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.

Minature Football Field

A little over a year ago I was at the Football Hall of Fame in Canton, Ohio. In a somewhat related note, Canton is also the home and burial place of President William McKinley. The inside of the hall was awesome. But outside had a miniature artificial turf football field!! I think it was 40 yards long. For the most part, my use of my lawn consists of watering it and mowing it. I thought how awesome it would be to turn it into something like that. I figured that could be kind of expensive. I looked online and there was and ad for 10x10 feet of artificial turf for $95. Maybe its doable after all. At the very least, it makes for a cool math problem.

Suppose I want a field of 30 yards. If you are just messing around with folks you don't want to be running 100 yards to score. I don't know if my yard is 30 yards long, but let's say it is. Now, how wide should it be? The real deal is 53 1/3 yards. So mine should be the solution to

100 : 53 1/3 = 30 : x

It actually works out cleaner solving it with fractions. Anyway, x = 16 yards.

For the area, I have 30x16 = 480 square yards. However, I need this in square feet. There are 9 square feet in a yard, so 480x9 = 4,320 square feet. This could get a little expensive.

The cost, y would be found with 100 : 95 = 4,320 : y

So, y = $4,104. Also, I would have to somehow get yard markers, etc painted it. Still, it might be worth it.

Below is a picture of the field in Canton. (Hard to get a good picture from ground level.)

Ryan Braun and Steroids

Ryan Braun is an outfielder for the Milwaukee Brewers. He won the MVP award in 2011. At the end of that season he was accused by Major League Baseball of taking steroids. However, he got off on a technicality. He was accused again in 2013. This one stuck and he was suspended for the rest of the season. That time he admitted it and took his punishment. He has played almost two full seasons since his suspension.

What is interesting in Braun's case is that he is still in the prime of his career. He is 31 years old. Many that have been accused of taking steroids were near the end of their career. If they did come back from a suspension, a decrease in their statistics could be because they are no longer using steroids or just because of father time. A decrease in Braun's statistics would seemingly be due only to him now playing clean.

To compare his statistics pre and post-suspension would be an interesting exercise. It wouldn't be helpful to look at the totals as he played almost seven seasons before the suspension and only two seasons after. However, you could translate those time periods into single 600 at-bat seasons. That is what I did. You can do so by looking at the grand totals and setting up proportions. Also helpful in this exercise is knowing that the definition of batting average is the number of hits divided by the at-bats.

Algebra students would have plenty of opportunity here to review proportions. Let me just skip the messy stuff and go right to the final stats.

Pre-Suspension Statistics:
At-bats 600, Runs 104, Hits 187, Doubles 38, Home Runs 34, RBIs 110, Batting Average .312

Post-Suspension Statistics:
At-bats 600, Runs 90, Hits 166, Doubles 33, Home Runs 26, RBIs 96, Batting Average .277

You could then ask students what they make of these statistics. Some, perhaps with some leading by you, might suggest looking at the percentage decrease. This turns out to be quite interesting. You can easily make the claim that a player is 86 to 87% as effective without using steroids. At least that seems to be the case with Braun in pretty much every area. I've compare post to pre-suspension statistics and changed them into percentages. Check this out:

Runs 87%, Hits 89%, Doubles 87%, Home Runs 76%, RBIs 87%, Batting Average 89%. Its kind of surprising these numbers are all in the same ballpark, so to speak.

Anyway this might make a good review of proportions, takes a topic they've all heard about, and gets students to do some statistical analysis.

Smartest Presidents

I saw an article online which listed the IQ's of each of our presidents. By their own admission, they were doing a bit of guesswork. Since IQ tests weren't developed until about the time of our 26th president, Teddy Roosevelt, there isn't a lot of hard data that can be used. I would think we would take these numbers with a grain of salt. I have some disagreements with a few of these placements and you probably do, too. In spite of that, here we go:

The Top 5 and their estimated IQ scores:

1. John Quincy Adams - 168.8
2. Thomas Jefferson - 153.8
3. John Kennedy - 150.65
4. Bill Clinton - 148.8
5. Woodrow Wilson - 145.1

And the bottom 5:

Andrew Johnson - 125.7
George W. Bush - 124.9
Warren G. Harding - 124.3
James Monroe - 124.1
Ulysses S. Grant - 120.0

We might well ask just how smart these guys really are. Since the mean IQ score is taken to be 100, just like Lake Wobegon, they are all above average. So, President Grant was above average. But was he just a little above or way above?

We could get an idea from looking at how many standard deviations away from the mean he is. Taking the standard deviation for IQ scores to be 16, we see that Grant is 20/16 = 1.25 standard deviations above the mean. Consulting a Z-score table, that puts him in the 89.4 percentile. Pretty good. Even if Ulysses wasn't the sharpest guy ever, it must take a certain amount of intelligence to get elected president twice and to win a war.

What about John Quincy? J.Q. is literally off the chart. Let's go with the runner-up Thomas Jefferson. He is 53.8/16 = 3.36 standard deviations from the mean. This puts him in the upper 99.96 percentile. He's smart. Not John Quincy Adams smart, but smart.

The complete list can be found at http://us-presidents.insidegov.com/stories