Monday Morning Math: Frieze Patterns

Good morning! One of my favorite kinds of symmetry is the symmetry of Frieze Patterns, which are patterns that repeat in a band, like at the top of a wall. Because the patterns only repeat in one dimension, the number of possible symmetries is fairly small. Plus, although there are formal ways to refer to the patterns, you can also use letters of the alphabet! (Can I demonstrate each of them with a single letter? typetypetype No I can’t for all, but I’ve seen two letter combinations in places, and that works very nicely.)

• One option is no symmetry except translation: LLLLLLLL or bbbbbbbbbb
• Another is vertical symmetry: AAAAAAAAAA or bdbdbdbdbd
• Another is horizontal symmetry: BBBBBBBBB
• Or, on a variation of horizontal symmetry, it could have glide symmetry (if there’s a reflection across the horizontal access and then a shift): bpbpbpbpbpbp
• And finally it could have rotational symmetry: SSSSSSSSSS or bqbqbqbqbqbqbq

So we’re up to 5 options, but we can combine them! If you think of there being three main types (vertical, horizontal/glide, and rotational), that suggests that there would be 2*3*2=12 frieze patterns (vertical or not, horizontal or glide or not, and rotational or not) but interestingly it turns out that if you have any two of the main types, you automatically get the third. That means that there are only two more possibilities, depending on whether you use horizontal or glide.

• Vertical, horizontal, and rotational: HHHHHHH
• Vertical, glide, and rotational: bdpqbdpqbdpq

A nice thing about frieze patterns is that you can see examples in many many historical objects: pottery, beadwork, knitting, painting, etc., This means you can look for examples in many time periods and locations to see real-life examples. Or, of course, you can just pick your favorite font and sort all the letters of the alphabet into their symmetry types.

Monday Morning Math: Four Colors Suffice

The Four Color Theorem is a theorem that says that every map on a plane can be colored, with different colors in adjacent areas, using just four colors: purple, gold, pink, and teal. Or whatever your favorite colors are. The known mathematical history of this theorem began exactly 171 years ago, and by “exactly” I mean exactly. Specifically, in 1852, a law student named Francis Guthrie was coloring a map of the counties of England (for…some reason? Maybe for some law assignment, maybe just for fun) and he noticed that he only needed 4 colors. Then he thought that might be true for all maps, and tried to prove it, but wasn’t happy with his proof. So he asked his brother Frederick, who offered to ask his math teacher. And Frederick did just that, on October 23, 1852. How do we know the date? Well, Frederick’s math teacher was August de Morgan, a famous mathematician, and de Morgan wrote a letter that day to William Hamilton, another famous mathematician, which began:

My dear Hamilton
A student of mine asked me to day to give him a reason for a fact which I did not know was a fact—and do not yet. He says that if a figure be any how divided and the compartments differently coloured so that figures with any portion of common boundary line are differently coloured—four colours may be wanted, but not more…

De Morgan’s letter to Hamilton, public domain

That was the start of the Four Color Conjecture making its way through the mathematical community! De Morgan wrote to more people, and people did work on it, but didn’t come up with a solution in the 1850s. Or 1860s. Or a hundred years after that. It wasn’t until the 1970s that Kenneth Appel and Wolfgang Haken came up with a proof, one that was so computer-intensive that for the next two decades, until there was another shorter proof, there was still question about whether Guthrie’s conjecture had been proven.

So let’s raise a glass/cup of coffee to the power of interesting observations, and the places they lead!

Sources:

• “Francis Guthrie: A Colourful Life” by Pieter Maritz and Sonja Mouton in The Mathematical Intelligencer.
• “9. Note on the Colouring of Maps” by Frederick Guthrie in The Proceedings of the Royal Society of Edinburgh.
• “The Four Color Theorem” in Wolfram MathWorld

Monday Morning Math: Knit Mobius

Good morning! We’re in fall weather, which is my favorite, and a good time to do some knitting, if you knit. And if you want to knit, I’d recommend a mobius strip.

A mobius strip is a ring with a twist in it (often called a half-twist because you twist it 180 degrees, not 360). I usually make them out of paper. This one was made from an envelope:

This one was made from an envelope, which conveniently had a different design on the inside so you can see the twist. One feature of the mobius strip is that the twist means that it has only one side because the two sides of the envelope join at the twist, and another is that it has only one edge, for the same reason. This means that if you want to knit a mobius strip, there are two different ways you could do it. The first is to mimic the paper version: knit a long strip and then join the ends with a twist. But the second way is to knit it as a mobius strip from the start, by knitting on circular needles and putting the twist in from the start:

The cool thing about this is that you are knitting from the middle out, taking advantage of that mobius one-edge structure, and if you use a variegated yarn (like above) the stripes automatically go along both-but-actually-one edge. I’d heard of this before but never made one, and in working on it I realized another cool fact: because of the twist, the same color appears as a knit stitch one one side of the band and as a purl stitch along the other side. I think this is why all the patterns I’ve seen recommend doing several rows of knitting and then several rows of purling, to make that knit/purl change part of a larger pattern. [Hmmm – I think of crochet as being a little more similar on both sides, so if you could join a row of crochet in the round it might be a little less obvious if you didn’t want to alternate. Maybe.]

I’m only knitting a small version right now, but if you make a mobius scarf or cowl – and you should! – one feature is that it will be someone less bulky than if you were to knit a more cylindrical version, because the twist inherent in a mobius strip becomes a fold that lets the figure like flat. There are instructions here, which is what I used along with the video below, to get started on my mini version. Happy knitting!

Monday Morningish Math: Mayan Numbers!

Good afternoon!  Alas, I did not get this put together in time for Monday Morning, at least on the East Coast, but it’s still Monday Math and I’m still happy to be thinking about all of you.  So let’s jump into it!

One of my favorite number systems is Mayan.  I like it because it only uses 3 symbols: a shell for 0, a dot for 1, and a line for 5.  These are combined in a straightforward way:

(CC by Neuromancer2K4 and then Bryan Derksen)

What happens when you get to 20?  The Mayans turned to place value!  They wrote vertically, so the numbers on the bottom row were the 1s and then the row above counted 20s.  that means this number 

CC by Ellen Souza

would be read as 1 twenty and then 3 ones, making it 23. 

And this number

CC by Ellen Souza

 would be 81, from 4 twenties and 1 one.  (Also CC by Ellen Souza)

OK, so with this you could count up to 399 (19 twenties and 19 ones), but what about larger numbers?  Well, in the most straightforward system the third row would be 400s, from twenty twenties, and then if you needed larger numbers you’d add another row above that for the 8000s.  But the Mayans also used a Long Count form, where instead of a 400 row the third row could count 360, which is only 18×20.  Why, you might ask?  The Long Count was used in calendars, and there approximately 360 days in a year. so it is likely (although not certain) that that is where it came from. That means that this number

might mean 5 four-hundreds, 0 twenties, and 12 ones (making it 2012) or might mean 5 three-hundred-sixties, 0 twenties, and 12 ones (making it 1812).   In the Long Count form a fourth place-value would be 20×360=7200, and each other one 20 times the previous.

You can see some actual examples in the roughly thousand-year-old Dresden Codex, like page 34 below

(This codex is probably worthy of its own Monday Morning Math, but a quick note is that it is from southeastern Mexico, and is named after the city it now is housed in.)

Enjoy, and Happy Monday!

Monday Morning Math: Cell size and number

Today’s topic comes from Tracy (thanks Tracy!), and is about cells. Ian A. Hatton at the Max Planck Institute for Mathematics in the Sciences, along with Eric D. Galbraith, Nono S. C. Merleau, Teemu P. Miettinen, Benjamin McDonald Smith, and Jeffery A. Shander found that there was an inverse relationship between the size of cells in the human body and their number: there are more small cells than large ones. This itself may not be too surprising, since it is more efficient for cells to be small than large. That’s because nutrients and gases have to pass in and out of cells, which uses surface area, and for larger cells the amount of surface area is smaller compared to the volume of the cell. (Math time! If you have a 1x1x1 cube it has a volume of 1 and a surface area of 6×1=6. If you have a 2x2x2 cube it has a volume of 8, which is 8 times as large as that 1x1x1 cube, but the surface area is 6×4=24, which is only 4 times as large as the 1x1x1 cube.

So, lots of small cells, fewer big ones. What Hatton and all found, and published just a few weeks ago in the Proceedings of the National Academy of Sciences, is that the total mass of different kinds of cells in the body is very similar. So if one cell type is half the size of another, then there are approximately twice as many cells of that type.

The author’s share that they don’t know why this happens, but human cells aren’t the only place that this half-the-size-means-twice-as-many pattern occurs. Plankton is another, and ocean life in general, and almost certainly more.

Tracy shared this article, which summarizes the findings.