It might seem a strange question to ask whether all functions are computable. We are used to seeing functions written down in mathematical language, making them automatically computable. For example, the quadratic function is obviously computable:

We can put any x into this function and calculate the answer. Or to take another example, the sine function is computable since we can write it as an infinite series:

We can put any x into this function and calculate sin(x) to any precision we like by including as many terms as we need in the infinites series. Surprisingly, however, there are an infinite number of functions that we can’t compute. The reason is that there are more functions than there are algorithms to compute them.

What does it mean to compute a function? Intuitively, it means that we can work out the answer on pencil and paper or we can punch numbers into a caculator to compute the function. More generally, a function is computable if we can write a computer program to compute it. For example, one python code to compute the quadratic function would be

def quadratic_function(x):     
    return x**2 + 5*x + 2

How Many Possible Computer Programs Are There?

Computer programs are strings of characters of finite, but arbitrarily large length. For example, the python program above could be represented as the string of characters:

def\space\quadratic_function(x):\enter\\tab\\return x**2 + 5*x + 2

where \space\ is the space character,\enter\ is a carriage return, and \tab\ is a tab. All computer programs can be written as strings of characters of finite length. How many possible computer programs are there? It would seem that there are an infinite number of compute programs, but we’ll need to be more precise about what we mean by infinity.

Countable Infinity

A set is countably infinite if it can be put in order and counted by the infinite positive integers. This definition immediately leads to surprising consequences. The set of even numbers is countably infinite since the even numbers can be put in order and counted by the positive integers.

Thus, it turns out that a subset of a countably infinite set is also countably infinite. When we are talking about infinite numbers, a subset of an infinite set has the same number of elements as the set itself.

What about the rational numbers? They are countably infinite as well, since we can put them in order and count them by the positive integers:

In this diagram, we order the rational numbers by following the arrows, excluding the rational numbers circled in green, since we already counted them previously. 1/1 would be the first element, counted as one, 1/2 would be second, counted as two, 2/1 would be the third element, counted as three, and so on.

Uncountable Infinity

What about the real numbers? Let’s assume we can order them by putting them into a one-to-one correspondence with the natural numbers as in the diagram below, in which we have circled the diagonal element in each real number.

Now let’s construct a new real number with the property that its first digit is different from the first digit of the first real number, its second digit is different from the second digit of the second real number, continuing in that fashion:

This new real number can’t be on the list. As a consequence, there are more real numbers than there are integers. Even if you think you’ve written down an infinite set of real numbers, there are always more. The set of real numbers isn’t countable—it’s uncountably infinite.

Is The Set of Possible Computer Programs Countably or Uncountably Infinite?

The set of all possible computer programs is countably infinite since we can order them. Every computer program is a sequence of character symbols of finite length. To order them, we can consider all legitimate computer programs that have one character, all legtimate computer programs that are two characters in length, etc. We can order them by placing the one character programs first (counting only those that are legitimate, if any, in the programming language we are using), then the two character programs next, and so on. Within the programs of character length N, we can also order them further by defining rules that order sequences of characters: if the first letter is “a”, it comes before any program that begins with any other letter or symbol, including “A” with the rule that lower case letters are counted before upper case letters. If both programs begin with “a,” then we proceed to the next character and apply the same ordering rules. In this way, we can associate every legitimate program with a positive integer. The set of all possible legitimate programs is countably infinite.

How Many Functions Are There?

The number of functions is uncountably infinite. To see this, let’s define a family of functions as follows: for every real number, the function is

where n is a positive integer and d_n is the nth digit in the infinite decimal expansion of the real number. We can define one such function for every real number, implying that this subset of all the functions is uncountably infinite. Since there are more functions than there are algorithms to compute them, there must be an infinite number of functions that are not computable.

Implications

Most real numbers are not computable, since there are an uncoutably infinite number of real numbers and a countably infinite number of computer programs. If a real number is not computable, that means that there is no algorithm to compute its value to arbitrary precision. The computable reals are countable, however, because there are as many of them as there are algorithms.

Chaitin’s Constant

Chaitin’s constant is the probability that a randomly generated computer program will halt. These probabilities are non-computable. They can be approximated, but not to arbitrary precision.

Zeros of Functions

Since most real numbers are non-computable, it should not be surprising to realize that there an infinite number of functions with some real zeros that cannot be computed.

In this post, we’ll see how the modern number system elegantly solves the problem (discussed in this post ) of how to write numbers as big as we please and how to calculate using pen and paper. The modern way we write numbers and do arithmetic is one of the most important inventions in the history of ideas. Indian mathematicians around the 7^th century AD saw that it was possible to write numbers using only ten symbols, a creative solution to a problem that had vexed mathematicians for thousands of years. We often think that mathematics is discovered, but how we chose to write down mathematical symbols was invented. There is no rule that says we must use ten symbols for numbers. Indian mathematicians invented that solution. When Arab mathematicians adopted the Indian numbers, they realized that they could easily write big numbers and that they could do arithmetic with pen and paper. That simple insight eventually led to an explosion of mathematical progress.

Modern numbers originated from the Indian number system

The Indian numerals, developed in the eight century A.D, were founded on two simple, but incredibly powerful ideas. First, you only need to group by ten using ten fundamental symbols, including the number zero. And second, you don’t need any symbols beyond these ten, since you can indicate other groupings such as the one hundred grouping or one thousand grouping by the position of each digit. So, you don’t need an L for 50 or a C for 100 as you would have in the Roman system. You also don’t need cuneiform symbols for one and ten as the Babylonians used for sexagesimals.

The Indian numbers, 1-9 and zero, originally looked like this.

Because these digits had to be hand copied into scholarly manuscripts, they would naturally change a little bit over time. Eventually, as they were copied over and over, they changed a bit after a few hundred years, with the changes indicated in red:

The horizontal line representing the number 1 eventually became vertical, probably because it saved space to write it that way. The other changes are very slight, and probably came about because it’s easier to make the symbol in one smooth motion without lifting the pen. So, the number for 2 likely came from drawing the top horizontal line from left to right, then moving the pen down diagonally to the left, and then making the bottom horizontal line. The modern three likely came about the same way: the three horizontal lines were drawn without lifting the pen. The numbers 4 and 5 are variations. Five came about by adding the curly cue to the bottom of the original 5, possibly as a flourish. The modern eight and nine seem to have developed from writing the number in one continuous circular motion.

During the Arab conquests, the Indian numerals were imported into Islamic science and mathematics and were used extensively by scholars. Arab mathematicians also developed the techniques that schoolchildren use today to do arithmetic by hand using paper and pen. But the Indian numbers never caught on with the general public in the Islamic world of the Middle Ages.

Europe eventually adopted the Indian-Arab number system

At the turn of the thirteenth century, the Indian number system was introduced into Italy by Leonardo of Pisa, also known as Fibonacci. Fibonacci was an Italian merchant and scholar who traveled extensively for business and for mathematical research. While visiting what is today Algeria, Fibonacci learned of the Indian numbering system and the arithmetic Arab mathematicians had developed. Fibonacci thought the Indian-Arab number system so superior to the Roman system and arithmetic with pen and paper so much better than the abacus that he resolved to replace the Roman numbers with the Indian-Arab numbers.

In 1202 AD, Fibonacci published a book, Liber Abaci, Latin for “The Book of Calculation,” although what it means more precisely is the book of calculation without the abacus. Fibonacci’s goal was to write a book that, while as logically rigorous as Euclid’s Elements, would also be an accessible and practical training manual for merchants and scholars. Fibonacci understood the value of marketing, even in mathematics, and so he filled the book with examples from the worlds of finance and trading.

Fibonacci began the first chapter with “The nine Indian figures are 9,8,7,6,5,4,3,2,1. With these 9 figures and sign 0, which the Arabs call zephyr, any number whatsoever is written, as is demonstrated below.” Fibonacci then explained how to do addition, subtraction, multiplication and division by hand, without the abacus, much as we learn to do it in elementary school today. The genius of the Indian system is that the positions of the numbers indicates whether they should be grouped by ones, tens, hundreds, thousands, or something bigger. Arithmetic now became possible without an abacus because we add, subtract, multiply, or divide numbers one position at a time. If we get a result that is too big or too small in the position we are focusing on, we carry the result over to the next position or borrow from the next position and then keep going. Because Arabic is read right to left, Arab arithmetic was done right to left, a custom we still maintain today.

For example, consider adding 45 and 76. 45 in the Indian-Arab notation means 4 tens plus 5 ones and 76 means 7 tens plus 6 ones. In the number 76, 6 is in the ones place and 7 is in the tens place. In the number 45, 5 is in the ones place and four is in the tens place. To add 45 and 76, we first add the 5 and 6 in the ones place. Moving right to left, we get 11, so we keep the 1 in the ones place and carry over ten, which means we carry over 1 ten into the tens place. In the second stage, we add the 1 ten carried over plus the 4 tens plus the 7 tens to get 12 tens. Then, we keep two tens from 12 in the tens place and carry over 1 hundred to the hundreds place. Overall, we have finished with 1 in the hundreds place, 2 in the tens place, and 1 in the ones place. Thus, 45 + 76 = 121. This simple calculation, that’s so easy using pen and paper, is very difficult to perform using Roman numbers, since there are no positions in the numbers indicating how counting should be grouped.

The Indian-Arab numbers introduced the concept of a ‘base’

Nowadays, we call this numbering system a base ten numbering system. The reason we use the term ‘base’ is that the Indian-Arab system is very flexible. It works with any grouping or ‘base,’ which is another word for number grouping. For example, in a base 2 system, there are two numbers, 0 and 1, and we indicate the value of the number by the positions of 0 or 1 in the number. In a base 2 number, reading from right to left, we have a ones place, a twos place, a fours place, an eights place, and so on. Each place is double the previous place. Base 2 numbers are also called binary numbers.

Base 2 or binary numbers

To illustrate how binary numbers work, let’s write 45 in base 2. It would be 101101 or 1 in the thirty-twos place plus 0 in the sixteens place plus 1 in the eights plus 1 in the fours place plus 0 in the twos place plus 1 one in the ones place, or 32 + 8 + 4 +1 = 45.

Arithmetic can be done in binary the same way as in base ten. Binary numbers are essential for computer arithmetic since numbers and calculations on computers are done in base 2. The beauty of Indian-Arab numbers is that we can use any base that is convenient. Computer scientists sometimes use base 16, which are called hexadecimal numbers. They work just like base 10 numbers, except they require 16 basic symbols, 0-9, and the letters a through f.

Indian-Arab numbers eventually replaced Roman numerals

Although Fibonacci’s book began to be used by Italian merchants almost immediately to train apprentices, it took 300 years for the Indian-Arab system to finally prevail over the Roman numbers. Many people resisted the new Indian-Arab numbers and the new arithmetic. Roman numbers had been used for over a thousand years without problems. Abacuses worked just fine. If it’s not broken, why fix it? Nevertheless, the advantages of the new numbers gradually won out and Roman numbers disappeared by and large by 1500 AD, although they are still used for limited purposes today, such as for numbering book chapters.

Decimals based on the Indian-Arab numbers replaced the sexagesimal fraction system

Despite his advocacy for a place-value numbering system using a base of ten, Fibonacci didn’t go to the final step to realize that fractions could also be written in base ten. In our modern numbering system, 0.345 is the same as , essentially equivalent to the sexagesimal system except that ten is used as the base rather than 60. Ironically, Fibonacci thought that if sexagesimal numbers weren’t broken, why fix them? Mathematicians from the time of the Greeks through the1500s in Europe still used them. The Egyptian astronomer Ptolemy as well as Copernicus were able to perform their calculations with acceptable accuracy using sexagesimals, so why change the system?

Just as the advantages of using the Indian base ten place-value system won out for numbers for everyday use, they also won out for fractions. The French mathematician Francoise Viete argued to use decimals exclusively by the latter part of the 16^th century. A little later, the mathematician Simon Stevin wrote a manual for how to do arithmetic with decimals, just as Fibonacci had done for natural numbers. People learned to do decimal calculations from Stevins’ book. By the beginning of the 17^th century the modern base ten numbering system was in place.

Some lessons from the history of numbers

Why did it take over 3000 years, from the time of the Babylonians to the end of the Middle Ages, for the modern numbering system to be developed? There are some important lessons here that we see over and over in the history of mathematics.

First, simple but incredibly powerful ideas, such as the base ten numbering system, were often overlooked by even the greatest of mathematicians. Archimedes missed it.

Second, even when these powerful concepts are discovered, they may not go anywhere on their own. Base ten numbers were developed in the eighth century in India but they didn’t instantly take over. Base ten numbers might have never taken over if the Arab conquests hadn’t brought the idea into the Arab world, where it moved further into Italy. But it took 800 years for that to happen!

Third, somebody needs to understand the significance of an idea, explain it, and push it forward. In this case, that somebody was Fibonacci. Today, to the extent that Fibonacci is known at all he’s generally recognized by mathematicians for a sequence of numbers he is famous for, the Fibonacci sequence. He’s also known among botanists and some gardeners since Fibonacci sequences show up in how plants form and develop. Fibonacci should be remembered for championing perhaps the most important innovation in numbers in history. But even Fibonacci didn’t see that fractions could also be included in a base ten numbering system, which took almost another 100 years to develop. This episode reminds us of another lesson we see over and over in the history of mathematics.

It’s very difficult to know in advance how important a mathematical discovery will turn out to be. Mathematicians usually work on a problem for its own sake. Sometimes the answer becomes vitally useful and sometimes it doesn’t. The proponents of Roman numbers in the late Middle Ages were wrong, but they weren’t terribly wrong. Base ten numbers were better, but they were not that much better than Roman numbers plus the abacus for most everyday purposes.

What the advocates of Roman numbers couldn’t see was that once numbers were severed from the abacus, new calculations that could never have been accomplished using Roman numerals and the abacus became possible. Fibonacci probably didn’t understand fully the profound significance of what he was advocating either. Base ten numbers catapulted mathematics dramatically forward in a startling way that most people at that time could not have foreseen.

1. We know that the earth rotates around its axis approximately once in 24 hours. Assuming the earth is a perfect sphere, what is the speed of a point on the equator?

Answer: The radius of the earth at the equator is r = 6341 km = 3,963 miles. The circumference of the earth at the equator is therefore 2x pi x r = 39,840 km = 24,900 miles. A point on the equator traverses this distance in 24 hours so the speed of the point is ~ 1038 miles/hour or 1,660 km/hour. This is not so fast compared to how fast jet aircraft can fly. The fastest jet plane on record at the moment is the NASA X-43, an experimental aircraft that reached a speed of Mach 9.6, or approximately 7,366 mph (11,854 km/h).

2. How fast does the earth travel in its orbit around the sun?

Answer: The distance of the earth from the sun is 93.5 million miles and it travels around the sun in 365 days. Hence, in one day, it will travel 2 x pi x 93.5/365 million miles/day. Converting this into miles per second gives the speed of the earth in its orbit as18.6 miles/second. This means that in 1 hour, the earth moves about 67,000 miles forward in its orbit.

3. How fast does the moon travel in its orbit around the earth?

Answer: The moon is at a distance of approximately 238,900 miles from the earth and its orbital period is 27 days. This gives its orbital speed as 0.64 miles/second. Each day the moon travels 1/27 th of its motion around the earth. This translates into a delay of approximately53 minutes each day. The moon reaches the same point in the sky after 24 hours and 53 minutes each day.

4. If you stood on the surface of the moon and looked up at the earth, want would be the most astonishing thing you would notice about it?

Answer: The moon keeps the same face turned towards the earth. The most astonishing thing you would notice is that the earth does not move at all, it just hangs in one spot in the sky. You would see it rotate on its axis once in 24 hours, but it would not move from this one spot. In fact, the earth is not visible at all from the side of the moon facing away from the earth (the so called “dark side”). The Soviet spacecraft Luna 3 took the first images of this side in 1959. A detail map of the far side was finally made in 1967 by a US uncrewed mission called Lunar Orbiter 5.

5. Does the earth rotate East to West or West to East?

Answer: West to East. Since the position of the sun is more or less fixed in one day, this make the sun seems to move East to West in the earth’s sky.

5. The axis of rotation of the earth tilts about 23 degrees from the plane of its orbit around the sun. This is the reason why polar regions on the earth have long nights in winter and long days in the summer. The planet Uranus has the most unusual axis of rotation tilt: the axis is almost in the plane of its orbit (~ 98 degrees tilt). How would the sun seem to move in the sky on Uranus (approximately) if you stood at the north pole of this planet?

Answer: Uranus rotates on its axis in ~ 17.25 hours and revolves around the sun in ~ 84 earth years. Since the axis of rotation of the planets always points in the same direction relative to the fixed stars (because of angular momentum conservation), the sun would be visible from its northern hemisphere for 42 years and invisible for 42 years. At the height of summer at the north pole, the sun would be high in the sky and then make slowly widening concentric circles in the sky each 17.25 hours, moving from the north to the south. These circles would reach the horizon in 21 years. After that, for 42 years, the sun would be invisible from the northern hemisphere and the concentric circular motion of the sun would be seen in the southern hemisphere, moving from the horizon up to the zenith for 21 years and then back down again for another 21 years. After that the sun would emerge again in the northern hemisphere at the horizon and start its journey up to the zenith, reaching it in 21 years.

6. a. What is the length of a day on the moon? b. And does the moon have earth and/or solar eclipses?

Answer: The moon’s rotation axis is about 6 degrees from its orbital axis so it rotates almost perpendicular to its orbital plane. The time it takes for the moon to complete an orbit around the earth is 27.32 earth days – this is the time it takes to come back to the position it had. However, the earth/moon system has moved forward around the sun so as seen from the earth, the time between two full moons is a bit longer – about 29.53 earth days.

a. Starting with the sun at the zenith at the lunar equator, as the moon moves around the earth, it will experience sunset after 7.38 earth days. Then there will be a period of darkness for 14.76 earth days after which the sun will rise again at the opposite spot from where it set and then take 7.38 earth days to reach the zenith again. Thus the moon has about two weeks (earth time) of daylight followed by about 2 weeks of darkness. For more northern or southern latitudes, the sun is lower down in the sky just like on the earth. At the lunar rotational poles, the sun makes a circle around the horizon tilted by ~ 6 degrees.

b. The moon does have solar eclipses, which happen when the earth blocks out the sun. But these are restricted to the side of the moon facing the earth. Curiously, these coincide with lunar eclipses on the earth, when the shadow of the earth blocks out the moon. But there are no eclipses of the earth on the moon, as these would require the sun to block out the earth, which cannot happen.

In 2019, Scientific American published Is Inequality Inevitable?, arguing that economies move inexorably towards oligarchy, with a single person ultimately owning almost all the wealth. The motivation for this claim was the so-called “yard sale” model. In this model, developed by a physicist, wealth changes hands during economic transactions when one party pays either more or less than an item is intrinsically worth. When the transactions in the model are simulated, the end result is that one agent ends up with all the wealth in the long run, even if all market agents are equally likely to overpay or underpay, and even if all market agents start with the same level of wealth.

The article suggests that an underlying complex model is at work that explains the tendency for concentration of wealth in the economic system; in reality the underlying model is just an application of the classic “gambler’s ruin” problem, although, oddly, the term “gambler’s ruin” is never mentioned in the article.

The yard sale model is equivalent to a bet in which each player randomly wins or loses a fixed sum of money on every turn. Let’s derive the probability that one player will clean out the other player in the long run, which will make the article’s argument more clear.

Suppose we have two gamblers, Alice and Bob. When they start, Alice has k dollars and Bob has m dollars. The total amount of money is N = m + k. They play a game of chance in which Alice has a probability p of winning on each turn while Bob has probability 1 - p = q of winning. If Alice wins on a turn, Bob pays her one dollar. If Bob wins, Alice pays him one dollar. They keep playing the game until one of the players wins all the money, N dollars. We are interested in the probability A_k that Alice will win all the money when she has k dollars. The probability that Alice wins all the money when she has k dollars is equal to the average of winning a dollar and thus winning all the money when she now has k +1 dollars or losing a dollar and winning all the money when she now has k - 1 dollars, i.e.,

We can rearrange this expression:

Thus,

Then we can write

In general,

Summing these expressions, we have

and so

We can evaluate the geometric series using the formula in Gyan’s previous post.

Since A_N = 1, we can solve for A₁ as

We still need to derive A_k. Recall that

Then

In general,

This formula is not defined when p = 1/2, since it is 0/0 at that value. However, we can use l’Hopital’s rule to derive the solution for this case.

The formula implies that if Alice and Bob both have a 1/2 probability of winning a dollar on each round and if they start with the same wealth, i.e., k = m, then N = k + m = 2k, and

Both Alice and Bob have a 50% chance of winning all the money and becoming the oligarch. Ultimately, the Scientific American article is making this mundane point.

Of course the economic validity of this mundane point depends on whether transactions can be viewed as gambles in which there is some objective value for a good or service that one of participants can get wrong. That view ignores perhaps a century of modern economics, which emphasizes that prices are determined in markets by the interplay of preferences, technology, and resource constraints. The “yard sale” model implicitly assumes that arbitrage opportunities—the ability to buy and sell the same good or service for different prices simultaneously—are everywhere. However, the empirical evidence suggests that true arbitrage opportunities are exceedingly rare.

Is It Better To Be More Skillful or To Have More Starting Wealth?

We can answer this question by using the general formula. Let’s assume that Alice is twice as skillful as Bob in winning a dollar on each round of the game, so p = 2/3 and q = 1/3. On the other hand, Bob has twice the starting wealth. If Alice starts with $10, Bob starts with $20. What is the probability that Alice will eventually win all the money? Using the formula

we see that Alice is almost certain to win all the money in the long run. It is better to be twice as skillful rather than twice as wealthy in games of chance.

Being More Skillful and Having More Starting Capital Is Even Better

When playing roulette in a casino, one the simplest bets you can make is an “outside bet” in which you bet that the ball will fall in one of the 18 red pockets or it will fall in one of the 18 black pockets. If the ball falls in a red pocket and a player had bet on red, the player wins a dollar from the casino. On the other hand, if the ball falls in one of the black pockets or in the 0 or 00 pockets, the house takes a dollar from the gambler. The 0 and 00 pockets give the house a slight edge.

What is the probability that the casino will clean out the player if he keeps betting? For the casino, the chance of winning is 20/38, since there are 20 ways for the casino to win out of 38 possibilities, counting the 0 and 00 pockets. In general, the casino will have much more capital to bet with than a typical gambler, given the casino’s size, and since casino’s impose table limits to constrain the gambler’s capital. Let’s suppose the casino has ten times the starting capital as the gambler, so that if k = $100, the gambler’s capital is $10. Using the formula, the probability that the casino will clean out the gambler, C₁₀₀ , is

A very slight advantage plus a large capital difference makes gambling an extremely profitable business for the casino.

Most of what you learn about arithmetic in the first stages of mathematics is memorization. You memorize the multiplication table and the steps to divide one number into another. By the time you are ready to leave elementary school, you take for granted that there is only one way to write numbers and one way to do calculations with numbers. The message most people get after they have taken their first mathematical steps is that the way we do it in math is the way it’s done!

That message starts many people down the road to disliking mathematics. By the time you leave elementary school, you may well understand why each step you take to divide numbers makes sense, but you may also have come away with the belief that math is a body of rules that you have to memorize. And that’s just how it is. For many people, their first and most important impression of math is that it’s logical, but also boring and tedious. There is nothing creative or inventive about it.

That impression is very wrong. The way we write numbers today and do arithmetic is not just how it is. For most of history, people used different methods to write numbers. Because of the way they wrote numbers, the calculations that we can do easily today, even if tediously, were much, much harder for them, and in most cases impossible. We take it for granted that we can do arithmetic with pencil and paper. Without our modern method for writing numbers, however, people in earlier times couldn’t write out calculations. They had to use counting machines. In addition, before modern numbers were invented, people couldn’t write large numbers. Writing numbers in terms of multiples of ten might seem simple and commonplace today, but it took thousands of years for people to figure out the advantages.

From 1500 BC, the Babylonian system didn’t change essentially for about 3000 years until the late Middle Ages in Europe when modern numbers were developed. During that time, although the symbols used for writing numbers changed, no real improvements were made in the western world. The Greek mathematician Archimedes tried to fix the problem of not being able to write big numbers. In doing so, he came close to inventing modern numbers. But he missed the key step and no further improvements were made for almost 2000 years after that.

Studying the history of numbers, in which we compare the present to the past, helps us to appreciate what marvelous inventions modern numbers and arithmetic are. Arithmetic may be tedious, but the historical alternatives are much worse. Understanding the history also reminds us that mathematical ideas never came easily. The Greek philosopher Aristotle, Alexander the Great’s tutor, told his pupil that “there is no royal road to learning.” There isn’t. When you find yourself bewildered by some mathematical idea, maybe even arithmetic, you shouldn’t lose heart if it takes you a few minutes, a few hours, or even a few months to figure it out. When our ancestors, including the greatest mathematicians from antiquity, struggled to understand the same idea you might find perplexing today, it usually took them a few hundred years and sometimes even a few thousand years before they got it. It took 3000 years before modern ways of writing and calculating with numbers were developed.

Primitive societies didn’t need to count

Primitive societies didn’t need to count. They had only three numbers, one, two, and many. Primitive humans didn’t need to count higher than two since humans share with animals ‘number sense,’ an ability to recognize when something is missing. In the eighteenth century, the Abipones, a tribe of South American Indians whose language had only three numbers, could notice when a dog was missing from a huge pack of dogs. They didn’t need to count the dogs.

When primitive people did use numbers, they often didn’t distinguish the number of the things being counted from the things themselves. Fiji islanders say 10 boats are a ‘bola,’ 10 coconuts are a ‘koro,’ and 1000 coconuts are a ‘saloro.’ We can see the vestiges of our own primitive past in modern English when we don’t separate the number from the things counted. We still call two singers a duet, two oxen a yoke, and two partridges a brace. We also do the same for the ‘many’ number. We say a group of lions is a pride, and a group of crows is a murder. Nowadays, we think of numbers as being different from what they count, but that’s not how our ancestors saw it.

When societies needed to count, they usually used groupings of five, ten, or twenty

Once people’s daily lives became too complicated to use number sense, they started counting. Gradually, they added more numbers to their languages. Once the numbers got too big to just add another word for the next needed number, people took the next natural step: they started grouping when they counted. If you’ve ever needed to count a large number of items, such as pennies, you know it’s hard to do without getting confused and losing your place. So, you’ll usually count in groups. You might for example count in groups of 20 pennies. That way, you can look at the height of each group and use your number sense to see if all the pennies have been accurately counted.

When people started grouping to help them count larger numbers, you see groups of 5, 10, and 20 popping up in most societies. Some used different groupings of course, but grouping by 5, 10, and 20 was most common, because people would naturally count using their bodies. Five fingers are on one hand, so five is a common grouping. Ten fingers are on both hands, so ten is a very typical grouping. And we have 20 fingers and toes, so twenty is a grouping we often see too. In Native American languages, for example, groups of 5,10, and 20 were used most often. Today, the grouping has standardized to 10 in our numbers, but we can still see remnants of the past in counting by groups of twenty.

For example, in modern-day French, counting is by tens after the number 10, which is ‘dix’ in French. Twenty in French is ‘vingt.’ Thirty is ‘trente.’ In French, you keep counting by tens until eighty, when suddenly you say ‘quatre-vingts’—which means four twenties. In old French, counting by twenty was more prevalent but all that remains of that past in modern French are the numbers 80, four twenties, and 90, quatre-vingt-dix, or four twenties and ten.

Counting by 20 is still seen in English too. In the Gettysburg Address, Lincoln famously said ‘four score and 7 years ago.’ Because a score is 20 years, Lincoln was saying 87 years ago. Score comes from Old Norse, ‘skor,’ and means notch or tally. The word originated from the practice of shepherds counting sheep by twenties and then putting a notch or ‘skor’ on a stick for each group of 20 sheep. We use ‘score’ all the time today when we talk about ‘keeping score’ or when we ask ‘what’s the score?’

The Babylonian number system

Although 5, 10, and 20 were the most common groupings used when counting, there were important variations among ancient peoples that still matter today. About 3500 years ago, the Babylonians grouped by 60 when performing mathematical and astronomical calculations, using a system called the sexagesimal numbering system. The sexagesimal system was similar to the modern system we use that is based on grouping by ten, except the base for sexagesimals was 60. Even though 60 was the base, the Babylonian sexagesimals also used a grouping of ten for the integers. Integers up to 60 were written using combinations of symbols for 1 and 10.

Like our modern numbering system, sexagesimal numbers were expressed in place-value notation. The place of the digit indicated the power of 60 that should be multiplied by the digit. There was no symbol for zero and no decimal point. As a result, the value of numbers had to be judged by the context.

The Babylonian sexagesimals used cuneiform

To see how sexagesimal numbers were written, we can look at the mathematical school clay tablet YC 7289, housed in the Babylonian Collection at the Yale Peabody Museum of Natural History.

The tablet depicts a square with two diagonals. On the side of one of the triangles bisecting the square is the number 30 in Babylonian cuneiform. Cuneiform is the system of writing used in some languages in the ancient Middle East. Written inside the square along the diagonal of the triangle are two further numbers. The diagram below shows how to decipher these numbers, which are very hard to read by modern standards.

The number 30, the length of the side of the square inscribed in the tablet, is written as three cuneiform symbols that represent 10 each. 3 X 10 = 30. The number 35, the last number on the lower right, is written as three symbols that represent 10 each plus five symbols that represent 1 each. 3 X 10 + 5 X1 = 35. Spaces between the symbols for counting the groups of ten and the symbols for counting groups of one separate the tens and ones places. Spaces also separate each full number. You read the numbers from left to right, starting with the tens place and then move to the ones place.

Since there is no symbol distinguishing the integer value of the number from the fractional part, such as the decimal point we have today, the Babylonians had to decide from the context if the number has a fractional part. On the tablet, some of the numbers represent fractions in a base 60 number system. For example, the sexagesimal number 1 24 51 10, the number on the diagonal of the square, should be very familiar to the modern mathematician. It’s a sexagesimal fraction in which the spaces between the last three numbers, 24, 51, and 10 indicate powers of 1/60. So, 1 24 51 10 should be understood as

If we write that out in decimal form, it’s 1.414213. To five decimal places, √2=1.4142135, showing that the number on the clay tablet is the square root of 2 written to surprisingly high accuracy. Just as a reminder of what the square root is, if you take the square root of any number, such as √x, and multiply it by itself, then (√x)² = x So, (√2)² = 2.

What do the other numbers mean. 42 35 25 is a sexagesimal fraction too:

But notice that

which is the second number on the Clay tablet. 42.4 is not exactly equal to the sexagesimal number 42.6 on the tablet because the Babylonian approximation to the square root of two is not as accurate as our modern decimal version.

An ancient tutorial on the Pythagorean theorem

The Clay tablet is actually an ancient mathematics lesson illustrating the Pythagorean theorem, which the Babylonians knew at least as early as 1500 BC. We’ll talk more about the Pythagorean theorem in the next chapter but for now we’ll state it: if you have a right triangle, then the sum of the squares of the sides equals the square of the hypotenuse. In terms of the Clay tablet, the sides of the right triangle are both equal to 30. That means the diagonal d can be written by the Pythagorean theorem as

Taking the square root of both sides, the diagonal equals

as the tablet showed.

The sexagesimal system was very advanced, allowing accurate computations for geometric and astronomical calculations.

How did the ancient Greeks write numbers?

The Greeks adopted a separate numbering system for everyday affairs and used the Babylonian sexagesimal system for mathematics and astronomy. The use of 60 in the Babylonian numbers is the reason the Greeks divided the circle into 60 parts and later Hipparchus, a Greek mathematician who is considered the founder of trigonometry, divided the circle further into 360 parts. That’s why a circle has 360 degrees today. We also have 60 minutes in an hour and 60 seconds in a minute for the same reason. The numbers 60 and 360 came to have special significance in ancient Greece and we see references to these numbers outside of mathematics. In the Odyssey, composed in the eighth century B.C., we are told that Eumaeus, Odysseus’s faithful swineherd, watches over 360 boars and 600 sows.

In ancient Greece, numbers for everyday use were based on the Greek alphabet rather than cuneiform. For example, the first letter of the Greek alphabet, Α, alpha, was associated with the number 1. The second letter of the alphabet, beta, Β, was the number 2. Each number from 1 through 9 was depicted by a letter from the alphabet. Separate letters were reserved for 10, 20, 30, and so on when counting by 10 from 10 to 100. Counting by thousands also had their own letters, with another letter reserved for 10,000. Ten thousand was called a myriad, a word we still use today to indicate a large number. The Greeks read numbers by adding up the value of each letter, subject to some other rules.

Problems with the Greek numbers

The numbering system used by the Greeks, and all numbering systems like it, had two practical problems. First, it was hard to write really big numbers, but the Greeks mostly didn’t need to write really big numbers. They thought there were magnitudes too big to count, such as the number of grains of sand in the world. The Greeks had a word that translates to ‘sand-hundred,’ which refers to an uncountable number. The 5^th century B.C. poet Pindar in his second Olympian Ode said ‘the sand escapes counting.’

Perhaps more important, you couldn’t do arithmetic with pen and paper with the Greek numbering system. The arithmetic you learned in elementary school--addition, subtraction, multiplication, and long division--was too hard to do when everyday numbers were written as the Babylonians, Egyptians, Greeks, and other civilizations wrote them. Instead, people used counting boards, which were later called abacuses, to do arithmetic. Counting boards were simple devices that people used to count out the answer to an arithmetic problem using pebbles, beads or something similar. Finger counting, in which people did arithmetic on their fingers, was also in widespread from ancient times to the Middle Ages.

Archimedes figured out how to write really large numbers

Archimedes, one of the greatest mathematicians of the ancient world, challenged the belief that there were some numbers too big to count. In his short work, “The Sand Reckoner,” probably the first popular scientific article in history, Archimedes set out to prove to the King of Syracuse, using arguments the King could follow, that the number of grains of sand that could fill up the universe could be counted. To count the grains of sand that would fill up the universe, Archimedes adopted Aristarchus’s theory in which the sun is at the center and the earth revolves around it. The stars in the theory are like the sun, but at great distances away. Even in the third century B.C., many Greek mathematicians and astronomers had a pretty good idea of how the universe was constructed, knowledge that would be re-discovered in Europe over a thousand years later.

Because the number of grains of sand were so large according to Archimedes’ calculations, he had to invent new ways of writing large numbers. The traditional Greek number system would not work. He started with the myriad, which was 10,000 in the Greek numbering system, and multiplied it by itself, creating a new counting group of 100 million. He then created additional groups based on the myriad multiplied by itself again and again. Continuing in this way, Archimedes defined groups for counting as large as 80,000,000,000,000,000. Archimedes ultimately estimated that the universe, which he thought was a giant sphere, had a volume that could contain about 10⁶³ grains of sand—10 multiplied by itself 63 times. That’s a very big number, but much too small an estimate given today’s astronomical measurements.

In building this new number system, Archimedes almost invented our modern number system. On the one hand, Archimedes demonstrated that number groupings can be as big as needed. If you start with a base grouping, the myriad, you could build up larger and larger groupings by multiplying the myriad by itself. That’s what we do today with the ten grouping. We keep multiplying 10 times itself to get larger and larger groupings—10, 100, 1000, and so on. When we write a number such as 245 today, we know that it means 2 X100 + 4X10 + 5 = 200 + 40 + 5 = 245.

However, Archimedes missed the other essential element of modern numbers, that a digit’s position in the number shows what counting group it belongs to. This is an odd oversight given the Greeks understood that the Babylonian sexagesimal system employed a place-value numbering system based on powers of the fraction 1/60. When we write the number 245 today, we mean that the 2 is in the hundreds counting place, so 2 means count two groups of one hundred or 200. The 4 is in the tens counting place, so 4 means count four groups of ten or 40. The 5 is in the ones place and so means count 5 things. This simple idea, which is so commonplace today, eluded even Archimedes, and was only re-discovered much later, in the 8^th century A.D., a continent away in India.

The Roman numbering system took over from the Greeks

When the Romans conquered the world, they replaced Greek numbers with a number system similar to the Greeks but with different symbols. They counted in groups of 5 and 10. So, for example, I was 1, II was 2, V was 5, X was 10, L was 50, and C was 100. The Roman number system had the same problems as the Greek system: you couldn’t easily write large numbers and you couldn’t do arithmetic by hand using pen and paper. However, those problems didn’t prevent the Roman numbering system from being used throughout the Middle Ages.

The difficulty with writing large numbers wasn’t a big problem for most people during the Middle Ages. When numbers were used for everyday affairs, such as quoting prices, levying taxes, or measuring weights, Roman numerals could express large enough numbers. When merchants in the Middle Ages needed to do calculations, such as adding up costs and revenues from sales, they would do all the arithmetic on an abacus and then write down the final result in Roman numerals. For everyday calculations, no one had a great incentive to improve the number system. For calculations with fractions, mathematicians and astronomers continued to use the sexagesimal system adopted from the Babylonians until the late Middle Ages.

Although the Roman numerals and sexagesimal fractions were replaced in the late Middle Ages, some ancient number groupings besides 60 and 360 still survive. The Egyptians divided the day into 10 parts but reserved a twilight period in the morning and a twilight period in the evening, making 12 in all. They divided the night hours into 12 also, based on observations of stars. That’s why we have 12 hours in the day and 12 hours of night. The Egyptians also divided the year into 12 months.

The number seven is an additional important grouping that originated in ancient times. The Babylonians revered the number seven, since it was the number of the sun, the moon, and five planets they could see without a telescope. Veneration of seven was passed down by the Babylonians, making its way into the Old Testament. The Book of Genesis advised that God created the world in 6 days and on the seventh day he rested. Ultimately, that’s why we have a seven-day week.

There is a famous problem (shown above) which is discussed extensively on the internet. It asks if it is possible to connect 3 houses to 3 utilities without the connection lines crossing. The answer is that this cannot be done. We will solve here the general case of n houses connected to m utilities and show that it is possible to connect them without lines crossing if and only if either m or n is less than 3.

The graph of n houses connected to m utilities is called a K_n,m bipartite graph. The K_4,3 graph is shown in the figure. Assume the utility lines do not intersect, so the graph of the lines from the houses to the utilities is planar. We have shown that for planar graphs, Euler’s equation V+F-E=2 fit true (see here for the relevant post on this substack). Here we have V = (n+m) and E = nm. Applying the Euler formula, we get F = nm-(n+m)+2. Since there are no lines connecting houses or utilities, there are no faces with three sides. This means that each face must have at least 4 edges. Thus 4F is a lower bound for the number of edges. Each edge is connected to 2 faces so the total number of edges is 2E. So, we must have 4F ≤ 2E. This gives 4nm-4(n+m)+8 ≤ 2nm which simplifies to nm – 2(n+m) + 4 ≤ 0 or (n-2)(m-2) ≤ 0. This is only possible if either n=2 or m=2. Hence the K_n,m graph is planar if and only if either n=2 or m=2. In particular, connecting 3 houses to 3 utilities without lines crossing is impossible.

A VIP shows up and demands a room. The manager asks Hilbert what to do. No problem says Hilbert. He suggests they move the person in room 1 to room 2, the person in room 2 to room 3 and so on. The VIP can occupy room 1. This is shown below.

But what if an infinite number of VIPs show up and they all must have rooms? Again, it is very simple for Dr. Hilbert. Move all the guest in room k to room 2k. The odd numbered rooms will become empty, and the infinite number of VIPs can move into them. This is because the set of even integers is the same size as the set of integers. An infinite subset of an infinite set can be the same size as the whole set.

And finally, what if an infinite number of VIPs show up, each with an infinite number of attendants, and all demand rooms? No problem says Hilbert. Here is how it would work: Put the VIPs in Prime numbered rooms 2, 3, 5, 7, 11, 13,… Next, put attendants of the room 2 VIP in rooms 2², 2³, 2⁴… attendants of room 3 VIP in rooms 3², 3³, 3⁴,… and so on.

Interestingly there will be an infinite number of empty rooms after these transfers. Any room whose number is the product of two different primes will be empty. Examples: Room 15 = 3 x 5, Room 21 = 3 x 7, Room 35 = 5x7,.. etc. Even after giving rooms to an infinite number of VIPs, each with an infinite number of attendants, an infinite number of rooms remain empty! Infinity is indeed a queer beast.

Roy Cleveland Sullivan was an American park ranger in Shenandoah National Park in Virginia. He holds the Guinness Book of World Record for most lightning strikes on a human. Seven lightning strikes were documented but he was probably struck even more times. In many of these strikes, his hair caught on fire, so he carried a water bottle to dowse himself. Sullivan believed the clouds actually followed him during a storm and had lightning rods installed on his four-poster bed. What is the probability of being hit by lightning 7 times in a lifetime?

According to the National Weather Service, the probability of being struck by lightning once in a lifetime is approximately 1/15300. The probability of one person being struck by lightning 7 times is p = (1/15300)⁷ = 5.1 x 10^-30. During the lifetime of Mr. Sullivan, the world population (say n) was between 2 and 3 billion. So, we can ask whether, in a world with a population of n = 2.5 x 10⁹, is it unusual to be struck by lightning seven times? Since p is very small and the world population n is a large number, we can use the Poisson distribution to estimate the probability that someone would be hit seven times. The Poisson parameter is m = np = 1.28 x 10^-20. The probability that at least one person in the world is struck by lightning seven times is 1-e^-m ~ m = 1.28 x 10^-20. This means that Sullivan’s being struck by lightning was an extremely low probability event.

Suggested level: middle school and up

Leonard Euler may have been the most productive mathematician of all time. Born in 1707, in his 76 years he produced a prodigious amount of original research and wrote textbooks in virtually every area of mathematics as well as in physics, astronomy, and mechanics. So far, 805 of his works have been translated by the Euler Archive, an online digital library dedicated to Euler. Besides his technical work, he wrote “Letters of Euler on Different Subjects in Natural Philosophy Addressed to a German Princess,” a popular account containing 200 letters on topics such as gravity, astronomy, why the sky is blue, why evil exists, and how sinners could be converted. The ‘Letters’ was his most widely-read book, becoming an international best seller. By the time he was in his early 50s, Euler had gone blind, but that didn’t slow him down. He still wrote about one paper per week as well as textbooks on algebra and calculus, continuing his amazing productivity up to his death in 1783.

In the 1700s, everyone thought that mathematics was purely a quantitative science. No less an authority than the Greek philosopher Aristotle had pronounced it so. Mathematics is concerned with measurement, numbers, and calculations. Euler thought that too. But because he was interested in so many areas, Euler sometimes took up questions that he himself thought weren’t necessarily mathematical. Euler amused himself by tackling the problem of how to classify polyhedra and in doing so he inadvertently kicked off new areas of mathematics—topology and graph theory—that are not strictly about measurement but still immensely important in mathematics today.

The Polyhedron

The polyhedron had been studied for thousands of years by the time Euler began to look at them. Polyhedron is a Greek word: poly means ‘many’ and hedron means ‘seats.’ So, a polyhedron is a 3-dimensional geometrical figure that has flat areas that would allow you to seat it on a table. The pyramid is an example of a polyhedron. You could seat it on a table on its square base, or you could seat it on one of its triangular sides.

You can build this pyramid out of polygons. Polygons are 2-dimensional shapes formed by connecting straight lines to each other. We can build the pyramid polyhedron by taking a square polygon and connecting each side of the square to the side of a triangle and then arranging the triangles so that they meet in a point.

There are an infinite number of polyhedra you could build from polygons. The Greeks noticed that there were some very special, beautiful polyhedra that could be made from polygons that all had the same shape and were symmetric: for each polygon, the length of each side of the was equal to every other side and the angles between all sides was the same.

The simplest polygon that fits this description is the square. The length of each side of a square is the same and all sides are at right angles to each other. The square creates the first special polyhedron, the cube. Nowadays, each side of a polyhedron is called a ‘face’ rather than a seat. A cube has six faces.

We can make three additional polyhedra from a triangle in which all sides of the triangle have the same length and all angles between each side are the same. The three polyhedra are the tetrahedron, the octahedron, and the icosohadron. The tetrahedron is constructed from four triangles. “Tetra’ means four in Greek. The octahedron is built from eight triangles. “Octa’ means eight in Greek—an octopus has eight legs. And the icosahedron is built from combining twenty triangles. ‘Icosa’ means twenty in Greek.

You can also make a polyhedron from a pentagram, which has five equal sides, with equal angles between each. This polyhedron that results is called the dodecahedron. Dodeca means twelve in Greek; a dodecahedron has twelve faces, each of which is a pentagram.

That’s five so far. How many more can we make? These polyhedra are special because they are only polyhedral that can be constructed from polygons that have equal sides and angles and with each face being the same polygon. That there are only five special polyhedra was discovered by the Greek mathematician Theaetetus, who lived at the same time as Plato. Plato was so impressed that he based his physical theories on the five special polygons in his dialogue “The Timaeus.” Today we call these special polyhedra the regular polyhedra.

The Greeks believed that there were four fundamental substances in the universe and that everything was made from them: earth, water, fire, and wind. Plato thought that the four substances must be made from the regular polyhedra, given their uniqueness. Plato decided that the earth was made from cubes, probably since dirt crumbled into pieces in his hands. Also, you can pack cubes together, making a hard material like rock or iron. Fire was made from the tetrahedron, since fire hurts like the prick you might feel from the tetrahedrons pointy edges. The icosahedron was associated with water: it’s nearly circular and so would readily flow. Plato assigned the octahedron to air, because it is more mobile than air but less sharp than fire. Plato thought the last polyhedron, the dodecahedron, arranged the constellations in the heavens.

The five regular polyhedra are known to this day as the Platonic solids. From ancient times through well past the middle ages, the Platonic solids were analogous to the modern concept of atoms, since they were the building blocks of everything that exists. At the end of the 16^th century, the German astronomer Johannes Kepler was tried to explain the solar system using the Platonic solids, which he thought would reveal God’s geometrical design of the universe. By that time, astronomers knew there were six planets. Kepler took each Platonic solid and constructed a circle within it and outside of it, leading to six circles that he thought were the planets’ orbits. Kepler’s theory was bound to fail, of course, since it implied that there were only five planets. As flawed as Kepler’s theory was, however, the three laws of planetary motion that came out of theory ultimately led to Newton’s mathematical theory of gravity, a stunning success.

By the time Euler started to examine polyhedra, over a hundred years later, Platonic solids were no longer important in physics, but they continued to be immensely interesting to mathematicians. While looking for some general way to classify them, Euler stumbled on an observation that was so simple it’s hard to understand why no one else hadn’t seen it in the over the 2000 years before Euler. Everyone else had studied polyhedra geometrically, looking at measurements of sides and angles. Euler focused how the parts of a polyhedron were put together without any measurement of the parts.

No one before had ever looked at the basic parts of a polyhedron or had even named all of them. Euler saw that every polyhedron is built from faces, edges, and vertices. The faces, which the Greeks called the ‘seats,’ are polygons. The edges are the lines where the polygons meet. And the vertices are the points where the edges meet.

The pyramid, for example, has four faces that are triangles and one face that’s a square—five faces. It also has eight edges and five vertices. Euler noticed that the number of faces F, minus the number of edges E, plus the number of vertices V, equals 2. Let’s check for the pyramid. F = 5, E = 8, and V = 5. 5-8+5 = 2.

What about the cube? It has six faces, 12 edges, and 8 vertices. 6 – 12 + 8 = 2. Astonishingly, all polyhedra, as long as they don’t have unusual features such as holes, satisfy the equation that Euler found, Euler’s polyhedron formula:

Proof That There Are Only Five Platonic Solids

Euler’s polyhedron formula gives us a new way, without using geometry, to prove that there only five platonic solids. Euclid’s classic proof used the geometric features of the polygons such as the lengths of the sides and their angles. With the Euler method, we only need to think about how the polyhedra are broadly constructed, how the faces, edges, and vertices fit together.

For any polyhedra, we know that F – E + V = 2. What else can we say about the faces, edges, and vertices for the Platonic solids? Let’s call m the number of edges of each face. One observation is that the number of edges of each face multiplied by the number of faces is always twice the total number of edges. That’s because each edge is where two faces meet. So, one condition for being a Platonic solid is

This condition contains the assumption that all faces on the polyhedron are the same in the sense that they all have the same number of edges. But the condition does not say anything about the geometry.

Let’s call the n the number of faces that meet at each vertex. If we take the number of faces n that meet at each vertex and multiply it by the number of vertices, that’s also twice the number of edges. That’s because any edge has two vertices. So, a second condition for being a Platonic solid is

Let’s take a moment and verify these observations for the cube. In the first condition, m, the number of edges per face, is 4. If we take each of the 6 faces and multiply by four edges, we get 6 X 4 = 24, which is twice the 12 edges. mF = 2E for the cube.

Now let’s take the number of faces that meet at each vertex, n, which is three for the cube. If we take three and multiply it by the number of vertices, which is eight, we get 24, twice the cube’s twelve edges. The double counting of edges happens because every time you count two vertices you count the edge between them twice. nV = 2E for the cube.

When you look at the other Platonic solids, you see the same pattern. The number of edges per face multiplied by the number of faces is twice the number of edges. And the number of faces that meet at each vertex multiplied by the number of vertices is twice the number of edges. This means that we can say for all the Platonic solids that

Solving for F and V

Let’s substitute F and V into Euler’s polyhedron formula, which holds for all polyhedra, including the Platonic solids.

Re-arranging

And, dividing through by 2E,

The number of edges is positive, so 1/E is positive, which means ½ + 1/E > ½.

That gives us the condition for a polyhedron to be a Platonic solid: one over the number of edges per face plus one over the number of faces that meet at a vertex must be greater than one half.

To find which Platonic solids are possible, we need to find all values of m and n that satisfy the condition that 1/m +1/n >1/2. Which m and n work? First, notice that m and n must be three or greater. m is the number of edges per face. A face is a polygon so we need at least 3 edges. n is the number of faces that meet at each vertex. If the number of faces that meet at a vertex is not at least three, we can’t have a three-dimensional figure.

The table below shows that the only combinations of m and n that will work are (m = 3, n = 3), (m = 3, n = 4), (m = 3, n = 5), (m =4, n = 3), and (m = 5, n = 3). For any other combinations of m and n,1/m + 1/n <= ½. For example, if m = 3 and n = 6, 1/3 +1/6 = 2/6+1/6 = 3/6 = ½, but we need it to be greater than ½.

How does that compare to the five Platonic solids? It checks out! Each of the (m,n) combinations corresponds to a Platonic solid. We have proved that there are only five Platonic solids using Euler’s polyhedron formula.

You might be thinking, “so what?” We have derived a different proof that there are only five Platonic solids, but there are always tons of different proofs for the same mathematical fact. There are scores of different proofs of the Pythagorean theorem.

This proof is fundamentally different though. Notice that we didn’t talk about numbers or measurement at all. We didn’t use the particular shapes of the faces—whether they were triangular or square. We didn’t use the fact that the polygons that make up the faces had to all have equal sides and equal angles. In Euler’s new perspective, what defined the Platonic solids was not their geometry, but rather how the pieces of the polygon, the faces, edges, and vertices, had to be put together. We used the fact that the number of edges per face had to be twice the number of total edges in the polygon; and we also used the fact that the number of faces that meet in a vertex must also be twice the total number of edges. Those conditions define the Platonic solids, but they are not geometric conditions. When we combined those conditions with Euler’s polyhedron formula, which applies to all polyhedra, we could prove that there are only five Platonic solids. Euler’s polyhedral formula sparked the beginning of a new branch of mathematics—topology.

Why is Euler’s Polyhedral Formula True?

Geometry comes from ‘ge,’ Greek for earth and ‘metria,’ Greek for measure. Geometry literally means measurement of the earth. Geometry is all about measuring the lengths, areas, volumes, and angles of geometric objects such as polyhedra. Topology comes from ‘topos’ which is place or location in Greek and ‘logos,’ which means study. The faces, edges, and vertices are like the places on the polyhedral. The topological property Euler discovered is that the number of faces minus the number of edges plus the number of vertices is always 2, no matter what the polyhedron looks like geometrically. That fundamental relationship between the places doesn’t change even if the polyhedra look very different.

Another way to think of topology is that it is the study of properties of geometric objects that don’t change when you stretch, deform, or bend them, as long as you don’t tear them: the ‘places’ of the geometric object keep their relationships with each other. We can see the topological nature of Euler’s polyhedron formula by working through why it must be true. Euler provided a proof, but that proof turned out to be wrong. Later on, many mathematicians proposed correct proofs. We’ll use a proof developed by 19^th century French mathematician Augustin-Louis Cauchy, since it illustrates the topological nature of Euler’s polyhedron formula.

The idea behind the proof is to simplify the problem by projecting the three-dimensional polyhedron into a two-dimensionsal polygon and then reducing the polygon to a triangle using a series of changes that don’t change the value of F – E + V for the polygon. The topological nature of the proof comes about because when we flatten the polyhedron into a polygon, we will have to stretch the edges and change the angles between them.

We reduce the polygon to a triangle because a triangle is simple. In two dimensions, Euler’s formula says that F-E+V = 1 for a triangle. That’s because every triangle has one face, three edges, and three vertices, so 1-3+3 = 1. If we can always reduce a polygon to a triangle without changing the value of F – E + V, then every polygon projected from a polyhedron must have F – E + V = 1, since a triangle does. Then, when we take the polyhedron back to three dimensions, we’ll add an extra face to get the third dimension, so that F-E+V = 2.

To see how this would work, let’s start with one of the simplest cases, the cube. Imagine that it is empty box. We take the top square off and then flatten it down into two dimensions, like laying out pizza dough on a table. As we are flattening the cube, we are allowed to stretch or contract the edges as needed and also change any angles to get it flat. We just can’t tear it.

Now we can work with the flattened version. In two dimensions, it has five faces, 12 edges, and 8 vertices: 5 – 12 + 8 = 1 as expected, but we want to develop a general method to reduce any two-dimensional polygon to a triangle so that we can prove that any flattened polygon has F – E + V = 1.

The first step is to change each face into a pair of triangles by drawing a diagonal across it. That operation does not change the value of F – E + V, since when we add the diagonal we have added one edge, but then we’ve also added another face to compensate.

Let’s add one diagonal, in red, to see what’s going on. When we add the diagonal to the top polygon of the flattened cube, we don’t change the number of vertices, but we do add an additional face and an additional edge. Since the additional edge is subtracted from the additional face in Euler’s formula, F – E + V doesn’t change. So, we can convert every face into a pair of triangles in the flattened cube without changing the value of F – E + V.

The first step then is to draw a diagonal across every face, shown in red. Once we’ve created two triangles in every face, the next step is to start removing triangles without changing the value of F – E + V. If we have any triangles pointing out so that only one side touches the polygon, we should remove them first. Removing them doesn’t change the value of F – E + V, since we remove 2 edges, 1 face, and 1 vertex. The face and the vertex is 2 and then we subtract two edges, producing no change.

Looking at the diagram, we don’t have any triangles sticking out. So, we then should remove as many triangles as we need that have one external side. That process also doesn’t change F – E + V. The reason is that we remove one edge and one face and they cancel each other in F – E + V. Our goal ios to remove two triangles to leave one outward pointing triangle, which we then remove.

To proceed, we remove two triangles with one edge on the outside of the polygon. Each triangle we remove is marked with an ‘x’ before we remove it.

Performing one more round:

And then we perform the final round:

We end up with a triangle, which we know has one face, two edges, and three vertices, F – E + V = 1 – 3 + 3 = 1.

Now, here is the crucial step to get back to three dimensions. We took one face off the top of the cube before we compressed it to two dimensions. Now imagine that we reverse the process and pull it back up to three dimensions. Nothing has changed. But now let’s put the top face back on. That adds one face. Therefore, for three dimensions

Of course, we did a simple case, the cube, in which we already know that F – E + V = 2. What we learned, however, is the general method for how to transform into a triangle any polyhedron compressed into a polygon, without changing the value of F – E +V. Once we put the polygon back into three dimensions and add back the top face, we will therefore always get F – E + V = 2.

The key pitfall in doing the proof is that you have to be careful about the order in which you remove the triangles. You should always remove any triangles jutting out before you do anything else. If you can’t remove any triangles jutting out, you must first remove enough triangles with one external face to get at least one triangle that is sticking out, which you then remove. If you get the order of triangle removal wrong, you can end up disconnecting the triangles and the proof will fail.

To see how this would work in a general case, let’s look at a another polygon in two dimensions. As we can see, the faces are much less regular. If we put one diagonal in the top face, we won’t get triangles. We’ll need to put several diagonals in many of the faces. Once we have triangles everywhere, we remove the triangles that are pointing out, labeled with an x, first. Then we remove triangles with one side on the exterior of the polygon until we get new triangles pointing out to remove. We keep going until we end with a single triangle.

Working through Cauchy’s proof gives us a feeling for the topological nature of Euler’s polyhedron formula. The formula shows that if we can identify the relationship between the ‘topos,’ the places on the shape, which are the faces, edges, and vertices, then those relationships are the same even if the shape looks very different geometrically.

Euler’s formula started as a way for Euler to classify polygons, but it’s much more general than that. We saw for example how it also applied to polygons, where F – E + V had a different value, one. Euler’s formula is now used as a method for classifying surfaces. All surfaces have an Euler characteristic, which equals F – E + V. For polygons, the Euler characteristic is one. For polyhedrons, the value of the Euler characteristic is 2.

Euler characteristics apply to other shapes too. The Euler characteristic of a sphere is also two, implying that topologically all polyhedrons are the same as spheres. You can see that intuitively if you can imagine deforming a polyhedron into a sphere by stretching and bending it without tearing it.

A problem in Philosophy is the problem of Free Will: Are our actions predetermined or can we choose to do what we please. In 1960, William Newcomb, a physicist at Livermore Radiation Lab in California, posed the following problem:

Suppose there are two boxes which you cannot see into. Box # 1 contains $1000. Box # 2 contains either $1,000,000 or nothing. You have two choices:

1. Take both Boxes

2. Take only Box # 2

But there is a catch. Yesterday, a Supreme Being (SB) with predictive powers, made a prediction about the choice you would make. If she predicted you would take both boxes, she put nothing in the second box. If instead, she predicted you would take only Box #2, she put the $1,000,000 into Box # 2. If you use a coin toss to decide what to do, she left Box # 2 empty. What should you do?

In the table above, we present this dilemma as a matrix game with the contents of the box as payoff. The SB has made her move (prediction and placement). What do you do?

What you do depends on what your belief system is. There are two ways to think about it.

Argument 1 (Believers): Suppose I take both boxes. Then the SB will have predicted I would do this and put nothing in Box # 2, so I will get $1000. If I take only Box # 2, the SB will have predicted this and put $1,000,000 into Box # 2. So I should take Box # 2.

Argument 2 (Non-Believers): The SB made her prediction yesterday and the money is already in the boxes. There is nothing the SB can do to change what is inside the boxes. If the $1,000,000 is in Box #2, it will not vanish if I take both boxes. So, I am better off taking both boxes, because even if the SB has not put the $1,000,000 in Box # 2, at least I will get $1000. So in either case, I should take both boxes.

Which Argument is correct depends on your level of belief. Suppose you think the SB is correct a fraction p times: The payoffs are :

A: both boxes: p x $1000 + (1-p) x $1,001,000 = $1,001,000 - $1,000,000 x p

B: only Box #2: (1-p) x 0 + p x $1,000,000 = $1,000,000 x p

Choice B is better than choice A unless p < 1,001,000/2,000,000 = 0.5005

How would you decide practically how often the SB is correct? You could watch 100 people play the game and record how often the SB made the correct prediction.

But you are a skeptic. You wish there was a way to see what was in the Boxes before you decided. Suppose the front of Box #1 was transparent and you could see the $1000 in it. However, the front of Box # 2 was opaque but the back was transparent and you had a friend who could see what was in that box. What would she tell you to do? But wait a minute! You already know what she would tell you to do. If there was nothing in Box # 2, she would tell you to take both boxes. If there was $1,000,000 in Box # 2, she would still tell you to take both boxes!

In either case, she would say “Take Both!” – you don’t benefit from her advice. You must decide for yourself!

Simpson's bias or Simpson’s Paradox is a statistical paradox first described by Edward H. Simpson (image above) in a 1951 paper titled “The Interpretation of Interaction in Contingency Tables.” This paradox may arise when making a decision either based on the global statistic over a set of numbers or using the same statistic over subsets of the numbers. The paradox is that the decision based on the global statistic is the opposite of the decision based on the statistic using subsets.

A classic case of Simpson’s Paradox is the Gerrymandering of voters by selectively redistricting to create a specific outcome. We will explain this with an example.

Consider the figure above which shows 50 voting districts. A red color represents voters who vote one way and blue who vote the opposite way. The group of 50 districts are to have five representatives based on division into five constituencies with one representative from each. A fair voting system would award three representatives to the blue districts and two to the red districts.

The figure above shows one possible fair division. However, it is also possible to group the districts into sets which would award all five to blue and none to red or the other way around.

This is shown in the two figures above and can be called the “Tyranny of the Majority” or the “Tyranny of the Minority.” These unfair ways of grouping illustrate Simpson’s paradox because the result is the opposite of what would be the outcome with representatives assigned based on the global voter distribution.

The major idea at the center of all biology is evolution, discovered simultaneously in the mid-1800s by Alfred Russell Wallace and Charles Darwin. The terms “Natural Selection” and “Descent with Modification”, coined by Wallace and Darwin respectively, completely changed our views about life on earth. When Darwin formulated his theory about “Descent with Modification” and Wallace about “Natural Selection”, all of the biology of genes and chromosomes and how they influence inheritance was unknown. Neither did Darwin and Wallace know about the experiments of Gregor Mendel, the monk whose studies on peas and flowers showed that information from one generation to the next is transmitted in particulate form, which we now know is due to the way genetic information is coded in chromosomes. When Mendel crossed pure bred pea plants with purple and white flowers to study inheritance patterns, he found that in the first generation (F1) they produced only purple flowers. When the F1 purple flowers were self-pollinated, they produced both purple and white flowers in the ratio of 3:1, purple:white. This led him to propose that the purple flower color in peas was a “dominant” trait, while the white flower color was a “recessive” trait. He also observed that other traits, such as seed shape or size, were not linked to flower color. His main conclusions were that in peas, traits such as flower color, pea shape etc. are inherited in distinct patterns and do not blend. Each trait is determined by factors (now known as gene sequences) and each factor comes from each parent.

We now know that the information in cells is coded in long linear helical chains called chromosomes made by attachments of four elemental chemicals or bases – adenine, cytosine, guanine and thymine (shortened to A,C,G,T). The genetic information contained in the sequence of bases on chromosomes code for the twenty amino acids which make up our bodies. The translation of the message on chromosomes to amino acid chains is done in units of three bases (called codons) using a Universal Genetic Code. This is accomplished by a complex cellular machinery which requires that first the base T or Thymine be transformed into U or Uracil before it can be translated into the appropriate amino acid. The Universal Genetic code is shown in the figure above. It is degenerate, with multiple codons coding for the same amino acid. Thus, both UUU, UUC (or TTT, TTC) code for the amino acid phenylalanine. And CGU (or CGT), CGC, CGA and CGG all code for the amino acid Argenine. In addition, there are two stop codons to tell the machinery where in the sequence to stop and one start codon AUG (Methionine).

The mathematical basis of how traits are inherited and how Mendel’s experiments can be understood was discovered by Wilhelm Weinberg and independently by Godfrey Harold Hardy using notions of probability in what has become enshrined as the Hardy-Weinberg Theorem. We will assume that we are dealing with a species such as humans, where each chromosome comes in two copies. Such species are called diploid. Since there are two sequences coding for a given protein (one on each chromosome), they could either be the same or different. We call each sequence on the chromosome coding for a protein an “allele”. Each protein in an individual is then defined by two alleles (is bi-allelic) and these two together are referred to as the individual’s genotype at that chromosomal position (locus). We also assume that the population consists of a single isolated species in a fixed geographic area. Under the assumptions listed below, we will consider a bi-allelic locus with alleles A and a (note that here A and a just represent the coding sequence and not bases or amino-acids). We will try to understand how the genomes of such a population get transformed between generations using the following simplifying assumptions:

· We consider only diploid organisms (those that have two copies of each chromosome). Hence, an individual will have genotype either AA or Aa/aA or aa at the locus.

· Reproduction is sexual (as opposed to cloning).

· Generations do not overlap (clean break between parents and offspring)

· All loci are bi-allelic (i.e., there are two sets of paired chromosomes).

· No sexual dimorphism. Frequencies of alleles are the same in males and females – which means these arguments do not apply to sex-linked chromosomes, which do not always come in pairs. In humans, this means that the arguments below do not apply to the X and Y chromosomes in Males.

· Mating is random without selection for traits.

· The population size is infinite so we can talk about frequencies.

· There is no migration or mixing and no mutations (no changes in genetic sequence between generations).

· There is no allele specific selection (no benefit from having some specific allele).

In the t th generation, let the individuals in the population have frequencies P, 2Q and R for the 3 possible genotypes: AA, Aa/aA, and aa respectively. These three genotypes are called homozygous wild type (AA), heterozygous Aa/aA and homozygous mutant (aa). By conservation of probability, P+2Q+R = 1. At time t+1, under random mating, the frequencies P’, 2Q’ and R’ of the three genotypes are given in the Table below. The parents are shown in the top row and first column, and each entry shows the resulting genotype(s) from the two parents along with its probability in brackets. The entries are obtained by looking at the row and column genotypes and finding all possible crosses. For example, crossing AA and Aa would result in equal amounts of AA, AA, Aa, aA or AA half the time and aA or Aa a quarter of the time. The probability of this cross in the population is PQ which is then split equally between these four possibilities.

The table above shows the frequencies in the next generation at time t+1 which are obtained by summing the cases which result in each of the three genotypes. Thus:

P’ = frequency of AA genotype at t+1 = P² + 2PQ + Q² = (P+Q)²

R’ = frequency of aa genotype at t+1 = Q² + 2QR + R² = (Q+R)²

Q’ = frequency of Aa genotype at t+1 = PQ + PR + QR + Q² = (P+Q)(R+Q)

Now consider what happens in generation t+2, with frequencies P’’, 2Q’’ and R’’ for the three genotypes. Using the same table but with P replaced by P’, Q by Q’ and R by R’, we have,

P’’ = (P’+Q’)² = [(P+Q)² + (P+Q)(R+Q)]² = (P+Q)² (P+2Q+R)² = (P+Q)² = P’ (since P+2Q+R = 1)

Thus, P’’ = P’. Similarly, we can show that R’’ = R’ and,

Q’’ = (P’+Q’)(R’+Q’) = [(P+Q)² + (P+Q)(R+Q)] [(R+Q)² + (P+Q)(R+Q)] = (P+Q)(R+Q)(P+2Q+R)²

= (P+Q)(R+Q) = Q’

Almost magically, the genotype frequencies became fixed after only one round of random mating. This is the Hardy-Weinberg theorem. It says that under the assumptions we made, genotypes in the population become fixed in one generation. Let p and q be the population frequencies of the “A” and “a” allele respectively. Since there are no mutations or migrations, these are fixed within our assumptions. In such a situation, the frequencies of the three genotypes AA, aA/Aa and aa would be p², 2pq, q² respectively. A locus where this is true is said to be in Hardy Weinberg Equilibrium.

It should be obvious that the Hardy Weinberg Theorem explains Mendel’s experiments succinctly. An allele “A” is said to be dominant over the allele “a” if it will force the expression of a phenotype (trait) in either the AA or the Aa genotype forms. A recessive allele is one which needs to be expressed on both alleles to show its phenotype. In Mendel’s experiment, the dominant purple color trait is coded in the AA or Aa/aA genotype while the white color is coded in the aa genotype. Offsprings of a pure cross AA and aa will be AA or Aa and always show the purple color, since it is a dominant trait. This means that , plants in generation F1 will be 50% AA and 50% Aa/aA. If these are then self-pollinated, the AA and AA cross will always have AA offspring with purple flowers. However, the Aa and Aa cross will have half expressing Aa (purple) and the other half expressing aa (white). Thus, on average across many experiments, we would get purple flowers (1 + ½) vs white flowers ½ the time or in the ratio of 3:1 purple to white.

The Hardy Weinberg Theorem is important because it resolves a criticism that Darwin had to contend with, which was the following: If phenotypic characteristics like color, height etc. are inherited from both parents, there should be an averaging effect over time – the so called “greying of the species”. After many rounds of random mating, everyone should become the same! However, this theorem shows that this cannot happen, because genotype frequencies (which determine traits or phenotype) remain fixed in a large, randomly mating population.

Some Consequences of the Hardy-Weinberg Theorem.

· A dominant allele will not spread. This is because the frequencies of the AA and Aa genotypes are fixed on average across the population, so the number of A alleles in the population are also fixed on average.

· A recessive allele a will never be lost even if the combination aa is lethal, so long as the heterozygous combination is viable and fertile. This is the case when P_aa = 0, P_Aa ≠0. Again, this is because the allele a will remain in the Aa genotype.

An example of such a situation is the case of Cystic Fibrosis (CF), a rare recessive disease caused by the presence of mutations in both copies of a single gene which codes for the CFTR protein (cystic fibrosis transmembrane conductance regulator). Among other serious problems, it causes frequent lung infections. The average life expectancy is between 37 and 50 years in the developed world and lung problems are responsible for death in 80% of people with this disease. The disease is most common in Northern Europeans and affects one in 2000 people. Thus, q² = 1/2000, which means, q = 0.02, p = 0.98 and 2pq = 0.04. This means that one in 25 people will have the mutation (be heterozygous for the disease) but will not be affected by it. The allele for the disease survives even though the homozygous mutant form is effectively lethal.

Suppose that the frequency of the “A” allele is p and of the “a” allele is q. Then the frequency of the AA, Aa, aa genotypes are p², 2pq and q² respectively. Since human females have two X chromosomes, the X linked genotype frequencies are in HWE in females but not in males, who have frequency p and q for the single allele A and a that they carry on their single X chromosome. If a recessive mutation on X causes disease then the disease will affect a fraction q² of females and a fraction q of males. The Male/Female disease susceptibility ratio is 1/q, which becomes large as qà0. Hence, the smaller the q, and the more likely it is to be mostly visible in males. Females are often carriers of X linked diseases and males are the sufferers. An example of an X linked disease in humans is Hemophilia, a disorder in which the blood does not clot properly and color blindness, both of which are much more common in males than in females.