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Creating Symmetry
The Artful Mathematics of Wallpaper Patterns
By Frank A. Farris PRINCETON UNIVERSITY PRESS
Copyright © 2015 Princeton University Press
All rights reserved.
ISBN: 978-1-4008-6567-3
CHAPTER 1
Going in Circles
Question: How do you make a circle (and other curves)? The ancients knew how to make a circle using a compass or its equivalent. I like to imagine an early genius tying a piece of charcoal to one end of a string of plant fiber and drawing the charcoal along a flat rock face while holding the other end of the string fixed. You can still do that, but suppose that we want to enjoy modern technology and draw a circle on a computer screen—a problem not faced in antiquity. How do we make a circle?
If you ask most people for the equation of a circle, you will probably hear
x2 + y2 = 1 or perhaps (x - h)2 + (y - k)2 = R2
for a circle with center (h, k) and radius R. This is fine, but it represents a static view of a circle, which is not the simplest way to direct the drawing of one.
The simplest way to instruct a machine to draw a circle uses a parametric form, also known as a vector-valued function:
γ(t) = (cos(t), sin(t))
for the unit circle and
γ(t) = (h + R cos(t), k + R sin(t))
for a more general one. In either case, as the time parameter t advances from 0 to π/2, we climb the upper-left quadrant of the circle in a counterclockwise fashion. It takes 2π units of time to return to our starting point and complete the drawing.
The given formula parametrizes the circle in one particular way. If we need to be more flexible, wanting to specify motion around a circle that starts at a particular point and goes a particular direction, we can tweak the formula, perhaps by swapping the coordinates or introducing minus signs, as in the following example.
Example: A Rolling Quarter
Place two quarters flat on the table with one above the other, both oriented right side up. If the top one rolls without slipping around the other, find a formula for the motion of the point that was originally at the bottom edge of the top circle.
Figure 1.1 shows the configuration before and after the quarter has rolled about 60°. We use a letter F, as well as some dashed construction lines, to help you follow the solution.
Solution. Suppose the quarter has radius 1 and has rolled so that the arc length rolled out on each quarter is t. The position of the center of the rolling quarter is then (2 sin(t), 2 cos(t)). This is what we meant by tweaking the formula: When no arc length has been rolled out (t = 0), this vector should be pointing straight up, and when t is small and positive, the x-value should be increasing. The formula matches.
Similarly, it takes a little work to figure out that the position of the rolling point relative to the center of the rolling quarter is (- sin(2t), - cos(2t)). Here are a few steps: In the diagram, the dashed lines might help you locate certain transversals. These are key to showing that the angle up from the vertical is twice the angle t.
Adding the two vector displacements, we find that the desired vector motion is
γ(t) = (2 sin(t) - sin(2t), 2 cos(t) - cos(2t)),
drawn on the top in Figure 1.2.
Beyond the Rolling Quarter. The path of the rolling quarter is an example of a curve called an epicyloid, the curve produced by rolling one circle around another. In general, if the fixed circle has radius a and the rolling circle has radius 1 , the formula for the moving point is
((a + 1) sin(t) - sin((a + 1)t), (a + 1) cos(t) - cos((a + 1)t)).
(You may wish to show this as an exercise.)
Figure 1.2 shows two examples, the first with both circles the same size and the second with one circle twice as large as the other.
Wheels on Wheels on Wheels
Let's think of the epicycloid as representing one particular kind of superposition of circular motions, where we insist that the circles roll without slipping. If we remove that restriction, we open the discussion to any sum of vector functions, each of which represents a circular motion, possibly tweaked to turn a different direction. I call this an instance of "wheels on wheels on wheels."
To create a particular example, I chose some more or less random wheels of different sizes and set them to turn at various rates, adding the vector displacements to form the function
μ(t) = (cos(t) + [cos(6t)/2] + [sin(14y)/3], sin(t) + [sin(6t)/2] + [cos(14t)/3]).
The first term in each component is our familiar unit circle; the other terms represent smaller wheels, one turning 6 times as fast, another 14 times as fast and altered somehow (the sine and cosine functions are swapped). The result is rendered in Figure 1.3. Take a moment to trace it with your eye and enjoy its dancing undulations, realizing that probably none of us has the patience to draw it without the aid of technology.
I hope that the figure—the "mystery curve"—surprises you. Nothing about the numbers 6 and 14 prepares for the evident rotational symmetry, which means that the figure is unchanged if we rotate it through 72°. Our next agenda item is to answer the question: What causes this curve to have 5-fold rotational symmetry?
Our computational paradigm. The formula for the mystery curve assigns one point of the Cartesian plane to every one of the infinitely many values of the time parameter t. When I ask a computer to draw the curve, modern software frees me from the need to figure out exactly how to instruct the computer to select a mere finite few time values for which it places blobs of ink on the page or lighted pixels on my screen. The details of computer graphics are lovely but are not the purpose of this book; here, I invite us to be consumers, rather than inventors of computer graphics.
In the instance of the mystery curve, a close examination of Figure 1.3 suggests that I could have been a slightly more demanding consumer: At a few places on the curve, I can detect that the machine has approximated the perfectly smooth shape by some line segments. Since I know the curve bends smoothly, I don't object to this imperfection. I enjoy the availability of technology that can do as well as it did. And if I wanted a finer rendition, I could have instructed the machine to use more points in its drawing routing.
From the relatively simply mystery curve to the most complicated color image in this book, the paradigm is the same: We look for mathematical objects, which we will call smooth functions, that have some symmetry that we wish to illustrate, perhaps the rotational symmetry of the mystery curve, perhaps something else. Directing computers to make images of the discovered objects is not really the interesting part of the process; it's the finding of the class of symmetrical things. This is what creating symmetry means to me: finding the formulas like the one for the mystery curve—with its enigmatic 6 and 14—that will display symmetrical images when rendered by software. It is not so much about the method of computer rendition but the mathematical theory of what makes things dance in the dazzling variety of patterned ways that we see in every tradition of decorative art.
Some Ancient Mathematics
The trigonometric functions are not the only way to parametrize the circle. There is some evidence that the ancient Babylonians knew a different way. It appears that they knew how to find pairs of rational numbers x and y that solve the equation x2 + y2 = 1, which, when we clear denominators, become what we know as Pythagorean triples.
Much has been written about the history of a mysterious tablet called Plimpton 322, which lists, without any commentary that we can understand today, a baffling array of Pythagorean triples [15]. Here we remark only that the vector function
γ(t) = (1 - t2/1 + t2, 2t/1 + t2) (1.1)
parametrizes the circle, as you can check with algebra, and that each coordinate is a rational number if t is rational. We could do the arithmetic by hand, but I found it simpler to ask a machine to plug in t = 54/125 to produce the Pythagorean triple
12,7092 + 13, 5002 = 18, 5412.
This fact was apparently known to the Babylonians about 3800 years ago!
CHAPTER 2
Complex Numbers and Rotations
Question: How do complex numbers make it easier to keep track of circles, curves, and rotations? The complex numbers, denoted C, are simply a way to express the Cartesian ordered pair of real numbers, (x, y), compactly as a single number
z = x + iy.
It is as if the symbol i creates an extra shelf on which to set the second number of the ordered pair; we call the part set next to i the imaginary part of z, written Im(z) and the other part the real part, Re(z). The plus sign is a hint that we will be doing algebra with these newly formed expressions. Indeed, to add two such expressions, we simply add the real and imaginary parts separately. To multiply by a scalar real number, we distribute the real multiple through real and imaginary parts.
To multiply complex numbers, use the definition
i2 = -1
and all the usual rules of algebra (multiplication is commutative and distributes over addition). Algebraic expressions using complex numbers serve us best when we clean them up by gathering the real and imaginary parts. For instance, check that
z2 = x2 - y2 + i2xy.
Reflection across the x-axis turns out to play a particularly important role in the complex numbers, so we give it a name. The number
[bar.z] = x - iy
is the complex conjugate of z = x + iy. Since conjugation simply negates the y- value of a number, we recognize it as reflection across the x-axis.
One quick way to see the utility of the conjugate is to compute the multiplicative inverse of a nonzero complex number z as
1/z = [bar.z]/z[bar.z] = x/[x2 + y2] – i[y/[x2 + y2]].
Another fruit of this computation is the fact that the norm of a complex number z = x + iy, which is the same as the distance from the Cartesian point (x, y) to the origin, is
√x2 + y2 = √z[bar.z].
If you skipped the preface, this might be a good time to return to the section "Ways to Read This Book." Some readers will be learning about complex numbers for the first time; others may enjoy seeing what I have to say about them; others may simply skip ahead.
Connections to Abstract Algebra and Number Theory. The complex numbers exemplify several famous algebraic structures, which we will meet as we go on. If you do not yet know the terms in the following sentences, rest assured that they will become familiar in subsequent chapters. When we think only of adding complex numbers, the set C has the structure of a group, which is a set closed under an operation that is nice in a certain technical sense. When we also consider that we can multiply complex numbers by real numbers, they take on the structure of a real vector space with a basis that consists of 1 and i. All this means is that every complex number is written uniquely as a sum of real multiples of 1 and i. When we include the multiplicative structure, C is a ring (closed under an additional operation, multiplication) and also a field, since every nonzero element has a multiplicative inverse. To study any of these abstract structures, it can be helpful to have C in mind as a concrete example.
We pause for some practice with complex multiplication, which allows us to mention a connection to number theory. Verify the following products:
(9 + i2)(9 - i2) = 81 + 4 = 85 and (7 + i6)(7 - i6) = 49 + 36 = 85.
It is curious that two different products give us the same real integer. The Gaussian integers are expressions of the form
a + ib, where a, b [member of] Z.
These Gaussian integers form a ring with identity 1. Investigating the similarities between this ring and the iconic ring of integers, Z, offers fine adventures in number theory.
The two rings are strikingly different in some ways. For instance, the only invertible elements in Z, called units, are 1 and -1, but the Gaussian integers have 1, -1, i, and -i. The fundamental fact about Z is that every integer has a unique factorization in terms of primes. We say that Z is a unique factorization domain. (Uniqueness is understood as being unique up to units, forgiving the double representation of 6 as 2-3 and (-2) · (-3).)
Our sample computation might suggest that 85 has multiple factorizations, but, in fact, the Gaussian integers are known to be a unique factorization domain. How can we reconcile this with our computation? Here is a suggestion: The conventional prime factorization of 85 is 85 = 5 · 17, but 5 and 17 can be factored further in the Gaussian integers, and so are not primes! (Hint: 5 = (2 + i)(2 - i).) To understand this better, show the four prime factors of 85, which are defined only up to multiplication by units; reassemble them in different ways to get 9 + 2i, and so on.
Making Circles and Curves with the Euler Formula
Using the new notation for Cartesian pairs, our parametrization of the circle becomes
γ(t) = (cos(t), sin(t)) = cos(t) + i sin(t).
If this seems unremarkable, a surprise is in store. The right-hand side of this equation has a celebrated, concise equivalent via the Euler formula. We take a circuitous route, perhaps imitating the style of Euler himself.
Euler knew that the trigonometric functions were not polynomials but found that they behaved like long polynomials—infinitely long. These are called power series, and they build on ideas of approximation. For instance, sin(t) is very like the polynomial t for small values of t, even more like t - t3/3!, and so on forever. Replacing the sine and cosine functions by their power series, which are known to converge for all values of t, we find that
γ(t) = 1 - t2/2! + t4/4! - ··· + i (t - t3/3! + t5/5! - ···).
In a flash of imagination, realizing things like i3 = -i and ignoring the convention about putting real parts first and imaginary ones last, we write this as
γ(t) = 1 + (it) + (it)2/2! + (it)3/3! + (it)4/4! + (it)5/5! + ···.
We recognize formal similarity to the real power series for et and define the symbol eit by the series (which can be shown to converge absolutely for all t),
eit = 1 + (it) + [(it)2/2!] + [(it)3/3!] + [(it)4/4!] + [(it)5/5!] + ···.
The definition is a responsible one, in that it generalizes the meaning of the real function et.
Without going too far astray, we can mention that this definition is the start of a long path of generalization; the next step would be to define eAt, where A is a matrix, using a power series. Definitions like these exemplify a very abstract idea: the exponential mapping from a Lie algebra to its Lie group. Let us return to safer ground and connect back to circles.
(Continues...)
Excerpted from Creating Symmetry by Frank A. Farris. Copyright © 2015 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
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