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 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

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.

golden rectangle 1

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:

golden rectangle 2

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.

golden rectangle 4

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

golden rectangle 3

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.

eadem mutata resurgo

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.

delosLater 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.

menaechmusThe 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.”

byrne title

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.

Feynman on Geometry

Richard feynmanFeynman (1918 – 1988) was one of the great physicists of the twentieth century. He worked on the Manhattan Project during World War II, then went on to help formulate the theory of quantum electrodynamics, for which he received the Nobel Prize. The following is from his Character of Physical Law.

To those who do not know mathematics it is difficult to get across a real feeling as to the beauty, the deepest beauty, of nature… It is reputed—I do not know if it is true—that when one of the kings was trying to learn geometry from Euclid he complained that it was difficult. And Euclid said, “There is no royal road to geometry.” And there is no royal road… If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in. She offers her information only in one form; we are not so unhumble as to demand that she change before we pay attention.

Fra Luca da Pacioli

The painting in the blog header is a detail from a portrait of Fra Luca da Pacioli (1466 – 1517), Italian Renaissance mathematician and Franciscan friar.


Pacioli is not known for any original discoveries. He’s most famous for his books, e.g., his Summa de Arithmetica, Geometria, Proportioni et Proportionalità (Summary of Arithmetic, Geometry, Proportion, and Proportionality), a textbook on mathematical knowledge at the time, and De Divina Proportione (On the Divine Proportion), a book on geometry. In the latter, he expounds on the golden ratio and its relationship to the Platonic solids. It was illustrated by his friend, Leonardo da Vinci:

pacioli polyhedron

The painting itself contains many interesting mathematical objects. The model hanging in the upper left-hand corner is a truncated rhombicuboctahedron, an Archimedean solid, i.e., a polyhedron whose vertices are all congruent to one another and whose faces are all regular polygons.

pacioli 4

In this case, the polygons are squares and equilateral triangles. Notice that the polygon is half-filled with water and that its panes show a reflection of the view through the window behind the painter.

Pacioli is making a demonstration out of Euclid on a piece of slate. A dodecahedron (a Platonic solid composed of regular pentagons, the construction of which is the climax of Euclid’s Elements), sits on the book to the right. The young man standing behind Pacioli has not been identified, but he’s conjectured to be Albrecht Dürer, a painter of the Northern Renaissance who spent some time in Italy and later wrote a book on geometry and perspective. Some of Dürer’s graphic works contain mathematical themes.

Mathematics is in many ways a very tradition-oriented subject. Any mathematician who receives a doctorate was advised in his or her dissertation by a senior mathematician in a master-pupil relationship. An advisor can have a profound influence over their students’ mathematical predilections, understanding, and outlook. The Math Genealogy Project allows one to trace any line of “ancestry” from pupil to master as far back as the record goes, or to start with a historical mathematician and trace their mathematical “progeny” down to modern times.

While reading about Pacioli’s life, I explored the various lines of his mathematical “descendants.” It is unknown who Pacioli’s advisor was; his sole student was a man named Domenico da Ferrara. Among Ferrara’s students was Nicolaus Copernicus, the great pioneer of modern astrophysics. I continued tracing this continuous line through the sixteenth, seventeenth, eighteenth, nineteenth, and twentieth centuries, meeting along the way such luminaries as Johann Pfaff, Carl Friederich Gauss, and Karl Weierstrass. And, much to my surprise, at the beginning of the twenty-first century, I arrived at…myself, your humble RGC math professor.

This doesn’t make me anything special, however, for Pacioli has had more than a hundred thousand descendants over the centuries!