## Reading the Comics, March 9, 2016: Mathematics Recreation Edition

I haven’t been skipping the comics, even with the effort of keeping up on the Leap Day 2016 A To Z Glossary. I just try to keep to the pace which Comic Strip Master Command sets.

The kids-information feature Short Cuts, by Jeff Harris, got ahead of “Pi Day” last Sunday. I imagine the feature gets run mid-week in some features, so that it’s better to run a full week before March 14th. But here’s a bundle of trivia, some jokes, some activities, that sort of thing. I am curious about one of Harris’s trivias, that Pi “plays an important role in some of the equations used in Einstein’s famous general theory of relativity”. That’s true, but it’s not as if general relativity is a rare appearance for pi in physics. Maybe Harris chose it on aesthetic grounds. General relativity has a familiar name and exotic concepts. And it allowed him to put in an equation that’s mysterious yet attractive-looking.

Samson’s Dark Side Of The Horse for the 7th of March made me wonder how many sudoku puzzles there are. The answer is — well, you have to start thinking carefully about what you mean by “how many”. For example: start with one puzzle. Swap out every appearance of a 1 with a 2, and a 2 with a 1. Is this new one actually a different puzzle? You can make a case for yes or for no. And that’s before we get into the question of how many clues to give to solve the puzzle. If I’m not misreading Wikipedia’s “Mathematics of Sudoku” page, the number of different nine-by-nine combinations of digits that can be legitimate sudoku puzzle solutions is 6,670,903,752,021,072,936,960. This was worked out in 2005 by Bertram Felgenhauer and Frazer Jarvis. They worked it out partly by logic, partly by brute force. Brute force is trying all the possibilities to see what works. It’s a method that rewards endurance. We like that we can turn it over to computers now. Or cartoon horses, whichever. They’re good at endurance.

Jef Mallett’s Frazz started a sequence about problem-writing on the 7th of March. Caulfield’s setup, complaining about trains and apple bushels, suggests he was annoyed by mathematics problems. I understand. Much of real mathematics starts with curiosity about something (how many sudoku puzzles are there?). Then it’s working out what computation might answer that question. Then it’s doing that calculation. And then it’s verifying that the calculation is right. Mathematics educators have to teach ways to do a calculation, and test that. And to teach how to know what calculation to do, and test that. That’s challenging enough. Add to that working out something to be curious about and you understand the appeal of stock setups. Maybe mathematics should include some courses in creative writing and short-short fiction. (Verification is, in my experience, the part nobody cares about. This is a shame. The hardest part of doing numerical mathematics is making sure your computation makes any sense.)

Richard Thompson’s Richard’s Poor Almanac rerun the 7th of March features the Non-Euclidean Creeper. It’s a plant perhaps related to the Cubist Fir Christmas tree and to the Otterloops’ troublesome non-Euclidean tree. Non-Euclidean geometry will probably always sound more intimidating and exotic. Euclidean geometry describes the way objects on the human scale behave. Shapes that fit on the table, or in your garden, follow Euclidean rules. But non-Euclidean isn’t magic; it’s the way that shapes on the surface of a globe work, for example. And the idea of drawing a thing like a square on the surface of the Earth isn’t so bizarre.

Paul Trap’s Thatababy for the 7th makes sport of geometry.

My love and I were talking the other day about Jim Toomey’s Sherman’s Lagoon. It’s a bit odd as comic strips go. It’s been around forever, for one, but nobody talks about it. It’s stayed reliably funny. Comic strips that’ve been around forever tend to … you know … not be. The strip’s done as a work-and-home strip except the cast is all sea life. And the thing is, Toomey keeps paying attention to new discoveries in sea life, and other animal research. And this is a fantastic era for discoveries in sea life, aside from how humans have now eaten all of it and we don’t have any left. I am not joking when I say the comic strip is an effortless way to keep up with new discoveries about the oceans.

I missed it when in December the discovery was announced to the world. But the setup, about the common name being given by a group of kids, is apparently quite correct. So we should expect from Toomey. (The scientific name is Etmopterus benchleyi. The last name refers to Peter Benchley, repentant Jaws novelist.) LiveScience.com’s article says lead author Dr Vicky Vásquez had to “scale them back” from their starting point, the “super ninja”. This differs from Hawthorne’s claim that the kids started from the “math stinks” shark, but it’s still a delight anyway.

## Earth Day

It’s a lovely day, so I felt like sharing these illustrations from the Life Through A Mathematicians Eyes blog. It’s simply superimposing graphs of equations over scenes of natural beauty, but that’s attractive enough.

When I say “graphs of equations”, I mean that we’re setting a coordinate system — here a Cartesian or rectangular one, one based on x- and y- and z- distances from some origin point — over space, and then drawing in white lines the sets of x- and y- and z-coordinates which make some equation true. That’s what we normally mean by saying “the graph of an equation”; it’s a drawing that shows when a relationship is true and when it is not.

I believe most of you know what Earth Day is celebrating ^_^ I think this is a great day and there are a lot of activities that could be done everywhere to celebrate it. You might think that there is not much I can say about Earth and mathematics, but if you are a math – lover like you already know a lot of mathematical shapes, patterns and constant that can be found in nature. If you want to see more about this check the photos from my album Math&Nature .

But today I want to talk a little about the idea of an American student, mathematician and photographer, Nikki Graziano. I believe the project is old, but still very interesting and perfect for today. She took photos of different natural forms and then found proper equations to explain those forms created by nature. Here are some of the images:

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## Making The End Of The World Quantitative

I haven’t forgot my little problem about working out where the apparent edge of the world was, from my visit to the Sleeping Bear Dunes in northern (lower) Michigan. What I have been is stuck on a way to do all the calculations in a way that’s clear and that avoids confusion. I realized the calculations were reasonably clear to me but were hard to describe because I could put into similar-looking symbols a bunch of things I wanted to describe.

So I’ve resolved that the best thing I can do is take some time to describe the things I mean, and why they’ll get the symbols that they do. The first part of this is drawing a slightly more mathematical representation of the situation of standing on top of the dune and looking out at the water, and seeing the apparent edge of the dune as something very much closer than the water is. This is what’s behind my new picture, a cross-section of the dune and a person looking out at its edge.

## The End Of The World, Qualitatively Explained

So I want to understand the illusion of being at the edge of the world at the Sleeping Bear Dunes in northern Michigan. Since I like doing mathematics I think of this as a mathematics problem; so, I figure, I need to put together some equations. Before I do that, I need to think of what I want the equations to represent, which is the part of the problem where I build a model of the dunes. In the process I should get at least a qualitative idea of the effect; later, I should be able to quantify that.

What’s a dune? Well, it’s a great whomping big pile of sand right next to the water. There’s more to a dune than that, but since all I’m interested in is how the dune looks, I don’t need to think about much more than what the dune’s shape is, and how it compares to the water beside it. If I wanted to understand the ecology of a dune, or fascinating things like how it moves, then I’d have to model it in greater detail, but for now I’m going to try out this incredibly simple model and see what it gets me.

## Just How Far Is The End Of The World?

I got to visit the Sleeping Bear Dunes National Lakeshore earlier this month. I thought I knew what dunes were, from the little piles of sand that accumulated on the Jersey Shore sometimes, but, no. These dunes, at the northern end of Michigan’s lower peninsula, look out on Lake Michigan from, at one spot on the Pierce Stocking Scenic Drive, about 450 feet above sea level. That’s a staggering height, that awed me, particularly as we approached it from the drive, and so had a previously quite nice enough drive through lovely forests come to a sudden, almost explosive, panorama of sand towering far above the ocean.

Maybe a quick working definition of a “scenic overlook” is a spot of land from which you can look noticeably down and see birds flying. This is vey scenic: had there been a Saturn V moon rocket, on the launchpad, at sea level, we would have been looking noticeably down at its escape rocket. For that matter, if there had been a Saturn V moon rocket, on top of which was somehow perched a Gemini-Titan rocket, we’d still be … well, we’d have to look up to see the crew in the Gemini capsule, but we would be about eye level with the top of the Titan booster, anyway.

Something that I imagine no picture except a three-dimensional one is going to capture, though, is the sense that one is standing at the edge of the world. From the top of the dune, the end of the sand seems to be very nearby, maybe a couple dozen feet off, and the water below is so clearly distant that it feels impossibly far away. Walking toward that edge makes the edge of the world recede, of course, but it never quite loses that sense of being on the precipice until quite far along.

Some mad souls do follow a trail all the way down to the beachfront of Lake Michigan; I wasn’t among them. The difficulty in walking back up — all on sand, on a pretty significant slope — from just walking a little near the edge and maybe dropping twenty feet or so in altitude convinced me not to carry on. I didn’t know it was a full 450 feet up, but it was obviously far enough.

The geometry of all this, though, has captivated me, and I hope to spend a couple essays here working out such questions as just how the optical illusion of this edge of the world worked, and how its recession works.