Three Ways To Get Dizzy

<FZzzzzzzzzzzzzzzzzzzzzzzttt!> “Urk … ulp … I need to sit down, quick.”

“Anne? Welcome back, the couch is over there. Goodness, you do look a little green. Can I get you something to drink?”

“A little cool water might help, thanks.”

“Here. Just sit and breathe. That wasn’t your usual fizzing sound when you visit my office. When you’re ready tell me what happened. Must have been an experience, considering some of your other superpower adventures. Where did you ‘push‘ to this time?”

“Well, you know when I push forward I go into the future and when I push backward I go into the past. When I push up or down I get bigger or smaller. You figured out how pushing sideways kicks me to alternate probabilities. And then <shudder> there was that time I found a new direction to push and almost blew up the Earth.”

“Yes, that was a bad one. I’d think you’ve pretty well used up all the directions, though.”

“Not quite. This time I pushed outwards, the same in every direction.”

“Creative. And what happened?”

“Suddenly I was out in deep space, just tumbling in the blackness. There wasn’t an up or down or anything. I couldn’t even tell how big I was. I could see stars way off in the distance or maybe they were galaxies, but they were spinning all crazy. It took me a minute to realize it was me that was spinning, gyrating in several ways at once. It was scary and nauseating but I finally stopped part of it.”

“Floating in space with nothing to kill your angular momentum … how’d you manage to stabilize yourself at all?”

“Using my push superpower, of course. The biggest push resistance is against the past. I pulled pastward from just my shoulders and that stopped my nose‑diving but I was still whirling and cart‑wheeling. I tried to stop that with my feet but that only slowed me down and I was getting dizzy. My white satin had transformed into a spacesuit and I definitely didn’t want to get sick in there so I came home.”

“How’d you do that?”

“Oh, that was simple, I pulled inward. I had to um, zig‑zag? until I got just the right amount.”

“That explains the odd fizzing. I’m glad you got back. Looks like you’re feeling better now.”

“Mostly. Whew! So, Mr Physicist Sy, help me understand it all. <her voice that sounds like molten silver> Please?”

“Well. Um. There’s a couple of ways to go here. I’ll start with degrees of freedom, okay?”

“Whatever you say.”

“Right. You’re used to thinking in straight‑line terms of front/back, left/right and up/down, which makes sense if you’re on a large mostly‑flat surface like on Earth. In mathspeak each of those lines marks an independent degree of freedom because you can move along it without moving along either of the other two.”

“Like in space where I had those three ways to get dizzy.”

“Yup, three rotations at right angles to each other. Boatmen and pilots call them pitch, roll and yaw. Three angular degrees of freedom. Normal space adds three x-y-z straight‑line degrees, but you wouldn’t have been able to move along those unless you brought along a rocket or something. I guess you didn’t, otherwise you could have controlled that spinning.”

“Why would I have carried a rocket when I didn’t know where I was going? Anyhow, my push‑power can drive my straight‑line motion except I didn’t know where I was and that awful spinning had me discombobulated”

“Frankly, I’m glad I don’t know how you feel. Anyhow, if measurable motion is defined along a degree of freedom the measurement is called a coordinate. Simple graphs have an x-coordinate and a y-coordinate. An origin plus almost any three coordinates makes a coordinate system able to locate any point in space. The Cartesian x-y-z system uses three distances or you can have two distances and an angle, that’s cylindrical coordinates, or two angles and one distance and that’s polar coordinates.”

“Three angles?”

“You don’t know where you are.”


~~ Rich Olcott

Throwing a Summertime curve

All cats are gray in the dark, and all lines are straight in one-dimensional space.  Sure, you can look at a garden hose and see curves (and kinks, dammit), but a short-sighted snail crawling along on it knows only forward and backward.  Without some 2D notion of sideways, the poor thing has no way to sense or cope with curvature.

Up here in 3D-land we can readily see the hose’s curved path through all three dimensions.  We can also see that the snail’s shell has two distinct curvatures in 3D-space — the tube has an oval cross-section and also spirals perpendicular to that.

But Einstein said that our 3D-space itself can have curvature.  Does mass somehow bend space through some extra dimension?  Can a gravity well be a funnel to … somewhere else?

No and no.  Mathematicians have come up with a dozen technically different kinds of curvature to fit different situations.  Most have to do with extrinsic non-straightness, apparent only from a higher dimension.  That’s us looking at the hose in 3D.

Einstein’s work centered on intrinsic curvature, dependent only upon properties that can be measured within an object’s “natural” set of dimensions.Torus curvature

On a surface, for instance, you could draw a triangle using three straight lines.  If the figure’s interior angles sum up to exactly 180°, you’ve got a flat plane, zero intrinsic curvature.  On a sphere (“straight line” = “arc from a great circle”) or the outside rim of a doughnut, the sum is greater than 180° and the curvature is positive.
Circle curvatures
If there’s zero curvature and positive curvature, there’s gotta be negative curvature, right?  Right — you’ll get less-than-180° triangles on a Pringles chip or on the inside rim of a doughnut.

Some surfaces don’t have intersecting straight lines, but you can still classify their curvature by using a different criterion.  Visualize our snail gliding along the biggest “circle” he/she/it (with snails it’s complicated) can get to while tethered by a thread pinned to a point on the surface. Divide the circle’s circumference by the length of the thread.  If the ratio’s equal to 2π then the snail’s on flat ground.  If the ratio is bigger than ,  the critter’s on a saddle surface (negative curvature). If it’s smaller, then he/she/it has found positive curvature.

In a sense, we’re comparing the length of a periphery and a measure of what’s inside it.  That’s the sense in which Einsteinian space is curved — there are regions in which the area inside a circle (or the volume inside a sphere) is greater than or less than what would be expected from the size of its boundary.

Here’s an example.  The upper panel’s dotted grid represents a simple flat space being traversed by a “disk.”  See how the disk’s location has no effect on its size or shape.  As a result, dividing its circumference by its radius always gives you 2π.Curvature 3

In the bottom panel I’ve transformed* the picture to represent space in the neighborhood of a black hole (the gray circle is its Event Horizon) as seen from a distance.  Close-up, every row of dots would appear straight.  However, from afar the disk’s apparent size and shape depend on where it is relative to the BH.

By the way, the disk is NOT “falling” into the BH.  This is about the shape of space itself — there’s no gravitational attraction or distortion by tidal spaghettification.

Visually, the disk appears to ooze down one of those famous 3D parabolic funnels.  But it doesn’t — all of this activity takes place within the BH’s equatorial plane, a completely 2D place.  The equations generate that visual effect by distorting space and changing the local distance scale near our massive object.  This particular distortion generates positive curvature — at 90% through the video, the disk’s C/r ratio is about 2% less than 2π.

As I tell Museum visitors, “miles are shorter near a black hole.”

~~ Rich Olcott

* – If you’re interested, here are the technical details.  A Schwarzchild BH, distances as multiples of the EH radius.  The disk (diameter 2.0) is depicted at successive time-free points in the BH equatorial plane.  The calculation uses Flamm’s paraboloid to convert each grid point’s local (r,φ) coordinates to (w,φ) to represent the spatial configuration as seen from r>>w.

Gin And The Art of Quantum Mechanics

“Fancy a card game, Johnny?”
“Sure, Jennie, deal me in.  Wot’re we playin’?”
“Gin rummy sound good?”

Great idea, and it fits right in with our current Entanglement theme.  The aspect of Entanglement that so bothered Einstein, “spooky action at a distance,” can be just as spooky close-up.  Check out this magic example — go ahead, it’s a fun trick to figure out.

Spooky, hey?  And it all has to do with cards being two-dimensional.  I know, as objects they’ve got three dimensions same as anyone (four, if you count time), but functionally they have only two dimensions — rank and suit.gin rummy hand

When you’re looking at a gin rummy hand you need to consider each dimension separately.  The queens in this hand form a set — three cards of the same rank.  So do the three nines.  In the suit dimension, the 4-5-6-7 run is a sequence of ranks all in the same suit.Gin rummy chart

A physicist might say that evaluating a gin rummy hand is a separable problem, because you can consider each dimension on its own. <Hmm … three queens, that’s a set, and three nines, another set.  The rest are hearts.  Hey, the hearts are in sequence, woo-hoo!> 


If you chart the hand, the run and sets and their separated dimensions show up clearly even if you don’t know cards.

A standard strategy for working a complex physics problem is to look for a way to split one kind of motion out from what else is going on.  If the whole shebang is moving in the z-direction, you can address  the z-positions, z-velocities and z-forces as an isolated sub-problem and treat the x and y stuff separately.  Then, if everything is rotating in the xy plane you may be able to separate the angular motion from the in-and-out (radial) motion.

But sometimes things don’t break out so readily.  One nasty example would be several massive stars flying toward each other at odd angles as they all dive into a black hole.  Each of the stars is moving in the black hole’s weirdly twisted space, but it’s also tugged at by every other star.  An astrophysicist would call the problem non-separable and probably try simulating it in a computer instead of setting up a series of ugly calculus problems.Trick chart

The card trick video uses a little sleight-of-eye to fake a non-separable situation.  Here’s the chart, with green dots for the original set of cards and purple dots for the final hand after “I’ve removed the card you thought of.”  The kings are different, and so are the queens and jacks.  As you see, the reason the trick works is that the performer removed all the cards from the original hand.

The goal of the illusion is to confuse you by muddling ranks with suits.  What had been a king of diamonds in the first position became a king of spades, whereas the other king became a queen.  You were left with an entangled perception of each card’s two dimensions.

In quantum mechanics that kind of entanglement crops up any time you’ve got two particles with a common history.  It’s built into the math — the two particles evolve together and the model gives you no way to tell which is which.

Suppose for instance that an electron pair has zero net spin  (spin direction is a dimension in QM like suit is a dimension in cards).  If the electron going to the left is spinning clockwise, the other one must be spinning counterclockwise.  Or the clockwise one may be the one going to the right — we just can’t tell from the math which is which until we test one of them.  The single test settles the matter for both.

Einstein didn’t like that ambiguity.  His intuition told him that QM’s statistics only summarize deeper happenings.  Bohr opposed that idea, holding that QM tells us all we can know about a system and that it’s nonsense to even speak of properties that cannot be measured.  Einstein called the deeper phenomena “elements of reality” though they’re currently referred to as “hidden variables.”  Bohr won the battle but maybe not the war — Einstein had such good intuition.

~~ Rich Olcott

And now for some completely different dimensions

Terry Pratchett wrote that Knowledge = Power = Energy = Matter = Mass.  Physicists don’t agree because the units don’t match up.

Physicists check equations with a powerful technique called “Dimensional Analysis,” but it’s only theoretically related to the “travel in space and time” kinds of dimension we discussed earlier.

Place setting LMTIt all started with Newton’s mechanics, his study of how objects affect the motion of other objects.  His vocabulary list included words like force, momentum, velocity, acceleration, mass, …, all concepts that seem familiar to us but which Newton either originated or fundamentally re-defined. As time went on, other thinkers added more terms like power, energy and action.

They’re all linked mathematically by various equations, but also by three fundamental dimensions: length (L), time (T) and mass (M). (There are a few others, like electric charge and temperature, that apply to problems outside of mechanics proper.)

Velocity, for example.  (Strictly speaking, velocity is speed in a particular direction but here we’re just concerned with its magnitude.)   You can measure it in miles per hour or millimeters per second or parsecs per millennium — in each case it’s length per time.  Velocity’s dimension expression is L/T no matter what units you use.

Momentum is the product of mass and velocity.  A 6,000-lb Escalade SUV doing 60 miles an hour has twice the momentum of a 3,000-lb compact car traveling at the same speed.  (Insurance companies are well aware of that fact and charge accordingly.)  In terms of dimensions, momentum is M*(L/T) = ML/T.

Acceleration is how rapidly velocity changes — a car clocked at “zero to 60 in 6 seconds” accelerated an average of 10 miles per hour per second.  Time’s in the denominator twice (who cares what the units are?), so the dimensional expression for acceleration is L/T2.

Physicists and chemists and engineers pay attention to these dimensional expressions because they have to match up across an equal sign.  Everyone knows Einstein’s equation, E = mc2. The c is the velocity of light.  As a velocity its dimension expression is L/T.  Therefore, the expression for energy must be M*(L/T)2 = ML2/T2.  See how easy?

Now things get more interesting.  Newton’s original Second Law calculated force on an object by how rapidly its momentum changed: (ML/T)/T.  Later on (possibly influenced by his feud with Liebniz about who invented calculus), he changed that to mass times acceleration M*(L/T2).  Conceptually they’re different but dimensionally they’re identical — both expressions for force work out to ML/T2.

Something seductively similar seems to apply to Heisenberg’s Area.  As we’ve seen, it’s the product of uncertainties in position (L) and momentum (ML/T) so the Area’s dimension expression works out to L*(ML/T) = ML2/T.

SeductiveThere is another way to get the same dimension expression but things aren’t not as nice there as they look at first glance.  Action is given by the amount of energy expended in a given time interval, times the length of that interval.  If you take the product of energy and time the dimensions work out as (ML2/T2)*T = ML2/T, just like Heisenberg’s Area.

It’s so tempting to think that energy and time negotiate precision like position and momentum do.  But they don’t.  In quantum mechanics, time is a driver, not a result.  If you tell me when an event happens (the t-coordinate), I can maybe calculate its energy and such.  But if you tell me the energy, I can’t give you a time when it’ll happen.  The situation reminds me of geologists trying to predict an earthquake.  They’ve got lots of statistics on tremor size distribution and can even give you average time between tremors of a certain size, but when will the next one hit?  Lord only knows.

File the detailed reasoning under “Arcane” — in technicalese, there are operators for position, momentum and energy but there’s no operator for time.  If you’re curious, John Baez’s paper has all the details.  Be warned, it contains equations!

Trust me — if you’ve spent a couple of days going through a long derivation, totting up the dimensions on either side of equations along the way is a great technique for reassuring yourself that you probably didn’t do something stupid back at hour 14.  Or maybe to detect that you did.

~~ Rich Olcott

Dimensional Venturing, Part 6 – Tiny Dimensions

“The Universe is much larger than is generally supposed.”  

What a great opening line, eh?  Decades later I still recall reading that in a technical paper about then-recent adjustments in the way astronomical distances were measured.

The authors didn’t know the half of it.  They were thinking in only three dimensions.  That’s so last-century.

If you read science articles in the popular press you’ve probably run into statements like this one from Brian Green’s article “Hanging by a String” in the January 2015 Smithsonian:

String theory’s equations require that the universe has extra dimensions beyond the three of everyday experience – left/right, back/forth and up/down…. [T]heorists realized that there might be two kinds of spatial dimensions: those that are large and extended, which we directly experience, and others that are tiny and tightly wound, too small for even our most refined equipment to reveal.

Tightly wound dimensions?  What’s that about?  And what’s it got to do with strings?

The “large extended” dimensions are the kind we discussed in Part 1 of this series.  The essential point is that (in principle) once you or a light ray start moving in a particular direction you can keep going in that direction forever.

Seems obvious, how else could it be?

tiny dimension 1Well, suppose that we bend one of those three familiar “large” dimensions around in a circle, as in the drawing to the right. Our little guy could walk straight out of the page “forever” in the X direction. He could walk straight up the page “forever” in the Z direction. However, if he tries to walk along the Y track perpendicular to both of those two, in a while he’ll wind up right back where he started.

That’s an example of a “tightly wound” dimension.

Because it makes the math easier, physicists usually don’t calculate the absolute distance traveled around the circle.  Instead they write equations that depend on the angle from zero as the starting point. Notice that 360 degrees is exactly the same as zero — that’ll be important in a later post here.  Anyhow, there’s reason to believe that the effective circumference of a “tightly wound” dimension is really, really small.

OK, having a closed-off dimension is a little strange but it’s just not real-world, is it?

tiny dimension 2Actually, our real world is like that but moreso. Look at this drawing where we’ve got a pair of perpendicular wound-up dimensions. The little guy on the Y track can go from Denver down to Mazatlan in Mexico and proceed all the way around the world back up to Denver. On the X track he’s going from Denver westward to Chico CA and could continue across the Pacific and onward until he gets back to Denver The only way he can travel in one direction “forever” is to go along the Z track, straight upward, and that’s why NASA builds rocket ships.

Back to the strings. Depending on which variety of string theory you choose, the strings wriggle in a space of three Z-style “extended” dimensions, plus time, plus half-a-dozen or more wound-up or “compactified” (look it up) dimensions.  If string-theory strings can wriggle in all those directions, then how much room does each one have to move around in?  We’ve all learned the formulas for area of a rectangle and volume of a cube — [length times height] and [length times height times depth].  To extend the notion of “volume” to more dimensions you just keep multiplying.

Back to the size of the Universe. You may think that just with straight-line space it’s pretty good-sized.  With those stringy dimensions in play, for every single cube-shaped region you pick in straight-line space you need to multiply that volume by [half-a-dozen or more dimensions] times [many possible angles] to account for all the “space” in all the enhanced regions you could choose from when you include those wound-up dimensions. The total multi-dimensional volume is very, very huge.

The universe is indeed much larger than is generally supposed.

Next week — buttered cats.

~~ Rich Olcott

Dimensional Venturing Part 5 – You Ain’t From Around Here, Are You?

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.

Mxycus 2Mxy 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).

The M*A*S*H signpost, now at the Smithsonian. Photo by Steven Williamson., in

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

Dimensional Venturing Part 4 – To infini-D and beyond!

apple plumNow that you’ve read my previous posts and have the 4-D thing working well, you’re ready to go for a few more dimensions.  Consider the apple that struck Isaac Newton’s head.  The event occurred in 1665, in England at 52°55´N by 0°38´E, roughly three feet above ground level.  The apple, variety “Flower of Kent,” weighed about 8 ounces and was probably somewhat past fully ripened.  Got that picture in your head?  You’re doing great.

Now visualize the apple taking thirty seconds to move twenty feet diagonally upward, northward and eastward as it morphs to an underripe 4-ounce Damson plum.

The change you just imagined followed an eight-dimensional path: three dimensions of space, one of time, one of weight, one for degree of ripeness, and two category dimensions, species and variety.

Length in a given direction is only one kind of dimension, as Sir Isaac’s example demonstrates.  A mathematician would say that a dimension is a set of values that can be traversed independently of any other set of values. A dimension can be confined to a limited range (360 degrees in a circle) or be infinite like … well, “infinitely far away.”  A dimension might be continuous (think how loudness can vary smoothly from sleeping-baby hush to stadium ROAR and beyond) or be in discrete steps like the click-stops on a digital controller.  The physicists are arguing now whether, at the smallest of scales, space itself is continuous or discrete.

Photo by Becky Ziemer

Color vision’s a good example of dimensions in action.  For most of us, our eyes have three types of cone cells, respectively optimized for red, green and blue light.  We see a specific color as some mixture of the three and that’s how the screen you’re looking at now can fake 16 million colors using just three kinds of color-emitting elements (phosphor dots in old-style TVs, LEDs in most devices these days).

Where did that 16 million number come from?  The signal-processing math is seriously techie, but at the bottom the technology uses 256 intensity levels of red, 256 levels of green and 256 levels of blue — each is a discrete dimension with a limited range.  Together they define a 256x256x256-point cube.  Any point in that cube represents a unique mix of primary colors.  One of the colors in the little girl’s hat, for instance, is at the intersection of 249/256 red, 71/256 green, and 48/256 blue.  The arithmetic tells us there are 16,777,216 points (possible mixed colors) in that cube.

Well, actually, there’s one more dimension to color vision because our eyes also have rod cells that simply sense light or darkness.  Neither brown nor grey are in the spectrum that cones care about.  A good printer uses four separate inks to produce browns and greys as mixtures of three dimensions of red-green-blue plus one of black.

So color is 3-dimensional, mostly.  But that’s just the start of color vision because most of us have millions of cone cells in each eye.  A mathematician would say that any scene you look at has that number of dimensions, because the intensity registered by one cone can vary in its range independently of all the other cones.

Ain’t it wonderful that you’re perfectly OK with living in a multi-million-dimensional world?

Next week – a word from the other side

~~ Rich Olcott

Dimensional venturing, Part 3 – Klein’s thingy

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.

Cut Torus
Adapted from “Torus”. Licensed under Public Domain via Commons –

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.)

IssueThe 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 KKleinthat 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.

Reverse Klein 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