Exercise 2.8.  Show that given a sphere with radius R, the surface area of a triangle T with vertices A, B, C and interior angles α, β, and γ inscribed on the surface of the sphere is R²(α+β+γ-π).

     We can start about by examining the surface area swept out by a single pair of great circles forming one of the interior angles of T.  For instance, the surface area swept out by the pair of great circles intersecting to form angle β:

     When β equals π, the area swept out covers the entire surface area of the sphere, or 4πR².  Therefore we can infer (dubious, see below discussion) the following relationship between the great circles intersecting to form angle β and the surface area A_β swept out by these great circles:

A_β = 4βR²

(1)

    

     Armed with this knowledge, let's look at all of the surface areas swept out by the great circles intersecting to form all of T's interior angles:

     Using equation (1), the total surface area swept out by the great circles intersecting to form the three interior angles for T is:

A_α,β,γ = 4R²(α+β+γ)

     Then note that the area of T, A_T, is swept out three times by each pair of great circles, and is mirrored on opposite poles of the sphere, so that triangle ABC is congruent to triangle A'B'C' on the sphere.  Therefore A_α,β,γ covers the entire surface area of the sphere plus extra area equal to a multiple of A_T:

4R²(α+β+γ)=4πR²+4A_T

       (2)

      

     Four times A_T because A_T is swept out two "extra" times on each pole of the sphere.  Using (2) it is easy to solve for the correct value of A_T.  (2) also indicates that the sum of the angles of a triangle inscribed on the surface of a sphere must always be greater than π.

Dubiousness...

      I'm not particularly happy with the way I just kinda closed one eye and went ahead with "inferring" expression (1) for the surface area swept out by the great circle intersections.  It seems right but I dunno whether there's something more I'm supposed to be doing so that I could nail that down with more rigor.  Or maybe that's par for the course given that Road is not after all a full-blown math course.  

     Anyway I got to the right answer without too much B.S. so I guess I should count myself lucky.  Working to finish up a beast right now that I may have leave unfinished and ask for help from whoever the hell might eventually one day actually read this stupid blog.  (hi, unlucky historian of the Internet reading this a thousand years in the future!)

Understanding Penrose's Discussion of the Dedekind Cut

What Does He Mean?

Penrose's discussion of what he calls the "Dedekind Cut" in Chapter 2 of Road is tantalizing but unfortunately brief to the point of being almost impossible for me to understand.   I was particularly confused by his reference to how irrational numbers such as the square root of 2 are defined by Dedekind cuts when "those to the left have no actual largest number and those to the right, no actual smallest one" (3.3).  Huh?

But with some reference to various pages on Wikipedia (which I fear I'll be making much use of as I slog through this tome) I believe I have pieced together what Penrose is talking about.  And it is quite interesting, and worth working through here.  

Basically, what Penrose seems to be talking about is defining irrational numbers as the "gaps" between rational numbers on the number line.  Let's define Q as the set of all rational (not real!) numbers.   We then "cut" Q into subsets A and B.  The trick is to define the point γ which divides A from B in such a way that γ belongs to neither subset.  Since A and B are subsets of Q and hence also comprised of rational numbers, but γ is not a member of A or B, γ cannot be a rational number.   

Suppose for example that you wanted to use such a cut to define the square root of 2.  You would cut Q as follows:

The notation is stolen from Wikipedia: the vee-shaped thingy means logical OR, and the wedge-shaped thingy means logical AND.  So A is the set of all rationals a in Q whose square is less than 2 OR all rationals less than or equal to 0.  Similarly, B is the set of all rationals b in Q whose square is greater than or equal to 2 AND all rationals greater than 0.  

Well what does this all add up to.  It is pretty obvious that all a must be less than γ, since we are either talking about negative numbers or positive numbers (including 0) whose square is less than 2.  It is not quite so obvious that all b must also be greater than γ. This is because the square root of 2 is not a rational number and hence cannot be a member of B.  All members of B must therefore have squares greater than 2.  

Looking at this we start to get some idea of what Penrose means by his mysterious references to "no actual largest/smallest numbers" in the two sides of the cut.  However I think it really helps to get a bit more precise in this case so that we can understand what is and is not being discussed.  Apparently Penrose is referring to the concepts of a supremum and infimum of a set.  You can define the supremum S (my symbol) of a set T of real numbers to be the smallest real number greater than or equal to every member of T; similarly, you can define the infimum I (again my symbol) of T to be the largest real number less than or equal to ever member of T. Examples:

We can now state much more concisely (and precisely, and clearly) that while γ is not a member of A or B, γ = S{A} and γ = I{B}.  The supremum and infimum for A and B respectively fall outside A and B: this is what Penrose is saying, I think (read: I hope).  

Additional Thoughts

Once I managed to piece all of this together in a reasonably coherent way I skimmed a page in Wikipedia concerning the relationship of suprema and infima to the completeness or lack thereof of sets.  I didn't really read closely but it was indeed interesting that it is possible to define subsets of rational numbers in a way that appears to violate the completeness requirements (since there are suprema and infima that fall outside these subsets, as we have seen).  So the irrationals arise almost as a necessity (?) in order to complete the set of rationals and form the larger (complete?) set of all real numbers.  Assuming that it even makes sense to talk about irrationals as a "necessary" complement to the rationals, one wonders whether there are other ways to cut the reals that "necessitates" complex numbers.  I dunno.  *shrug*

Halp!

Putting together entries with equations in them such as the last post is proving to be a MAJOR pain. Been spending some time online looking for a fix that would ideally provide fairly quick translation from TeX source to text and image files friendly for this blog format...

Exercise 2.4

Exercise 2.4.  Show that the projective representation of a hyperbolic straight line may be obtained from the conformal representation of a hyperbolic straight line by expansion from the center by a factor of M, and that:

Solution:

Above is a conformal pro jection on a minor circle c, with radius r , that we project onto a projective representation on the great circle C, which has radius R and origin O. Let r_c be the Euclidean distance from O to any point on the conformal repreesentation on C , and let b be the Euclidean distance from r_c to its corresponding point on the pro jective representation, so that the additive Euclidean distance from O to any point on the projective representation is b + r_c. Let M be the expansion factor from the Euclidean conformal distance to the Euclidean projective distance, so that:

                   (1)

Now, it is evident from the similarity of the two triangles that:

The problem is how to restate d in terms of R, r_c,  and b.  Given that the segment extending from O and joining d is equal to R, by the Pythagorean Theorem we know that 

So we rewrite the previous expression as 

And solve for b:

With this we can rewrite (1) and solve for M:

Let's See How It Goes...

I've been working on and off again on Roger Penrose's book Road to Reality which gives a nice, meaty introduction to the mathematical underpinnings of modern physics.  I've always been a physics-wannabe but never quite had the time or the talent to make it into a profession.  SO, it's been fun recently to begin working through the exercises Road in a last-ditch effort to really get a grasp of what has been going on for the last, erh, century or so of science.

This means working up solutions of lots of math problems, solutions which I either cannot completely get or which I think I can get but for which I need people to check on to point out potential problems.  This blog will hopefully be a useful place to put forward to the Internet questions about problems  I cannot solve and/or requests to check my solutions of problems.  I definitely welcome any constructive criticism folks better informed than I might have concerning the way I am approaching/solving these problems.  I just hope that folks can remain reasonably polite and remember I am coming from a completely non-expert background. Please keep this in mind when providing feedback.  It's also ok to find my ideas hilariously wrong: hopefully this'll provide some amusement for my betters.:)

Anyway, here goes!