers is closest to the Golden
Ratio? 1.16; 1.29; 1.62; 1.98.
3. Approximating again. Which of the following objects most closely
resembles a Golden Rectangle? A 3 5–inch index card; an 8.5 11–
inch paper; an 11 14–inch paper; an 11 17–inch paper.
1
have the same
4. Same solution. Why does the equation ϕ 1 ϕ
ϕ
1
solution as the equation ?
1
ϕ 1
5. X marks the unknown (ExH). Solve each equation for x:
1
2x
x
2
1
b.
c. 3 x a.
x 1
1
3 x−4
x 1
2
II. Solidifying Ideas
6. In search of gold. Find at least three examples of Golden Rectangles
in your surroundings. If possible, include photographs or sketches
and estimates of the ratio of base to height for each example.
7. Golden art. In the masterpiece Paris Street; Rainy Day by Gustave
Caillebotte (1877) shown below, find as many Golden Rectangles as
you can.
(Gustave Caillebotte, A Paris Street, Rain, 1877.
Art Institute of Chicago, U.S.A. Erich Lessing/Art
Resource, NY.)
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8. A cold tall one? Can a Golden Rectangle have a shorter base than
height? Explain your answer.
9. Fold the gold (H). Suppose you have a Golden Rectangle cut out of a
piece of paper. Now suppose you fold it in half along its base and then
in half along its width. You have just created a new, smaller rectangle.
Is that rectangle a Golden Rectangle? Justify your answer.
10. Sheets of gold. Suppose you have two sheets of paper, an unmarked
straightedge, and a pair of scissors. Explain how you can use one of
the sheets of paper and the straightedge to construct a perfect Golden
Rectangle on the other sheet. (Hint: You may cut the first piece of
paper.)
11. Circular logic? (H). Take a Golden Rectangle and draw the largest
circle inside it that touches three sides. The circle will touch two
opposite sides of the rectangle. If we connect those two points with
a line and then cut the rectangle into two pieces along that line, will
either of the two smaller rectangles be a Golden Rectangle? Explain
your reasoning.
12. Growing gold (H). Take a Golden
Rectangle and attach a square to the
longer side so that you create a new
Attach a
larger rectangle. Is this new rectangle
big square.
a Golden Rectangle? What if we
repeat this process with the new,
large rectangle?
13. Counterfeit gold? Draw a rectangle
with its longer edge as the base (it
could be a square, it could be a long and skinny rectangle, whatever you
like, but we suggest that you do not draw a Golden Rectangle). Now,
using the top edge of the rectangle, draw the square just above the
rectangle so that the square’s base is the top edge of the rectangle. You
have now produced a large new rectangle (the original rectangle
together with this square sitting above it). Now attach a square to
the right of this rectangle so that the square’s left side is the right
edge of the large rectangle. You’ve constructed an even larger
rectangle.
New
square
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Newer
square
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Now repeat this procedure—that is, append to the top of this huge
rectangle the largest square you can and follow that move by attaching
the largest square you can to the right of the resulting rectangle.
Start with a small rectangle near the bottom left corner of a page and
continue this process until you have filled the page. Now measure
the dimensions of the largest rectangle you’ve built and divide the
longer side by the shorter one. How does that ratio compare to
the Golden Ratio? Experiment with various starting rectangles. What
do you notice about the ratios?
14. In the grid (S). Consider the 10 10 grid at left. Find the four
points that, when joined to make a horizontal rectangle, make a
rectangle that is the closest approximation to a Golden Rectangle.
(Challenge: Suppose the rectangle can be tilted.)
15. A nest of gold. Consider the figure of infinitely nested Golden
Rectangles on page 269. Suppose we remove the largest square, and,
with the rectangle that remains, we enlarge the entire picture so that
its size is identical to the original rectangle. How will that enlarged
picture compare to the original figure? Explain your answer.
III. Creating New Ideas
16. Comparing areas (ExH). Let G be a Golden Rectangle having base
b and height h, and let G be the smaller Golden Rectangle made by
removing the largest square possible from G. Compute the ratio of
the area of G to the area of G. That is, compute Area(G)/Area(G).
Does your answer really depend on b and h (the original size of G)?
Are you surprised by your answer?
17. Do we get gold? Let’s make a rectangle somewhat like the Golden
Rectangle. As before, start with a square; however, instead of cutting
the base in half, cut it into thirds and draw the line from the upper
right vertex of the square to the point on the base that is onethird of the way from the right bottom vertex. Now use this new
line segment as the radius of the circle, and continue as we did in
the construction of the Golden Rectangle. This produces a new,
longer rectangle, as shown in the diagram. What is the ratio of the
base to the height of this rectangle (that is, what is base/height
for this new rectangle)? Now remove the largest square possible from
this new rectangle and notice that we are left with another rectangle.
Are the proportions of the base/height of this smaller rectangle the
same as the proportions of the big rectangle?
18. Do we get gold this time? (S) We now describe another construction
of a different type of rectangle. It is exactly the same as the Golden
Rectangle except that, instead of starting with a square, we begin with
a rectangle whose base is twice as long as its height. Now connect the
midpoint of the base to the upper right vertex with a line, and use this
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line as the radius of the circle and continue as we did in the
construction of the Golden Rectangle. This produces a
new, longer rectangle, as shown in the diagram. What is
the ratio of the base to the height of this new big rectangle
(that is, what is base/height for this new rectangle)? Now
remove the original rectangle. This gives us a new, smaller
rectangle. Are the proportions of the base/height of this
smaller rectangle the same as the proportions of the big rectangle?
Experiment with starting rectangles of differing proportions.
19. A silver lining? (H) Consider the diagonal in the Golden Rectangle
shown here and draw in the largest square possible. Notice that one
edge of the square cuts the diagonal into
two pieces. What is the ratio of the length
of the entire diagonal to the length of
the part of the diagonal that is inside the
square? That is, compute the length of
the entire diagonal divided by the length
of the part of the diagonal that is inside
the square. Surprised?
20. Cutting up triangles. Draw any right triangle. Find a way of cutting
up that triangle into four identical triangles such that each one is
identical in shape and proportion to the original large triangle except
that it is scaled down to one-fourth the area.
IV. Further Challenges
21. Going platinum. Determine the dimensions of a rectangle such that,
if you remove the largest square, then what remains has a ratio of
base to height that is twice the ratio of base to height of the original
rectangle.
22. Golden triangles. Draw a right triangle with one leg twice as long
as the other leg. This triangle is referred to as a Golden Triangle.
Suppose that one leg has length 1 and the other has length 2. What is
the length of the hypotenuse? Next draw a line from the right angle
of the triangle to the hypotenuse such that the line is perpendicular
to the hypotenuse. Now cut up the larger of the two new right
triangles into four triangles (see Mindscape III.20,“Cutting
up triangles”). Show that all five triangles are the
same size and are Golden Triangles. We will use
1
this neat cutting up of the Golden Triangle
in Section 4.4, “Soothing Symmetry and
Spinning Pinwheels.”
2
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Geometric Gems
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