Old Memories About Trisecting Angles

This is just some remembering. When I was in middle school (about 12 or 13 years old), I was convinced I could find a way to trisect an angle. Of course, this is known to be impossible, by an argument that I now (but didn’t, at the time) understand. That didn’t stop me from trying, though.  Here’s the construction I spent most of my time on:

Conjecture for trisecting an angle

I started out with an angle, which is the two outermost lines on the given drawing.  Using the compass, I marked off points equally far along the sides.  I then constructed their midpoint.  Then I drew three overlapping circles centered at those three points, as shown.  The intersection points of these circles furthest from the vertex of the angle were used to draw my “trisecting” lines.

Of course, this doesn’t really trisect an angle, since it’s a construction with a compass and straight edge, and no such construction can possibly trisect an angle.  But if you didn’t know that, there are a few sanity checks you might perform.

• Does it appears to work?  Yes, it does, as it turns out.  In fact, I performed this construction fifty or sixty times, and measured the result with a protractor.  Each time, it was close enough to trisected that I chalked up the difference to human error.  (Having just performed the construction now using the Kig software, to capture the image above, I can now see that the result is visibly off for extremely small angles; but of course these are the ones that I was least able to check without reasonable computer software at the time.)
• Does it work for cases that are easily calculated?  Yes!  A few easy examples: If you start with a 180 degree angle, you get back 60 degree angles… perfectly trisected.  A zero-degree angle is vacuously correct.  It’s a slightly more involved argument, but the construction also trisects a 90 degree angle correctly.

At this point, one can’t be blamed for getting a little excited.  We have a construction that gives a division of an angle into three parts.  The parts always seem to look equal, and are always close enough to be within reasonable measurement error.  It works for the cases we’ve checked so far.  One can’t be blamed for shifting gears a little here, gaining a little confidence, and looking a little less for a counter-example, and a little more for a proof.

Of course, those familiar with the impossibility of this task will be looking for the case where we start with a 60 degree angle.  And, of course, it turns out not to work.  In particular, the “trisected” angle works out to about 19.1066 degrees.  Close, but not quite.

So what am I really constructing?  Well, something pretty ugly.  I’m leaving out the math (basically, just pick out a couple triangles, and apply known triangle relations such as the law of sines and thelaw of cosines), but, the answer in Maxima notation (and simplified as best as Maxima can) is -asin((sin((3*%theta-5*%pi)/6)*abs(sin(%theta/2)))/sqrt(-2*sin(%theta/2)*cos((3*%theta-5*%pi)/6)+sin(%theta/2)^2+1)).  Yeah, wow.  But how could we have fooled ourselves into thinking that mass of complicated stuff is actually theta / 3?  Well, take a look at Maxima’s plot of this angle versus theta (that starting angle):

Graph of the constructed angle versus the original

Ifthe construction were correct, that graph would be a straight line passing through (3,1).  As it turns out, it may be somewhat difficult to tell the difference in this graph!  So while this construction is wrong, it’s also remarkably close.

Code for Manipulating Graphs in Haskell

Here’s a bunch of code I wrote, most of it about a year ago, for doing things with the graphs from the Data.Graph module in Haskell.  The choice of functions, from among the many generally useful functions acting on graphs, comes from a specific project.  The actually functionality is pretty generic, though.  So I’m just throwing this out there. If someone else wanted to package it and throw it on Hackage (likely with a different module name), they would be welcome to do so.

Re: Conditional Convergence, or when Addition Isn’t Commutative

I’m writing in response to an apparently deleted blog post that was submitted to Reddit about conditional convergence.  The post was incorrect… well, perhaps nonsensical would be a better phrase.  But even though I’m more of an algebraist, conditional convergence is still an interesting idea.  So I’m taking my response there, and turning it into a new post.

Most people probably know that the infinite series (1 + 1/2 + 1/3 + 1/4 + …) diverges.  Even though the numbers in the sum keep getting smaller and approach zero the further you get out the sequence, you can still add up enough terms of this sequence to surpass any number you like.  It just might take a long time.  Now here are a couple other simple consequences of that.

• Even if you throw away a bunch of terms from the beginning of the sequence, it still diverges.  Hopefully, this is obvious.
• Even if you only add up every other term, it still diverges.  (Proving this by contradiction is fun if you haven’t done a lot of basic analysis.  All you need to know is that if each term of one series A_n is closer to zero than the corresponding term of a series B_n, and B_n converges, then A_n converges, too.)

Okay, now lets consider the sequence we’re really interested in:

1 – 1/2 + 1/3 – 1/4 + 1/5 – 1/6 …

The difference between this sequence and the original is that every other term is negative.  Now, it’s not hard to see that this series actually converges.  But what does it converge to?  Well, in the order it’s given here, it converges to something between 1/2 and 1.  But the interesting point here is that I can rearrange the terms of this sequence to give it any convergence behavior I want.  I can make it converge to any real number x. I can make it diverge toward positive infinity.  I can make it diverge toward negative infinity.  I can even make it jump around back and forth between 0 and 1 but without actually converging anywhere.  And with all of these different convergence behaviors, I am still going to use the same terms of this series… just add them in different orders.

Making The Series Converge to x: Suppose I choose a real number x, and I want my series to converge to x.  Easy.  For simplicity, I’ll assume x is positive, but it’s easy (and obvious) to modify this for x negative, as well.  I’ll start out just taking the positive terms of the sequence… 1 + 1/3 + 1/5 + 1/7 + …, and I’ll keep going that way until the sum is strictly greater than x.  (How do I know that it will eventually become strictly greater than x?  Because I am just adding up every other term of that first sequence, and we agreed that sum diverges to infinity.  So I’ll reach x in some finite number of terms.  As soon as the sum surpasses x, I’m going to start including negative terms… – 1/2 – 1/4 – 1/6 … and so on.  I’ll continue that way until the sum drops back down below x again.  Then I’ll swap back to the positive terms, and so on.

I claim this uses all the same terms as the original series… but instead of converging to something between 1/2 and 1, it converges to any real number x.  (If x is negative, the process is the same, except you start with negative terms.  And if x is zero, you start with whichever you prefer.)  Here’s why that is.  Suppose you’re looking for term n of the original series.  In the rearranged series, it won’t occur at position n.  But every time I switch directions, I’m taking at least one term from that direction… so term number n is guaranteed to occur somewhere in the first n direction changes.  Each direction change is only finitely long (as we saw above; this is a consequence of the fact that the strictly positive series diverges), so term n occurs at some finite index in the sum.  So every term appears, and by the way I chose terms, I know I didn’t introduce any new ones, or duplicate any of them.  The terms are the same; only the order differs!

But this new series definitely converges to x.  Right at each direction change, we overshoot x by a little bit.  But the amount that we overshoot x in the sum is always no more than the last term picked in the sequence… and as we choose more terms both on the positive and negative sides, the largest unpicked term is decreasing.  So over time, we start overshooting by smaller and smaller amounts, and the sum converges.

Making the Series Diverge to Positive Infinity: I can also make the same series, with the same terms, increase without bound, thereby diverging to positive infinity.  The trick it to use the negative terms, but just use them much less often.  So add positive terms until you get to, say, 2.  Then negative terms until you get back down to one.  Then positive terms until you reach 3.  Then negative terms until you’re back down to 2.  Keep that up and you get a divergent series.  Change the numbers a bit, and your series diverges to negative infinity instead.

You can even keep the sum from approaching infinity, while also preventing it from converging to any particular finite sum.  Add positive terms up to 1.  Add negative terms until you get back to zero.  Positive terms back up to 1.  Repeat.  The details of proving that you’ve got the same terms, and that the series actually shows the desired behavior, are similar to the first case.

So that’s conditional convergence.  You really can take a series that conditionally converges, and make it show any behavior you like simply by rearranging the terms.

Needing Intuition in Math(s): one example

I’m helping a few people with abstract algebra at the moment, and I came to this realization.  Most people learning abstract algebra, as far as I can tell, have no idea why homomorphisms and factor groups are sensible things to think about.  They quickly come to understand the idea of a group, and enough varied examples are usually given that they can see how the idea of a group applies to a number of things.  They quickly come to terms with subgroups, though the idea looks rather trivial to them.  Then you get to homomorphisms and factor groups; at this point, most classes run out of intuition and just jump in for some unmotivated mathematical constructions.

I’m not quite sure why this is, honestly.  Anyone with the slightest modicum of mathematical curiosity probably has thoughtn about factor groups since they were seven or eight years old.  In the context of integers and addition, most children realize on their own (whether it’s taught to them or not) that the sum of two even numbers, or of two odd numbers, is even, while the sum of an even number and an odd number is odd.  This is, of course, a factor group.  Students who are presented with the mathematical definition of a factor group should first have, in their set of mental tools, this simple intuitive definition:

Factor Group: For any group (G,*), a factor group is a group that is obtained by being sufficiently sleep-deprived (or perhaps drunk, depending on the university) that one can’t tell the difference between some members of the original group, and then trying to write down a group table.

Of course, one then goes on to point out that sometimes this works, but sometimes it doesn’t.  If one looks at the integers and only sees “even” or “odd”, then it works.  If one looks at the integers and only sees “negative” or “non-negative”, then it doesn’t work, since the sum of a negative number and a positive number could be either negative or positive.  It then becomes natural to ask when it works, and when it doesn’t.  This provides a justification, then, for nailing down the abstract definitions, defining normal subgroups, proving that the factor group is well-defined when modding out a normal subgroup, and so on.  First, though, the student needs to be convinced that these are natural things to think about.

(A quick aside: I’m not pulling out the idiotic canard that students need to be convinced that mathematics is useful in “the real world” or anything so ridiculous as that.  But even the purest mathematicians are more interested in answering questions that naturally arise than those that seem quite arbitrary.)

Speaking of defining normal subgroups, it is really inexcusable how many students have never even noticed the close relationship between normal subgroups and commutativity.  Sure, everyone knows that all subgroups of an abelian group are normal; but this seems to be treated as a sort of occasionally useful curiosity.  Few students are even exposed to the simple fact that normality of subgroups is inextricably entwined in the degree to which the subgroup commutes with the surrounding group.

Case in point: one of the equivalent conditions for normality that is often taught is this: Let H be a subgroup of G.  Then H is normal in G if and only if for any h in H, and any g in G, the product (g’ h g) is in H (where by g’, I mean the inverse of g).  This statement is absolutely correct… but it is utterly useless to a student who will most certainly not recognize (g’ h g) as a statement about the commutativity of h and g.  So first, it helps to show a few other results:

1. The following are equivalent: (g’ h g) = h, and gh = hg.  The proof is trivial is both directions, and students can likely figure it out on their own.  But, and this is the important part, students will likely not recognize this simple fact if they haven’t seen it before.
2. Let H be a subgroup of G.  Then H is a subgroup of Center(G) if and only if for any h in H, and any g in G, (g’ h g) = h.  This is just applying the previous result and the definition of a center.

From this point, the path to the earlier condition is clear.  We’re really talking about commutativity, but of course our ultimate goal is to be able to blur our eyes (or get sleep-deprived, or drunk) such that we can’t tell the difference between the members of H.  Then one may start with #2 above, and simply remove the distinction between members of H — so instead of looking for (g’ h g) = h, we look for (g’ h g) being “close enough” to h… in other words, it should be in the same subgroup.  One may now recognize that normality really is a weaker analogue of commutativity.

So the literal point here is that students ought to be taught these two intuitions.  The larger point is to wonder why it’s apparently considered appropriate to teach abstract algebra without teaching these two intuitions.