Nineteenth-Century Monsters

The chimeranineteenth century was a time of change, when long-held assumptions about human society and the surrounding world were being questioned and overturned. It shouldn’t come as a surprise that it brought about an upheaval in the foundations of mathematics as well. It was an era of controversy and bitter contention. One of the main points of issue was cardinality.

The theory of sets and the idea of cardinality were first made rigorous by the mathematician Georg Cantor (1845 – 1918). At the time his work provoked widespread hostility, but posterity has vindicated him. As David Hilbert famously remarked:

No one shall expel us from the Paradise that Cantor has created.

Now his theory of sets forms part of most elementary education preparation programs.

Simply put, two sets are said to have the same cardinality if they can be paired off with one another without using anything more than once or leaving anything out. If a set has the same cardinality as the set

\lbrace 1, 2, 3, \dots \rbrace

of natural numbers, then it is said to be countable. This means that there is a numbering scheme that eventually numbers every object in the set. I discussed this in my post on counting without numbers.

For instance, consider the set of rational numbers (numbers that can be written as fractions) on the line segment from 0 to 1 on the number line. Every number in this set can be written as a/b, where ≤ b. Consider the following list:

\left\{ 0, 1, \frac{1}{2}, \frac{1}{3}, \frac{2}{3}, \frac{1}{4}, \frac{2}{4}, \frac{3}{4}, \frac{1}{5}, \frac{2}{5},\frac{3}{5},\frac{4}{5},\frac{1}{6},\frac{2}{6},\frac{3}{6},\frac{4}{6},\frac{5}{6},\frac{1}{7},\dots \right\}

It should be clear that every rational number between 0 and 1 will eventually appear on this list. Of course, there are many repeats. For instance, the rational numbers 1/2, 2/4, 3/6, and so on are all equal to each other. So remove any unreduced fraction from the list. This deletes 2/4, and 2/6, and 3/6, and 4/6, and so on.

\left\{ 0, 1, \frac{1}{2}, \frac{1}{3}, \frac{2}{3}, \frac{1}{4}, \frac{3}{4}, \frac{1}{5}, \frac{2}{5},\frac{3}{5},\frac{4}{5},\frac{1}{6},\frac{5}{6},\frac{1}{7},\dots \right\}

This lists every rational number exactly once. Now number off down the list. This shows that the set is countable.

On the other hand, the line segment from 0 to 1 is not countable. No matter how we try to pair the segment off with the natural numbers, there will always be infinitely many points of the segment left out. Intuitively, this is hardly surprising: the line segment is a continuum of points, and we wouldn’t expect that we could list them like we can the natural numbers. This idea of there being different grades of infinity was surprising to Cantor’s contemporaries. Even philosophers and theologians took note of it.

Given that the line segment is “more infinite” that the set of natural numbers, we might ask if there exists an uncountable set of points that isn’t a continuum. It turns out that there is: the Cantor ternary set, introduced by Georg Cantor in 1883. Take the line segment from 0 to 1 and remove the middle third, resulting in the line segments from 0 to 1/3 and from 2/3 to 1. Do the same for each of these segments, and so on, ad infinitum.

Cantor set

The resulting set is uncountable, contains no segments, and in fact has measure 0.

Cantor also showed that the line segment (which has topological dimension 1) has the same cardinality as the unit square (which has dimension 2). The notion that spaces of different dimension could be paired off in this way was so shocking that Cantor himself disbelieved the result for several years. One mathematician remarked:

It appears repugnant to common sense. The fact is that this is simply the conclusion of a type of reasoning which allows the intervention of idealistic fictions, wherein one lets them play the role of genuine quantities even though they are not even limits of representations of quantities. This is where the paradox resides.

Cantor’s ideas were savagely attacked by many of his fellow mathematicians; he was denounced as a “scientific charlatan” and a “corrupter of youth.” This exacerbated the bouts of mental illness he dealt with throughout his life, and he spent much of his time in sanatoriums.

Inspired by Cantor’s discovery, the Italian mathematician Guiseppe Peano set out to find a continuous mapping from the segment to the square that would cover the entire square. He did this in 1890 through an arithmetic description, and the curve was later interpreted geometrically as the limit of an iterative process. The result was the first known plane-filling curve. Another, better-known example was discovered by David Hilbert in 1891:

Hilbert Curve 3

Although none of the iterative steps are self-intersecting, their limit contacts itself at every point. There is no continuous pairing of the line segment with the square. (Cantor’s mapping, while a true pairing, jumps around the square, whereas Hilbert’s mapping, while continuous, revisits each point multiple times.)

Another famous “pathological” object from the end of the nineteenth century is the Koch curve, named for its discoverer, the Swiss mathematician Helge von Koch (1870 – 1924). Beginning with a line segment, replace the middle third with the two legs of the equilateral triangle erected upon it. Repeat this process for each of the four segments that result, and so on, ad infinitum.

Koch curve

Each iterative step is a simple polygonal curve of dimension 1. But the limit of these steps—the Koch curve—is something else entirely. It is a continuous curve with an infinite length. For, if the length of the initial segment is 1 unit, then the length of the second step is 4/3 units, and that of third, 4/3 ⋅ 4/3 = 16/9 units, and so on. This sequence of lengths tends to infinity.

Each portion of the curve winds back and forth infinitely many times; in fact, each portion resembles the whole. More specifically, the Koch curve contains four copies of itself, each scaled down by a factor of one third. It is a continuous curve whose derivative exists at no point; this amounts to saying that at no point is there a single tangent line. The existence of such curves was profoundly disturbing to the mathematicians of the late nineteenth century; in 1893, the mathematician Charles Hermite wrote:

I turn away with fear and horror from this lamentable plague of functions with no derivatives.

If the above process is applied to each side of an equilateral triangle, then the figure known as the Koch snowflake results.

Koch 2

It is possible to compute the area of the snowflake. Taking its initial side length to be 1, its initial area is √3/4. Adding on the three triangles at the second step, the area becomes

\frac{\sqrt{3}}{4}+ 3 \cdot \frac{\sqrt{3}}{4} \cdot \left(\frac{1}{3}\right)^2

At the third step, we add on 12 triangles, each of side length 1/9, so the area becomes

\frac{\sqrt{3}}{4}+ 3 \cdot \frac{\sqrt{3}}{4} \cdot \left(\frac{1}{3}\right)^2 + 12 \cdot \frac{\sqrt{3}}{4} \cdot \left(\frac{1}{9}\right)^2

Continuing this pattern, we find that the area is

\frac{\sqrt{3}}{4}\left(1 + \frac{1}{3} \left(1 + \frac{4}{9} + \frac{4^2}{9^2} + \frac{4^3}{9^3} + \cdots \right)\right) = \frac{2 \sqrt{3}}{5}

On the other hand, we’ve seen that the perimeter tends to infinity. So here we have a figure with a finite area and an infinite perimeter.

Plane-filling curves and the Koch snowflake are brought together in a beautiful way by a family of curves designed by Benoit Mandelbrot in the 1970s.

Snowflake Peano (Fifth Step)

The limiting curve fills the Koch snowflake. At each stage the figure consists of a single open, simply connected polygonal curve. In principle this entire curve can be traced on paper without lifting one’s pencil or crossing a line, beginning at the lower left-hand point of the star and ending at the lower right-hand point. Filling in the polygonal curve produces a closed polygon. As Mandelbrot says:

This advanced teragon, shown as boundary between two fantastically intertwined domains serves better than any number of words to explain what plane-filling means.

The word teragon, coined by Mandelbrot, comes from the Greek roots tera, or “monster,” and gon, or “corner.”

All the animations in this post were created by me using GeoGebra software.


Euclid Alone Has Looked on Beauty Bare

Euclid alone has looked on Beauty bare.
Let all who prate of Beauty hold their peace,
And lay them prone upon the earth and cease
To ponder on themselves, the while they stare
At nothing, intricately drawn nowhere
In shapes of shifting lineage; let geese
Gabble and hiss, but heroes seek release
From dusty bondage into luminous air.

O blinding hour, O holy, terrible day,
When first the shaft into his vision shone
Of light anatomized! Euclid alone
Has looked on Beauty bare. Fortunate they
Who, though once only and then but far away,
Have heard her massive sandal set on stone.

—Edna St. Vincent Millay

GeoGebra Art

In the classes I teach I often use GeoGebra for demonstrations; I assign GeoGebra constructions as homework, and our senior math majors use it in their oral presentations. GeoGebra, if you don’t know, is a Java-based dynamic geometry program freely available on the web. GeoGebraTube has thousands of applets developed by members for use in the classroom.

Now, GeoGebra is used mainly by high school teachers, and most of the applets available online are developed with that in mind. But the emphasis on algebra and coordinates by state curricula means that the program’s possibilities are vastly curtailed in common practice.

In trying to explore the software’s capabilities, I’ve set myself the task of constructing some of the fractals described in Mandelbrot’s Fractal Geometry of Nature. Images of my first attempts are shown below. At this point I’m not going to go into what makes a fractal a fractal, contenting myself merely with the following quatrain, which comes from Jonathan Swift:

So, Nat’ralists observe, a Flea,
Hath smaller Fleas that on him prey,
And these have smaller Fleas to bit ’em,
And so proceed ad infinitum.

Click on an image to view the slideshow and read the captions. Interactive versions can be found by going to my GeoGebra profile page. I won’t expand on my cleverness in the succinct programming that produced these images, but some of them are extremely clever.

The Quadrivium

Again, to start afresh, since of quantity one kind is viewed by itself, having no relation to anything else, as “even,” “odd,” “perfect,” and the like, and the other is relative to something else and is conceived of together with its relationship to another thing, like “double,” “greater,” “smaller,” “half,” “one and one-half times,” “one and one-third times,” and so forth, it is clear that two scientific methods will lay hold of and deal with the whole investigation of quantity; arithmetic, absolute quantity, and music, relative quantity.

And once more, inasmuch as part of “size” is in a state of rest and stability, and another part in motion and revolution, two other sciences in the same way will accurately treat of “size,” geometry the part that abides and is at rest, astronomy that which moves and revolves.

…In Plato’s Republic, when the interlocutor of Socrates appears to bring certain plausible reasons to bear upon the mathematical sciences, to show that they are useful to human life; arithmetic for reckoning, distributions, contributions, exchanges, and partnerships, geometry for sieges, the founding of cities and sanctuaries, and the partition of land, music for festivals, entertainment, and the worship of the gods, and the doctrine of the spheres, or astronomy, for farming, navigation and other undertakings, revealing beforehand the proper procedure and suitable season, Socrates, reproaching him, says: “You amuse me, because you seem to fear that these are useless studies that I recommend; but that is very difficult, nay, impossible. For the eye of the soul, blinded and buried by other pursuits, is rekindled and aroused again by these and these alone, and it is better that this be saved than thousands of bodily eyes, for by it alone is the truth of the universe beheld.”

–Nichomachus of Gerasa, Introduction to Arithmetic

liberal arts

The Eccentric Circle

When you were in high school, you undoubtedly encountered conics at some point. The most familiar example is the parabola, introduced in algebra courses as the graph of the equation y = x2 or, more generally, y = ax2 + bx c. You may have learned how to find the x– and y– intercepts, the coordinates of the vertex, and so on. These are all useful skills–although I personally have never used them outside of a classroom–but they tell you very little about the parabola as a geometrical object. The other conic sections–the ellipse and the hyperbola–don’t lend themselves to a description of the form y = f(x), hence are usually not studied at all. This is a terrible misfortune, as they are part of the treasure that comes down to us from the ancient geometers, and figure largely in physical sciences such as optics, acoustics, and celestial mechanics.

The most famous way to describe conics is as slices of cones, hence the name. In plane geometry, however, it is more convenient to think of them in a different way. We start with a line l (the directrix) and a point F not on the line (the focus). We select an eccentricity ε > 0. The conic is the locus of all points P such that the ratio of PF to the perpendicular distance from P to l is ε. If ε < 1, the conic is an ellipse; if ε = 1, a parabola; and if ε > 1, a hyperbola.


Here we see the eccentricity vary from 0 to ∞. The focus and directrix are in red. The parabola is the “boundary” between the family of hyperbolas and the family of ellipses, and occurs when the eccentricity is 1.

You may have observed that the ellipse becomes more and more “circular” as we slide the eccentricity toward 0. When the eccentricity reaches 0, then the circle collapses to the point F itself. But if we “blow up” the curve as the eccentricity shrinks to 0, then the limiting curve is indeed a circle. So we may view the circle as part of the same family of curves.

Many of the important properties of circles generalize to more universal properties shared by the other conics. For instance, from any point in the exterior of a conic there are precisely two tangents; or again, at each point on a conic there is exactly one tangent. It is possible to prove these properties by analytical methods, but this lacks geometrical finesse, involving arguments that would have been alien to the Greeks. The most excellent way to prove many of the basic theorems about conics is to use the eccentric circle.

eccentricConsider the conic with focus F, directrix l, and eccentricity ε. The interior of the conic is defined as the set of points P for which PF is less than ε times the distance to l. Let O be any point not in the interior and not on the directrix, and let s be its perpendicular distance from l. Construct the circle centered at O with radius εs.

Now let E be any point on the circle. Let R be the point where the line EF intersects l. Let P be the point where the line through F parallel to the radius intersects OR. An argument from similarity shows that P is on the conic.

Now imagine letting E traverse the circle. Then the conic is swept out by the point P as shown.


Here we have an ellipse. If the conic is a parabola, then the directrix is tangent to the eccentric circle; if the conic is a hyperbola, then the directrix intersects the eccentric circles at two points, and the two branches of the hyperbola correspond to the two parts of the circle.


The eccentric circle is what allows us to prove theorems concerning tangent lines. We already know that from any point in the exterior of a circle there are two tangents to the circle. Given an eccentric circle centered at O, the two tangents from F to the circle produce two tangents to the conic as shown.


Similarly, to show that there is exactly one tangent at a point of the conic, construct the eccentric circle at O. This circle passes through F, and there is exactly one tangent to the circle at F; this tangent produces the tangent at O.


Polygons and Divisibility

In the last post we introduced cyclic groups. The cyclic group Cn is generated by S, where S represents clockwise rotation through 360°/n (or 1/n of a revolution). It includes S2, which is 2/n of a revolution, and S3, which is 3/n of a revolution, and so on, all the way up to Sn-1. Notice that Sn = I is a complete revolution, which gets us back where we started.

More generally, suppose p is a whole number. Divide p by n to obtain the quotient q and remainder r. We can write p ÷ n = q R r or p = q ⋅ n + r. So Sp = Sqn+r = (Sn)q ∘ Sr. In other words, Sp consists of q complete revolutions composed with r/n of a revolution. It follows that Sp is the same as Sr. So, if we want to see what Sp does, we really only need to pay attention to the remainder of p ÷ n.

Suppose n = 8 for example. The group C8 is the group of rotations of the regular octagon, generated by a 45°-rotation S.

octagon symmetry

What transformation is (say) S29? Well, 29 ÷ 8 = 3 R 5, so S29 amounts to 3 complete revolutions composed with 5/8 of a revolution, or rotation through 5 ⋅ 45° = 225°. In other words, S29 = S5.

Now, we’ve noted that, if m is a divisor of n, then Cm is contained within Cn. This can be seen in the following way. Write n = m ⋅ q. Then Sq represents 1/m of a revolution. For instance, if n = 8, then we can take m = 4. Writing 8 = 4 ⋅ 2, we see that S2 represents one quarter of a revolution, i.e., a rotation through 90°.

polygon inscribed 2

This observation amounts to the following fact: a regular polygon with m sides can be inscribed in a regular polygon with n sides if and only if m is a divisor of n. (We regard a line segment as a “polygon” with two sides.) To draw the polygon with m sides, connect every qth vertex, where n = m ⋅ q.

For instance, the divisors of 6 are 2 and 3, so we can inscribe an equilateral triangle and a bisecting line segment in a regular hexagon. The triangle is constructed by connecting every second vertex. Or again, the divisors of 12 are 2, 3, 4, and 6, so we can inscribe a hexagon, a square, an equilateral triangle, and a bisecting line segment in the regular dodecagon. The square is constructed by connecting every third vertex, since 12 = 4 ⋅ 3.

polygon inscribed

The divisors of 15 are 3 and 5, so we can inscribe a regular pentagon and an equilateral triangle in the regular pentadecagon. The pentagon is constructed by connecting every third vertex, since 15 = 5 ⋅ 3.

So, if m is a divisor of n with n = m ⋅ q, then 〈Sq〉 = Cm. We’d like to answer the more general question now: If p is any whole number, then what does the group 〈Sp〉 amount to? In other words, if we keep composing p/n of a revolution with itself, what cyclic group do we obtain?

Let’s consider the example of n = 8 again. Take p = 3, and write T = S3. Then T is a 135°-rotation, or 3/8 of a revolution. Let’s start composing T with itself. To begin with, T2 is 6/8 of a revolution, or S6. Then T3 is 9/8 of a revolution, which is the same as 1/8 of a revolution. So T3 = S. Next, T4 is 12/8 of a revolution, which is the same as 4/8 of a revolution, or S4. Continuing like this, we find that the group generated by T consists of I, S3, S6, S, S4, S7, S2, and S5, in that order. So 〈T〉 = C8.

Now let’s take p = 6. Write U = S6. The group generated by U consists of S6, S4, S2, and I. We don’t obtain the entire group in this case. In fact, since S6 is a 270°-rotation, S4 is a 180°-rotation, and S2 is a 90°-rotation, we see that 〈U〉 = C4.

If we check each of the elements of C8, what we’ll find is the following. The group can be generated by each of S, S3, S5, and S7. The transformations S2 and S6 only generate C4. The transformation S4 generates C2. And the transformation I generates the trivial group C1.

Now, if p and n are whole numbers not both zero, then their greatest common divisor d is the largest whole number that divides both p and n. For instance, the greatest common divisor of 6 and 8 is 2, while the greatest common divisor of 12 and 30 is 6. In general, the group generated by Sp is the same as the group generated by Sd where d is the greatest common divisor of n and p. So, writing n = m ⋅ d, we find that 〈Sp〉 = Cm. If p and n share no common divisors larger than 1, then they are said to be relatively prime; in this case, m = n, and Sp generates the whole cyclic group.

Consider the case n = 12. Then S, S5, S7, and S11 each generate C12 since 12 shares no common divisors larger than 1 with 1, 5, 7, or 11. The transformations S2 and S10 each generate C6 since the greatest common divisor of 12 and 2, or 12 and 10, is 2. The transformations S3 and S9 each generate C4 since the greatest common divisor of 12 and 3, or 12 and 9, is 3. The transformations S4 and S8 generate C3. The transformation S6 generates C2. And I generates C1.

Or again, consider n = 15. Then S, S2, S4, S7, S8, S11, S13, and S14 each generate C15. Next, S3, S6, S9, and S12 each generate C5. The transformations S5 and S10 generate C3. And I generates C1.

Pick a single vertex of the n-sided polygon. Imagine repeatedly applying a rotation Sp to it (where p is less than n) and connecting each pair of consecutive points with a line segment. This amounts to connecting every pth vertex. If p happens to be 1 or n – 1, then we obtain the polygon itself. But if p is between 1 and n – 1, then we obtain either an inscribed polygon or a star. Also, the figure produced by p is the same as that produced by n – p; the direction is just reversed. It should also be clear that an inscribed star has n points if and only if p and n are relatively prime.

Everyone knows that there’s one 5-pointed star; this corresponds to p = 2 (or p = 3), because we draw it by connecting every second (or third) vertex.

star 5

Next, there are no 6-pointed stars because each of 2, 3, and 4 share common divisors with 6. But there are two 7-pointed stars, corresponding to p = 2 (or p = 5) and p = 3 (or p = 4).

star 7

We draw the first by connecting every second vertex, and the second by connecting every third vertex. There is only one 8-pointed star, corresponding to p = 3 (or p = 5).

star 8

It’s drawn by connecting every third vertex. If we connect every second vertex, we wind up with a square; if we connect every fourth vertex, we obtain a bisecting line segment. Next, there are two 9-pointed stars, corresponding to p = 2 (or p = 7) and p = 4 (or p = 5).

star 9

In general, if n happens to be a prime number, hence has no divisors other than 1 and itself, then there are (n – 3)/2 different n-pointed stars. For instance, there are (11 – 3)/2 = 4 different 11-pointed stars.

star 11

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.

pentagon symmetry 1

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.

pentagon symmetry 2

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.

Dihedral other

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.

Cyclic other

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.

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:

triangle symmetry 1

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

triangle symmetry 2

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:

square 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:

square symmetry 2

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

square symmetry 3

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:


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.


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.mobius 1Imagine 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:

rp2 construct 1Imagine making a purse out of ABCD by inflating the middle while drawing B and D toward one another.

rp2 construct 2Next, 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.

rp2 construct 3The 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.

rp2 construct 4To 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.

rp2 mobius 1In 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.

rp2 mobius 2Our 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.