Rational Trigonometry

I made a page about rational trigonometry, an idea proposed by N.J. Wildberger in his book about divine proportions.

Before you go to the page, try to find the height h of the following triangle, and the length of the angle bisector d, both as algebraic expressions, not approximations.

New page about installing Latex

I created a page with a description of the installation of Latex for Windows.

Math and the Press

Just recently I heard a radio interview with some German professor of math about votes and seats, and the problem to compute the distribution of the seats from the votes. The radio woman correctly identified the problem as the fractions, which occur when you multiply the percentage of votes by the total number of seats. After all, you cannot send 0.2 politician to the parliament. Funny remark, indeed.

The professor started to explain, that these fractions have to be used somehow to divide the remaining seats under the parties, threw in the previously used d’Hondt procedure, and both agreed that it would be too complicated to explain that here and now. The moderator admitted that he would not know exactly anyway.

The professor then continued that we have now a new algorithm which gives each vote the same weight, as he said, as it should be, he said. The surprising solution is to round to the next integer. That, he said, is much more fair than the d’Hondt procedure, which always rounds down. Both did not bother to confront the listener with the problem what to do if the number of seats assigned that way does not happen to be the number needed. But we learned that it is fair, and no longer hurts the small parties more than the bigger ones.

To demonstrate the problem of rounding down small versus large numbers, the professor used a striking example. If we would round down his income to the next million, it would hurt him much more than if we did the same with Bill Gate’s income. I assume the professor makes more than half a million, maybe by radio interviews. For otherwise, a better rounding can only help Bill Gates.

So, if you want to learn how useful math really is, listen to the radio!

The iterated Sine

I am a bit surprised that none of my readers like to solve problems. OK, then here is a proof. I adopt the notation for the iterated sine.

It is a simple problem to show that goes to 0 decreasing. So there is a k such that

Since all are monotone by induction, we get

Now, all we need to show is for all k

But this is the same as , which easily follows by induction on k.

The same can be done with any other monone  function of the form

For the iterated function, we always get

for . An example of such a function, which is easy to compute is

Since you are so fond of my problems, here is another one, I cannot solve yet myself. Numerical experiments show that

exists. Can you prove that? And what is that limit?

A Challenge

Find out, how

behaves. Here, I do not mean the n-th power of sine, but the sine function applied n times. Actually, I do not like to write the n-th power this way, but perfer to write the power after the bracket of the argument.

Three Cylinders, Continued

What is the volume of the intersection in the last blog posting? To compute that we decompose the intersection into 7 parts, three caps, and one cube, using PovRay again. That took me another hour.

The top cap can be integrated along its center, and the cube has side length . If I did not a mistake the volume is

This is a bit more than 4.

Three intersecting Cylinders in Povray

Can you imagine how three cylinders, each aligned along one axis, intersect? It is very difficult. So I took Povray for a the job. Here is the result.

I am absolutely no expert for Povray, more a beginner. It took me two or three hours studying the documentation and introduction pages in the net, before I got my result.

However, maybe I should use this tool more often. It might even be a good idea to add an export for Povray to the Euler Math Toolbox. If you look at my page about 3D geometry, you will find other programs to do 3D stuff. But none would produce the result as nicely as Povray.

Since you probably still cannot imagine the intersection, here is the union of the three cylinders. I hope this helps

Here is the code. It took me the most time to find out how to increase the ambient light. This is done by the assumed_gamma parameter below. I also used the 640×480 rendering with anti-aliazing. For the blog, the image was cropped and saved as PNG by Photoshop.

#include "colors.inc"
#include "stones.inc"

camera
{ location <5, 3, -3>
look_at <0, 0, 0>
angle 30 // opening angle of the lens
}

global_settings
{ ambient_light White
assumed_gamma 1.4
}

background
{ color Gray }

light_source
{ <2, 4, -3>
White
}

#declare cylobject = cylinder
{  <-2,0,0>, <2,0,0>, 1 }

intersection
{ object
{ cylobject
pigment { Red }
}
object
{ cylobject
pigment { Green }
rotate 90*y
}
object
{ cylobject
pigment { Blue }
rotate 90*z
}
}

Computer Algebra

I am currently teaching a class on software for schools. The main topic is computer algebra, along with a short introduction to LaTeX, and to dynamic geometry. As an algebra system, I use Maxima via my own program Euler Math Toolbox. Now, at the end of the class, I am somewhat disappointed about the progress of the students. Why is this so difficult for them?

The main reason might be the structure of the class, which is only 2 hours of presence time, with little or no homework. This, of course, is far too little to learn a new language, like LaTeX, or the syntax of a computer algebra program. I would have had much more success with dynamic geometry, but this was not my main point in this class.

But in this blog, I want to point out, that teaching math using computer algebra systems, or even using math via computer algebra systems, is much different from any other math activity in school or university. First of all, it involves talking a new language to the computer. Next, you have to listen to and understand the answers of the computer. So one of the main problems is a communication problem.

So, if I do this class again, I will concentrate on only one aspect, like the computer algebra, a less ambitious aim. Then I would start with small problems of the current mathematical world of the students, so they can relate the one language to the other, the content they know to its representation in the algebra system. Moreover, they should be able check the answers of the system, via a plausibility check, simpler problems, graphical representations of the result, or using the system itself.

The Pillars of Mathematical Thinking

I am not very acquainted with the philosophy and psychology of the learning and doing of math. However, it gets more and more apparent to me that mathematics is built on three fundamental brain activities. Each of them has nothing in common with the other, and the links between them are very loose.

• Visual - where I include all internal or external views and images with mathematical content. The most obvious examples are graphs of functions, and geometric figures. But you may also include Venn diagrams in set theory, commutative diagrams in algebra, or any arrangement of mathematical objects in diagrams, matrices, graphs etc. In calculus, include vector fields, tangent planes, gradient vectors, or the derivative of curves depicted as vectors.
• Arithmetic – including symbolic algebra, and calculus. There is not much to explain here, since we all have been tortured with pages and pages of computations during our school days. Arithmetic is, at least on a lower level, what a clever program can do for us.
• Logic – as the general question, why mathematical things are as they are, and why they have to be that way. Logic is necessary as soon as we ask for the general result, instead of a specific one. The answer is derived by showing that certain assumptions necessarily lead to certain consequences, in each and every model example one could come up with. Logic is also used and abused to construct things, which go beyond intuition.

It seems to me that each of the above techniques have to be learned separately. Doing math only by images and pictures does not help in computing things, and it can only be a vague guide in proving. Logic alone is bound to fail, if there is no imagination of the general picture. E.g. in synthetic geometry, a geometric picture helps to understand the situation, but does not automatically yield a mathematical proof. In topics like set theory or set theoretical topology, you are left with logic, and there is only little visual guidance.

It is obvious that logic and visual mathematics are related to different brain activities. But I think that arithmetic is also something different. Since we have now computers which can do a lot of arithmetic, it seems to be an activity on a lower level. It is guided by logic, or at least it should be guided by logic. For we want to know why we are allowed to perform a specific arithmetical step. Of course, arithmetic can also be done on a very high level. Then it is an artful juggling with patterns rather than a blind schematic activity. In any case, it is connected to pattern recognition, with patterns in formulas of non-geometric content.

If that is so, it is no surprise that mathematics is such a difficult object. The need to learn three different and distinct techniques, and to switch between them in a masterful way is clearly a challenge. Good mathematicians are good in all three areas in a balanced way. Students concetrate too much on one aspect. Not only is mathematics based on these three different pillars, but it is also a very huge amount of content, a big house to live in, split in areas inhabited only by experts. Each semester I am surprised how much information I pass to the students. It cannot be expected that they know and master each definition and theorem, not even the best students can.

School mathematics was long built on arithmetical skills too much, which is what the not so brilliant students seem to like. Things are changing, however. With the explosion of the computer power available to students, a simple schematic problem solving strategy is no longer adequate. At least, we should pose more open problems, where there is a choice of solutions, even if each of them is still schematic. Moreover, open problems force the student to make a visual sketch of the situation. If the problem is such, that it also forces a logical decision about the right or wrong way, even better.

And finally, all arithmetical activity in schools should be linked to logic. If possible, force students to reflect about each step of a computation. Let them add the correct implication during the computations. Contemplate with them, if a verification by insertion of the result into the question is absolutely necessary, or if it is just a way to make sure the result is correct. This is far more important than double underlining the result.

Billard

Long time ago, I did a notebook for Euler Math Toolbox, which simulates Billard on a circular table. However, I want to quizz you.

How many paths are possible from one given point to another point inside the circle, reflected at the boundary exactly once? Can you compute them exactly? Can you construct them with Compass and Ruler?