The gentleman depicted is Ludwig Boltzmann, the father of statistical mechanics, and the man who cast the second law of thermodynamics as a law of disorder. Informally, this law says that an isolated system will tend to become more disordered. A physicist refers to increasing disorder as increasing entropy. When Boltzmann was working on developing statistical mechanics, a colleague, Joseph Loschmidt, pointed out that if you reversed the momentum of every particle in a system, it would then decrease in disorder. In frustration Boltzmann replied “YOU reverse the momenta!” which shut Loschmidt up, but he was not wrong. The problem is that reversing the momenta of all the particles is incredibly hard and so incredibly unlikely. Disorder does not inevitably increase, but it almost always increases. This is why unbreaking a vase is only possible if the number of pieces are small and the breaks neat. It is why you cannot unfry an egg at all. What on earth does this have to do with Donald Trump?
Presidential Medal of Freedom recipient Katherine Johnson has passed away. She was an integral part of a team of African-American women, depicted in the movie Hidden Figures, that did for NASA’s moon missions what we now do with large digital computers. Occupy Math’s earlier post on Katherine Johnson mentioned many other women who were ignored while making contributions to mathematics. Our culture actively tells women that they cannot do mathematics both because they intrinsically lack talent and because it is not “sexy”. In spite of this false propaganda, women regularly appear in the top ranks of mathematical talent and contribution. Perhaps the most frustrating part of this is that many of them will never be known. This post will look some of the cases where credit was given where credit was due — eventually.
Occupy Math has done several posts that use a part of mathematics called graph theory. Occupy Math was trained in graph theory and his university is finally getting a course in graph theory which he gets to teach next winter! In the last ten years Occupy Math has taught reading courses in graph theory four times, because students needed the material to do their research. This is done as an overload, extra work with no comp time or added pay. Graph theory is really useful, showing up in everything from urban planning to particle physics, and getting to teach it to a whole class again will be wonderful.
A graph is just dots (called vertices) with some pairs of dots connected by lines (called edges). The graph at the top of the post is remarkable, for reasons we will get into a little later. A list of Occupy Math’s posts that use graph theory appears near the end of this post. Graph theory is one of the simplest types of advanced math to learn. In Cookies or Calculus, Occupy Math argues that graph theory is easier than calculus and — except for STEM students — more useful than calculus. Graph theory includes the study of networks, like contact networks in an epidemic, or influence networks in social or business situations. Graph theory is useful in some types of conflict resolution. This post is the first in a series that will introduce the power and beauty of graph theory.
John Conway was a character, a genius, an eccentric, and one of the greatest mathematicians in history. He died on Saturday April 11th of the coronavirus at the age of 82. He was a professor at Princeton and worked in games, abstract algebra, combinatorics, the theory of computing, and many other fields. Occupy Math has attended lectures by this great man and will recount the experience with one of them in the post. Fair warning: Occupy Math is going to cover three of Professor Conway’s many achievements and they are pretty deep. To combat this, there are lots of pictures, some tales of Conway’s personal eccentricity, and the story of Conway’s part in solving a single problem that took over a century to complete. One of Occupy Math’s most useful ideas was based on one of Conway’s algorithms — used in a way that would probably have appalled the professor.
One problem with math education is that access is very uneven. In theory, the internet can level things out a bit. This post is the next in Occupy Math’s series of activities that parents and teachers can use for enrichment and enhancement. For free.
Here is an example of a “guess-the-next-number” puzzle — “what is the next number in the sequence 2,5,8,11,14,?” The answer is 17. The student should figure out that the terms of the sequence increase by three every time, and 14+3=17. This post is about constructing this sort of puzzle with some notes on how to make harder and easier puzzles. Puzzles like this are good arenas to practice math skills. They can be structured as contests which is motivational, and with the information in this post, they can be tuned to your student’s needs.
If you play role-playing games, there is a good chance you have seen dice with the shapes above. They have 4, 6, 8, 12, and 20 sides. Why did those five shapes get chosen? All the sides are regular polygons and, without the numbers, there is nothing to tell one side from another. This makes the dice as fair as possible. Here is the reason why those shapes were chosen: there are only five shapes in the universe that have those properties! The formal rules for these shapes — to make them fair dice — is that:
- All the faces must be the same regular polygon — with all the sides and angles the same,
- The faces touch one another only at the edges of the polygons, and,
- The same number of faces meet at each corner of the die.
It turns out that there are three more of these shapes, if you allow an infinite number of faces. Intrigued? Read on.
You get very little credit for being wrong. This post is going to look at a couple of situations where being wrong was incredibly important and useful. This usefulness arises from the different way that mathematics treats being wrong, in one special way. Two claims that are supported by mathematical reasoning are these: there are no trees on earth, there is no multi-cellular life on earth. The post looks at why these claims are worth considering, since they are obviously false, and then ends showing an example of how being wrong is actually a core technique in math.
Occupy Math and his editor debated this post quite a bit. If you get through this post with no issues, without needing to re-read paragraphs, there is a decent chance you are a natural mathematician. More people should be able to get through the post, however, because Occupy Math’s editor made him revise this post more times than any other, ever. If you’re not feeling up to a challenge today, read the last paragraph and go make some hot chocolate or whatever your comfort beverage is.
We are used to multiplying numbers, as in 5×7=35, but there are many different kinds of multiplication in mathematics. That immediately causes a serious problem in understanding: people are so strongly conditioned to thinking of multiplication as something you do with, and only with, numbers that multiplying other types of objects is just weird. In math, multiplication can be any way of taking two objects A and B and getting back a third object C. We use the same notation: C=A×B but the symbol × has thousands of different possible meanings. The good news? This post is only looking at one of them. Occupy Math picked out this one, particular type of “multiplication” because it shows up all over both abstract mathematics and in natural science.
This week on Occupy Math, we look at Venn diagrams. Most students encounter these early in their education and they seem pretty simple. In this post we not only look at the math, but humor and the use of Venn diagrams as a creative prop. We will also see that there are still graduate research level questions about Venn diagrams. A Venn diagram takes different sets of objects and diagrams them as circles — or other shapes — with objects in both sets appearing in the intersections of the circles. Optionally, the area of different parts of the diagram can give you information about the size of the set of things in that part of the diagram. The example at the top of the post is about the “positive whole numbers”. This is called the universal set, which is fancy talk for everything the diagram covers. The special sets shown in the diagram are the even numbers and the prime numbers. Since all the primes except 2 are odd, the only thing in the intersection of the even and prime numbers is 2, which the diagram shows.
Thousands of years ago, on the shores of the Aegean Sea, a man named Eratosthenes wrote out the numbers 2, 3, 4, 5, 6, … History does not record where he stopped. He skipped 1 because it is strange and special. He circled two and crossed off every other number. The next number not yet crossed off was three. He circled three and crossed off every third number. He kept at this, and the circled numbers were the primes: 2, 3, 5, 7, 11, 13, 17, 19, and so on. His technique for finding prime numbers is called the Sieve of Eratosthenes and it represents the beginning of our species’ fascination with strange patterns in sequences of numbers. There is an enormous repository of knowledge about sequences of numbers that, like the primes, mean something, even if we do not always know what they mean yet. That is the topic of today’s post.
This week’s post is about the Online Encyclopedia of Integer Sequences (OEIS). This website is an incredibly powerful tool for research mathematicians, but it can also be very helpful to students or people trying to solve simple counting problems. It is a searchable list of counting sequences. The picture at the top of the post is a pictorial demonstration that the number of ways to fill a 2-wide rectangle with dominos can be done one way with one domino, two ways with two dominos, three ways with three dominos, and five ways with four dominos. The sequence 1,2,3,5 are the first four terms of the counting sequence for the problem of filling a two-wide rectangle with dominos. If we type “1,2,3,5” into the search bar of the online encyclopedia we get:
A000045 Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.The cool thing is that the Fibonacci numbers are the counting sequence for the dominos-in-a-2-wide rectangle problem. There are eight ways to fill a 2×5 rectangle with dominos, which is the next term in the Fibonacci sequence. The “A000045” is the serial number of the sequence in the encyclopedia. But that’s not all we get! Click through the link, clear the search bar, and type in 1,2,3,5, and scroll for a while. The encyclopedia lists a huge number of other things that the Fibonacci numbers count. After that list are references that source the information and relevant web links.