Fractal Video

I saw an interesting video on fractals. It was produced by NOVA called Fractals: Hunting the Hidden Dimension. Being for the general public, it, of course, didn't get too hard core with the mathematics, but it didn't completely back away from it each.

Because of that, a few of the applications left one with some questions. Such a case was when someone in the video said, "Fractals are important in code breaking." Then they leave it there because to try to explain it would cause most people's heads to explode. In a lot of cases, to have to skip over the math is somewhat unsatisfying, but probably pretty much unavoidable.

Interesting to me was how Benoit Mandlebrot first got involved with applications of fractals. As computers were just starting to communicate, there were problems. Computer data was being sent over telephone lines. However, it often wasn't getting through as intended.

Benoit B. Mandelbrot, then an employee of IBM, noticed a certain pattern of interference over, say, a ten minute span. He then noticed that same pattern would appear if he looked at maybe a five minute span, then a two and a half minute span, etc.

The video called it, "self-similarity". The fact that this self-similarity was taking place, told him this situation could be modeled with fractals.

It is a good video. It is from 2011, so not too out of date. Students no doubt will be chagrined at how excited the math nerds in the video get over these fractals. Even with that - a good video.

Statis Pro Baseball One Last Time

I know my blog has been a little heavy with the baseball applications. Specifically with regards to the best game ever made - Statis Pro Baseball. One more week, then I'll move on. This and other older games are great, though, for math applications because its right there in front of you. All the computer games have the statistics / mathematics hidden away in the computer program running it.

This is application is actually from a different game that I played once with a friend of mine. At the time I thought it was kind of ingenious, although I'm not sure I put a lot of thought into how they did it. Each baseball player had a card with spinner which represented statistically what you could expect from him in an at-bat.

If the first batter up was Ty Cobb, I would take his card, spin the spinner and see what he did. It might land on a colored section of card marked "Out". How did they come up with the colors on the cards anyway? Let's make Ty's situation real simple and divide it into sectors for hits and outs. For his career he 4,189 hits in 11,434 times at bat. That makes a batting average of 0.366. This means of course that he gets a hit 36.6 percent of the time which would be a sector of 36.6% of 360 degrees. This is a sector of 131.76 degrees. His chance of going out would be a different colored sector of 360 - 131.76 = 228.24 degrees.

There were more divisions than just hits and outs. Although it has been a while, I'm sure there were singles, doubles, triples, home runs, outs, and walks at least. To build the circle would mean finding percentages, changing them to degrees of a circle, dividing up the circle, and coloring and labeling the sectors. (The picture is not what the spinner looked like, obviously, but that's the idea.)

Whoever came up with it, I thought it was a pretty good game. It also didn't last, but that's the way it goes, I guess.

So, this isn't a high level math application, obviously, but I think an interesting one, and it reviews, protractor use, percentages, and circles.

Computer Baseball

Last week I wrote about Statis Pro Baseball - a game I played growing up. Drawing cards numbered from 11 to 88 determined how a player did in a particular at-bat. A computerized version of this can be used making use of the same randomness as drawing from the deck of shuffled cards.

Let's take one random baseball player. How about Babe Ruth? I was recently in his boyhood home / museum in Baltimore. It is really cool. Anyway, let's take his 1927 season.

 That year, Ruth in 540 at-bats had 95 singles, 29 doubles, 8 triples, and 60 home runs. The percent of each of these types of hits out of 540 at-bats is as follows:

Singles:         17.6%
Doubles:         5.4%
Triples:            1.5%
Home Runs:   11.1%

Now, we could use a random number generator, available on many calculators or on-line and simulate any number of at-bats. Adding together the above percentages and change them to numbers from zero to a thousand, we get the following:

Singles from 0 to 176
Doubles from 177 to 230
Triples from 231 to 245
Home Runs 246 to 356
Everything 357 and above will be an out

Running the random generator for twenty numbers and giving the results:

791 - Out
365 - Out
258 - Home Run
320 - Home Run
494 - Out
75   - Single
929 - Out
842 - Out
461 - Out
536 - Out
210 - Double
388 - Out
836 - Out
914 - Out
214 - Double
812 - Out
509 - Out
978 - Out
955 - Out

20 At bats, 1 Single, 2 Doubles, 0 Triples, 2 Home Runs

Actually, a little sub-par for the Babe. I'll spare you all the numbers, but I did it two more times just for fun, and here are the results.

20 At bats, 0 Singles, 3 Doubles, 0 Triples, 4 Home Runs

20 At bats, 3 Singles, 3 Doubles, 0 Triples, 2 Home Runs

I think this is a pretty good Algebra I application covering ratios and to get them thinking about the law of large numbers

Statis Pro Baseball

As a kid/adult I had what was called a Statis Pro Baseball game. It came in a box and you could recreate games with actual player statistics. It has since gone out of business, but some addicts still make the player cards and put them on-line. I'm reminded of this game because I recently saw sets for the 55 World Series (Yankees vs. Mets) and the 1919 World Series (Reds vs. White Sox (a.k.a. the Black Sox)) You just take the players' statistics and convert them to numbers the game uses. It used the numbers 11 to 88. I'm not certain why those numbers specifically, but that is what they did.

This is a fun (for fans of this kind of thing) math application involving ratios. So for example, say you have a batter, Johnny Baseball, that was up to bat 428 times and he had 101 singles. Of the 78 numbers used in the game (11 to 88), how many would be used to represent the singles?

The proportion 101/428 = x/78 gives a value of x = 18.4. So we round that off to 18 and for the Statis Pro game that is represented by the first 18 numbers - 11 to 28.

Suppose Johnny had 13 doubles. So, use the proportion 18/428 = x/78. This gives x = 3.3. So another three units is represented by the next three numbers, which would be 29, 30, and 31. Continuing in that manner, you could keep going and compute the numbers for triples, home runs, strike outs, walks, being hit by a pitch, or making an out. All that info would be on his player card.

You could divide things differently, of course. The "hit by pitch" category could be combined with walks. The game actually divides singles by singles to left, center, or right field. Depending on where the ball is hit helps to determine how far base runners can advance. (If you don't get that, don't worry about it.)

So how were those numbers used anyway? Besides the player cards there was another groups of cards marked randomly 11 to 88. Suppose Johnny Baseball is up to bat and his number drawn is a 17. Since 17 is between 11 and 28, he hit a single. If he was up and drew a 31, he hits a double.

It gets a little more complicated than than, but not too bad. You have to figure in not just the batter's card, but the pitcher his is facing has a card that describe how well he pitches. That has to be taken into account as well. Despite all this complexity, a 9 inning game can be played in an hour. So individual at-bats take well under a minute. That is much better than the eternity they take in real games.

So, I've probably done enough reminiscing right now, but I think the idea behind this is pretty valuable. Computer games today depend on this kind of randomness. We'll check that out next time.

Space Equation

I try to focus on high school math applications in this blog. Therefore, this picture probably doesn't quite fit. However, it is such a cool picture that I wanted to put it in. I saw it, or at least a portion of it in the September, 2016 issue of Reader's Digest. I went looking for the picture and found an expanded view of it on-line at http://rarehistoricalphotos.com/nasa-scientists-board-calculations-1961/. The Reader's Digest piece said that it was taken on October 10, 1957. These are equations related to satellite orbits. The picture was taken six days after the launch of Sputnik, putting the USSR up 1-0 in the space race. That seemed to get things going in the United States. NASA was created the next month and two months later, the U.S. had launched its own satellite.

Initially, I thought the photographer did some kind of time-lapse photography and these were all the same guy. Although, one person did the writing - astronomer Samuel Herrick - these are all different scientists. Its been my observation that everyone from the 1950s looked more or less the same. I think that is the reason for my confusion.

At the above website, I got some more information about the photo. The point was made that there are no calculations here - just equations that they might use. That makes sense being at the start of the space race and smack dab in the middle of the cold war. So no top level information was being given away in this photo.

Usually I think its a poor idea to present applications that are over the heads of students, but I think an exception could be made here. There is virtually no calculus here and concepts in trigonometry, "e", etc would be recognizable to many high school students.

The article ended with this:  For a complex equation that deals with time-steps and feeds back on itself, the prominent scientists of NASA would have “math parties”!!! [exclamation points, mine]. Everyone would master one part of the equation. Then the first guy would do his part and hand it off to the next guy and so on. Eventually the final guy would go back to the first person and give him the new inputs for 1ms [microsecond?] further in time. After a few hours you could have a nice neat graph of everything over a 1-2 second period. That is how the first nuclear reactors, nuclear bombs and a lot of aerospace calculations were done.