Unveiling Pythagoras: How Circle-Based Area Transformations Reveal the Truth

I recently had a discussion about the Pythagorean theorem and how to understand it better. I realized that the role of the circle as a fundamental element is often underrated. You can hardly find any proof where the right-angled triangle is placed within a circle to support the reasoning. Instead, the triangles are usually presented in a plane without reference. However, when you position the triangle inside a circle, it becomes much easier to visualize how the relationship holds as the right-angle vertex moves along the circle.

The key idea to understand is that in the following image, for a unit square, the green and blue areas are always equal to cos²(α), while the orange and red areas correspond to sin²(α). It is evident that the square, with an area of 1, is the sum of its left and right sections.

By opening the GeoGebra resource [Pythagoras proof in a circle (opens a new tab)], you can slide the right-angle vertex (red point) anywhere along the upper half of the circle. 

Law of sines, chords and similar triangles

Recently, I picked up my pencil and notebook and started to draw circles and triangles again. Swiftly, by drawing similar triangles circumscribed by circles, I got interested in their proportionalities, leading me to the law of sines.
For any triangle ABC, where a is the length of the side opposite to angle A, b the length of the side opposite to angle B and c the length of the side opposite to angle C, the law of sines states that:

where d is the diameter of the circle circumscribing ABC, as traced in Figure 1.

Figure 1: Arbitrary triangle ABC circumscribed in its circle with diameter d

My search in literature and web left me somewhat frustrated, for different reasons:

• one often omits to mention the diameter d, in its statement and even in its proof,
• one rarely develops this very elegant statement to closely related geometrical principles, like the intercept theorem or similarity transformations
• the historical background of the law of sines couldn't be checked easily from its original sources.

Why we should not leave out the diameter d of the circumscribed circle

The law is often stated in the reciprocal form and leaving out the diameter.

while in this case, we should really write:

We should have in mind that the sine of an angle is the length of the chord of the same angle inscribed in a circle of unit diameter, as you can teach children with spaghetti.

The law of sines can be understood as a statement relative to proportions between:
- chords traced in a circle with unit diameter,
- and similar chords traced in a circle scaled up by a factor d.

For example, we could trace a circle with unit diameter tangent at A inside the original circle, see Figure 2. With the notation used in this figure, the scale factor d can then be read out easily as different ratios: d = a/a₀ = b/b₀ = c/c₀.

Figure 2: Triangle ABC and tangent unit circle in A

This is one of the intuitions behind the law of sines. and we could view it as a natural law of similarity: "Corresponding lines in similar figures are in proportion, and corresponding angles in similar figures have the same measure."

Of course, if we want to set up a formal proof, we can deduce it from other laws :

1. The inscribed angle:

When inscribed in the same circle, all angles, subtending arcs of the same measure, are equal.

In Figure 2, BC being of the same length as DE, the angles  and are equal.

Therefore sin A = "opposite side over hypotenuse" = a/d.

Q.E.D.

This is also visually explained at "Better explained" or for those who read French "Blog de maths".

2. Central and inscribed angle, with Pythagoras:

This proof involves an additional notion: the central angle.
One can find it on other good sites:
Pat Ballew's blog
Math less travelled

3. Using the height of the triangle
This proof comes in different variations, either through expressing the area respective to the different heights (as given on wikipedia), either through expressing one height as ratios with two different sides (this is the academic proof, example here). Not my favorite one, as it doesn't give any insight in the scale factor d. If you don't need to pass exams, but doing math for fun, please forget this one!

Homothetic transformation with scale factor d

A homothety is a transformation where a geometric entity is transformed a similar version with a scale factor. In Figure 2, we represented the homothety from a circle of unit diameter towards a circle with diameter d. The inscribed lines, triangles and other polygons undergo the same scaling. And thus, we can complement the law of sines with a list of other ratios that also equal d.
In Figure 3, I draw the unit circle at an arbitrary place in space. Then joining similar points. The intersection of the lines is the homothetic center O.

Figure 3: Triangle ABC as a homothety from A₀B₀C₀ centered in O

Now, any line passing through O and intersecting with the small circle, will also intersect with the large circle at similar points (example D and D₀, C and C₀, B and B₀, etc.) The ratios of various line segments that are created if we trace pairs of parallels from these points will be the same, for example:
CD/C₀D₀ = BD/B₀D₀ = AD/A₀D₀ = OD/OD₀ = d
This is the intercept theorem.

Curiously, when searching on the web, both laws, the law of sines and the intercept theorem aren't often associated, while they are, in my opinion, stemming from the same basic principle of conservation of proportions.

I refer to two interesting posts that are related:
At Math is fun: Theorems about Similar Triangles
At Girls' Angle: Do you believe this?

Historical background

And in the history of geometry?

I've looked up sources about Apollonius of Perga, Ptolemy, Regiomontanus, Viete, Coignet (http://logica.ugent.be/albrecht/math/bosmans/R007.pdf), Simson (Elements of the conic sections), Jakob Steiner, but couldn't always find the original sources. I would be interested to have access to them.

The same for a paper by Richard Brandon Kershner. "The Law of Sines and Law of Cosines for Polygons." Mathematics Magazine, vol. 44, no. 3, 1971, pp. 150–153. www.jstor.org/stable/2688227.

Dreaming in Geneva - FQXi essay

The theme for this year's FQXi contest topic is "Questioning the Foundations: Which of Our Basic Physical Assumptions are Wrong?". I had some difficulty to start with this topic (I didn't seem to be the only one, see Ajit Jadhav's blog here and there). I had a lot of things to say about what has gone wrong with physics, which assumptions had to be reconsidered. So, since the opening of the contest, I regularly put some ideas in a draft, being confident that I would be able to arrange them into a coherent thesis for the essay. However by the 20th of August (ten days before closing), I still didn't know how I could write them together into an essay without being suspected of "trotting out my pet theory" (see warning in the Evaluation Criteria).

My "pet theory" is simple: the fundamental entity in physics is "THE quantum particle" which you can represent as an arrow (a vector, a ket). From the mechanical interactions between such rod-like particles, you may deduce all of physics, provided that you assume some complementary parameters (such as the velocity at which two particles fly one from another = c, the length of the rod = Bohr diameter of hydrogen). No mass, no force, no charge, etc. Just paths of rotating arrows that interact with each other through contact (collision). This is the way I reason about photons, electrons, quarks, fields, waves, etc. But I can't reasonably write it that way in an essay. I would need to recall a lot of history of science. So I chose to bring up some ideas that have emerged in history of science that we could reconsider, not necessarily in the same way, but gaining insight with hindsight.

Also I prefer to avoid abstract mathematics when talking physics. Mathematics is just a language, very convenient though, but really just a language that can hinder us in our intuitive understanding. Instead of math, scientists could as well use words, fantasy, dreams, pictures, poems maybe. It is an art and sometimes it is necessary to change the expression of this art. I hope you'll enjoy my dreaming in Geneva.

Forwards multiplying, backwards dividing

My last post on the Morley triangle theorem got encouraging feedback, namely that it showed the living and breathing side of geometry. Although there are a lot of results discovered in previous, sometimes ancient times, the future of geometry is alike the future of life. You can construct its future in many directions, using different languages, without being constrained by impossibilities which show up on some paths. In case of dead ends, it’s up to us to step back, reexamine the fundamentals, and take another path that has not yet been explored.

One of those geometrical impossibilities which many people know of, is the division of an arbitrary angle by 3, with only a compass and an unmarked ruler. My advice is: don't try it using the geometry you learned at school, you'll be caught in a dead end. Of course, you might try it for some time to get experience with it, experiencing by yourself the hopes and frustrations that generations of mathematical inquirers have felt, but don't expect to break through in this way. In order to bypass the impossibility, you need to step back and try it differently. That's how inquiring minds discovered physical tools or paper folding manners that allow to trisect an angle.

Another way to explore trisection is to reformulate the problem. Dividing an angle is the inverse operation of multiplying an angle. So, if we want to solve problems involving the trisection of an angle, we might first focus on solutions to the problem of tripling an angle. This may sound trivial, so trivial that hardly anyone emphasizes this point. Before learning the operation of division, we should first learn how to multiply. Through multiplication, we advance constructively from a unit towards a product of factors. Once we know how we got that product through multiplication, we can divide backwards the product through factorization.

Multiplying geometrically means generating one length (or surface or volume) from another. If we do this recursively, we get a series of successive powers. For example, with straight lines, using the Thales intercept theorem, we can mark successive powers of a number on lines, see Figure 1. If OA'/OA = x and OA is our unit length and OA' = OB, than you can verify that OB = x, OC = x², OD = x³, OE = x⁴, etc.

So forwards, we multiply each time by x. If we go backwards in the series we divide each time by x and we needn't stop at our initial unit length, but we may divide indefinitely and recursively by x. So the geometric representation of multiplication gives us insight on how to divide an initial length OA by x.

We can do something similar with circular arc lengths. For example, for the recursive doubling of an arc, we can proceed as illustrated in Figure 2.

1. Define a unit arc from an origin O on a circle of center P. Draw also a little circle of diameter OP.
2. If from the endpoint 1 of the unit arc, we draw a radius towards P, it intersects the little circle at the (red) point illustrated on the figure. If we double the (red) chord from O to that (red) point, it will end up at length 2 on the large circle.
3. If from the (red) endpoint 2 of the arc, we draw the (red) radius towards P, it intersects the little circle at the (blue) point illustrated on the figure. If we double the (blue) chord from O to that (blue) point, it will end up at at length 2*2=4 on the large circle.
4. Proceeding further (blue, green, violet...) in the same manner will mark successive powers of 2 on the large circle (as well on the small circle).

If we reverse the direction of this recurrence, we can halve the length of the arc indefinitely. The proof of this recurrence can be given using the isosceles triangles, two sides of which are successive chords.

Update 19 November 2016: the conjectured process below is incorrect. This is not the correct way to multiply angles. It only works for doubling and quadrupling angles (may other powers of 2, I didn't check).

By adapting the geometric configuration of the two circles, we can do the same for the multiplication of an arc by any integer number, forwards. There is no impossibility to multiply an angle by an integer. Reversing the direction of the recurrence, it becomes therefore possible to access to the division of an arc by any integer number, and henceforth any rational number. As food for thought, the beginning of the conjectured process is illustrated in Figure 3.

1. Draw a circle of diameter QP.
2. A circle of center P and with radius smaller than QP defines the origin of the arc at one of their intersections O.
3. Define a unit arc from origin O on the circle of center P.
4. If from endpoint 1 of the unit arc, we draw a radius towards P, it intersects the other circle at the (red) point illustrated on the figure. If we prolongate the (red) chord from Q to the (red) point, it will end up at length x on the arc.
5. If from (red) endpoint x of the arc, we draw the (red) radius towards P, it intersects the other circle at the (blue) point illustrated on the figure. If we prolongate the (blue) chord from Q to that (blue) point, it will end up at length x*x=x² on the arc.
6. Proceeding further (blue, green...) in the same manner will mark successive powers of x on the circle.

Playing with angles

A few months ago, I prepared this post but didn't feel ready to publish it. It raised more questions than it answered, so I wanted to have a little more background about it. Hopeless wish! There's so much about angles that I don't fully understand. With pi-day approaching, I'm also looking for some inspiration about circles and angles, so I thought publishing might get me in the direction...

I really became aware of the difficulty to divide an angle into three equal angles (a.k.a. angle trisection) some three years ago. It's bloody hard to find general procedures to divide precisely a circular arc into three equal arcs. There are some tricks that are well documented on the web, but you won’t find a simple geometric procedure using only a compass and an unmarked ruler. It has even been "proven" impossible. We only know of some "cheating" tricks like using a marked ruler (for Archimedes' trisection), origami or mechanical tools that will allow you to trisect precisely an angle. And there is an equilateral triangle that appears "by miracle" when you trisect all angles of a triangle, a result found by mathematician Frank Morley (because he himself tried to better understand or crack that impossibility?). So the angle trisection really became to intrigue me and I progressively dug into it.

One of the questions that has obsessed me is : How do we characterize an “Angle”? I’ve already written a post on that question, pointing out the utility of the chord subtending the circular arc of a circle of unit diameter. The measure of this chord is what we call the sine of the subtended angle. It characterizes the angle and it greatly facilitates our way to count angles, for example by progressing stepwise with chords in a circle.

Roughly speaking there are two ways to manipulate angles with circles. Either you use central angles, meaning that you fix the vertex and you radiate towards the circle in order to subtend an arc, or you use inscribed angles, meaning that you put the vertex of the angle on the circle. Figure 1 shows the equivalence between both methods. The central angle of a subtended arc in a circle of radius 1 is equal to the same angle when it is inscribed in a circle of diameter 1. This equivalence between central and inscribed angles is another way of stating that the measure of a central angle is exactly twice the measure of the inscribed angle, the factor 2 expressing the ratio between diameter and radius.

Figure 2 and 3 illustrate some advantages of inscribed angles. You can put the vertex anywhere on the circle: if the subtended circular arc (or its chord) remains constant in length, the angle remains the same. For the case when the vertex is located between the two endpoints of the subtended arc, you have to add the parts at each side of the vertex.

This flexibility of inscribed angles opens possibilities that remain hidden for central angles. If you combine different circles at intersections with straight lines, you’re simply calculating with angles. Figure 4 illustrates an alternative way to count angles. Starting with an angle α inscribed in a circle of unit diameter, if we draw a unit circle symmetric to the chord of the subtended angle, we can read off an angle (α+α) at the other side of the circle. Adding another circle results in an angle (2α+α) at the far side, etc. When the span of the angle exceeds the height of the unit circle, we can use extra circles in order to take over the continuation of the subtended arc (see the angles > 6α on Figure 4. So this simple procedure allows us to read off integer multiples of any angle.

The interesting thing is that wherever we shift a unit circle in this setting, the opposite subtended arcs differ always by an angle amount α, as shown in Figure 5 (in fact the circle needn't be aligned on the axis). So this allows us to add or subtract any angle to another angle.

2011 and prime number sieves

I'm a slow blogger. It's already been more than a week that we entered the 2011th year of the Christian era. Interestingly 2011 is a prime number. Many bloggers have mentioned the fact that the number 2011 is also the sum of 11 (a prime number) consecutive primes, see for example at Republic of Math, at Pat'sBlog, at The Math less Traveled, at Lucky's Notes, at the Pattern Connection ... and I apologize to those whom I forget, there must be many more.

I would state the obvious if I said that primes are special numbers. But what makes primality so special? Don Zagier, an eminent specialist, said: Upon looking at these numbers, one has the feeling of being in the presence of one of the inexplicable secrets of creation. That's awesome, isn't it? In antiquity, the Greeks already tried to understand the patterns underpinning the prime numbers. A way to visualize those patterns is through prime number sieves. Eratosthenes is said to have proposed the first sieve, the Eratosthenes sieve. (Hey, have you noticed that Eratosthenes was born 2287 years ago? So we are in year 2287 of the Eratosthenian era. Also a prime number). But there are other sieves. More of that later.

Prime numbers occur in nature in cyclic processes. Imagine that, somewhere in the universe, there's a star with a huge number of planets, all of them having the same orbiting speed. They are numbered from 1 to N, planet n°1 being the inner planet and the diameters of their circular orbits are as following:

• The diameter of planet n°2's orbit is twice the diameter of planet n°1's orbit.
• The diameter of planet n°3's orbit is three times the diameter of planet n°1's orbit.
• The diameter of planet n°4's orbit is four times the diameter of planet n°1's orbit.
• And so on, the unit length being the diameter of planet n°1's orbit, the nth planet has an orbit of diameter n, n being an integer between 1 and N.

Imagine also that at some timestamp zero, all the planets are aligned at the same side of the star (yes I know, this is very improbable, but it's just a thought experiment). We define the orbital period of the first planet as unit time. "Prime times" occur when two and only two planets are aligned on the initial direction with the star: the outer planet than has a prime distance to the star (the closest planet being always planet n°1). I should make an animation to visualize this effect. As making an animation is very time-consuming, I looked for other ways to illustrate this principle and ended up with another sieve: a circle sieve. Instead of drawing concentric circles for the orbits, I shift each circle such that all are tangent at the origin (for example the location of the center of the star, see Figure 1). For each period of a planet, I successively copy its circle and place it tangentially to the right (see Figure 2 for the first periods of planet 1 and 2).

Now, as each planet is orbiting at the same speed, the first alignment along the initial direction will occur at timestamp 2. Planet 1 will have circled twice around the star and planet 2 will have circled once. Circles 1 and 2 touch at abscissa 2. Number 2 is a prime number. The next alignment on the axis will occur for planet 3. Circles 1 and 3 touch at abscissa 3 without circle 2 (see Figure 3). Number 3 is a prime number.

The next alignment will occur for planets 1, 2 and 4. Number 4 is not a prime. But number 5 is prime. We can see that easily if we draw only circles 2 and 3 (leaving out the unit circles), see Figure 4. Prime numbers can only occur at vacant places along the axis. The symmetries that show up along this axis help to design primality tests. Any prime number greater than 3 must be of the form 3*2n±1. This formula is a condensed form of the intersection of odd numbers (noted 2n +1) and non-multiples of 3 (noted 3n+{1,2}) and generates all primes between 3 and 5².

The succession of circles n°5 together with n°2 and 3 sieves out all primes between 5 and 7², see Figure 5.

I let you try it further with circles n°7. You will see patterns with concentric circles become more apparent. I like this geometric variant of Eratosthenes' sieve because symmetries can be caught with the eye and suggest prime generating formulas. For example, I came across two simple prime generating polynomials (which must have been noticed by other people because they are less powerful than Euler's P(n) = n² − n + 41):

n²+9n+1

n²+21n+1

This way of sieving has been described earlier and more extensively by physicist Imre Mikoss. Check his work at "The Prime Numbers Hidden Symmetric Structure and its Relation to the Twin Prime Infinitude and an Improved Prime Number Theorem".

Happy prime year!

Math and Physics: Creative Arts?

My favorite approach towards math is through drawing geometric figures like circles, triangles, squares... They link the mathematical abstraction to the physical reality, whatever that may mean. They help to visualize patterns that may remain hidden when described with symbols. Doodling geometric figures allow digressive exploration of otherwise marked out paths and thus favor creative and curiosity-driven approaches towards math. Although I am not a mathematician, I practice math for my job and as a hobby. I subscribe to views expressed by G.H. Hardy in his Mathematician's Apology

I am interested in mathematics only as a creative art.

or by Paul Halmos in his Mathematics as a Creative Art

Mathematics is far closer to an art than it is to the business of equation-solving.

or by Paul Lockhart in his Mathematician's Lament

I’m just playing. That’s what math  is— wondering, playing, amusing yourself with your imagination.

Interestingly, in these quotes, math could be replaced by physics. Concerning me, it would likewise express my interest in physics. What if all of math and physics could be expressed through art? Well, it should.

Geometric representations of higher degree binomials

A binomial is a mathematical expression with two terms, say x and y. The simplest binomial is the sum of those two terms x+y. Powers of this binomial (x+y)ⁿ are ruled by the so-called binomial theorem, which states that the coefficients of the expansion terms x^py^q are given by the Pascal triangle entries. So we have:
(x+y)⁰ = 1
(x+y)¹ = 1.x + 1.y

(x+y)² = 1.x²+ 2.xy + 1.y²

(x+y)³ = 1.x³+ 3.x²y + 3.xy² + 1.y³

(x+y)⁴ = 1.x⁴+ 4.x³y + 6.x²y² + 4.xy³ + 1.y⁴

and so forth ...

But where do these coefficients come from? You can just be satisfied by the result algebraic calculation, or you could look for geometric representations. For instance, a nice geometric explanation is given at the www.mathaware.org site.

Geometric proof of square binomial (original at The Geometry of the Binomial Theorem, Math Awareness Month site)

Geometric proof of cubic binomial (original at The Geometry of the Binomial Theorem, Math Awareness Month site) 

For the second degree binomial, the Pascal triangle entries emerge naturally as the number of squares or rectangles with area given by the expansion terms x^py^q. For third degree binomials, they emerge as the number of cubes or parallelepipeds with volume given by the expansion terms. And for higher degree binomials, they emerge as the number of hypercubes and hyperparallelepipeds given by the expansion terms, so an Euclidean geometric representation for those higher degree binomials seems impossible.

This suggested impossibility prompted me to have a closer look at geometric representations of binomial expansions. As a matter of fact a number to the fourth power x⁴ is just a number, but it is also a square x⁴ = (x²)². And there is no impossibility in representing numbers and squares geometrically, so the suggested impossibility is only an impossibility along the common line of thought, which sees cubic expansions as volumes. Along another line of thought, which seems to have been unnoticed, cubic expansions can be seen as areas, and then geometric representations of higher degree binomials become possible. The following figures illustrate this fact for the special case where x+y is normalized to 1 (for arbitrary x+y, one just has to rescale the figure each time, the pattern remains the same).

For the binomial (x+y)¹ = 1.x + 1.y, we can divide a unit square into two rectangles, one rectangle of area x and one rectangle of area y, see Figure 1.

For the binomial (x+y)² = 1.x²+ 2.xy + 1.y², we can divide the unit square into two squares, one of area x² and one of area y², plus two rectangles of area xy, see Figure 2.

For the binomial (x+y)³ = 1.x³+ 3.x²y + 3.xy² + 1.y³, we can divide each of the previous squares and rectangles into proportions x and y, giving eight rectangles, one of area x³, one of area y³, plus three rectangles of area x²y and three rectangles of area xy², see Figure 3.

For the binomial (x+y)⁴ = 1.x⁴+ 4.x³y + 6.x²y² + 4.xy³ + 1.y⁴, we can again divide each of the previous squares and rectangles into proportions x and y, giving one square of area x⁴, one of area y⁴, four squares and two rectangles (giving six) of area x²y², four rectangles of area x³y and four rectangles of area xy³, see Figure 4.

And we could go further indefinitely, doodling areas of incrementing powers, just in 2D, without any reference to unintuitive hyperspaces.

Volume of a bead vs. volume of a sphere

Imagine you drill a hole through the center of a sphere. The remaining object (a bead, if we think of a little pearl with a hole) has an interesting property as pointed out by Pat Ballew on his blog at the end of this post. Whatever the radius of the initial sphere, the volume of the bead depends only on its height h (see following figure taken from Pat's blog). More precisely, its volume is the same as the volume of a sphere with diameter h. A proof of this property has been given by Pat in a later post.

As I was thinking about it, I thought about another way to prove it, inspired by Mamikon's visual calculus method.

If we think of a plane parallel to the axis of the bead and tangent to its inner cylindrical surface, the intersection of the bead and the plane is a disk of diameter h. If we now rotate the plane around the axis for a whole turn such that it remains tangent to the inner surface, the disk will also rotate a whole turn, as if it sweeps the volume of a sphere of diameter h.

Post scriptum. Afterwards I found a drawing of such a bead in Mamikon's original notes (see second page of his drawings).

Alternative Pythagorean quadruples and other extensions to Pythagoras theorem

The vertices of an arbitrary triangle can be disposed onto two concentric circles such that the base is the diameter of the first circle (which I call the base circle) and the opposite vertex is on the second circle (which I call the leg circle). As there are three bases, there are generally three ways to arrange this setting. I mentioned in my preceding post that we can apply the Pythagorean-like relation a²+b²=c²±2t² to this triangle, where c is its base, a and b are the legs and t is the tangent to the inner circle emanating from the outer circle. For clarity, I reproduce an illustrative figure from my preceding post, for the case where the inner circle is the base circle.

Now, like for the classical Pythagoras relation, there are many features that can be investigated regarding this alternative relation. For example, I would like to better understand the relation of this 2D formula with the Cartesian 3D variant of Pythagoras: x²+y²+z²=d², which is analogous to the case where the leg circle is smaller than the base circle (see Figure 5c of preceding post). Another interesting aspect is the fact that the square of the tangent t² (multiplied by π) determines the area of the annular ring delimited by the base and the leg circles, as stated by Mamikon Mnatsakanian. This fact offers possibilities for areal representations of the squares or circles related to the sides of the arbitrary triangle, like we are acquainted to do with the squares related to the sides of a right triangle. Also, what would be interesting to develop is its relation with the law of cosines (as noticed by Pat Ballew) and the angle subtended by sides a and b. I guess this will leave enough stuff for future posts or conversations, or even papers in specialized journals.

As a conclusion, let me mention Pythagorean numerological features which provide stuff for entertaining puzzles. Maybe you know the Pythagorean triples, those sets of integer numbers (n,m,l) that verify n²+m²=l² and of which (3,4,5) is the simplest instance. The 3D Pythagoras relation allows an extension to Pythagorean quadruples, sets of integer numbers (n,m,l,k) that satisfy n²+m²+l²=k² and of which (1,2,2,3) is the simplest instance. I couldn't resist to look for some integer quadruples that satisfy a²+b²=c²+2t². Among them I found two nice quadruples with successive a,b,c :

(7,8,9,4) for which 7² + 8² = 9² + 2 × 4² = 113

(35,36,37,24) for which 35² + 36² = 37² + 2 × 24² = 2521

A Pythagorean relation for any triangle?

Physics makes extensive use of the Pythagorean law relating the squares of the sides of a right triangle. The well-known a² + b² = c² relation is of special interest for the determination of distances and lengths of vectors, but also for energy conservation laws and Lorentz transformations. There are various relations that resemble the Pythagoras law, but none of them seems to have the usefulness of Pythagoras’ original one, as well as the “beauty” originating from its sole quadratic terms. When the vertex opposite to the hypotenuse runs on the circle determined by the right triangle, we have the Pythagoras relation illustrated in Figure 1.

Some time ago, I was made aware of another interesting Pythagorean law, discovered by Nguyen Tan Tai and which I mentioned in a previous post. For any triangle with sides a, b and c, if the vertex opposite to the base (say c) runs on any fixed circle centered at the center of c, we have the relation:

a² + b² = c² + constant

In order to have a better understanding of this quadratic relation, I tried to re-derive it in my own mental representation. I will call the fixed circle with the running vertex C the “leg circle”, because it is determined by the vertex that is common to legs AC and BC of the triangle (see Figure 2).

When the leg circle is larger than the base circle, the legs AC and BC intersect the base circle and we can decompose the arbitrary triangle into two right triangles, for example as illustrated in Figure 3, the right triangles ABD and BCD.

With the Pythagoras relation we than have:

AC² + BC² = (AD + CD)² + BC²
= (AD² + CD² + 2.AD.CD) + (BD² + CD²)
= AD² + BD² + 2.CD² + 2.AD.CD
= (AD² + BD²) + 2.(AD + CD).CD
= AB² + 2.AC.CD

But the product AC.CD is the power of point C with respect to the base circle, i.e. for every point C on the leg circle, AC.CD is constant and equal to the square of the tangent ray CT (see Figure 4).


If we use another notation, AC = a, BC = b, AB = c and CT = t, the relation becomes a nice equation with only quadratic terms:

a² + b² = c² + t² + t²

One can verify that if the leg circle has same size as the base circle, we retrieve the original Pythagoras relation :

a² + b² = c²

And if the leg circle is smaller than the base circle, we have:

a² + b² + t² + t² = c²

where t is now the tangent ray to the leg circle emanating from any point of the base circle.

We thus have a Pythagoras-like relation for any triangle given its base and leg circles, as illustrated by Figures 5a, 5b and 5c.

Lost theorem about angular proportions

Last week, I came across a so called missing theorem about angular proportions in a triangle, discovered (or rediscovered) by Leon Romain. This triangle construction is presented by user Linelites on youtube.

For such a "Romain triangle", the missing theorem states that, if one of the inner angles is twice another inner angle, one has the property a² = bc +c², where a, b and c are outlined in Figure 1.

The sine chord pattern presented in the Teaching sine function with spaghetti provides helpful insights for the missing theorem. In this pattern, all angles at the intersection of chords are integer multiples of a chosen unit angle or its complement, modulo 90°. All segment lengths in this pattern can therefore be written as sums or differences of cosine and sine products and ratios of that angle. Figure 2 pictures some measures of sine chords in a circle of unit diameter, for an arbitrary angle θ.

Figure 3 shows a Romain triangle for angle α in this pattern. There are numerous other Romain triangles in this pattern. Can you figure them out? With the help of the measures pictured in Figure 2, one can follow visually the elements of a proof for the missing theorem.

Side b is the sine of the chosen angle θ (Figure 4):

b = sinθ

Side c times sin3θ equals b times sinθ (Figure 5):

c = sin²θ/sin3θ

Side a equals b cosθ minus c cos3θ (Figure 6)

a = sinθ cosθ - sin²θ cos3θ /sin3θ

Working out a² and bc +c², one finds that they are both equal to 2 sin²θ cosθ / sin3θ.

The interesting thing is that the sine chord pattern hosts plenty of "missing theorems" about angular proportions, which are only waiting to be (re)discovered.

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Update (June 7, 2010): I corrected Figure 6, which held a wrong term.

Teaching sine function with spaghetti

Sine functions appear everywhere in physics and mathematics. This seems to be related to the circular symmetry of space and to the periodic behavior of dynamical processes. The usual definition is that the sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse in a right triangle (definition at wikipedia). The sine is also often represented as the ordinate of a point running on a unit circle centered at the origin. Any right triangle can be put in that setting. The length of the blue segment on Figure 1 illustrates this definition for an arbitrary angle α, in a circle of unit radius.

It is of interest to have in mind other geometric representations of the sine function. This helps to associate figures with sine and cosine operations or identities. The red segment in Figure 1 shows such an alternative representation, because the sine is also a chord of a circle of unit diameter. When we increment the angle by steps α, one endpoint of the chord advances on the little circle, while the other endpoint remains fixed at the origin, as illustrated by the animation in Figure 2.

When it represents the sine of angle α, the chord in the little circle spans an angle 2α. It is therefore a direct way to the angle multiplication and division by 2. Moreover, the endpoints of this chord can be placed two by two in any other direction on this circle. This enables one to find other variations on the recurrence pattern of the angle in the circle. During one of my circle drawing sessions, I was surprised by the stepwise zigzagging pattern of Figure 3.

Teaching the sine function with this zigzag pattern is particularly suited for daily life situations. Children like it, especially when you explain it at the table.

Keeping track of the circle for integral representations of π

The fundamental constant π is characterized in many ways. Historically, it all began from tentative measurements of the circle's perimeter or area and gradually shifted into more advanced mathematics, in such a way that the link between the circle and modern characterizations of π faded away, see for example the formulas presented at Wolfram MathWorld or Wikipedia.

In the preceding posts, I mentioned infinite products as approximations for π. These may be seen geometrically as exhaustion methods, where the area of a polygon approaches the circular area alternately from above, from below, from above, from below, etc.

There are also integral representations of pi. In such integral representations, π appears in the quantitative value of the integral of a mathematical function. Visually, this is often represented as the area delimited by the bounds of the function. However, the relation with the circle is lost, when viewed under Cartesian coordinates. For example, the graph of the simplest instance of the Cauchy-Lorentz distribution, f(x)=1/(1+x²), "has nothing at all to do with circles or geometry in any obvious way" as quoted from last Pi-day Sunday function from Matt Springer's Built on Facts blog.

In order to view the role of the circle in integral representations of π, we need to switch to alternative ways to visualize math functions. As an example, let's take the constant function y=f(x)=2. The function f maps an element x from a domain to the element y of the target. In this case, for every x, the target y has the constant value 2. With Cartesian coordinates, we are used to represent this function as a horizontal straight line, like in Figure 1a (click on the figure to view it enlarged). If however we write it as R=f(r)=2, where the function f maps any circle of radius r of the domain to a target circle of radius R=2, the same function can be viewed as a circle of constant radius, like in Figure 1b. So the same function f can be equally well viewed as a straight line or as a circle (x, y, r or R are only dummy variables).

Now if we take another example, the linear function, y=f(x)=2x, we are often used to view it in Cartesian coordinates as a straight line with slope 2, like in Figure 1c. In the circular representation R=f(r)=2r, this works however differently. Because we are relating circles of the input domain to other circles of the target, for each circle of radius r, we need to draw the target circle of radius 2r. A single line won't do. For one value of r, we need to draw two circles. If we use blue circles for elements of the input domain and red circles for elements of the target, we could visualize it for successive values of r as an animation like in Figure 1d. In that way, we view the progression of the target circle as the input circle becomes larger.

Unlike the Cartesian representation which shows the progression of a function in a static graph, this circular representation needs a dynamic or recurrent process to get grip of the progression of the function. Therefore it isn't very adapted for illustrations in print media. On the other hand, it has the advantage of keeping track of the geometrical form of the circle. And that's exactly what we need in order to perceive the circular nature when π shows up in mathematical functions. The relation of the integral of the Cauchy-Lorentz distribution f(r)=1/(1+r²) with the circle can then be seen with the help of the geometric counterparts of arithmetic operations like addition, squaring and dividing. A convenient procedure is illustrated in the successive steps of Figure 2.

Step 1. Draw the input circle of radius r and the reference circle of radius unity.

Step 2. Determine r².

Step 3. Add 1 to r².

Step 4. Invert (1+r²). We now have the target circle of radius R=1/(1+r²).

Step 5. Find the target ring related to the input ring ranging over [r, r + dr]. This yields a ring of width dr/(1+r²). The location of this ring depends on the relative progression rates of r and r² (I've not yet found a straightforward explanation for this determination).

Step 6. Integrate dr/(1+r²) for r running over all space. For r becoming larger and larger, the summed area tends towards the area of a circle of radius 1. For the positive half plane, this corresponds to the π/2 value found analytically.

The tricky step seems to be the way how to relate the progression between r and 1/(1+r²) in steps 5 and 6. One can verify for example the value of the integral at intermediate steps. For the integral from r=0 to 1, the value in the positive half plane must be π/4, which can be verified on the figure below.

In order to gain more insight on π, it could be of interest to develop skills for this circular representation.