(w, x, y, z) in Four Dimensions

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(C)Copyright 2014, C. Burke.

I may make a poster of this even if my students get neither the math nor the TV reference. They need to learn about both.

As far as the math goes, there are two points on the co-ordinate plane to start. Let's call the first one Point A, with coordinates (-5, 2). The other point we'll call Point B, was at (2, 4) -- so it wasn't a rotational image of Point A. Point B has up and moved itself, which points don't usually do -- but then, they don't sing TV theme ditties, either. The image of Point B is at a translation of up 3 and 5 to the right, or T_{5, 3}, which puts it at (7, 7).

In the second image, B' moves from Quadrant I to Quadrant IV. It's not a reflection over the x-axis (although it could be represented as a glide reflection), so we'll call it another translation. This translation appears to be 3 to the left and 12 down, which we represent with negative numbers, T_{-3, -12}. This puts B'' at (4, -5).

Things get interesting with the third illustration as the point moves two more times, this time moving in three dimensions. We don't have any scale for the z-axis, so let's label the original plane as z = 0 and then increment by 1 as we move up. Using that frame of reference, B'' is located at (4, -5, 0). It then gets translated to B'''(-6, -6, 1), which is a transformation of T_{-10, -1, 1}, and finally to B''''(2, 5, 2), a transformation of T_{8, 11, 1}.

Keeping in mind that this panel shows a Composition of Transformations, we denote it in reverse order, the translation of the translation, using the form

T_{8, 11, 1} o T_{-10, -1, 1}

The final frame shows a tesseract. I don't get to use those very often, but they are fun to draw (and to say). A tesseract is a way of representing hypothetical four-dimensional spatial coordinates. It might be better represented with a three-dimensional model, in the same way that three dimensions can be modeled in a two-dimensional image, but since I don't have an actual three-dimensional comic, this is the best I can do.

We use the last three letters of the alphabet for the usual three dimensions, (x, y, z), so we're out of letters. How do we represent the fourth spatial dimension? We can use w for it, but it seems awkward using (x, y, z, w) (and maybe people who work with higher mathematics than I do actually do this -- I don't know), but it would seem easier and more logical to use (w, x, y, z), putting it first like move up an office tower increases the first digit of the room number (and different corridors might represent increments of the second digit). Because we can't tell where the point is coming from nor do we have any scale for the new position, we'll skip writing a transformation statement for it. (Or I could leave it as an "obvious" "exercise for the reader".)

One final note and I'll end my seemingly endless mind-dump of this stream of thought: I keep referring to four spatial dimensions. This is to distinguish from the usual four dimensions we live in. Originally, the comic read "(w, x, y, z) in 4D space-time", but something about that bothered me. First, I don't know that tesseracts would be used to represent a time dimension mixed with three spatial ones, although I don't see a reason why they couldn't be. Moreover, the entire point of the comic was to use "(w, x, y, z)", which wouldn't make sense if we were using time, which would more logically be represented as t, which would give us (x, y, z, t), space plus time gives us space-time.

And that doesn't sound like a radio station. Certainly not one Dr. Johnny Fever would work at.

Keeping Time With Music

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(C)Copyright 2014, C. Burke.

Doesn't anything before 11 or after 8 in the morning?

She cried, "Moe! Moe! Moe!" Nyuk, nyuk! Wiseguy!

(x, why?) Mini: Fun With Tetrominoes

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(C)Copyright 2014, C. Burke.

Decline to recline? The Tetromino's "no"'s.

For more whimsical pondering on these fab four-square puzzle pieces, check out my recent column How Many Colored Tetrominoes?.

(x, why?) Mini: On a Plane!

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(C)Copyright 2014, C. Burke.

Or maybe get Dierks on a plane? Don't try this at home, especially if the plane isn't level.

Shakespearean Inequalities

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(C)Copyright 2014, C. Burke.

I'd estimate it as Horatio of 2 to 1.

Remembering 9-11 and a Long Walk Home

I've pretty much used up the math metaphors for the Twin Towers years ago, so I don't have another comic commemorating the event. Maybe next year. On the other hand, I do have something that I wrote a few years ago in another forum. People from around the country participated, and I was one of the "locals" to the Event there. My story is just one of the millions that could be told that day, and it's hardly one of the more frightening, nor one of bravery and heroism. It's just a first-hand report.

My moderator of the forum asked me the following year if I would mind reposting it. Of course not, and I did. In the years since, she's had my permission to post it if I'm not on by a certain time to do it myself.

My report is here. Just one from the crowd. If anyone wishes to add their stories, please use the comment section. Thank you.

---

As requested (and as promised):

This was originally posted (I think) in 2005. It's been an annual repost since then.

At about 8:40am on that morning, I was walking into the Court building on Adams Street (actually, the Court St. entrance) in Downtown Brooklyn to start serving jury duty. As luck would have it, I had been halfway to the train station before I’d realized that I’d left my Walkman on the kitchen table, so I didn’t bother going back for it. I figured I’d just get a newspaper. Besides, I didn’t know how well I’d be able to pick up AM inside the building anyway.

I sat in a dark room watching a video on How to Be a Good Juror, oblivious to what was going on right across the river. We were told to relax in the room that they have, and I looked out the window at the Marriot Hotel. Traffic on Adams St was snarled, not moving. Must’ve been an accident on the Brooklyn Bridge, I thought. (It was a block away.)

People were standing around outside the hotel. Must be waiting for a tour bus or something. What did I know.

Fire trucks and ambulances started flying by on the wrong side of Adams Street, which had no traffic. Okay, traffic doesn’t come into Brooklyn much in the morning, but something was odd here. I had been facing 180 degrees from where I needed to be looking.

Finally, they had made an announcement. America was at war, under attack. The World Trade Center and the Pentagon had been destroyed. They were trying to get coverage on the TV sets in the jury rooms. I don’t think they succeeded. Even if they wanted to, only CBS would be available because it still broadcast from the Empire State Building.

People were beside themselves, many broke down, everyone was rushing for the payphones. I met a woman who had been listening to her radio. She let me share her earbuds. She was shaken and needed a cigarette. I don’t smoke, but I walked her to the smoking room. (There was one on the floor. Quite a few people were there.)

We were dismissed. Not much was going to get done in the Court building that week. Groups of people huddled outside with questions, comments, gossip and hearsay. Some of the lawyers said they saw it happen. What kind of plane was it?

Does anyone know if the trains are running? What about the buses? No trains. No LIRR. A few buses and they’re all packed. It was time to start walking and no one wanted to walk alone. We walked in groups.

Dust was falling from the sky in downtown Brooklyn like a dirty snow that was covering the cars. Papers fell too. We started walking up Atlantic Avenue. People were wandering around with their cell phones out trying to get a signal. no luck.

We took a turn down Third Avenue. I needed to. I wanted to stop at my mother’s house. It was a good resting point for me. The group I’d tagged along with decided to join me. One guy stopped in a hardware store for masks and passed them out.

When we passed Third St and reached the Gowanus Canal, we had our first real look. It was like a scene out of a bad movie. The skyline was there. But the Towers were missing. Just a terrible column of smoke and a cloud drifting our way.

We didn’t stay long. We kept walking. I made it to my mother’s house and said good-bye to the others. Some were walking all the way to Staten Island. One who had joined our group had walked over the Brooklyn Bridge — after having walked down 50 floors of Tower 1. God was looking out for him.

I watched some of the coverage until the trains were running again. I took one that left me about a mile or so from my inlaws, the meeting place for the rest of the family. I stopped in at St. Athanasius on the way. I hadn’t been there since a wedding about 15-20 years earlier. I stayed for a little while and walked the rest of the way.

Thankfully, my wife, who worked at the foot of the Brooklyn Bridge on the Manhattan side had evacuated immediately before the trains had stopped running.

It’s not a walk that I’ll ever forget.

---

Mom isn't there anymore and my "resting place" won't be around much longer, either, sadly. Just more things to remember about that day.

(x, why?) Mini: HOTS!

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(C)Copyright 2014, C. Burke.

Should've thought that out more ... at a higher order.

How Many Colored Tetrominoes?

Question: How many different colored tetrominoes are there if we allow only four colors total?

Second question: What the heck is a tetromino?

Dominoes are a great game with rectangle tiles, composed of two adjacent squares with certain numbers of pips on them. A tetromino is a group of four adjacent squares, each sharing at least one side with at least one other square. In other words, those little falling shapes made popular in the game Tetris, and all of its knock-off variations, as seen below:

There are five basic arrangements, if you allow for reflection. (That is, if you allow for picking a piece up and flipping it over.) If you only allow for rotation, then there are seven shapes, each of which can be designated a letter of the alphabet to describe it.

In most games, the shapes are different colors because a) it's a great visual, and b) it's a clue to the player that, say, a "J" is falling not an "L". Ditto for the "S" and "Z" pieces.

As with any successful game, there have been many imitations and variations. Even games that are somewhat unrelated produce their own variations, which are suddenly similar to Tetris. I've seen a few of these where the pieces, for a multitude of reasons, are multicolored instead of monocolored, as shown below:

This lead me to thinking about the number of possible colored blocks that could fall in the game of varying shapes and color schemes. My only arbitrary limit was that each block had to contain each of the same four colors. (Naturally, I picked red, yellow, green, and blue, pretty much by default.)

First instinct is to use the Counting Principle: the number of shapes times the number of four-color arrangements. That would give us (7) X (4!), or 7 X 24 = 168.

Unfortunately, first instincts may put you on the right track, but they sometimes leave things out. In this case, we can't forget about rotational symmetry.

The I piece (the line) has rotational symmetry of Order 2. If you rotate it 180 degrees, it looks the same. That makes it the same twice in one 360-degrees rotation. The S and Z pieces also have Order 2 symmetry. The O piece (the square) has rotational symmetry of Order 4. If you rotate it 90 degrees, it looks the same. That makes it the same four times in one 360-degrees rotation.

So we have 3 shapes with 4! color variations, 3 with 4!/2 variations and 1 with 4!/4 variations: 3 X 24 + 3 X 12 + 1 X 6 = 72 + 36 + 6 = 114 possible tetrominoes.

For some reason, I feel better knowing this, and maybe it won't distract me so much next time I play one of these games on the train where I have no wifi. (A Tetris Offensive, perhaps?)

(As always, you're free to correct my math. In the event of an actual mistake, I'll edit my work and pretend I have no idea what you're talking about.)

Closed!

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(C)Copyright 2014, C. Burke.

Everyone goes to Six. But he still have to follow the Four.

This would be a Math Movie Quote, but it doesn't have a number. Previous quotes came from the American Film Institute Top 100 Quotes list, but Casablanca has so many good lines that they only picked a few to include.

By the way, Casablanca, like many other wonderful movies, is celebrating its 75th anniversary, but I guess I missed it on TV. If you have the chance, see it in a movie theater. Seriously, the first time I saw the film was 25 years ago, on its 50th anniversary re-release. My girlfriend (now my wife) found out two years earlier that I'd never seen the movie and she banned me from watching it on TV (where it would've been chopped up for commercials) because she knew it would be re-released.

That same year, I saw Gone With the Wind for the first time as well -- at Radio City Music Hall, where they showed the restored edition. The movies saw a Golden Year in 1939, which gave us The Wizard of Oz and Mr. Smith Goes to Washington among many others.

Roman LOGIC

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(C)Copyright 2014, C. Burke.

LOG 99 = 1996/1000.

Quick bit of research revealed that while the Romans had neither fractions (as we know them) nor decimal points (nor place values, for that matter), they did have expressions for certain parts of whole. In particular, they had words for 1/12, 1/24, 1/48, 1/72, 1/144, and 1/288. The names for the latter three are very similar according to one source, unfortunately, searching on the word for 1/72 (scriptulum), I was informed by several other sites that that word actually meant 1/288.

Whichever word it was, the Romans had a system for "one off", by putting a prefix like "de" in the front, it would mean one 1/72 off the whole, which would mean 71/72. In this case, whichever that word was, one plus that fraction would be the closest approximation to log₁₀ 99.

Well, it would be if they'd had logarithms.

Sure! If they actually had them, I'd love for you to post a link in the comment section! Thank you for asking!

Mini: Favorite Part of School

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(C)Copyright 2014, C. Burke.

BSOQ: Big Square On Quad.

Cybermen

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(C)Copyright 2014, C. Burke.

I'm waiting for the NFL to be Upgraded.

More good-natured fun can be found in yesterday's column Real-World Math: Bushels of Fun.