OK, I’ll admit it, back in the day I read a lot of comics.  Even then, though, I was skeptical — “Wait, how could Superman just pick up that building?  It’d fall apart!”

But I was intrigued by one recurring character, Mr Mxyzptlk, a pixie-like “visitor from the 5th dimension.”   His primary purpose in life (other than getting us to buy more comics) seemed to be to play tricks on or otherwise torment Our Hero.

Mxy wasn’t the only comics character coming in “from another dimension.” It seemed like the entire Marvel team (both sides) was continually flickering out of and into our universe that way. How often did Jane Grey die and then somehow get cloned or refreshed?  (BTW, if the accompanying cartoon is a little obscure, show it to the friendly clerks at your local comics store — it may give them a chuckle.)

But my question was, where was that dimension Mxy came from?  I got an answer, sort of, when our geometry teacher explained that a dimension is just a direction you could travel.  Different dimensions are directions at right angles to each other.  She was right (see my first post in this series), at least in the context of then-HS math, but that explanation opened an editorial issue that’s never been properly settled.

A dimension is a direction, not a location.  You can’t be “from” a fifth or sixth or nth dimension any more than you can be from up.  If there is a spatial fifth dimension, we’re already “in” it in the same sense that we’re already somewhere along east-to-west and somewhen along past-to-future.

What’s going on is that for the purpose of the story, the authors want the character to come from somewhere very else.  We often associate a place with the direction to it — the sun rises in the east, Frodo departs to the west,  Heaven is up, Hell is down — but those are all directions relative to our current location.  We even associate future times as being in front of us and past times behind us (there’s that 4th dimension again).

But a place is more specific than a direction — to navigate to a certain there you need to know the direction and the distance (or another quantity that stands in for a distance).  That matters.  Jimmie Rodgers sang, “Twelve more miles to Tucumcari” as he kept track of the distance left to go along the road he was traveling.  Or away from the town, as it turned out.

Physicists have lots of uses for the combination of a direction and a magnitude, so many that they gave the combination a name — a vector.  The vector may represent a direction and a distance, a direction and the strength of a magnetic field, or a direction and any quantity that happens to be useful in the application at hand.  A wind map uses vectors of direction and wind speed to show air flow.  Here’s a very nice wind map of the US, and I love NOAA’s wind map of the world.  Vectors will be real useful when we start talking about black holes.

OK, so Mr Mxyzpltk (the spelling seemed to vary from issue to issue of the comic) comes from somewhere along a fifth dimension, but they never tell us from how far away.

Next week –As Steve Martin said, “Let’s get small, really small.”

~~ Rich Olcott

The Klein bottle is one of the most misunderstood objects in popular math.  Let’s start with the name.  When he initially described the object Herr Doktor Professor Felix Klein said it was a surface.  Being German and writing in German, he used the word Fläche (note the two little dots over the a).  When his paper was translated to English, the translator noticed the shape of the thing but didn’t notice those two little dots.  He misread the word as Flasche, meaning flask or bottle, and the latter word stuck.

So who was Herr Klein and what was it that he wrote about?  One of the world’s foremost mathematicians in the last quarter of the 19 Century, he specialized in geometry, complex analysis and mathematical physics.  Among his other accomplishments was that as director of a research center at Georg-August-Universität Göttingen in 1895 he supervised Germany’s first Ph.D. thesis written by a woman (Grace Chisholm Young).

As a small part of one of his many papers he noted that a cylinder could be deformed in two different ways to connect its two ends. The first way is to simply bend it around in a circle to form a torus (or bicycle tire or doughnut, depending on how hungry you are.)

The other way is more of a challenge — bringing one end to meet the other from inside the cylinder.  The problem is that in the context of Klein’s work, he wasn’t “allowed” to pierce the cylinder’s surface to get it there.  Klein’s solution was simple  — swing it in through the fourth dimension.  Sounds like a cheat, doesn’t it?

Actually, the cheat is in the way that the Klein bottle is usually represented.  When the glass artisan created this example, he probably brought one tube up from the bottom, sealed it to the side wall, blew a hole in the side wall at that location, and then brought the outside tube around to join there.  That hole would not have satisfied Herr Klein.

If you’ve looked at my previous post in this series you probably have a pretty good idea of how he would have preferred his Fläche to be depicted — as an animation which exploits time as the fourth dimension.  So here you have it.

Suppose the figure’s wall sprouts from a bud somewhere near the intersection point.  After the figure has grown for a while, the earliest section of the wall begins to recede, disappearing like the Cheshire Cat but leaving its ever-expanding smile behind.  By the time the growth front gets to where the bud was, there’s nothing there to intersect.

If you opt to build the “bottle” in 4-space, there’s no problem getting those two ends of the cylinder to join up.  A shape that’s impossible to build in three dimensions is easy-peasy (with a little planning) in four.

Yeah, yeah, the Klein bottle has lots of interesting properties, like not having an inside, but we’ll defer talking about them for a while.

Next week, getting past that pesky four-dimension limitation.

~~ Rich Olcott