At a forum in Palm Springs, California, in 2022 January, Professor Whalen just gave two wonderful talks for a lay audience on great works of literature by Homer (Oμηρoς), Vergil, and Shakespeare. These works have great historical importance and literary value. When we were chatting afterward he asked me what I do and I said I was a mathematician. He said when mathematicians and physicists talked about their work what he found most interesting, surprising, and pleasing was the prominence of aesthetic beauty, a beauty they were reticent to explain. Maybe they were shy, maybe they felt an English professor wouldn't understand, maybe they wanted to keep their appreciation to themselves, or maybe they really didn't understand the beauty of their work. In any case, I claim my work is aesthetically beautiful and, if I'm going to claim my work is aesthetically beautiful, then it is incumbent upon me to be able to explain that beauty, even to a non-mathematician. (Here is the Wikipedia page on beauty in mathematics.)

     Like most disciplines, mathematics is cloaked in inclusion and exclusion, smart people who know the notation and rules are included and everybody else is excluded. Making fun of that attitude in one of our professors we joked among ourselves, "I have the funniest joke to tell, but first let me explain the notation." Picking up a math text or paper and just reading it is much harder than being part of a common group that understands it.

     It's not just mathematicians. Picking up the Odyssey and reading it isn't easy either. First of all, the original is written in ancient Greek, not a language most of us know. Even in translation it's not an easy work to follow and there are a lot of cultural assumptions that require education and intellect to understand. Reading Homer makes the movie "O Brother, Where Art Thou" a lot funnier and easier to understand. Come to think of it, seeing the movie "Zero Hour" sure makes the movie "Airplane!" a lot funnier.

     Please be patient that this exploration of the beauty of mathematics doesn't follow a linear, sequential curriculum. As I focus my vision on one beautiful part of a mathematical mind I'm like a dog seeing a squirrel when something else comes in the side window. I hope my not-always-direct presentation gives some insight into the beauty I see when I do mathematics.

     PHYSICAL BEAUTY

     The most accessible beauty in mathematics is visual. There are mathematical forms that are pretty to look at. As when I'm in a beautiful place like Bryce Canyon I wonder sometimes if the beauty is inherent in the thing itself or just in my mind as I look at it. Then I realize it doesn't matter as I can experience the beauty no matter where it comes from.

     Even with no mathematical knowledge, two-dimensional polygons and three-dimensional polytopes are interesting to look at. The symmetries of triangles, squares, pentagons, hexagons, et cetera are interesting. Pentagons connect us to the spooky occult as well as the space aliens on the old TV show "The Invaders." Triangles, squares, and hexagons can tesselate a flat plane surface repeating to infinity. There are tesselations with repeating multiple shapes fitting perfectly together to cover the plane to infinity. Just to make it interesting, there is almost always some exception to prove the rule. In this case there are non-regular tesselations that are non-repeating in that the arrangement of their shapes is not exactly the same in two places. One doesn't have to understand any mathematics to appreciate the aesthetic beauty of polygons, but a little insider mathematical knowledge and understanding makes the vision more interesting and intense.

     Even the simplest three-dimensional shapes have some recognizable beauty. A four-sided triangular-pyramid tetrahedron has fascinating symmetry and square pyramids are made famous in Egypt. A cube is fascinating in its three-fold and four-fold symmetries and the interesting two-dimensional patterns of six paper squares that can fold up to form a six-sided cube. The small-dot pips on dice have their own pattern and structure as opposide sides add up to seven. If we take a cube and file down the corners partway we get a figure with six squares and eight triangles and if we file the corners down further we get just eight triangles in an octahedron. Once we add the more-complicated icosohedron and dodecahedron we have all five platonic solids, named for the Greek philosopher Plato. These five three-dimensional shapes are beautiful even without any mathematical knowledge, more beautiful when one understands their structure. Only cubes can tesselate a three-dimensional space without leaving any gaps.

     Spirals are interesting, some with even spacing that look like they're expanding or contracting when they're rotating and some with proportional logarithmic spacing so they look the same at any bigger or smaller scale. A three-dimensional version of a spiral is a spring-shape helix, made famous by Watson's and Crick's double-helix DNA. A so-called spiral staircase is really a helix and so is the Guggenheim Museum in New York City.

     In their petals and spirals daisies follow a mathematical construct called the Fibonacci sequence 1,1,2,3,5,8,13,21,34,55,89,... where each number is the sum of the two before. A Google search tells me they have 21, 34, 55, and 89 petals and their inner spirals have consecutive Fibonacci numbers in their opposite directions. The ratios of Fibonacci numbers converge to the golden ratio φ=1.618034 which is said to be the most aethetically-pleasing ratio for a rectangle.

     I will take a moment to explore fractals, as in the picture at the top of this essay, as mathematical art because they got a lot of attention. These are mathematical figures with an ambiguous dimension that often have the same or similar structure at large and small scale. Take an equilateral triangle Δ, put a pointy hat on each of the three sides, put pointy hats on all twelve sides, ad infinitum, and the result is a fractal. I don't know of a lot of places where fractals have great mathematical significance, but the concept of zooming in with never-ending complexity manifests itself in some real things. Think of a coastline on a map with its twists and turns, we look a little more closely, there are smaller twists and turns, we look more closely still, and there are yet more smaller twists and turns. I guess an actual coastline stops at the molecular-atomic level, but a mathematical fractal goes on like that forever.

    

    

    

     DUALITY

     One can draw a triangle, put a dot it the middle of each of the three sides, connect those dots, and have a smaller triangle. One can do the same thing with a square to get a smaller square or a pentagon to get a smaller pentagon. We can think of this as shaving down the corners until the centers of the polygon sides are new corners. This sort of dual polygon might be exciting to a six year old playing with pencil and paper.

     Consider a four-sided triangular pyramid tetrahedron with four triangular faces, six linear edges, and four corner vertices. Do the same shaving-down game on the corners and, lo and behold!, we get a smaller tetrahedron with corners where the faces used to be. This might entertain us even when we're eight or nine. Along the way we get a shape with four hexagons and four triangles, a little more interesting, but that's not where I'm going with this discussion.

     Let's do the same thing with a cube with six square faces, twelve linear edges, and eight corner vertices. Now we don't get another cube. Instead, ah so!, we get a shape called an octahedron with eight triangular faces, twelve linear edges perpendicular to the original cube edges, and six corner vertices. Along the way we get a shape with eight triangles and six smaller, rotated squares, again interesting but off my topic here. The cube with faces-edges-vertices 6-12-8 is the dual of the octahedron 8-12-6.

     There are two more three-dimensional polyhedra. One is a dodecahedron with twelve pentagonal faces, thirty linear edges, and twenty corner vertices. They used to make desk sculpture calenders in this shape, twelve faces, twelve months, one month on each face. Its dual is an icosohedron with twenty triangular faces, thirty linear edges, and twelve corner vertices. The dodecahedron 12-30-20 is the dual of the icosohedron 20-30-12. Besides the physical beauty of these five platonic solids, the duality relationships are wonderful.

     In addition to the duality-symmetry of these shapes, there a rule that edges+2=faces+vertices. Put more succinctly E+2=F+V or F-E+V=2 for any three-dimensional polyhedron. The analogous rule for two-dimension polygons is kind of stupid and obvious, edges=vertices, E=V.

     If one can imagine four dimensions, maybe late at night after a few drinks, the same rules apply. There are six regular polyhedroids with similar duality relationships from three-dimensional cells, two-dimensional faces, one-dimensional edges, and zero-dimensional corner vertices.

	3-D cells	2-D faces	1-D edges	0-D vertices	dual
Pentahedroid	4 tetrahedra	6 triangles	6 edges	4 vertices	itself
Tesseract	8 cubes	24 squares	32 edges	16 vertices	16-cell
16-cell	16 tetrahedra	32 triangles	24 edges	8 vertices	tesseract
24-cell	24 octahedra	96 triangles	96 edges	24 vertices	itself
120-cell	120 dodecahedra	720 pentagons	1200 edges	24 600 vertices	600-cell
600-cell	600 tetrahedra	1200 triangles	720 edges	24 120 vertices	120-cell

     Okay, this is sounding too abstract. Let's consider something more down to earth. Consider a college student Mary selling food at night. (We had those when I was in college.) Mary has a cart big enough for ten personal pizzas or twenty bagels or any combination in between. Let's simplify the problem by removing the element of chance and we'll assume her pizza market is limited to exactly eight and her bagel market is limited to exactly sixteen. She makes three dollars $3.00 on each pizza and one dollar $1.00 on each bagel. We're looking for a mathematically optimal solution to this linear program.

     We can write the equations this way if she puts P pizzas and B bagels on her cart.

(market for pizza)         P≤8
(market for bagels)       B≤16
(room on the cart)       (2×P)+B≤20
(objective function)       maximize profit=(3×P)+B

We can display this as a tableau with a poorly rendered graph.

                                               Bagels
                                                16 |------
            Pizzas Bagels  -1                      |       \
PizzaMarket   1      0      8  = -mktPizza         |        \
BagelMarket   0      1     16  = -mkt              |         |
CartCapacity  2      1     20  = -cart             |         |
              3      1      D  = profit (max)      |         |
                                                   +-------------Pizzas
                                                             8

     Here's the interesting part. We can create a different set of equations reading down the columns instead of across. We have three variables PizzaMarket, BagelMarket, and CartCapacity.

(pizza)     PizzaMarket+(2×CartCapacity)≥3
(bagel)     BagelMarket+CartCapacity≥1
(objective function)     minimize (8×PizzaMarket)+(16×BagelMarket)+(20×CartCapacity)

Here's the cool, pretty, amazing part of all this. These strange dual variables PizzaMarket, BagelMarket, and CartCapacity are shadow prices, the incremental value of the constraints. Think of somebody trying to buy the business by paying enough for each resource to make it worthwhile for somebody running the business to sell it. Those resources are pizza market, bagel market, and cart space.

     The original primal problem has an optimal solution of eight pizzas and four bagels with a value of $28.00 and the dual shadow-price problem has an optimal solution of $1.00 for each PizzaMarket and $1.00 for each CartCapacity with the same value of $28.00. When the problem is linear like this one, where twice as much of any resource costs twice as much, and both primal and dual problems have a solution, the primal and dual optimal values are the same.

     ISOMORPHISM AND EXTENSION

     Some things are the same when they don't look the same. Let me start with a silly example. We take cards numbered One through Nine face up on the table. You take a card, then I take a card, then you take a card, then I take a card, until one of us has three cards that add up to fifteen. I take the Eight, you take the Five, I take the Two, you take the Four, I take the Six, and now I can stand up, jump on top of the table, and claim victory. I have Two, Six, and Eight, you have Four and Five, and the remaining cards are One, Three, Seven, and Nine. If you take the One, then I can take the Seven to go with my Six and Two. If you take the Seven, then I can take the One to go with my Six and Eight.

     Gosh, what a complicated game. Now I pull out my Math Mind and arrange the nine numbers in a magic square where every row, every column, and both diagonals add up to fifteen. It's also important that those are the only triples that add up to fifteen. Now when I take the Eight on my first turn, I'm putting an X in the upper-left square. When you take the five, you're putting an O in the center square. My choice of Two is an X in the lower right. I trust the reader doesn't have to be a mathematician to recognize the children's game that is isomorphic to this perplexing add-up-to-fifteen game.

     In the case of a map of fifty states or thirty-nine counties in Europe, it's not that hard. But we can extend the map-coloring concept to vastly more complicated problems. When we were assigning 312 radio channels to 5000 radios, we had rules of which pairs of radios were too close to use the same channel, essentially the same combinatorial problem. Kenneth Appel's proof of the four-color theorem didn't help me much, but there were complex computational methods that did help me find solutions about twice as good as the previous state of the art.

     This is a beautiful thing. We took a geometric map-coloring problem and extended it to a radio-telephony-technology solution using the same mathematical constructs.

     PROVING A LOT FROM A LITTLE

     The structure of classical mathematics is to take an interesting field of study and to reduce it to as few basic premises as possible.

     The classic example is Euclid and his geometry. Think of all the geometric knowledge we have and think of all we do with it. We have the pattern of Euclid's Elements (Στοιχεια) where he codifies all the various observations and practices of geometry into five postulates in Book 1 from which he derives twelve more books of theorems telling about geometry useful for surveying, planning, and other stuff. What he didn't know in 300 B.C.E. was how far that same geometry would go in understanding abstractions in algrebra, number theory, calculus, and more-advanced mathematics.

     It's a logic-math game of Jenga where the smartest people over thousands of years struggle to keep the entire stack of mathematics upright with the absolute minimum axiom structure, getting the most from the least.

     As I recall learning in primary school, Euclid was convinced that his Fifth Postulate was a logical consequence of the first four and was, therefore, not needed as a fundamental axiom of two-dimensional plane geometry. It says that, given a line and a point not on that line, there is one and only one line through that point and parallel to the given line. Before we wave our hands and cough at how simple the idea is, it took two millennia to answer that question and the answer was far more interesting than the question.

     The geometry we learned in primary school kept the notion of parallel lines without question. Around two hundred years ago Carl Friedrich Gauss and Georg Friedrich Bernhard Reimann discovered a geometry that didn't satisfy the Fifth Postulate. He realized the surface of a sphere is a geometry with points, lines, shapes, and curves, all except there are no parallel lines, all geodesic lines cross. Nikolai Ivanovich Lobachevsky figured out a saddle shaped geometry where there are lots of lines through a point that don't interesect a given line. The blinding, beautiful insight was that these non-Euclidean geometries were complete geometries in every other aspect.

     I think of a conceptual funnel where we have a vast, complicated reality that we boil down to a small number of fundamental, mathematical axioms and then we derive a rich collection of conclusions. In the case of geometry, we have designers, architects, surveyers, and planners all in need of rules of land and space, we can agree on four or five basic postulates, and, voila!, we have all this geometric insight. Euclid thought the funnel apex, in my thinking, was four postulates and then Gauss and Reimann found all five were required to make it work.

     Similarly, we have lengths, widths, heights, sizes, weights, volumes, amounts, et cetera and we need a mathematics for calculation. There are rules of algebra around addition and multiplication like A+0=A and A×1=A for all numbers A (other than zero) and there is always a number B so that A×B=1 and B×A=1. There are rules for counting numbers like 1, 2, 3, 4,  5 where there is no B satisfying A×B=1 unless A is also 1. There are rules for rational numbers that are ratios of two counting numbers like 3⁄7 or 13⁄4 (as long as we don't divide by zero). The beautiful thing about numbers is that we don't stop there. It turns out that if we want a number A so A²=A×A=2, then we're not going to find it in the rational numbers. We have to add rules for numbers that are irrational, that solve algebraic equations but are not ratios of whole numbers. An algebraic number is the solution of some rational polynomial like A³+(4×A²)−(7×A)−2.5=0. We shorten the notation by leaving out the × multiplication signs A³+4A²−7A−2.5=0.

     Even the algebraic numbers aren't enough for mathematics. There are numbers as basic as the circumference of a circle that are not only irrational but also not algebraic, not the solution of any rational polynomial equation. We call these numbers transcendental and the field we mathematicians use that includes all these numbers is called a real field or the real numbers.

     In case that's not enough of a mind bender, Leonhard Euler suggested that we could have an imaginary number i where i²=i×i=−1. I lied a bit two paragraphs ago because not all polynomial equations have real-number solutions. Some of them require this extra Euler number i and the field that includes all algebraic numbers and all transcendental numbers and everything is the field of complex numbers. A complex number is A+Bi where A is the real part and B is the imaginary part.

     These are simple intuitive steps that require, pardon my pun, complex mathematical solutions. Once we grasp the complex numbers derived, again, from about a dozen axioms, we can derive an incredibly rich mathematical structure. René Decartes figured out that the two dimensions of the complex numbers are analogous to the two dimensions of Euclidian geometry. Think about that for a minute. A point in a plane becomes a number in a mathematical algebra!

     The same is true in physics. Isaac Newton had three laws of motion, a law of gravity, and not much else. These laws explain motion and forces from dust particles to galaxies. Maxwell came up with some similarly-simple laws about electricity and magnetism. These laws explain electricity from thunderstorms to appliances, magnetism from compasses to powerplants, and radiation from radio and microwaves to light and X-rays.

     The only catch is that Maxwell's equations have an absolute reference to the speed of light. Physicists realized there was a problem with an absolute speed in a universe with lots of relative motions. It took geniuses like Albert Einstein to figure out a way that relatively-moving observers could agree on an absolute speed of light. There are a lot of complicated equations in the Theory of Relativity, but the basic notions are beautiful in that they reject time and mass as absolute so the speed of light can be absolute. The explanatory and predictive powers of this theory are enormous.

     Another example of a lot from a little is a mathematical field called topology. Topology rejects geometric shape and retains only fundamental character, so they say a topologist can't tell a coffee cup from a doughnut as they both have a single hole. There is the notion of two shapes being transformable into one another as we can imagine a coffee cup transforming continuously into a doughnut. There are topological equivalences that are not transformable as a simple knot can't be transformed into its equivalent mirror image.

     All of topology comes from a basic notion of an open set where a set is an open set if every point in the set has every near-enough point in the open set. That sounds a bit complicated, but we can simplify it by saying an open set is its interior without its boundary. A universe set and its collection of open sets form a topology and all the rest of the cool stuff comes from that basic notion. The proofs of these things are anything but simple, but, again, we get a lot of mathematical insight from a small number of axiomatic hypotheses.

     We play the same game in designing laws and legal contacts. We look for the smallest list of requirements to define a code of conduct that works over the largest set of circumstances. Our American Constitution defines the government for a vast country in just a few pages from a short list of values and principles. There is great beauty in that.

     SIMPLICITY

     Sometimes a simple idea works wonderfully well and enlightens us about the universe in some way. If I take a surface with hills and valleys, then I can calculate a slope at every (x,y) point. If I want to know how much energy it takes to move a heavy object from a valley to a hilltop on a given path, then I can multiply the weight of the heavy object times the slope at every point along the way (times the cosine of the angle) to calculate the amount of work it will take. It turns out the answer is the same, independent of the path, just by knowing the altitude difference between the start and the end. Physicists say there is a gravitational potential at each location and that describes completely the work required from Point A to Point B. The work required from Point A to Point A would have to be zero.

     There is a similar potential function for voltage fields so electrically-charged objects (electrons, for example) going through a voltage potential take or give energy and that energy amount is defined completely by the potential.

     It doesn't always work. A magnetic field doesn't have the same potential function and moving a magnetic object around a path from Point A to Point A can absorb or produce a non-zero amount of energy. (That energy comes from somewhere else and I'm going to "wuss out" and say that the electrodynamic physical science is complicated enough that I'll leave it at that.)

     Another example of a different kind of potential function is called a logarithm. (Math people don't usually call logarithms potential functions, but go along with me here for the sake of aesthetic beauty.) If I say log 1 is zero, log 10 is one, log 100 is two, log 1000 is three, log 10000 is four, et cetera, then I can add the logarithms to multiply the numbers, so rather than do the arduous calculation 10 × 100=1000, I can do the much-simpler calculation 1+2=3. Okay, it's not much simpler for powers of ten, but maybe, just maybe, we can assign fractional logarithms for numbers that aren't powers of ten and hope it all works out. The square root of ten √10 is about 3.162 so 3.162× 3.162=10, so we can say log 3.162 is ½. Since two to the tenth power 2¹⁰ is about 1000 and log 1000 is three, maybe log 2 is about 0.3. We can do all kinds of approximating and inferring to get logarithms for all numbers, but do they really work out? Are there logarithm values for all positive numbers so log (A×B)=log A+log B for all A and B values? This time the answer is yes and that's how a slide rule works (for those of us old enough to remember slide rules). The wooden or plastic slides perform mechanical addition on scales marked in exponential values so the result is a mechanical multiplying machine. (By the way, log 2 comes out to 0.301029995.)

     For numbers "close to one" then we can use A−1 as a logarithm for A so 1.02×1.03=1.05 as 0.02+0.03=0.05 and 1.01×0.97=0.98 as 0.01-0.03=−0.02. This trick only works for numbers near one and these are clearly not the same logarithms described above. 10^0.01 is 1.02329, nowhere near 1.01, but there is a number e where e^0.01=1.01 and e^–0.02=0.98 where e is about 2.7182818284590. It's another irrational, transcendental number just like π=3.14159265358. These natural logarithms are interesting in many ways and, when combined with the imaginary number i=√−1, they are connected to the sines and cosines we supposedly learned in high-school trigonometry class.

     I thought Paul Erdös said, "Elementary does not mean easy," but I can't find that quote on my Internet searches. I found a quote from Richard Feynman, "I am going to give what I will call an elementary demonstration. But elementary does not mean easy to understand. Elementary means that very little is required to know ahead of time in order to understand it, except to have an infinite amount of intelligence."

     Just because we can write something in a small number of symbols doesn't mean it's elementary or easy. A classic example of elegence is an equation e^πi=−1. It's just five or six symbols but there's a lot of mathematics there. We extended our rational-fraction numbers to irrational-transcendental numbers like e. We extended our integer-exponent A⁴=A×A×A×A to fraction exponents with roots so A^3/2=√A×A×A and even to A^π with a sequence of ever-closer approximations, A³, A^22/7, A^355/113, et cetera. It's a non-trivial leap to the notion of an imaginary exponent i=√−1, but it works out to be sines and cosines from trig class in high school. Putting all that together gets −1 for e^πi, but after the time and effort to get all the background, the equation's simplicity and elegance is lost on all but serious math weenies like me.

     An example of successful simplicity out of complexity is Georg Cantor's definition of infinity ∞. If we define infinity as just bigger than any number, then we have all kinds of problems. Is ∞+∞ bigger than just ∞ and then is ∞×∞ even bigger? Cantor's answer was to define infinity as a counting cardinality of a set of numbers. The cardinality of the set {1,2,3,4,5} is five, duh. But the cardinality of the set of all integers {1,2,3,4,5,...} is bigger than any integer. Cantor called that ℵ_o or "Aleph null" as he proved this is the smallest of these transfinite numbers. He said two of these transfinite numbers are equal if they are cardinalities of sets that can be put in one-to-one correspondence, so the cardinalities of {1,2,3,4,5} and {A,B,C,D,E} are the same, both five, and the cardinalities of {1,2,3,4,5,...} and {2,4,6,8,10,...} are the same, mapping N to 2×N, so there are no fewer even numbers than all numbers.

     If we have two disjoint sets with no members in common, then the cardinality of their union is the sum of their cardinalities, so the union {1,3,5,7,9,...}∪{2,4,6,8,10,...} has cardinality ℵ_o+ℵ_o, the same as {1,2,3,4,5,...} ℵ_o. So transfinite number can have strange notions like ℵ_o+ℵ_o=ℵ_o, or even ℵ_o×ℵ_o=ℵ_o. (The multiplication isn't obvious and I'll skip that for now.)

     The cool thing is that Cantor showed that there actually are bigger infinities, that the number of real numbers is actually more than the number of integers, that there is no one-to-one mapping of integers to real numbers. There's a hierarchy ℵ_o, ℵ₁, ℵ₂, et cetera and the mathematical question of whether there is a transfinite more than integers and less than reals turned out to have an interesting answer. There is no proof either way from the mathematical structure we have, just as there is no proof of Euclid's fifth, parallel-line postulate from his first four.

     I think this is a beautiful thing and, yet, simple enough that I was able to explain it to a high school class.

     NEGATIVITY AND EXTENSIONS

     The notion of something being "impossible" sounds impossible. After all, don't we tell people anything is possible? Well, maybe, but sometimes we have to outside the agreed-upon box to get there.

     Is there a rational number A where A²=A×A=2? A rational number is a ratio of two whole-number integers like 13⁄9. Well (13⁄9)² is 169⁄81=2.08642, close but not quite 2.0, no cigar. We can get closer and closer, but there really is no rational number solving A²=2. Yes, there is such a number, but it's not rational. It's impossible to express the square root of two √2 as a ratio of two integers.

     Similarly there is no expression for the ratio of a circle's circumference to its diameter π as a ratio. The fraction 355⁄113=3.14159292 comes close to 3.14159265, but, again, no stogie for the hopeful smoker. The value of π exists, it's a real number, just not a ratio.

     There are classic geometric examples of impossibility. Using a compass and straightedge (not a ruler, just a straightedge) we can start with a square and construct a square with twice the area. In other words, we can multiply a given distance by √2. Given an angle, we can construct an angle half as large. It turns out we can't construct a cube with twice the volume, we can't create a distance cube-root-of-two ∛2 bigger than a given distance with these tools. We also can't trisect a given angle with these tools and we can't come up with a circle with the same area as a given square which would require constructing π.

     The notion that a twenty-degree angle is not "constructable" doesn't mean we can't get 20° angles, only that we need something more sophisticated than these basic tools to do it.

     In his heart of hearts Euclid believed he could prove his Fifth Postulate that given a line and a point not on the line there was one and only one line through the given point parallel to the given line. Two millennia later mathematicians found he was wrong, there is no proof because there are geometries without it.

     When he studied infinity in the form of his transfinite numbers Georg Cantor showed there are more real numbers than rational or algebraic numbers, but he could neither prove nor disprove that there were any transfinite numbers between them. It doesn't seem so hard, either there's a set with strictly more than the rationals and strictly less than the reals or there isn't. In 1878 Cantor believed there was no such in-between set or number and in 1963 Paul Cohen proved that this hypothesis was not provable from the underlying mathematics. One could assume yes or one could assume no and both would generate good, consistent, logical mathematics. It ties into the Axiom of Choice that says if I have a collection of non-empty bins that it is possible to make a selection of exactly one object from each bin. This is trivial if the bin collection is finite, but things get interesting and weird if there are an infinity of bins.

     Another impossible one is Fermat's Last Theorem. There are oodles of whole-number integers where A²+B²=C². For example, 3²+4²=5² and 12²+5²=13², but there are no examples where A³+B³=C³. Pierre de Fermat said there were no integer examples where Aⁿ+Bⁿ=Cⁿ for any whole-number-integer value of n≥3. In 1637 he teased the world by saying he had a marvelous proof that wouldn't fit in the margin where he doodled the so-called theorem and in 1993 Andrew Wiles proved it with mathematics that weren't even a gleam in the eyes of mathematicians three centuries earlier.

     From the four-color map theorem discussed above it is similarly impossible to make a flat map of five contiguous countries with each touching the other four. If one of those "countries" is the ocean, then it's the same as four countries on an island each bordering the other three and each with beachfront.

     There is great beauty not so much in the impossibility of doing these things but in the understanding of what one has to learn to know that they are indeed impossible.

     ABSTRACTION

     There is a field symbolic logic where we take all meaning out of logical proofs and deal only with the symbols themselves. A relationship "A implies B" A⇒B normally has the meaning, "If A is true, then B is also true." We think of this as a connection between A and B. The statement "All sheep have four legs" is logically the same as "If X is a sheep, then X has four legs." We can have syllogisms, kind of like mathematical-logic haiku.

All Pennsylvanians are Americans.
George is a Pennsylvanian.
Therefore George is an American.

We have relationships among relationships. If all sheep have four legs, then we have the contrapositive that all creatures not having four legs are not sheep. Both of these are satisfied if the creature is not a sheep or has four legs.

     We make the leap from reality to symbolic logic when we strip away meaning and figure out what we can glean just from symbolic manipulation. The relationship A⇒B is equivalent to notB⇒notA, never mind what A and B might mean. The relationship A⇒B is also equivalent to notA OR B. The field of symbolic logic has lots of manipulative rules for logical expressions where the meaning is removed and only the abstract logic is left.

     Logic is about truth-vs.false and often does not extend to probabilities or evidence. The statements "All sheep have four legs" and "All non-four-legged creatures are not sheep" are logically equivalent. Seeing a two-legged, not-a-sheep gorilla supports the latter but not the former. If I go into court trying to prove every member of the Perkins family was at Moe's Bar on January 10, I don't think the judge will be happy if I present as evidence a list of non-Perkins people who were at a different place that night.

     Given a logical expression like (A AND notB) OR not(not B AND notC) there is a reasonable question of whether there is some kind of combination of true-false values for A, B, C, et cetera where the expression is true. One can walk through all 2^N combinations false-false-false, false-false-false, false-true-false, false-true-true, true-false-false, true-false-true, true-true-false, and true-true-true for A, B, and C, not so terrible for three variables but very terrible for thousands of variables. Without some structure knowledge of the question, there isn't an alternative to some painful enumeration. The solution is exponentially like 2^N time consuming as the problem grows.

     There are problems that are simpler to solve. Their solution times grow as polynomials like N² as the problem grows. If I have a map, then figuring out if can be colored in two colors is easy. We look for every vertex where three or more countries meet and, if every one of those has an even number of countries, then the map can be colored in two colors. If I have a network of roads or paths meeting in intersections, then can I walk each road exactly once? Again the solution is easy. If every intersection has an even number of roads, or every intersection except two, then there is a Eulerian path where each road is walked exactly once. We can take some liberty and say the polynomial-time N² problems are easy while the exponential-time 2^N problems are hard.

     Here's the cool part. Of all of these problems, the logical-satisfiability problem above is the hardest and there is a whole collection of problems similarly-exponentially hard. While two-coloring a map and finding an Eulerian route are easy, three-coloring a map and finding a Hamiltonian route that visits each intersection exactly once are hard. Within a polynomial-time "easy" conversion, they're just as exponentially hard as logical satisfiability. There's beauty in being able to catalogue problems by their complexity.

     Map coloring combinatorics where the map is not necessarily on a flat surface or globe is, alas, expontially hard in general and that was precisely the problem I faced in finding the minimum number of radio channels needed to serve a city with cellular telephone service. The abstraction of coloring paper maps to assigning radio channels got me from one difficult situation to another, but at least I had some useful mathematical insights about map coloring in these higher dimensions.

     In addition to geometries on surfaces like a plane or a globe, or three-dimensional geometries like outer space, there are finite geometries that are physically non-geometric but follow all of Euclid's postulates. We can extrapolate all the physical properties of geometry to abstract notions that have no physical meaning.

     MUSIC

     The connection between mathematics and music comes up so frequently in conversation that I feel I must address it here. There is much about music that is mathematical.

     Rhythm is clearly fundamentally mathematical. Classical rhythms such as two, three, and four beats per measure extend easily to five and six beats per measure and, with more difficulty, to seven beats and beyond. There is arhythmic music without any periodic or regular timing. I find such pieces unsatisfying, but others may be more open minded.

     Pitch is full of mathematical structure. From the rational intervals of an octave 2-to-1, a fifth 3-to-2, a fourth 4-to-3, and a major third 5-to-4 we get a lovely "just" scale that doesn't transpose well. Once we settled on a seven-note scale (suggesting the name "octave" is a misnomer) and twelve chromatic tones, the way to transpose from key to key without retuning is to use the twelfth root of two 2^1/12 and to live with the inaccuracies.

     There are more-complex issues like what starting frequency to use. Classical tuning was C-256 which was raised a bit to A-440. The piano has its own oddities of tuning from the different tuning of the multiple strings for each note to change the tone decay to the "longer" octaves, more than 2-to-1, to make the overtones fit neatly with the higher notes.

     The Circle of Fifths for major and minor scales has significant mathematical structure.

     While all of this is mathematical, it's hardly more essential mathematics than the thermodynamics and heat equations around the culinary art of cooking a meal. I'll make two arguments that the connection between mathematics and music isn't as real as some would like to believe.

     First, I gain little mathematical insight from music and, conversely, little musical insight from knowing more mathematics. While topology, algebra, geometry, and analysis give insights to the others, knowing the Circle of Fifths or the non-harmonic structure of the piano doesn't help me prove theorems. Similarly, attempts to build music directly from mathematics has resulted in renditions of pitch sequences based on the digits of π and similarly-awful sound experiences.

     Second, music and mathematical geniuses aren't the same people. There isn't a list of great composers with mathematical theorems named after them. The Hungarian Jewish stock of humanity produced the greatest mathematical minds of the Twentieth Century and the only musical member of that extraordinary community was Maestro Eugene Ormandy.

     TRUTH

     I'd like to touch on the topic of mathematical truth. In a mathematician's world, a statement is "true" if it follows logically from foundational axioms, so the same statement that is true in Euclid's geometry may not be true in Reimann's geometry. The statement that there is a square root of two √2 is true in our full-blown real numbers but not if we're only looking at whole number integers or even rational numbers that are ratios of integers.

     Even in physical reality truth is more elusive than the absolutist-mind would like to believe. My world is flat when I'm flying my airplane only a few hundred miles on a map. Its actual curvature doesn't help me navigate and I can fly constant-heading rhumb lines with impunity. Make the voyage like longer airline flights and rhumb lines are poor approximations of the shortest route on the globe. Now my world is round because a flat earth no longer gets a satisfactory approximation to the best answer. "They tell me the Flat Earth Society has members around the globe."

     When I'm observing astronomy I use Ptolemy's model of a geocentric Earth-centered solar system with its inherent cycles and epicycles because that's how I align my telescopes on the ground. When I'm plotting orbits I use Copernicus's heliocentric Sun-centered solar system with all the planets in orbit around the sun. (I learned about nine planets and have a hard time thinking of Pluto as not a planet. Now that my Star-Wars-universe home on Alderaan isn't there anymore, I keep a home on Charon to escape the summer heat on Earth.) In fact, both of these models are wrong and the actual barycenter of the solar system is a spot somewhere outside the sun itself because Jupiter's mass and distance are enough to make the true center somewhere in the sun's corona that all the planet and asteroids orbit around.

     So how do mathematically-minded people deal with conflict? Is all truth relative? Not at all. Once we accept a set of premises or axioms as fact, either temporarily in mathematical study or permanently in social or political context, then the conclusions are true and absolute.

     It's inconvenient that some of the simplest notions become mathematically complicated. That the number two doesn't have a rational square root, that the notes of a musical scale can only be evenly spaced by the twelfth root of two, that the ratio of a circle's perimeter to its diameter is a complicated number like π=3.14159265359, that the base of natural logarithms is another complicated number like e=2.718281828 are deeply rooted in our mathematical universe. Axiomatic structures that simplify these complexities are out there, but they are so far from anything practical or physically real that we don't use them, at least not when we're plotting or planning anything useful.

     Just because physical reality is relative, or so said Albert Einstein, doesn't mean there aren't things that are true and false about it. There is a notion that events in space and time are absolute but the distances in space and time between them are not. J.B.S. Haldane said the universe is not only queerer than we imagine—it is queerer than we can imagine. Here it is, I found it on Wikipedia, so it must be absolutely true, "My own suspician is that the universe is not only queerer than we suppose, but queerer than we can suppose."

     I live my life in this universe of truth and I can assure those who doubt my comfort that I'm not only comfortable with the notion and flexibility of mathematical truth, but that I'm constantly awed by its aesthetic beauty.

     NUMERICAL COMPUTATION

     Those I know who study classic literature never got to meet Homer, Virgil, or Shakespeare, to pick the three authors in Professor Whalen's presentations. Similarly, I never met the founders of numerical computation, John von Neumann and Alan Turing. However, I did get the opportunity to sit in the classroom with their intellectual progeny as teachers, Forman S. Acton for von Neumann and Jim Wilkinson for Turing.

     There is a chasm, often ignored in the classroom, between deriving a mathematical model for a problem and calculating the right numerical answer, especially in a reasonable amount of time.

     In the right-answer category there are clever ways to avoid taking small differences of large numbers, but most of these fall in the category of crafty rather than elegant or beautiful.

     On the other hand, there are wonderfully attractive notions of aesthetic mathematics in finding the way to get the right answer with less work. Let's take a simple problem of sort a list in order, maybe alphabetical, maybe numerical, maybe some other less-obvious criterion given to us. On first glance one would think it would be required to compare every pair of items in the list. If the list has one hundred items to sort, then that would require 1+2+3+...+99+100=5050 comparisons. In fact, it can be done with a maximum of 2100 comparisons. That's the number in the list, 100, times the number of binary decisions needed to cover 100 choices times three. Since two to the seventh power 2⁷=128 is enough for 100 choices, it takes 100×7×2=1400 pairwise tests or less. That reduction from ½N² to 3N log N is a huge win for larger lists.

     At the risk of dragging my reader through some technical mud, I'll explain the heapsort algorithm I use to sort lists. Consider a mathematical binary-tree structure with a single node at the top and each node can have one or two children going down the page. (I draw my trees with the "root" at the top because I tend to think of them from top down.) A heap is a specific binary tree where every parent node is larger-valued (further down the sorted list) than any children it might have. There is no comparison claim between the children or between parts of the tree that aren't parent and child. Furthermore, the binary tree is minimum depth in that we fill all 2^N places on the tree at level N before adding any children at the next level down N+1.

     So I start with my unsorted list, grab the first item, and build a one-item heap with it. Then I take the second item and make it a child of the first item. I have to test them against each other to make sure the larger-later item is the parent and the smaller-earlier item is the child and to switch them around, if necessary, to make them that way. The third item becomes the second child of the heap, again requiring a single comparison of two items. Building a heap is only N log N comparisons because each new addition has to be tested until it's smaller than its proposed parent and that could happen log N times. (Mathematically-savvy people will please forgive me for a little hand-waving here. My description may not be precise, but the math ends up being right.)

     The last item in the sorted list is the top of the heap because the largest-last item can't be anybody's child. Then I take the last item on top of the heap, take it off the top of the heap, put it at the end of my sorted list, and figure out which of its children is larger to make that child the parent. I have to check both children, so that may require two comparisons for each of the potentially log N times I have to pick a new parent. The new top of the heap is the new largest-last item, so I can put that one at the end, just before the last one I took from the top, and keep repeating the process until the heap is gone and all I have is, voila!, the sorted list.

     The ultra-cool part of building this heap-sort algorithm is that the computer memory for the heap and the list co-exist perfectly as the non-heap members of the list fit perfectly at the end after the heap members of the list. By having the children of node K being 2×K and (2×K)+1, it all fits perfectly and the calculations are fast. There are sneaky tweaks to make the calculation closer to 2N log N than 3N log N, but I'll stop here and go back to my appreciation of the beauty and wonder of computational algorithms.

     There are many computational challenges where understanding the true, physical nature of a problem is the key to computing the right answer quickly. There are others where the computational essense has little or nothing to do with physical reality.

     My reader might want to look at me at this point and say, "Adam, how can you compare this with the beauty of the `Mona Lisa' or the `Odyssey'?" I answer that the beauty is there and just as profound. The only catch is that one has to know the background. Is that so terribly different that the inculturation required really to get the messages from Leonardo and Homer?

     CULTURE

     There is a culture of common knowledge among mathematicians and physicists and other geeks. Just as a side reference to "a petty pace" evokes recollection of Macbeth, a reference to a particular number or equation connotes a mathematical mindset.

     Don was a senior at Princton while I was a freshman and we were both in the Computer Center (back when there were separate computer centers) and he was trying to compute an interesting infinite fraction, the limit of 1/2, (1/2)/(3/4), ((1/2)/(3/4))/((5/6)/(7/8)), ..., and the computer gave an unexpected number. Don said the computed value was "Point seven oh seven" (0.707) and my immediate response was "Point seven oh seven one oh six seven?" as in 0.7071067, the square root of one half. Neither of us mentioned one half or the square root, we both had the common knowledge that √½ is about 0.7071067 and that there was no reason obvious to either of us this fraction should converge to that value. I'll make the stronger statement that it was obvious there was no reason this fraction should converge to that value. We both shared a common body of knowledge and understanding, a "tribal knowledge" of mathematics, that made our conversation elliptical (as in brief, not as in oval shaped), just as medical doctors talk in their own code for brevity's sake.

     What made Ramanujan so amazing is that he read a bunch of books and was able to engage in discourse with Oxford professors without having been part of the mathematics community. I wonder who could read Shakespeare's work in isolation and jump into a conversation of Shakespeare scholars without missing a beat.

     Not all this common knowledge is intellect expanding. I have interviewed several mathematical-scientific candidates where I'm explaining the retail relationship between price P and quantity Q. I point out that economic elasticity E=βP means the calculus derivative dQ/dP=−βQ and wait for the flash of insight that almost never comes. Then I say, "Surely you know the solution of dY/dX=−βY and the candidate jumps up and says, "Oh, yes, it's e to the minus beta X," Y=e^−βX. When it's X and Y it's obvious but when it's P and Q it's obscure.

     When I was in Rome I bought my uncle a copy of Macbeth in Italian as he was a Shakespeare scholar and he knew Italian. I figured it was a cute gag, but he said he enjoyed it thoroughly. I raised an eyebrow, Spock style, and he said, "I know the play, but the introduction referred to well-known and well-read Continental authors and playwrights instead of English ones." I suppose it might have been Dante Alighieri and Miguel de Cervantes instead of Geoffey Chaucer and Thomas Kyd. He found being immersed in a different cultural community in the Italian introduction to Macbeth fascinating.

     I find the same thing when I discuss mathematical concepts I know well in a different context such as engineering or another branch of mathematical study.

     CONCLUSION

     I hope I've communicated some of the aesthetic and human beauty of mathematics without immersing and drowning my reader in technical details. Mathematical study goes back to the ancient Greeks as does classical literature and there is much beauty from then until now.

     Once we make the leap from visual or audible beauty to beauty in the flow of words in literature, it's not a huge leap to the more-abstract beauty of mathematics and science.