Torus Formulas

I've had a week to reflect on the torus. Most of my reflections have been in the past few minutes leading up to my current act of typing, but I like to think my subconscious has been mulling it over.

Remember that last week we learned that the plural of torus is tori. True, even though my spell checker has put a squiggly red line under it.

I did think of another application. Remember a torus is a donut shape. Inner tubes of tires are also tori. So the square inch amount of rubber in the tire can be found with the surface area formula and the air in the tire would be found with the volume formula.

Also, there are apparently applications I would never have come up with.

In our model of cosmometry, the torus is the fundamental form of balanced energy flow found in sustainable systems at all scales. It is the primary component that enables a seamless fractal embedding of energy flow from micro-atomic to macro-galactic wherein each individual entity has its unique identity while also being connected with all else. 

This is from the website http://cosmometry.net/the-torus---dynamic-flow-process. I do not know what this all means. I'm not even sure what all the individual words mean, but it certainly sounds very important.

Last time I listed the formulas for the torus. I found there are other ways to find volume and surface area. Instead of using the variables in the way we used last week, these formulas use r, the distance from the center of the torus to the inner edge and R the distance from the center to the outer edge. 

V =  1/4(pi)^2(r+R)(R-r)^2

S.A. = (pi)^2(R^2-r^2)

(Again, I apologize for my inability to write exponents any other way.)

Beside finding volumes and surface areas of donuts, inner tubes, and various balanced energy flows (?) there is another nice application here. To find the ratio of surface area to volume of any three-dimensional shape is an important concept. To do so with the above formulas is especially cool as it simplifies down a lot.

Donut / Torus

I thought donuts would be an interesting topic. They are actually the mathematical shape called a torus. I first heard of this sitting in an undergraduate math class. Our professor told us we might try to find out about the torus before the next class. I actually looked it up. I was the only one in the class to do it and was able to talk about it the next time we met. I'm sure I got labeled as a nerd at that point. I wouldn't mind that, but when a room full of mathematicians think you're a nerd, that is probably an especially bad sign.

It turns out there is a lot I didn't know about this topic. Like the plural of torus. (It's tori, with a long i sound.) Is it donut or doughnut? (The consensus by those that decide these things seems to be "doughnut" although they seem to put up with "donut". "Donut really didn't come into regular use until Dunkin' Donuts started up in the 1950's.) I thought maybe this had a tie-in to the car, but no. It is spelled "Taurus" and I'm guessing has to do with the zodiac sign.

Imagine two circles linked as a chain. If one makes a full lap following the path of that first circle, we have a torus. Let's say the moving circle is radius r and the stationary circle has radius R.

The surface area is S = 4(pi^2)Rr. (Sorry, I do not know how to make my blog write the pi symbol or how to do exponents.) The derivation of this formula is more easily seen if written S = (2(pi)r)(2(pi)R). It is the circumference of the moving circle taking the a path along the circumference of the big circle.

The volume is V = 2(pi^2)(r^2)R. While this is the simplified version, again it is easier to see where it comes from by writing it differently: V = ((pi)(r^2))(2(pi)r). It is the area of the moving circle again taking the a path along the circumference of the big circle.

What can we use these formulas for? Not important, but there are a lot of them - donuts. Important, but none actually exist - the space station shown in the movie 2001. In the picture notice that it seems more of a rectangle than a circle on the outer edge. I think that is still a torus. The torus definition from different sources I found say "a closed curve", "a closed curve, especially a circle", or simply "a circle".

Enough for now. We'll look into this topic more next time.

The Sophomore Jinx

The sophomore jinx, or sophomore slump, takes place when the second round is not as good as the first. The second album, the second season, doesn't seem to be quite as good. They got everyone's hopes up after a great debut. What's with that?

Does it really happen, anyway? Maybe its just hit or miss. The Grammy Award for Best New Artist in 1964 was a group called the Beatles. Good call. But the year before, the Best New Artist was Ward Swingle. First let's look at the case for there being such a thing as a sophomore jinx. With a quick look at the internet one can find plenty of examples that seem to support this idea.

Album sales by some pretty well known names:

Terence Trent D'arby - Album Number One - 12 million, Album Number Two - 2 million.
Spin Doctors - Album Number One - 5 million, Album Number Two - 1 million.
Christopher Cross -  Album Number One - 5 million, Album Number Two - 500,000 thousand.
Hootie and the Blowfish - Album Number One - 16 million, Album Number Two - 3 million.

You get the idea. Aaron Gleeman in an article titled The Sophomore Slump looked at all of the Rookie of the Year award winners, comparing their first and second seasons by using a baseball statistic called win shares. He found that 73 of the winners got worse in season two, while only 37 improved. Four stayed the same.

Rick Sutcliffe was the National League Rookie of the Year in 1979. Overall, he had a fine career, winning 179 games. His first year he won 17 games and lost 10. He gave up about three and a half runs a game. Next year he won 3 and lost 9 and gave up about five and a half runs a game.

This so-called sophomore jinx, can be explained at least in part statistically with the concept of the regression to the mean. The dictionary says, "In statistics, regression toward (or to) the mean is the phenomenon that if a variable is extreme on its first measurement, it will tend to be closer to the average on its second measurement.

If we flip a coin 100 times and get 63 heads, would we do better next time? Yes, maybe. But probably not. But if we got 40 heads on a first try, chances are, next time we'll see an increase. In either case we go back toward or regress toward the mean.

The examples we have seen have something in common. All of these first years were very good. They got our attention. People are wondering what they'll do for a follow up. All of these burst upon the scene with a great debut. Perhaps they were far above what their usual production would be. That can happen, but chances are in any given effort, we will do what our historical average would suggest.

Are there cases where the sophomore slump doesn't happen? Consider the baseball player that has a first year that is a bit below what he is capable of. Likely, he will improve the next year. The public really didn't notice his first year because it was nothing spectacular. We we're all paying attention to the Rookie of the Year winners. 

Batting Average

Algebra I students start out solving one-step equations. A good application is finding a baseball player's batting average. The batting average is the ratio of hits to official at-bats. An at-bat that is not "official" would refer to getting on base by means other than your batting ability. Being walked or being hit by a pitch is not counted. Batting average is expressed as a decimal rounded to thousandths place. A person getting one hit in four at-bats is hitting 0.250, pronounced "two fifty". A person going two for three is batting 0.667, pronounced "six sixty-seven". Do not get mathematically correct and pronounce this "six hundred sixty-seven thousandths". Expect blank stares or ridicule if you do. Incidentally, when baseball people are talking about Ted Williams being the last person to hit four hundred, they mean the last person to have his hit to at-bat ratio being greater than or equal to 0.400.

Since baseball's regular season just ended a couple days ago, let's take some batting averages from the last season.

Miguel Cabrerra won the American League batting title by having 145 hits in 429 at-bats. Students could use the formula BA = H/A to get his batting average (.338). Or, given his batting average was .338 and he was at bat 429 times, how many hits did he get? Or, Cabrerra was had a batting average of .338 with 145 hits. How many times was he up to bat? This also leads to an opportunity to talk about round off error as the last question could be answered by saying he was up to bat 428.99 times.

The National league batting title was won by Dee Gordon, having a .333 average by having 205 hits in 615 at-bats. He edged out Bryce Harper who was 172 for 521 for a .330 average. Actually, going into the final day of the season, they were tied. Each had a .331 batting average. OK, not exactly tied. Harper had an average of .33075 and Gordon was at .33061. On the final day of the season, Gordon was 3 for 4 and Harper was 1 for 4, giving Gordon the title.

Leaving out some information and being creative, a person could probably come up with a number of algebra problems from that scenario.