Solutions to Exercises of section 12.6 in A Survey of Geometry

In a blog entropy of 20 June 2019 we talked about cross ratios in the Gauss plane, the topic of section 12.6 of Howard Eves' A Survey of Geometry. One thing that distinguishes this text is the wonderful set of exercises the author includes. They give the reader the chance to deepen their understanding of the theory presented in the book while extending the results covered by the author. In many cases, readers don't have time to explore these carefully selected exercises in full.

Although the real fun and true value of exercises is in doing them oneself, not everyone has the time to do so. Since I enjoy working out such things and in this case did so, I am including this link to them  Solutions to Exercises of section 12.6 in A Survey of Geometry  in case someone might like to learn more about cross ratios. There a few results in this set which are pretty useful results in themselves. My favorites were the Constructions in 12.6-14 and the Theorems that follow from Schick's Theorem in 12.6-16.

I hope that there are no typos or other such nuisances and apologize in advance if there are. If there are questions or concerns about any solution, I would welcome comments in that regard.

Entropy

If you look up the meaning of "Entropy" with a google search you will likely be no more sure of what it is after reading the multitude of descriptions offered. That is not to say they aren't all true, it is just that it is not completely clear what the English words mean. Here are ten examples:

 1. The entropy of an object is a measure of the amount of energy which is unavailable to do work.

 2. Entropy is a measure of the number of possible arrangements the atoms in a system can have.

 3. Entropy is a measure of uncertainty or randomness.

 4. Physics: a thermodynamic quantity representing the unavailability of a system's thermal energy for conversion into mechanical work, often interpreted as the degree of disorder or randomness in the system.

 5. Lack of order or predictability; gradual decline into disorder. "a marketplace where entropy reigns supreme"

 6. Entropy is a measure of the energy dispersal in the system. We see evidence that the universe tends toward highest entropy many places in our lives. A campfire is an example of entropy. The solid wood burns and becomes ash, smoke and gases, all of which spread energy outwards more easily than the solid fuel.

 7. Entropy is a measure of the random activity in a system. The entropy of a system depends on your observations at one moment. How the system gets to that point doesn't matter at all.

 8. Entropy, the measure of a system's thermal energy per unit temperature that is unavailable for doing useful work. Because work is obtained from ordered molecular motion, the amount of entropy is also a measure of the molecular disorder, or randomness, of a system.

 9. Here are two examples:  Low entropy: A carbon crystal structure at a temperature near absolute zero. ... High entropy: A box filled with two elements in their gaseous state, both of which are noble gases, heated to a very high temperature, with the gas "not very dense".

 10. Entropy is one of the consequences of the second law of thermodynamics. The most popular concept related to entropy is the idea of disorder. Entropy is the measure of disorder: the higher the disorder, the higher the entropy of the system. ... This means that the entropy of the universe is constantly increasing.

 So, if you are like me, after reading such prosaic descriptions of what entropy is, you are looking for a mathematical description that satisfies. In particular, something precise, something that you know how to measure. In fact, part of the difficulty with understanding entropy in lay terms is that it is a mathematical construction. Physics uses mathematical constructions to help explain things. Electric fields are an example. What is an electric field? Describing the theory of electricity (and magnetism) in terms of Electric (Magnetic) fields gives us a model we can "picture" in our minds that helps to describe the properties of electrons moving under the influence of uneven charges. Another example of a mathematical construct is energy. What is energy? It is often described as an "ability to do work". That is, energy (whatever it is) is expended in doing work. It is conserved in its many forms. That is, it cannot be created or destroyed. But what is it? A mathematical construct.

 The point is, sometimes, as in the cases of electric fields, energy, and entropy, there are mathematical functions of well known entities which satisfy certain useful properties. By giving names to these functions, we can then use them in our study of those entities. Such mathematical functions are not like the real things that exist in nature like atoms, but they are functions of such real things which prove useful. When we then go back to describe these concepts in non-mathematical terms, they lose the exactness of their true definition. If you really want to understand a mathematical construction, you have to understand the mathematics that gave rise to it.

 I undertook this quest to try to understand what entropy is. I read a book by Enrico Fermi, listened to some lectures from the Khan Academy, reviewed some Calculus, and in the end started to get an idea of what lies beneath the descriptions of entropy such as those above.

 Here's the bottom line: There are two ways to describe the mathematical entity called entropy. The classic version arose out of studies about heat and work. The statistical one arose out of statistical mechanics.

 Classic: The change in entropy of a system when it transforms from equilibrium state A to equilibrium state B is the sum over all the heat sources T of ratios Q/T where Q is the amount of heat the system absorbs from a source at temperature T.

 Statistical: The entropy of a thermodynamical system in equilibrium state A is proportional to the logarithm of the number N of dynamical states that give rise to that thermodynamical state where the constant of proportionality is the ratio of the gas constant R to Avogadro's number A.

 Okay, so now we know what entropy is!! You can read all about it in a summary paper I wrote after my study. Although there is nothing new in the paper, I have tried to iron out a few of the wrinkles and fill in a few gaps I ran across in my readings to make it a little smoother reading for you. The paper is pretty self-contained if you have studied a little chemistry and calculus.

Oriented Angles and Segment Lengths

In Volume 2, Chapter 12 of Howard Eves' "A Survey of Geometry" he introduces the basic notions of the Geometry of Complex Numbers. Everyone gets a small dose of this in high school but not a lot. Oftentimes the next time one runs into complex numbers is a Complex Analysis course in an undergraduate or graduate course. After reading Eves' treatment in his "Survey" I wish I had had this deeper dig into this topic to better appreciate the analytic course. That is, a deeper understanding of the algebraic and geometric properties of the Inversive plane gives one a better appreciation for the properties of analytic functions.

Like many topics in mathematics, the modern approach in school is often to emphasize breadth at the cost of depth in introducing new concepts. The result is that one can achieve advanced placement standing in Calculus with a very elementary depth of knowledge about plane Geometry. Often concepts are presented without an appreciation for the historical development of that concept. This can make it hard for any but the most advanced to fully grasp the subtleties of the things they learn. Unless students are presented with and then work through a wide assortment of problems they will never run against the "special cases" that make them fully appreciate the results they are learning.

In Eves' Chapter 12, section 6, he looks at the cross ratio of four points in the inversive plane. The cross ratio is shown to have a useful connection to homographies which basically account for all circular transformations in the plane (i.e. transformations that map circles into circles). In fact, the Moebius theorem says that a circular transformation must be a homography or an antigraphy. A homography is a transformation of the inversive plane onto itself defined as z' = (az+b)/(cz+d) where (ad-bc) is not zero. An antigraphy is just the product of a homography and a reflection in the real line. (Replace z with its conjugate.)In his Theorem 12.6.9, Eves establishes the result that the homography determined by the three distinct pairs of finite corresponding points Z1,Z1'; Z2,Z2'; and Z3,Z3' can be expressed in terms of the cross ratio (z'z1',z2'z3') = (zz1,z2z3) where (ab,cd) = (a-c)(b-d)/(a-d)(b-c).

The point of this entry is much simpler than than the (motivating) discussion above. It turns out that in the cross ratio discussion of section 6, Eves draws on the idea of oriented angles and segment measures in a way that is very useful but possibly a little confusing if one is not tuned into the details of the argument. To fine tune the reader's attention I have included a short paper that lays out in detail the definitions of oriented angles and segment measures to make the arguments on cross ratio representations most clear. For anyone interested the paper is Oriented Angles and Segment Lengths.

Why did I win so many tennis matches against an opponent with whom I felt so evenly matched point by point?

I played tennis for nearly two decades with a friend who shared the same tennis spirit with me. He was a reliable (on time and eager to meet) partner and we played on public courts nearly all through the year. Only injuries, rain or temperatures below 25 would keep us off the courts. In the early days I was a beginner and he patiently played with me as I developed my skills. But over the final few years, even though we felt evenly matched point by point, I seemed to win a surprising number of matches (best of three). In fact, even the three set match became somewhat of a rarity. So the question became, how evenly matched were we point by point? From a mathematical point of view we could try to answer the question a posteriori. That is, by noting my winning percentage in terms of sets, deduce what my per point advantage might be.

In the paper Tennis Probabilities we calculate the probabilities of winning a game, set, match given the probability of winning any given point. The paper includes a table of results from which one can work in reverse. Thus, if I was winning 81.5% of the sets we played, I could read from the table that my probability of winning any one point must be .55, etc. This 55% chance of winning a point is not much better than 50-50 so it never felt like a big advantage......but the numbers tell a different story!

Circle Inversions and Applications

There were a lot of geometric discoveries made in the 19th century that not many people know much about. By "not many" I mean not only those who end their mathematical studies in high school or even college, but even many mathematicians who spend their limited time and energy in other areas. Thanks to some though we do have some excellent texts that capture some of these results...one being Dan Pedoe's "Geometry A Comprehensive Course". In the last entry (Dec 7, 2017) we gave a detailed answer to an introductory exercise from Pedoe's book which opened up a whole universe of interrelated circles. Since then, having delved further into this book, this blogger learned about a 19th century geometric tool that gives insight not only regarding that problem we analyzed using primitive means, but offers a way to solve many problems in a way that we might just not be able to get our heads around using only primitive tools. This "new" tool is that of "inversion in a circle". These inversions transform circles and lines in the planes into other circles and lines while keeping the angles of intersection of any two such figures (and thus tangency and orthogonality).

In the paper Circles: Inversions and Applications , we have presented an introduction to this topic, proved the necessary results needed to establish the tools and worked many examples and exercises using the tools. Taken in its entirety it will give the reader a firm grasp of inversion as a tool to use in everyday geometric situations.

This blogger wants to send kudos to Dan Pedoe for his excellent book of which this is but a small slice.

It is a shame that the standard mathematical curriculum jumps away from geometry so soon when there is so much more to explore. I have run into many people who say they loved Geometry but began to lose interest in subsequent courses (how might that be possible?). Although the Greeks did amazing work in Geometry it should be no surprise that the subsequent centuries would add to that heritage. Well, here at oriolescience we like to offer up a bit of what you may have missed.

One solution we explain in detail in the paper is the Steiner Problem. We give the theoretical solution and work out a detailed example. This in itself is worth the reader's time and attention.

Exercise 0.9 from Dan Pedoe's Geometry A Comprehensive Course

The following problem was posed as Dan Pedoe's Exercise 0.9:

If (z-a)/(z-b) = r(cos(θ) + i sin(θ), show that the curves r = constant, and θ = constant, for fixed complex numbers a and b and varying Z, are circles of orthogonal coaxal systems, the one which has the points a and b as limiting points, and the other of circles which pass through these two points.

Since this problem occurred in an introductory chapter of the text, my expectation was that it was going to be a straightforward application of his quick review on complex numbers. It proved to be something more interesting.

In fact, establishing the part when r is constant is pretty straightforward once you recognize that the set of points whose distance to two fixed points is a constant ratio r is in fact a circle.

But the case for fixed θ proved the more interesting/complicated.

I recommend that anyone who has read this much should try to work it out before reading the solution linked below. I would love to see another approach to the solution if you find one.

For anyone who is interested, here is a link to a paper which provides what I hope will be a pretty self-contained solution intended for a reader who has studied senior high school level math. It includes a little background material for anyone who might be a little rusty.

Note: One point not explicitly mentioned in the paper is that when θ is held constant and r varies, θ presents itself as the angle between the perpendicular bisector of the segment ab and the line joining the center of the so-called θ-circle and the point a (or b). Thus the changing θ-circles can be visualized for θ in [0,π/2].

Descartes Circle Theorem - Beecroft Configurations

This entry is an expanded exposition of the rest of the material in Chapter 1.5 of Coxeter's "Introduction to Geometry" where he uses Beecroft's observations to prove the Descartes Circle Theorem.

Given four circles E_i, i = A, B, C, D mutually tangent to each other at six distinct points of tangency, we define the bend , e_i, of each circle as plus or minus the reciprocal of the corresponding radius r_i, i = A, B, C, D where bend is negative if the circle surrounds the other three and positive otherwise. That is:

e_i = ±1/r_i i = A, B, C, D where e_i < 0 if E_i surrounds the other three circles and positive otherwise. The bend of a straight line is taken as 0.

Descartes Circle Theorem: 2(Σe_i²) = (Σe_i)²

This tells us that if we have the radii of three tangent circles we can obtain the fourth using a simple quadratic in the unknown bend.

It is Coxeter's genius to mark the trail of a proof using the most concise and well thought out diagrams and text to make his exposition fully informative to one who is willing to use the guideposts to complete the trail. In this case the proof is not direct. It uses a result of an English "amateur", Philip Beecroft, who rediscovered Descartes work some 200 years later and carried it further. In particular, Beecroft observed that the quartet of circles E_i give rise to another quartet of circles H_i which are also tangent to each other at the same six distinct points and intricately related to the originals. Define H_i for i= A, B, C, D, as the circumcircle of the triangle made up of the three intersections points of its three complementary E-circles, that is, the triplet other than the E_i^th, and let n_i be the bend of H_i. Then H_i is either the incircle or an excircle of the triangle formed by the centers of the complementary circles. In particular:

• Case A: Suppose E_B surrounds E_C and E_D; then H_A is the B-excircle of ΔBCD. (Of course case A might have E_C or E_D) surrounding in which case H_A would be the C-excircle or D-excircle of ΔBCD.)
• Case B: Suppose E_B, E_C and E_D are externally tangent; then H_A is the incircle of ΔBCD

Similarly, we define H_B, H_C and H_D. Then these H-circles are four mutually tangent circles with the same six points of tangency as the corresponding E-circles. Let n_i, i = A, B, C, D be the bends of the H-circles. Here is a jump to a diagram of one such configuration.

Consider case A: We learned in the previous two blog entries that e_B = -1/s, e_C = 1/(s-d), e_D = 1/(s-c) and n_A = ±(1/r_b) where s is the semiperimeter of ΔBCD with sides of length b, c, d and r_b is the radius of the B-excircle. (Whether n_A is positive or negative depends on the particular configuration of circles we are working with. See note CASEA.)

Now consider case B: Since E_B, E_C and E_D are externally tangent we know that e_B = 1/(s-b), e_C = 1/(s-c), e_D = 1/(s-d). Also we know that n_A = ±(1/r) where r = radius of the incircle of ΔBCD. See note CASEB.

Recall from our previous blog entry that r² = (s-b)(s-c)(s-d)/s and r_b ² = s(s-c)(s-d)/(s-b).

We use the identity: uv + uw + vw = (1/u + 1/v + 1/w)uvw to get e_Be_C + e_Be_D + e_Ce_D =

• Case A: (-s + s - d + s - c)(-1/(s)(s-d)(s-c)) = (s - c - d)(-1/(s-b)r_b²) = 1 / r_b²
• Case B: (s - b + s - c + s - d)(1/(s-b)(s-c)(s-d)) = s (1/r²s) = 1/r²

Since in case A, n_A² = 1 / r_b² and in case B n_A² = 1 / r², we have in both cases that:

(1)

e_Be_C + e_Be_D + e_Ce_D = n_A²

By symmetry between the E-circles and the H-circles, the relationship holds in the other direction:

(2)

n_Bn_C + n_Bn_D + n_Cn_D = e_A²

Both hold (again by symmetry in the indices) for all combinations of the subscripts. Thus

(e_A + e_B + e_C + e_D)² = e_A² + e_B² + e_C² + e_D² + 2(e_Ae_B + ... + e_Ce_D) = e_A² + e_B² + e_C² + e_D² + n_A² + n_B² + n_C² + n_D²

(3)

(Σe_i)² = Σe_i² + Σn_i² and by symmetry (4) (Σn_i)² = Σe_i² + Σn_i²

Thus (5)

(Σe_i)² = (Σn_i)² and since the sum of the bends of four mutually tangent circles is positive, this gives us (6) Σe_i = Σn_i > 0

Note that

(e_A + e_B + e_C - e_D)(e_A + e_B + e_C + e_D) = (e_A + e_B + e_C + e_D)² - e_D² = e_A² + e_B² + e_C² + 2(e_Ae_B + e_Ae_C + e_Be_C) - e_D² = e_A² + e_B² + e_C² - e_D² + 2n_D² = (n_Bn_C + n_Be_D + n_Ce_D) + (n_An_C + n_Ae_D + n_Ce_D) + (n_An_B + n_Ae_D + n_Be_D) - (n_An_B + n_Ae_C + n_Be_C) + 2n_D² = 2(n_An_D + n_Be_D + n_Ce_D) - 2n_D² = 2n_D(n_A + n_B + n_C + n_D) But Σe_i = Σn_i So (7) e_A + e_B + e_C - e_D = 2n_D This holds for all combinations of the subscripts and switching the e's and n's. Thus, (e_A + e_B + e_C - e_D)² = 4n_D² (e_B + e_C + e_D - e_A)² = 4n_A² (e_C + e_D + e_A - e_B)² = 4n_B² (e_D + e_A + e_B - e_C)² = 4n_C² Adding all four of these equations and simplifying we get 4Σe_i² = 4Σn_i² So (8) Σe_i² = Σn_i² Thus (9) (Σe_i)² = Σe_i² + Σn_i² = 2Σe_i² And this establishes the Descartes Circle Theorem.

CASEA: H_A is the B-excircle of ΔBCD.

• Case A.1: Suppose E_A is on the far side of line CD from B. Then H_B, H_C, and H_D are surrounded by H_A so n_A = -1/r_b
• Case A.2: Suppose E_A is on the same side of line CD as B. Then H_B, H_C, and H_D are externally tangent to H_A so n_A = 1/r_b

CASEB: H_A is the incircle of ΔBCD.

• Case B.1: Suppose E_A surrounds E_B, E_C and E_D. Then H_B, H_C, and H_D are all externally tangent to H_A so n_A = 1/r_b
• Case B.2: Suppose E_A is externally tangent to E_B, E_C, and E_D. Then H_A surrounds H_B, H_C, and H_D so n_A = -1/r_b

An exercise from Coxeter section 1.5

Coxeter refers to the octet of E-circles and associated H-circles as a Beecroft configuration. We now consider Exercise 9. For any four numbers satisfying k + l + m + n = 0, there is a "Beecroft configuration" having bends e₁ = k(k + l), e₂ = l(k + l), e₃ = n² - kl, e₄ = m² - kl, n₁ = l² - mn, n₂ = k² - mn, n₃ = m(m + n), n₄ = n(m + n). [Hint: Express e₃, e₄, n₁, n₂ as rational functions of e₁, e₂, n₃, n₄]

Solution to Exercise 9

We have seen above that equations such as

e_Be_C + e_Be_D + e_Ce_D = n_A²

hold for all such combinations of e's and n's. Thus, we can write (identifying subscripts A, B, C, and D with 1, 2, 3, 4 and assuming the variable indices i, j, k, l are distinct elements in {1, 2, 3, 4})

n_i(n_j + n_k) = - n_jn_k + e_l²

n_i = (e_l² - n_jn_k)/ (n_j + n_k)

By using (7) twice we see that e_i + e_j = n_k+ n_l, so we have

n_i = (e_l² - n_jn_k)/ (e_i + e_l)

Using this and the corresponding equation for e_i, we get the rational functions suggested in the hint.

n₁ = (e₂² - n₃n₄)/ (e₁ + e₂)

n₂ = (e₁² - n₃n₄)/ (e₁ + e₂)

e₃ = (n₄² - e₁e₂)/ (e₁ + e₂)

e₄ = (n₃² - e₁e₂)/ (e₁ + e₂)

Since we are trying to show the configuration is possible as stated, we can assume, without loss of generality, that e₁ = k(k + l), e₂ = l(k + l), and n₃ = m(m + n). We know k + l + m + n = 0 and e₁ + e₂ = n₃ + n₄. Therefore, k(l + k) + l(l + k) - m(m + n) = n₄. That is (l + k)² - m(m + n) = n₄ and since (l + k) = -(m + n), we have (m + n)² - m(m + n) = n₄ so n(m + n) = n₄. Now using the rational functions established previously we have n₁ = (l²(k + l)² - mn(m + n)²)/(l(k + l) + k(k + l)) = l² - mn. Similarly, n₂ = k² - mn, e₃ = n² - kl, e₄ = m² - kl.

Note: Suppose (l + k) = 0. Then e₁ = e₂ = 0 by assignment and (m + n) = 0 because k + l + m + n = 0. Therefore, n₃ = n₄ = 0. In this case, E₁, E₂, H₃ and H₄ are circles with bend 0, i.e., straight lines with E₁ parallel to E₂ (tangent at infinity) as are H₃ and H₄. In this case e₁ = k(k + l) = 0 = e₂ = l(k + l) and n₃ = m(m + n) = 0 = n₄ = n(m + n). Now k = -l and m = -n mean that e₃ = e₄ = n₁ = e₂ = k² + n² so circles of radius 1/(k² + n²) as half the distance between E₁, E₂ and H₃, H₄. Using the point at infinity as one of the points of tangency we get a (degenerate) configuration where E₃, E₄ are tangent to each other as well as E₁, E₂. H₃ contains the intersections of E₁E₄, E₂E₄ and E₁E₂ (infinity), etc. Similarly H₁ and H₂ can be drawn. Note: if k² + n² = 0 then k = l = m = n = 0.

Finally, (as in Coxeter's solution) the parameterization of the problem can be obtained directly from a configuration as follows: Given e₁, e₂, n₃, n₄, set e₁ = k(l + k), e₂ = l(k + l), so e₁ + e₂ = (k + l)², (k + l) = ±√[e₁ + e₂]. Then k = ±e₁/√[e₁ + e₂] and l = ±e₂/√[e₁ + e₂]. Similarly m = ±n₃/√[e₁ + e₂] and n = ±n₄/√[e₁ + e₂] where we used e₁ + e₂ = n₃ + n₄. By choosing, say, +, +, -, - we get k + l + m + n = [(e₁ + e₂) - (n₃ + n₄)]/√[e₁ + e₂] = 0.

The following diagram shows a Beecroft configuration with k = 5, l = 4, m = 3, and n = -12 It has values e₁ = 45, e₂ = 36, e₃ = 124, e₄ = -11, n₁ = 52, n₂ = 61, n₃ = -27, n₄ = 108. E₁ has center at the (0,0) and E₂ has center on the x-axis. E₄ surrounds the other E-circles.