And we had another of those peculiar days where a lot of strips are on-topic enough for me to talk about.
Eric the Circle, this one by Kyle, for the 26th has a bit of mathematical physics in it. This is the kind of diagram you’ll see all the time, at least if you do the mathematics that tells you where things will be and when. The particular example is an easy problem, a thing rolling down an inclined plane. But the work done for it applies to more complicated problems. The question it’s for is, “what happens when this thing slides down the plane?” And that depends on the forces at work. There’s gravity, certainly . If there were something else it’d be labelled. Gravity’s represented with that arrow pointing straight down. That gives us the direction. The label (Eric)(g) gives us how strong this force is.
Where the diagram gets interesting, and useful, are those dashed lines ending in arrows. One of those lines is, or at least means to be, parallel to the incline. The other is perpendicular to it. These both reflect gravity. We can represent the force of gravity as a vector. That means, we can represent the force of gravity as the sum of vectors. This is like how we can can write “8” or we can write “3 + 5”, depending on what’s more useful for what we’re doing. (For example, if you wanted to work out “67 + 8”, you might be better off doing “67 + 3 + 5”.) The vector parallel to the plane and the one perpendicular to the plane add up to the original gravity vector.
The force that’s parallel to the plane is the only force that’ll actually accelerate Eric. The force perpendicular to the plane just … keeps it snug against the plane. (Well, it can produce friction. We try not to deal with that in introductory physics because it is so hard. At most we might look at whether there’s enough friction to keep Eric from starting to slide downhill.) The magnitude of the force parallel to the plane, and perpendicular to the plane, are easy enough to work out. These two forces and the original gravity can be put together into a little right triangle. It’s the same shape but different size to the right triangle made by the inclined plane plus a horizontal and a vertical axis. So that’s how the diagram knows the parallel force is the original gravity times the sine of x. And that the perpendicular force is the original gravity times the cosine of x.
The perpendicular force is often called the “normal” force. This because mathematical physicists noticed we had only 2,038 other, unrelated, things called “normal”.
Rick Detorie’s One Big Happy for the 26th sees Ruthie demand to know who this Venn person was. Fair question. Mathematics often gets presented as these things that just are. That someone first thought about these things gets forgotten.
John Venn, who lived from 1834 to 1923 — he died the 4th of April, it happens — was an English mathematician and philosopher and logician and (Anglican) priest. This is not a rare combination of professions. From 1862 he was a lecturer in Moral Science at Cambridge. This included work in logic, yes. But he also worked on probability questions. Wikipedia credits his 1866 Logic Of Chance with advancing the frequentist interpretation of probability. This is one of the major schools of thought about what the “probability of an event” is. It’s the one where you list all the things that could possibly happen, and consider how many of those are the thing you’re interested in. So, when you do a problem like “what’s the probability of rolling two six-sided dice and getting a total of four”? You’re doing a frequentist probability problem.
Venn Diagrams he presented to the world around 1880. These show the relationships between different sets. And the relationships of mathematical logic problems they represent. Venn, if my sources aren’t fibbing, didn’t take these diagrams to be a new invention of his own. He wrote of them as “Euler diagrams”. Venn diagrams, properly, need to show all the possible intersections of all the sets in play. You just mark in some way the intersections that happen to have nothing in them. Euler diagrams don’t require this overlapping. The name “Venn diagram” got attached to these pictures in the early 20th century. Euler here is Leonhard Euler, who created every symbol and notation mathematicians use for everything, and who has a different “Euler’s Theorem” that’s foundational to every field of mathematics, including the ones we don’t yet know exist. I exaggerate by 0.04 percent here.
Although we always start Venn diagrams off with circles, they don’t have to be. Circles are good shapes if you have two or three sets. It gets hard to represent all the possible intersections with four circles, though. This is when you start seeing weirder shapes. Wikipedia offers some pictures of Venn diagrams for four, five, and six sets. Meanwhile Mathworld has illustrations for seven- and eleven-set Venn diagrams. At this point, the diagrams are more for aesthetic value than to clarify anything, though. You could draw them with squares. Some people already do. Euler diagrams, particularly, are often squares, sometimes with rounded corners.
Venn had his other projects, too. His biography at St Andrews writes of his composing The Biographical History of Gonville and Caius College (Cambridge). And then he had another history of the whole Cambridge University. It also mentions his skills in building machines, though only cites one, a device for bowling cricket balls. The St Andrews biography says that in 1909 “Venn’s machine clean bowled one of [the Australian Cricket Team’s] top stars four times”. I do not know precisely what it means but I infer it to be a pretty good showing for the machine. His Wikipedia biography calls him a “passionate gardener”. Apparently the Cambridgeshire Horticultural Society awarded him prizes for his roses in July 1885 and for white carrots in September that year. And that he was a supporter of votes for women.
Ashleigh Brilliant’s Pot-Shots for the 26th makes a cute and true claim about percentiles. That a person will usually be in the upper 99% of whatever’s being measured? Hard to dispute. But, measure enough things and eventually you’ll fall out of at least one of them. How many things? This is easy to calculate if we look at different things that are independent of each other. In that case we could look at 69 things before there we’d expect a 50% chance of at least one not being in the upper 99%.
It’s getting that independence that’s hard. There’s often links between things. For example, a person’s height does not tell us much about their weight. But it does tell us something. A person six foot, ten inches tall is almost certainly not also 35 pounds, even though a person could be that size or could be that weight. A person’s scores on a reading comprehension test and their income? But test-taking results and wealth are certainly tied together. Age and income? Most of us have a bigger income at 46 than at 6. This is part of what makes studying populations so hard.
T Shepherd’s Snow Sez for the 26th is finally a strip I can talk about briefly, for a change. Snow does a bit of arithmetic wordplay, toying with what an expression like “1 + 1” might represent.
There were a lot of mathematically-themed comic strips last week. There’ll be another essay soon, and it should appear at this link. And then there’s always Sunday, as long as I stay ahead of deadline. I am never ahead of deadline.