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.

It From Bit or Bit From It? FQXi essay

2013 FQXi essay contest is announced. Topic "It From Bit or Bit From It?" This topic is somehow connected with the 2011 topic "Is Reality Digital or Analog?", not my favorite one, as I have grown my conviction that the it from physics is what really what underlies reality. Talking about "Bits" then just talking about data, information, sensations perceptions, formulations that originate from the "It"

I've been writing FQXi essays three times in row:

• "Ordinary Analogues for Quantum Mechanics" in 2009, for the "What's Ultimately Possible in Physics?" essay.
• "Reality Will Ultimately Be Analog and Digital" in 2011, for the "Is Reality Digital or Analog?" essay.
• "Dreaming in Geneva" in 2012, for the "Which of Our Basic Physical Assumptions Are Wrong?" essay.

Never in the prizes, but enjoying the writing. This time, I think I'll pass my turn as I need to finish my PhD thesis before this summer, which enables me to apply some ideas of my 2009 essay to some unsolved experimental issues in semiconductor nanophysics. A question of focus.

For those who'll compete, enjoy and good luck!

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.

Auguste Bravais - 201st birthday anniversary

On August 23, 1811, during a relatively calm period of Napoleon's reign, Aurélie-Adelaïde Thomé, spouse of physician François-Victor Bravais, gave birth to Auguste in Annonay. Annonay is located just south east of the Pilat massif, in the French mild climate department of the Ardèche.

Some 30 years earlier, the people of Annonay witnessed the first public hot-air balloon flights, as it was the hometown of the Montgolfier brothers. Buth both brothers died before Auguste was given the privilege to nest in Annonay. As last one, Joseph-Michel died almost one year earlier. Auguste surely benefited from the scientific entrepreneurial spirit of that town.

Stanislas College caption

He was sent to Paris for his studies, first at the college Stanislas. And consequently was admitted to Polytechnique.

Auguste Bravais is best known for pointing out that there are in total 14 types of crystallographic lattices. His ordering and denomination of lattices is still in use today.

In his young years, his main interest was in meteorological observations. At age 10, he climbed alone the Pilat mountain hoping to better understand cloud formation. Later in his life, together with two other scientists, he participated in the first scientific mission at the top of the Mont-Blanc, as well as in numerous observations on the Faulhorn with his brother Louis.

With Louis, he also shared a passion for botany, which was given to them by their father. Together, they investigated the arrangement of leaves on the stem of plants, which shows Fibonacci series in their spiraling. They came to the conclusion, that the leaves were never really growing vertically of each other. There was a prevalent tendency that two successive leaves were following each other on a spiral at 137.5 degrees (or, which is the same, at 222.5 degrees counter-wise). This result they published in 1835.

In 1868, Wilhelm Hofmeister gave an explanation for that angle, now known as Hofmeister's rule: as the plant grows, each new leave originates at the least crowded spot. A very natural law...

Sadly, for the last ten yours of his life, he lost his intellectual capacities, being aware that he could not fulfill the redaction of all his scientific work. He was said to start work at 4 o'clock in the morning with a lot of caffeine  The lack of sleep surely didn't arrange things. He died March 30, 1863 near Versailles.

Louis de Broglie - 120th birthday anniversary

Exactly 120 years ago, on August 15th, 1892, Louis de Broglie was born in Dieppe, a little town on the coast of Normandy. De Broglie is one of my favorite physicists because he has tried to conciliate quantum theory with intuition. He entered the physics stage after the first World War, where he had served as radiographer on the Eiffel tower. That stimulated his interest in electromagnetic radiation questions. At that time, it became clear that electromagnetic radiation could be explained as well by wave mechanics (constructive and destructive interference as evidenced by Thomas Young in 1803), as by a collection of particles (photoelectric effect explained by Albert Einstein in 1905). Louis de Broglie made an important following step: if light had dual wave-particle behavior, matter also should have that duality.

De Broglie tried to interpret this duality as phase matching between a particle embedded in a wave, the pilot wave. There should be phase matching between both: "les photons incidents possèdent une fréquence d’oscillation interne égale à celle de l’onde (my translation: the incident photons have an internal oscillation frequency equal to that of the wave)". He saw photons, as well as electrons, as little clock-watches embedded in their wave. I am sure this intuition will lead to new physics in the future, because this aspect of duality has hardly been investigated, see Couder's bouncing droplets in pilot wave. Personally I am working with this pilot wave idea in order to explain some properties of quantum dots.

As Louis de Broglie lived his last years in a little town, Louveciennes, that is close to where I live, I had a walk there today. Maybe I could find some place related to him. Unfortunately, I didn't find the exact location of his residence  (please drop a comment if you know). But surely the scenery of the pictures below near to the royal residence of the Manoir du Coeur Volant must have been very familiar to him.

Manoir du Coeur Volant

Abreuvoir of Marly-le-Roi

Royal Domain of Marly-le-Roi

Commemoration plaque of the Manoir du Coeur-Volant

Electronic vibrations in ski poles

Two weeks ago, I was skiing in the Vosges mountains. The ski resort Lac Blanc is crossed by 400 kV high tension transmission lines which run over the ski trails at a height about 8 - 10 m. While waiting under them, I had the surprise to "feel" electronic vibrations in the ski pole with the tip of my fingers, as if bunches of electrons were running back and forth on the surface of the pole. Experimenting a bit with them, I noticed that the ski pole had to be planted in the ground (or the snow in this case) to set up these vibrations. For ski poles where the tip was isolated with a plastic material, there were no such vibrations. Also, this worked whatever the orientation of the pole, parallel or perpendicular to the transmission lines, which I found quite surprising. Whatever, I tried not to dwell too long under those lines, not sure to which extent these transmission lines affected my neuronal electrons ;-)

Spinning dancers around poles

Some ideas for a spinning dancers choreography, representing spinning electrons with spin up and down, inspired by Pauli exclusion principle: http://commonsensequantum.blogspot.com/2010/10/explaining-electron-spin-and-pauli.html

All electrons spinning at same speed:

With an excited electron, spinning twice the speed of other electrons. At that speed, it doesn't disturb the dance:

To be continued with perturbing dancers representing laser light.