A Geometry Thing That’s Left Me Unsettled


I came across a little geometry thing that left me unsettled, even as I have to admit it’s correct. Start with a two-dimensional space, or as the hew-mons call it, a plane. Draw a square with sides of length two and centered on the origin. So it has corners at the points with Cartesian coordinates (+1, +1), (+1, -1), (-1, +1), and (-1, -1). Around each of these corners draw a circle of radius 1.

There is some largest circle that you can draw, centered on the origin, the dead center of the square, with Cartesian coordinates (0, 0), and that just touches all of the corners’ circles. It has a radius of a little under 0.414.

Now think of the three-dimensional analog. Three-dimensional space. Draw a box with sides all of length two and centered on the origin. So it has corners at the points with Cartesian coordinates (+1, +1, +1), (+1, +1, -1), (+1, -1, +1), (+1, -1, -1), (-1, +1, +1), (-1, +1, -1), (-1, -1, +1), and (-1, -1, -1). Around each of these eight corners draw a circle of radius 1.

There is some largest sphere that you can draw, centered on the origin, the point with Cartesian coordinates (0, 0, 0), that just touches all of the corners’ circles. It has a radius of a little under 0.732.

Think of the four-dimensional analog. This is harder to sketch. But a four-dimensional hypercube, with each side of length 2 and centered on the origin. So it has corners at the points with Cartesian coordinates (+1, +1, +1, +1), (+1, +1, +1, -1), (+1, +1, -1, +1), (+1, +1, -1, -1), and you know what? Will you let me pretend we listed all sixteen corners? Thanks. Around each of these corners draw a circle of radius 1.

There is some largest hypersphere you can draw, centered on the origin, the point with Cartesian coordinates (0, 0, 0, 0), and that just touches all of these corners’ circles. It has a radius of 1.

Keep going. Five-dimensional space, with corners like (+1, +1, +1, +1, +1). Six-dimensional space, with corners like (+1, +1, +1, +1, +1, +1). Seven-dimensional space. And so on.

Eventually, the space is vast enough that the radius of this largest-touching hypersphere is bigger than 2. That is, reaching out more than twice as far as the original box goes, this even though the corner hyperspheres line the edges of the box, and touch one another along those edges.

Non-Euclidean geometry has the reputation of holding deep, inscrutable mysteries. To say something is a non-Euclidean space, outside of a mathematical context, is to designate it as a place immune to reason and beyond human comprehension. This is not such a case. This is a completely Euclidean space; it’s just got a lot of dimensions to it. Strange things will happen.

Another weird, but to me not so unsettling matter, concerns the surface (or hypersurface) area and the volume of these spheres. The circumference of a unit circle is, famously, 2π. The area of a unit sphere is 4π. For a four-dimensional hypersphere the surface area is a bit bigger yet. And bigger again for five and six and seven dimensions. But at eight dimensions the surface area starts shrinking again, and it never grows again. Have a great enough number of dimensions and the unit hypersphere has almost zero surface area. The volume of a unit circle is π. Of a unit sphere, \frac43 \pi . For a four-dimensional hypersphere, \frac12 \pi^2 . For a five-dimensional hypersphere, \frac{8}{15}\pi^2 . It is never so large again; for six or more dimensions the volume starts to shrink again. As the number of dimensions of space grows, the volume of the unit hypersphere dwindles to zero.

You know, that’s unsettling me more now that I’m paying attention to it.

Reading the Comics, April 27, 2016: Closing The Month (April) Out Edition


I concede this isn’t a set of mathematically-themed comics that inspires deep discussions. That’s all right. It’s got three that I can give pictures for, which is important. Also it means I can wrap up April with another essay. This gives me two months in a row of posting something every day, and I’d have bet that couldn’t happen.

Ted Shearer’s Quincy for the 1st of March, 1977, rerun the 25th of April, is not actually a “mathematics is useless in the real world” comic strip. It’s more about the uselessness of any school stuff in the face of problems like the neighborhood bully. Arithmetic just fits on the blackboard efficiently. There’s some sadness in the setting. There’s also some lovely artwork, though, and it’s worth noticing it. The lines are nice and expressive, and the greyscale wash well-placed. It’s good to look at.

'I've got a question, Miss Reid, about this stuff we're learning in school.' 'What's that, Quincy?' Quincy points to the bully out the window. 'How's it gonna help us out in the real world?'
Ted Shearer’s Quincy for the 1st of March, 1977, rerun the 25th of April, 2016. I just noticed ‘Miss Reid’ is probably a funny-character-name.

dro-mo for the 26th I admit I’m not sure what exactly is going on. I suppose it’s a contest to describe the most interesting geometric shape. I believe the fourth panel is meant to be a representation of the tesseract, the four-dimensional analog of the cube. This causes me to realize I don’t remember any illustrations of a five-dimensional hypercube. Wikipedia has a couple, but they’re a bit disappointing. They look like the four-dimensional cube with some more lines. Maybe it has some more flattering angles somewhere.

Bill Amend’s FoxTrot for the 26th (a rerun from the 3rd of May, 2005) poses a legitimate geometry problem. Amend likes to do this. It was one of the things that first attracted me to the comic strip, actually, that his mathematics or physics or computer science jokes were correct. “Determine the sum of the interior angles for an N-sided polygon” makes sense. The commenters at Gocomics.com are quick to say what the sum is. If there are N sides, the interior angles sum up to (N – 2) times 180 degrees. I believe the commenters misread the question. “Determine”, to me, implies explaining why the sum is given by that formula. That’s a more interesting question and I think still reasonable for a freshman in high school. I would do it by way of triangles.

BEIRB -OO-O; CINEM OOO--; CCHITE O--OO-; RUSPRE O---OO. Two, Three, Five, and Seven will always be -- ----- -----.
David L Hoyt and Jeff Knurek’s Jumble for the 27th of April, 2016. I bet the link’s already expired by the time you read this.

David L Hoyt and Jeff Knurek’s Jumble for the 27th of April gives us another arithmetic puzzle. As often happens, you can solve the surprise-answer by looking hard at the cartoon and picking up the clues from there. And it gives us an anthropomorphic-numerals gag for this collection.

Bill Holbrook’s On The Fastrack for the 28th of April has the misanthropic Fi explain some of the glories of numbers. As she says, they can be reliable, consistent partners. If you have learned something about ‘6’, then it not only is true, it must be true, at least if we are using ‘6’ to mean the same thing. This is the sort of thing that transcends ordinary knowledge and that’s so wonderful about mathematics.

Fi's STEM talk: 'Numbers are reliable. They're consistent. They don't lie. They don't betray you. They don't pretend to be something they're NOT. x and y, on the other hand, are shifty little goobers.'
Bill Holbrook’s On The Fastrack for the 28th of April, 2016. I don’t know why the ‘y’ would be in kind-of-cursive while the ‘x’ isn’t, but you do see this sort of thing a fair bit in normal mathematics.

Fi describes ‘x’ and ‘y’ as “shifty little goobers”, which is a bit unfair. ‘x’ and ‘y’ are names we give to numbers when we don’t yet know what values they have, or when we don’t care what they have. We’ve settled on those names mostly in imitation of Réné Descartes. Trying to do without names is a mess. You can do it, but it’s rather like novels in which none of the characters has a name. The most skilled writers can carry that off. The rest of us make a horrid mess. So we give placeholder names. Before ‘x’ and ‘y’ mathematicians would use names like ‘the thing’ (well, ‘re’) or ‘the heap’. Anything that the quantity we talk about might measure. It’s done better that way.

Reading the Comics, October 5, 2015: Boxes and Hyperboxes Edition


I’ve got more mathematically-themed comic strips than this to write about, but this should do for one day’s postings. Motley did give me the puzzle of figuring out whether the character’s description of a process could be made sensible, which is a bit of extra fun. Boxes and cubes come up in three of the comics, too.

John McPherson’s Close to Home for the 3rd of October drops in the abacus as a backup for the bank’s computers. It’s a cute enough idea. Deep down, I admit, I’m not sure that an abacus would be needed for most of the work a teller has to do during a temporary computer outage, though. Most of the calculations to do would be working out whether there’s enough money in the account to allow a given withdrawal. That’s database-checking, really. Also I’m not sure that’s a model of abacus that’s actually been made, but if I understood what was wanted, then in some ways wasn’t the artwork successful?

Larry Wright’s Motley Classics for the 3rd of October is a rerun from the same day in 1987. Debbie gives the terribly complicated instructions on how to calculate a tip. I’m not sure how tip-calculating got to the pop culture position of “most complicated thing people do with mathematics that isn’t taxes”. Probably that it is a fairly universal need for mathematics that isn’t taxes (and so seasonally bound) explains it. I think she’s describing a valid algorithm, though, if we make some assumptions about her pronouns.

Suppose we start with the price P. Double that and move the decimal one place over, to the left I suppose, and we have 0.20 times P. Suppose that this is the first answer. If we divide this first answer by four, then, this second answer will be 0.05 times P. And subtracting the second answer from the first is, indeed, 0.15 times P, or fifteen percent of the original price. While correct, though, it’s still a lousy algorithm. Too many steps, too much division, and subtraction is a challenge. Taking one-tenth the price plus half a tenth would be numerically identical and less challenging. Taking one-sixth the price would be a division, yes, but get you to near enough fifteen percent with only one move.

Mark Pett’s Lucky Cow for the 4th of October, another rerun, shows off one of the silly semantic-equation games that mathematics majors sometimes play. Forgive them. There’s a similar argument which proves that half a ham sandwich is greater than God. It all amounts to playing on arguments which might (not always!) be correct in form but have things with silly meanings plugged into them.

Stephan Pastis’s Pearls Before Swine for the 4th of October gives Pig the chance to panic. It’s another strip about the difference between what “positive” and “negative” mean in inference testing, and so in medical testing, versus the connotations of “good” and “`bad” they have. I’ve explained this before, in other Reading the Comics essays, so I’ll spare the whole thing. But in short, “positive” in this case means “these test results are so far away from normal values that it strains plausibility to think it’s normal”. “Negative” means “these test results are not so far away from normal values as to strain plausibility to think it’s normal”.

Geoff Grogan’s Jetpack Jr for the 5th of October draws a hypercube as the box little alien Jetpack Junior arrived in. Well, these are some of the common representations of how a four-dimensional cube would look in our three-dimensional space (and that, rendered on a two-dimensional screen). The difficult-to-conceptualize part is that in the cube, seen in the middle third of the strip, every one of the red lines is the same length, and is perpendicular to all its neighbors. The triptych of shapes are all the same four-dimensional cube, too, just rotated along different axes by different amounts.

All my old links to play with hypercube rotations seem to have expired or turn out to be Java applets. Here’s a page that offers a couple of pictures, though. It has a link to an iOS app that should let people play with rotating a four-dimensional hypercube. Might enjoy it. I think this is the first time Jetpack Jr as such has got around here. It used to run as Plastic Babyheads from Outer Space, with a silly overarching story about aliens with plastic baby heads, ah, invading. I don’t think that made the Reading the Comics roster, though, unless some of the aliens mentioned pi, which they might have done.

Charles Brubaker’s Ask A Cat for the 5th of October I think is another debuting strip around here. It’s about the problem of Schrödinger’s Cat, a thought-experiment designed to show we don’t really understand what the conventional mathematical models of quantum mechanics mean. In at least some views, the mathematics of quantum mechanics suggests we could have an apparently ridiculous result: something big, like a cat, that we expect should work like a classical-physics entity, behaving instead like a quantum-mechanical entity, with no definable state. The problem has been with us for eighty years and isn’t well-answered, but that happens. Zeno’s paradoxes have been with us three thousand years and are still showing us things we don’t quite understand about divisibility and continuity.

Anthony Smith’s Learn to Speak Cat for the 5th of October is a completely different cat comic strip that I think is making a debut here. This is more a matter of silly symbolic manipulation than anything serious, though.

Tom Toles’s Randolph Itch, 2 am from the 5th of October is a rerun from 1999. And it shows a soap-bubble cube. Soap bubbles allow for some neat mathematics. They act like animate computers working out the way to enclose a given volume with the least surface area. A web site written by Dr Michael Hutchings at the University of California/Berkeley describes some of the mathematical work involved. Surprising to me is that it was only in the 1970s that the “double bubble conjecture” was proven. That’s a question about how to cover a given volume using two bubbles. The answer is what you might get from playing with soap bubble wands, but it took about a century of working on to prove. Granting, mathematicians did other things with their time, so it wasn’t uninterrupted soap-bubble work. Hutchings includes some review of the field as it existed in the early 2000s, and lists three open problems. The first of them is one that’s understandable even without knowing more mathematical lingo than what R3 is. (And folks who’re hanging around here know that by now.) Also it has pictures of soap bubbles, which are good for a lazy Friday morning.

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