# Rotational Ornaments

An isometry is a transformation of the plane that takes each line segment to a congruent line segment and each angle to a congruent angle. Basic types of isometries include reflections, rotations, and translations.

In the last last post, we saw that a symmetry of a geometrical figure is an isometry that preserves the figure. A group is a collection of symmetries satisfying: (1) if a symmetry T is in the group, then its inverse is also in the group; and (2) if T and U are in the group, then the composition U ∘ T is also in the group.

We discussed the dihedral groups D3 of the equilateral triangle and D4 of the square. Now consider the set of symmetries that preserves the regular pentagon.

In this case we have five reflections: R1, R2, R3, R4, and R5. In addition, we have clockwise rotation S through one fifth of a complete revolution, or 72°, and its compositions with itself: S2, S3, S4. So the dihedral group D5 consists of ten symmetries: I, S, S2, S3, S4, R1, R2, R3, R4, and R5. Once again, we can write D5 = 〈R1,R2〉, where R1 and R2 are reflections across adjacent lines making an angle of 36° with one another. For instance, R1 ∘ R2 is a 72°-rotation.

Now imagine extending the sides of the pentagon as shown.

This “breaks” the symmetry. No longer can we reflect. However, the rotations still preserve the figure. So the symmetry group now consists of five symmetries: I, S, S2, S3, and S4. This is referred to as the cyclic group C5. It’s generated by S, so we can write C5 = 〈S〉. The cyclic group is a subgroup of the dihedral group.

In general, the cyclic group Cn is generated by a rotation through 360°/n, or one nth of a revolution, while the dihedral group Dn is generated by two reflections whose lines make an angle of 360°/2n. Cyclic and dihedral groups abound in ornamental symmetry; Owen Jones’ 1910 Grammar of Ornament contains a wealth of examples.

The group D1 contains a single reflection; it may be thought of as the set of symmetries of the letter V. This is bilateral symmetry, as exhibited in the following Egyptian ornament.

The group D2 contains two reflections across perpendicular lines; it may be thought of as the set of symmetries of the letter H. We see examples of it in the following Greek and Pompeian ornaments.

Notice that if we ignore the color in the Pompeian ornament, then we have a D4 symmetry instead. A true example of D4 symmetry is given in the following Persian ornament.

Larger dihedral groups are rarer but still occur. The following ornaments are, respectively, Assyrian, Chinese, and Byzantine; they have symmetry groups D5, D8, and D22.

Ornaments that exhibit only a cyclic symmetry generally have some kind of overlapping knot pattern or swirl element which prevents reflection. The cyclic group C1 is the trivial group consisting of just the identity.

The cyclic group C2 contains a single rotation through 180°. This may be thought of as the set of symmetries of the letter N. The following Celtic ornament has C2 symmetry; the overlapping of the knot prevents its having D2 symmetry.

The cyclic group C4 is generated by a 90°-rotation. Examples include the following medieval European and Moorish ornaments.

The group C6 is generated by a 60°-rotation, while C8 is generated by a 45°-rotation; these are exhibited in the following medieval European and Arabian ornaments.

In the latter case, we have to ignore the box design around the circular ornament. If we don’t, then the symmetry group is C4 rather than C8. In general, Cn contains Cm as a subgroup if and only if m is a divisor of n. So C8 contains C1, C2, and C4. Similarly, C6 contains C1, C2, and C3. We’ll return to this in the next post on the topic.

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# Symmetry Groups

Think of a spacial arrangement of geometrical elements. It could be a polygon, or a honeycomb lattice, or some other arrangement. A symmetry is a rigid transformation that preserves the arrangement.  Consider an equilateral triangle, for instance:

There are six “moves” that would preserve the triangle. To begin with, we could reflect it across any of the three axes of symmetry.

In addition, we could spin it through 120° around the center, clockwise or counterclockwise. And, finally, we could just leave it be.

Symmetries have the following property: if we perform two symmetries in succession, then the composition obtained is also a symmetry. For instance, if we reflect the triangle across two lines of symmetry, then this amounts to a 120° rotation. In general, if T and U are two symmetries, then we’ll use the notation U ∘ T to represent the symmetry obtained by performing first T and then U. We can think of this as a “multiplication” of symmetries. Be careful, though: the order matters. The symmetry U ∘ T is not necessarily the same as T ∘ U.

To make all this clearer, let’s consider another example. A square has four lines of symmetry:

Reflection across any of these lines preserves the shape. Label the four reflections as R1, R2, R3, and R4. We can also rotate the square clockwise around the center through 90°, or 180°, or 270°. If S is the symmetry that rotates the square through 90°, then S ∘ S = S2 is rotation through 180°, and S ∘ S ∘ S = S3 is rotation through 270°. And, finally, there’s the identity symmetry, which just leaves the square as it is. We’ll call that I. Notice that S4 = I, since S4 amounts to a 360° rotation.

So we have eight symmetries in all: I, S, S2, S3, R1, R2, R3, and R4. As mentioned above, we can view composition of symmetries as a kind of “multiplication” on this set. However, it isn’t commutative: the order matters. To see this, consider S and R1. Let’s see what S ∘ R1 does to the square:

This amounts to the transformation R4. Now let’s see what R1 ∘ S does:

This amounts to R2. So S ∘ R1 = R4 while R1 ∘ S = R2. It follows that S ∘ R1 ≠ R1 ∘ S.

What is true is that the composition of any two symmetries in our set is also in the set. The set of symmetries is “closed” under composition. In addition, each symmetry has its inverse, which carries the shape back to the original position. For instance, R1 is its own inverse, since R1 ∘ R1 = I. The same is true of all the reflections. On the other hand, the inverse of S is S3, since S ∘ S3 is a 360° rotation, which amounts to I. And, finally, the inverse of S2 is itself, since two 180° rotations amount to a 360° rotation. If we wanted to, we could make an 8 × 8 multiplication table for this set of symmetries. Each row and each column would contain each symmetry exactly once.

Another interesting thing to note is that we could use a pair of reflections to obtain the other symmetries. Take R1 and R2, for instance. Notice that S = R1 ∘ R2. We can compose S with itself to obtain the other rotations: S2 = R1 ∘ R2 ∘ R1 ∘ R2 and S3 = R1 ∘ R2 ∘ R1 ∘ R2 ∘ R1 ∘ R2. Finally, R3 = R2 ∘ R1 ∘ R2, while R4 = R1 ∘ R2 ∘ R1. So we’ve written every symmetry as a product of R1 and R2.

A group is a set of symmetry operations satisfying the following two requirements: (1) the composition of two operations in the group is also in the group, and (2) for each operation in the group, the inverse of the operation is also in the group. The group we’ve been discussing is the dihedral group D4. It has order 8, meaning that it contains 8 operations: I, S, S2, S3, R1, R2, R3, and R4. We say that D4 is generated by R1 and R2, and write D4 = 〈R1,R2〉.

In coming posts, we’ll see how the theory of groups can be used to classify ornamental symmetries.

# Unity and Infinity

In recent weeks, Yahoo! News, that worthy and reliable news source, ran an article purporting to list the eleven most beautiful mathematical equations (ten having been too few). Alas, of the list, five are not mathematical, and one is not an equation. However, among the remaining five is the following:

$1=.999\ldots$

where the 9’s are understood to keep going forever. In my experience, this equation prompts varied responses. The most common is sheer disbelief. “How could these two be equal?” people ask. The answer to this is that, if they aren’t equal, then we should be able to find their difference:

$1 - .999\ldots = \;?$

But these doubters of the mysteries of unity and infinity can’t state the answer. Generally they’ll admit that it must be a number smaller than every other number, but they still insist that it can’t be zero. What’s interesting is that they’re generally comfortable with the equation

$\frac{1}{3} = .333\ldots$

After all, a calculator tells us it’s so. But if we multiply both sides by 3, then don’t we get the original?

It isn’t surprising that our equation seems a little dubious. What this really stems from is not understanding what infinite decimal expansions like .333… and .999… actually mean. Now, terminating decimal expansions are easy to understand. For instance,

$.9 = \frac{9}{10} \;\;\;\;\;\; .99 = \frac{9}{10} + \frac{9}{100} \;\;\;\;\;\; .999 = \frac{9}{10} + \frac{9}{100} + \frac{9}{1000}$

But by what right can we write down something like

$.999\ldots = \frac{9}{10} + \frac{9}{100} + \frac{9}{1000} + \cdots$

where the sum just keeps going and going? How can we add infinitely many numbers? Does it even make sense to talk about something like that?

To answer this, let’s consider the sequence of partial sums:

$.9 = \frac{9}{10} \;\;\;\;\;\; .99 = \frac{9}{10} + \frac{9}{100} \;\;\;\;\;\; .999 = \frac{9}{10} + \frac{9}{100} + \frac{9}{1000}$

and so on. Notice that each term in the sequence is a bit larger than the preceding, and that all are less than 1:

$.9 < .99 < .999 < .9999 < \cdots < 1$

Modern mathematics defines the meaning of the infinite decimal expansion as follows: the value of the expansion is the number x such that, for any positive number ϵ, no matter how small, our sequence of partial sums is eventually in between x – ϵ and x. Put in another way, the value is x if, for any small number ϵ, the difference between x and the nth partial sum is smaller than ϵ if n is large enough.

So, you see, the admission that the difference between 1 and .999… is a number “smaller than every other number” means that .999… is in fact equal to 1.

# Orientability

In topology, a surface is said to be orientable if the notion of right-handedness makes sense on it. Consider the sphere, for instance. If we write the letter R on the outside of the sphere, then a person on the inside would view it as Я. We could call the first the right-handed view, and the second, the left-handed view. It’s consistent across the sphere in the sense that, no matter how we slide the R around on the surface, it always looks like R to someone on our side (the outside).

A surface is nonorientable if it isn’t possible to decide in a consistent way what “right-handed” means at each point. The most famous example is the Möbius band, constructed by gluing the two ends of a rectangular strip after giving it a half-twist.Imagine writing R on the Möbius band. Slide it around the band once. When it comes back to where it started, it looks like Я. If we slide it around a second time, it looks like R again. Unlike the cylinder, which has two sides, the Möbius band only has one side, and we can’t distinguish between right-handed and left-handed.

The Möbius band is a surface with boundary: its boundary is the single loop that runs around the band twice. To construct a nonorientable surface without boundary, we’ll have to use our imaginations a bit. We start with a square, ABCD:

Imagine making a purse out of ABCD by inflating the middle while drawing B and D toward one another.

Next, attach edge AB to edge CD. Glue A to C, then zip up the pair of segments, attaching B to D at the end. This produces something like the following.

The edges AD and CB are still loose. The rim of the hole in the picture represents AD. A is at the bottom, D at the top. The segment CB is on the back. The vertical “seam” is the edge AB, now identified with CD.

The next goal is to attach AD to CB without crossing the membrane between them. This is impossible in three dimensions, but don’t worry about that. Just imagine attaching AD to CB through the membrane, resulting in two intersecting membranes, where we simply agree that we can only get from one membrane to the other by going all the way around as shown.

To see why this surface, known as the projective plane or RP2, is nonorientable, simply imagine placing our R on the loop and running it from one side to the other. In fact, RP2 is the basic nonorientable surface. Any other nonorientable surface without boundary can be constructed by surgically joining copies of RP2.

The Möbius band can be obtained from RP2 as well. Recall that the Möbius band has a single boundary loop. Imagine shrinking this loop to a point as though it were a drawstring. Then the surface we end up with is RP2.

Similarly, if we punch a hole in RP2, we obtain the Möbius band. To see this, imagine cutting the hole out of the very top of our model.

In terms of the original square, this amounts to snipping off the corners at B and D. If we stretch the square along the edges left by our cuts, we obtain a strip as shown.

Our construction of RP2 attaches D – A – B to B – C – D. If we follow through with that now, what we obtain is the Möbius band.

# Beauty of Harmony

The scientist does not study nature because it is useful to do so. He studies it because he takes pleasure in it, and he takes pleasure in it because it is beautiful. If nature were not beautiful it would not be worth knowing, and life would not be worth living. I am not speaking, of course, of the beauty which strikes the senses, of the beauty of qualities and appearances… What I mean is that more intimate beauty which comes from the harmonious order of its parts, and which a pure intelligence can grasp. (Henri Poincaré, Science and Method)

# Eadem Mutata Resurgo

A golden rectangle is a rectangle ABCD whose dimensions are such that, if a square ABPQ is inscribed as shown, then the smaller rectangle PCDQ cut off is similar to the original.

Because the sides of the smaller rectangle are in the same ratio as those of the original, the process may be repeated, thereby cutting off yet a smaller golden rectangle. This could be repeated ad infinitum, thus:

It’s easy to find the ratio of the sides. If we take x = BC and y = AB, then the ratio of x to y is the same as the ratio of y to x – y. So x/y = y/(x – y). Writing x = ϕy, we have ϕ = 1/(ϕ – 1), or ϕ2 – ϕ – 1 = 0. Solving for ϕ yields (1 + √5)/2, or about 1.618. We call ϕ the golden ratio.

Now, if we take our sequence of squares and construct a quarter-circle in each square, then we obtain a spiral shape.

It’s often claimed that this models the shell of the chambered nautilus and other objects found in nature.

This isn’t really the case, however. Part of the reason is that the “spiral” is not a true spiral. It’s pieced together from a sequence of quarter-circles, each of which has a different radius. In other words, the radius of curvature jumps discontinuously as we move around the spiral. The shell of a nautilus, on the other hand, is a logarithmic or equiangular spiral. The radius varies continuously and by a geometric progression as we turn around the spiral.

We can still use the golden rectangle to construct a logarithmic spiral. If we draw two successive diagonals as shown

then they intersect at the point that the sequence of squares converges upon. Call this point O. As we look closer and closer at O, we have an infinite sequence of rectangles, each of which is similar to the original. For instance, if we were to expand the picture to make SU as long as BC is now, then it would look exactly the same, complete with the infinite sequence of squares spiraling in to O. The picture is self-similar. Each of the four line segments emanating from O touches infinitely many corners in the sequence of golden rectangles. For instance, the segment from O to B touches B, S, and so on—the points are too close together to label—and each occupies the same corner in its respective golden rectangle.

The point O is the fixed center of our true spiral. The segment lengths OB, OC are in a golden ratio, as are OC, OD, and OD, OQ, and so on. Furthermore, these successive pairs all meet at right angles.

Imagine tracing out the spiral starting at A. The point O is the center of the spiral. As we turn, the spiral draws closer and closer to O, in such a way that its distance decreases by a factor of ϕ with each quarter-turn. Thus, if the distance from O to A is 1, then, as we turn clockwise from OA to OP, the distance of the spiral from O should decrease to ϕ-1. As we turn from OP to OR, the distance should decrease to ϕ-2. And so on. Conversely, if we turn in the counterclockwise direction, then the distance of the spiral from O increases by a factor of ϕ with each quarter-turn.

The golden spiral thus constructed is closely approximated by the artificial spiral we constructed above. The latter is sometimes incorrectly called the golden spiral, but is actually known as the Fibonacci spiral. The golden spiral is but one example of a logarithmic spiral; it is based on the factor ϕ, but any other factor s > 1 could be used instead. In terms of polar coordinates, this would be parametrized by r(θ) = α ⋅ s2θ/π, where α > 0 is an arbitrary scaling factor.

The logarithmic spiral was studied by the Swiss mathematician Jacob Bernoulli (1654 – 1705), who called it the Spira miribilis—the “wonderful spiral”—because of its property of self-similarity. He was so taken with this property that he requested a logarithmic spiral to be incised on his epitaph, together with the motto EADEM MUTATA RESURGO (“though changed, I remain the same”). The craftsmen misunderstood and put a simple Archimedean spiral instead.

# Counting without Numbers

Imagine a person living at the dawn of civilization, a goatherd, let’s say, dwelling somewhere in the Fertile Crescent. Every day the goatherd lets his animals out of their pen into the pasture so they can graze. When evening comes, he opens the gate and calls to his goats, and they return.

One day the goatherd notices that the herd seems to take up less space in the pen. He begins to worry that he may be losing some goats to thieves or wild animals while they’re out in the field grazing. How is he to make certain?

One obvious suggestion might be to count the goats. That’s what you or I would do. But our goatherd is living at a time when there was no systematic way to count.

Think about this. The English language has proper names for the first twelve counting numbers: one, two, three, and so on, up to twelve. Beyond that, we use the base ten numeration system to label the numbers. For instance, twenty-seven is two tens and seven ones. Three hundred and forty-five is three hundreds, four tens, and five ones.

This machinery originated in India in fairly recent times, only one or two thousand years ago. Our goatherd has no such system. If he wants to label the numbers, he just has to make up proper names for them, and there’s only so many proper names you can come up with. For all we know, his culture may not even have a word for two; the aborigines of Australia are said to have words only for one and many. It would be about as reasonable to ask our goatherd to invent a numeration system on the spot as it would be to ask him to build a computer from scratch. So, how is he to keep track?

Here’s an idea. He could gather a big heap of pebbles and get a large basket. As the goats go out in the morning, he puts one pebble in the basket for each animal that passes him. Once the pen is empty, he knows he has exactly as many pebbles in the basket as goats in the pasture. In other words, he knows that he could pair off the goats and the pebbles without leaving anything out.

Then, when the herd returns in the evening, he can remove one pebble for each goat that passes. If he runs out of goats first, he knows he has a problem. If he runs out of pebbles first, well, he knows that nature has taken its course.

This assignment of pebbles to goats is known as a one-to-one correspondence. Various peoples of antiquity actually did use such methods to keep track of amounts. The ancient Sumerians are said to have used baked clay tokens rather than pebbles for their accounting. They would then seal the tokens in a clay pouch, and put as many marks on the pouch as there were tokens inside. Eventually they decided to do away with the tokens and just use the marks. And the first numeration system was born.

You see, whenever we count, we are establishing a one-to-one correspondence between a list of numbers and a group of objects. The set of counting numbers may thus be viewed as a universal, abstract set of “pebbles.” Instead of pairing goats with pebbles, we pair goats with numbers, and pebbles with numbers. This involves a profound leap in human thought. The same idea forms the foundation of the modern theory of number as formulated by the great German mathematician Georg Cantor (1845 – 1918). It is to Cantor that we owe the knowledge that there are different kinds of infinities, and that the set of real numbers is more “numerous” than, say, the set of counting numbers.

The child psychologist Jean Piaget (1896 – 1980) studied the role of one-to-one correspondence in early childhood development. In The Child’s Conception of Number, he describes several stages. First, the child compares groups of objects by noting their spacial arrangement or extension, much as our goatherd did when he observed the size of his herd in the pen. This frequently leads to incorrect responses. Later, the child may be brought to recognize the equivalence of two sets through observing a pairing. But it is not until the child realizes that anything done respectively to the two groups can be undone, thus restoring them to the paired arrangement, that they arrive at a true grasp of counting. In group-theoretic terms, we would say that the child has to recognize that the operations performed on the sets are invertible.

So here we have a remarkable parallel between the origins of counting at the dawn of civilization, the theoretical foundation of sets and numbers, and the development of the conception of number in the human mind.

# Capax Universi

The functionary is trained. Training is distinguished by its orientation toward something partial, and specialized, in the human being, and toward some one section of the world. Education is concerned with the whole: whoever is educated knows how the world as a whole behaves. Education concerns the whole human being, insofar as he is capax universi, “capable of the whole,” able to comprehend the sum total of existing things. (Josef Pieper, Leisure, the Basis of Culture)

A thinking reed.—It is not from space that I must seek my dignity, but from the government of my thought. I shall have no more if I possess worlds. By space the universe encompasses and swallows me up like an atom; by thought I comprehend the world. (Blaise Pascal, Pensées)

# The Altar at Delos

An anonymous tragic poet of Greek antiquity once represented the legendary king Minos as erecting a tomb. Minos, dissatisfied with the size of the tomb, which measured 100 feet each way, decided to double its volume by doubling each of the dimensions. No one today knows who this poet was; his work has not survived. The only reason we know about him is that he became notorious among the Greek mathematicians for the error in his reasoning. For, if the height, breadth, and depth of the tomb were all doubled, then the size of the tomb would be octupled, not doubled, because 2 × 2 × 2 is 8.

Later on, the same problem of “doubling the cube” arose in a religious quandary. The tiny island of Delos was held as sacred by all the Greeks; during historical times, it was revered as the birthplace of the god Apollo. According to the story, a terrible plague afflicted the people of the island, and they sent representatives to Delphi, the oracle of Apollo on the Greek mainland, to ask the god’s advice. In order to end the plague, the oracle replied, the island’s craftsmen had only to double the size of the cubical altar of Apollo.

This advice, while easy to state, was not so easy to carry out. The problem is to find a new side length that exactly doubles the volume of the cube. The Delians weren’t sure how to solve it, so they sent to the philosopher Plato for help. Plato explained that the oracle’s purpose wasn’t so much to double the size of the altar as to shame the Greeks for their ignorance of geometry. He then handed the problem over to his colleagues at the Academy. Archytas, Eudoxus, and Menaechmus each provided independent solutions.

The solution of Menaechmus involved mean proportionals. Suppose that the dimensions of the altar are a × a × a. Suppose further that we can find mean proportionals x and y between a and 2a so that a < x < y < 2a and a/x = x/y = y/2a. Cross-multiplying, we obtain x2 = ay, y2 = 2ax, and xy = 2a2. Combining these, we find that x3 equals axy, which equals 2a3. So, if x is the side of the new altar, then the volume of the new altar is twice the volume of the old altar. The goal, then, is to find x and y so that x2 = ay and y2 = 2ax. Taking a = 1, we have x2 = y and y2 = 2x. These are the equations of two parabolas in the plane as shown. The coordinates of the intersection point are (x,y).

This may seem simple enough to someone who’s taken a course in algebra. But the Greek mathematicians had no algebra or coordinate geometry; at the time Menaechmus provided the solution, they were completely unaware of the conic sections (ellipses, parabolas, hyperbolas) and their properties. In fact, this is widely regarded as the beginning of the study of conic sections. The celebrated treatise of Apollonius was written a generation later.

For all their interesting properties, however, conic sections remained a mathematical curiosity for nearly two thousand years…until the Renaissance, when it was discovered that they model projectile motion and planetary orbits. The parabola is still used today to determine the shape of satellite dishes.

# Byrne’s Euclid

In 1847, an eccentric new edition of Euclid’s Elements was published in Britain. Designed by the otherwise obscure mathematician Oliver Byrne, it replaced letter variables with color diagrams and symbols “for the greater ease of learning.”

History has tended not to agree with Byrne on the pedagogical success of his edition; the market apparently didn’t, either, for the edition didn’t sell well, and its extravagant expense sent the printing firm into bankruptcy.

Despite all of this, it is a true delight to read. A facsimile edition has been published, but a complete scan of the book is available online at this link. Byrne’s Euclid has been called “one of the oddest and most beautiful books of the century.” It was featured at the Great Exhibition of 1851, and has been seen as an anticipation of the Bauhaus school of design.