Introduction: Tokyo Games
The Olympic Games are now in full swing. But as I mentioned in my last post, the pandemic continues to cast its shadow over the competitions in Tokyo. And even before the coronavirus, some people have complained about the increasing cost and hassle of the Games, to the point that they ask, should the Olympics be abolished.
Notice that many sports already have their own international tournaments -- and in many cases, these tournaments are more prestigious than their Olympic counterparts. The most well-known example, of course, is soccer. More fans pay attention to the World Cup than the Olympic tournament -- and compared to even continental championships like the recently completed Euro Cup and the ongoing CONCACAF Gold Cup, the Olympics are an afterthought. So if the Olympics were abolished, there are still major international tournaments for soccer teams.
As a former high school track athlete, I often pay special attention to Track and Field. There are World Championships in Track and Field, ordinarily held in odd-numbered years so that they don't clash with the Olympics. (This time, worlds will be in 2022 due to the Olympics this year.) But unlike with soccer, most fans don't pay attention to Track and Field until the Olympics -- we watch Usain Bolt in Beijing, London, and Rio, but not at worlds. So if the Games were abolished, there would still be worlds in Track and Field, but hardly anyone would watch them.
This year, I've turned on the Olympics from time to time. The sports that have aired during my viewing sessions include beach volleyball, swimming, skateboarding, synchronized diving, mens basketball, and mens triathlon. Swimming is in the same boat as Track and Field -- there are World Championships in odd-numbered years, but how many people watched Michael Phelps or Katie Ledecky during those odd-numbered years? So abolishing the Olympics would relegate swimmers to relative obscurity.
As for basketball, this sport ought to be more like soccer, since there's also a basketball World Cup held every four years. But the 2019 World Cup was mostly ignored by Americans, whose team lost unceremoniously to France in the quarterfinals. So international basketball -- just like track and swimming -- is also ignored until the Olympics. Then again, basketball -- like soccer -- has domestic leagues (like the NBA) that fans do enjoy outside of international competition. Basketball players don't need the Games to be famous.
But consider an athlete like Simone Biles. During the runup to the Rio Games, some commentators were already proclaiming her to be the greatest gymnast of all time -- the GOAT -- but others were wondering, how can this "overhyped" gymnast be the GOAT when she hadn't done anything yet? Of course, she had dominated the World Artistic Gymnastics Championships the three previous years, but her sport is just like track and swimming -- only the Olympics matter.
The prime of a gymnast's career is often considered to last eight years -- from her sixteenth birthday to her twenty-fourth. Indeed, the minimum age to compete as a gymnast in the Games is 16, and while gymnasts often do participate past their mid-20's (including a 46-year-old Uzbek who was at Tokyo this year), they usually aren't competitive.
Since a gymnast's prime lasts eight years, we expect there to be two Olympics during that time. But for Biles, who was born on Pi Day 1997, there was only edition of the Olympics during her prime. She was too young for the London Games, and the Tokyo Games are slightly beyond her prime years.
Had Biles been born a mere three months earlier (as calendar years count rather than birthdays), she likely would have participated in the London Games, and possibly would have dominated. Then she wouldn't have been disparaged as a "nobody" leading up to Rio. Then London and Rio would have been her two Olympiads -- dominating there would have cemented her reputation as the GOAT. She then could retire after the 2019 Worlds, especially once the pandemic began.
Instead, 1997 turns out to be a terrible year for a gymnast to be born, especially one who has her eyes set on becoming the GOAT. Since Biles was too young to compete in London, she ends up pushing herself to compete in Tokyo, where the Games are delayed due to the pandemic. She ends up competing past her 24th birthday, making her older than most of her opponents.
And her age showed -- during last night's team competition, Biles struggled. Team USA had to settle for the silver medal, and gymnast herself specifically cited having mental issues. She said that she wasn't having fun in Tokyo -- and I bet she would have had fun in London had she been allowed to compete.
And all of this is because Biles was under so much pressure to win at the Olympics -- because her victories at worlds aren't considered good enough. If the Olympics were abolished, she would definitely be the GOAT based on her performances at worlds and other competitions -- but then she wouldn't be as well-known, since her sport would largely be ignored.
Molly and the Mathematical Mysteries 5: Escape the Boiler Room
Let's begin Cheng's fifth adventure:
"Molly finds the right door, but it swings shut behind her with a loud CLANG. It won't open again! She'll have to find another way out of here and on through the rest of the house."
There is another note for Molly on this page:
"The door needs steam to help it open. Fold the pipes into cubes to make them connect up. To open the exit door, find the net that joins up with no overlaps or gaps."
This puzzle is similar to the Hidden Shapes maze, but I was a bit more confused. It took me some time to figure out that the folding needs to be three-dimensional. The key word in the clue is "cubes." Here the author explains:
"In this room, the folding shapes are made from six squares, which fold up to make cubes. You might wonder what other flat patterns of six squares fold up to make a cube. Not all of them will work...but if it does, it is called a net. We can look for nets of other 3D shapes, too. There are 11 possible nets that make a cube."
Recall that nets appear in Lesson 9-7 of the U of Chicago text, on making surfaces. They are also emphasized in the Common Core Standards, in our Math 6 classes.
And there's also a challenge for the reader. It's to identify which of two possible patterns of squares is really the net of a cube. While I enjoy this type of Geometry problem, it's not possible for me to post such visual problems to the blog.
Molly and the Mathematical Mysteries 6: Carpet Weaving
Let's begin Cheng's sixth adventure:
"The steam-controlled door bursts open...The next room Molly enters is filled with wonderful tapestries. There are carpets all over the floor, the walls, and even the ceiling!"\
"Weave the strips over and under so that only shades of green are showing. It should match the pattern on the loom! If you do this correctly, a door will magically appear. Push it to enter the next room!"
I can see what weaving needs to be done -- indeed, back when I was a young first grader, all of us created an art design based on weaving. The problem is that these aren't my own strips of paper, but strips connected to the page in the book. I fear that by trying to weave the strips, I might accidentally rip them and ruin it for the next reader -- perhaps a young girl waiting to be awed and inspired. So instead I leave the weaving strips alone.
Here is the explanation from the author:
"Take a look at the hanging carpets above. The pattern only shows each shade of green or pink once in each row and once in each column. Both carpets do this, but the patterns are different. This is called a Latin square."
The most common Latin square in our daily lives, by the way, is a Sudoku puzzle. Unlike the 4 * 4 Latin squares in this book, a Sudoku is a 9 * 9 Latin square that satisfies other properties as well (such as the 3 * 3 sub-square rule).
And there's also a challenge for the reader:
"Look closely at other hanging carpets on the wall. Can you find somewhere Molly hasn't visited yet? Perhaps that's where she's headed next!"
OK, I recognize the Hidden Shapes maze, impossible shapes, a self-similar tree fractal, and an Escher impossible staircase. So we'll have to wait until next time to find out Molly's next destination.
Lemay: More on Chapter 16
In each of these posts, I'm continuing to return to some of the Lemay chapters that we passed over. We already spent two posts on Chapter 15, and so I want to look at Chapter 16.
In this chapter, Lemay introduced packages and interfaces. I'm hoping to write code to implement a Quadrilateral Hierarchy using interfaces. In C++ we'd use multiple inheritance to code this hierarchy, but according to Lemay, this is not possible in Java. So we must use interfaces instead.
Let's start by writing a basic quadrilateral class. What information do we need to keep track of in order to determine our quad? If this had been a triangle instead, we might keep track of the three sides -- then we could use the Law of Cosines to find each angle. Or we might keep track of ASA instead -- then the third angle is found using Triangle Sum and then the other two sides using the Law of Sines. Come to think of it, any triangle congruence theorem can be used -- SSS, SAS, ASA, AAS.
This suggests that for quads, we should use a quadrilateral congruence instead. We've discussed these on the blog before -- two such congruences are SASAS and ASASA. But which is better? To answer this, we must keep in mind that we're eventually going to write interfaces for all of the special quads in the hierarchy -- isosceles trapezoids, kites, parallelograms, and so on.
Let's try SASAS first. How can we tell, given SASAS, that a quad is a trapezoid? Well, the sides don't help us, but the angles do -- if two adjacent angles are supplementary, then it's a trapezoid. (This is the Trapezoid Angle Theorem, Lesson 5-5, U of Chicago text.) Notice that the two A's in SASAS need not be the supplementary ones -- that is, if in Quad ABCD the given parts are AB, Angle B, BC, Angle C, CD, then B and C aren't necessarily supplementary angles. It could be that A and B are supplementary (and hence C and D as well) instead. But that might not matter for the program just yet -- when implementing a trapezoid, we might force B and C to be the supplementary angles.
Now let's consider isosceles trapezoids. And that's where our problems begin -- if ABCD is an isosceles trapezoid with B and C as the supplementary angles, then the congruent sides are BC and AD -- rather than the desired given AB and CD (unless the isosceles trapezoid happens to be a rectangle). Given the SASAS condition, we're more likely to make AB and CD the congruent sides, which then makes B and C be congruent angles. It turns out that this is indeed sufficient for ABCD to be an isosceles trapezoid (with the proof given years ago on the blog).
But then we'd be in an odd situation -- for trapezoids, B and C are supplementary, while for isosceles trapezoids, B and C are congruent. So our computer would conclude that an isosceles trapezoid isn't a trapezoid necessarily, despite the name. This is the opposite of what we want.
The solution is to use ASASA instead of SASAS. The three given angles are now A, B, C, and the given sides are AB, BC. This solves the problem -- for trapezoids, either A, B or B, C are supplementary, and for isosceles trapezoids, A, C are supplementary with B congruent to either A or C. We also proceed down the hierarchy -- for parallelograms, A, C are congruent with B supplementary to both, and for rectangles, A, B, C are all 90 degrees. We can see why ASASA is superior to SASAS -- the angles provide us more information (particularly about parallel sides) than the sides, and so it's better to be given more angles.
This leaves us with the kite and the figures below it in the hierarchy In fact, rhombi (respectively squares) are easy -- we just use parallelograms (respectively rectangles) with AB = BC. As for kites, sufficient information for a kite is to have A, C congruent and AB = BC. The only problem is that we could have a kite with angles B, D congruent and BC = CD instead. This is analogous to what would have happened with trapezoids under SASAS, and our solution here is similar -- when implementing a kite, we force A, C to be the congruent angles and B, D to be the ends.
OK, so let's start writing code. Since we're learning about packages, let's write our Quad class as part of a quadrilateral package. All five side and angle measures will be of type double. Let's start by having accessor methods for each of the variables, and a constructor:
public class Quad {
private double A, AB, B, BC, C;
public double getA () {return A;}
public double getAB () {return AB;}
public double getB () {return B;}
public double getBC () {return BC;}
public double getC () {return C;}
public void setValues (double X,double XY,double Y,double YZ,double Z){
A = X;
AB= XY;
B = Y;
BC = YZ;
C = Z;
}
}
Notice that there's one more method we can easily add here -- a getD method. This is easy since the angles of a quad add up to 360:
public double getD () {return 360-A-B-C;}
Conclusion
Right now (as of the time stamp of this post), I'm watching the Olympics. There is an interview on right now with three gymnasts -- the three teammates of Simone Biles. Once again, they are discussing the problems their famous teammate is having with the competition this year.
I look forward to watching Track and Field, which will start in a few days. Five years ago at Rio, the Olympics started later in August, and many of track finals were past the first day of school. I missed many of the races because I was busy at the old charter school, trying to begin my career as a math teacher. Many people will watch our sport for ten days -- and then ignore our sport until the Paris Games in 2024, despite there being World Championships in 2022 and 2023.
Some people think that the Tokyo Games should have been delayed yet again to 2022. Of course, there will also be the Beijing Winter Games next year -- but then again, having the summer and winter Games in the same year was the norm through 1992. Some people think that all Olympics should be cancelled until the pandemic is "over" -- but when will that be? As I wrote in my last post, even the vaccines aren't enough to end the pandemic. There might be a new Greek-letter variant of the coronavirus that breaks out just in time to ruin the Paris Games in 2024, or even our local Olympics here in Southern California in 2028. If the pandemic doesn't complete end until 2031 (as I feared in my last post), then all Olympics would need to be cancelled until the Brisbane Games in 2032.
Or, as some people have suggested even before the pandemic, the Olympics simply should be abolished once and for all. So instead of ignoring Track and Field for the next three years, sports fans would be free to ignore our sport forever.
Some anti-Olympians might point out that if a fast sprinter wants eyeballs, then he should just quit Track and Field and become a running back in the NFL, where he'll get all the attention he wants. But that doesn't really help our female Olympians. The Games are their only stage -- if we abolish the Olympics, then thus ends their path to fame and fortune.
I've been told that when I was a young three-year-old, I was excited to hear about and watch the Olympics -- the last time they were held in my hometown. And so I, as a former high school track athlete, will continue to defend the Games.
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