Proof without Words: The Law of Cosines

(I really need to put this onto a poster.)

Posted 17 July, 2012 by Blue in Trigonography

(Almost) Everything you need to remember about Trig …

The “Complete Triangle” figure —which represents the six trig functions as lengths of segments— is something of an obsession of mine, and has been for some time.

It’s the basis of my triple-ricochet company logo and name of my iOS-specific brand, tricochet, and it’s the subject of my very first iOS app (one of the very first in the App Store when it opened), Trigger … which is actually the latest evolution of what was once a HyperCard stack I designed on my Mac Classic. The Complete Triangle and I are very old … er, um … I mean, very old friends.

My fascination with the Complete Triangle as a pedagogical aid is that it encodes a great deal of information about the trigonometric functions, not the least of which is the reason “tangent” is called tangent. (I hate that textbooks define tangent as “sine-over-cosine”, because it strips the word of its meaning and adds to the perception of mathematics as a field of arbitrary buzzwords. But I digress …) The seven similar right triangles that make up the Complete Triangle neatly illustrate many of the fundamental trigonometric identities, such as

$$\sin^2 + \cos^2 = 1 \qquad \text{and} \qquad \frac{\tan}{1}=\frac{\sin}{\cos}$$

(While “tangent is sine-over-cosine” is a lousy definition, it’s a perfectly-good theorem.)

As a dynamic figure with an adjustable angle, the Complete Triangle aids intuition about the overall nature of each individual trig function, such as its range, and relations among them, such as their relative sizes. I could go on and on.

In fact, I did.

With the original release of Trigger, I drafted a 12-page(!) discussion of the Complete Triangle and various trigonometric properties that can be observed within it: “(Almost) Everything You Need to Remember about Trigonometry, in One Simple Diagram”. While I still haven’t gotten around to finalizing the text, my goal is eventually to turn the document into an interactive iBook.

Obviously, I highly recommend working this stuff into any trigonometry curriculum; when I’ve given guest lectures at schools on this topic, students have raved —especially when the lectures came right before exam time— about how the figure is not just a handy mnemonic, but a true encapsulation of much of what they’d been studying, and about how “it all makes sense now”.

Posted 17 July, 2012 by Blue in Trigonography

About the “Game-ifying Math” Category

I’ll edit this later with a proper description of the “Game-ifying Math” category.

Posted 16 July, 2012 by Blue in Game-ifying Math

About the “Harmonious Figures” Category

These articles derive from work I did for my masters thesis on “affinely-regular” figures, which are things like regular polygons and Platonic solids after the application of an affine (or just linear) transformation. The primary result there was that one can decompose any figure (say, a polyhedron) into a “sum” of affinely-regular versions of that figure: a random quadrilateral is the “sum” of parallelograms. Even better, it turns out that one can decompose an affinely-regular figure into a “sum” of actually-regular versions. (See my post on the Theorem of Barlotti.) Ultimately, these two results imply that any figure is the “sum” of regular versions of itself: squash and stretch and distort a cube any way you like (while keeping the edges straight), and it’ll be the sum of un-distorted cubes of various sizes and orientations. Very cool!

“Harmony” is a term that generalizes the basic notion of regularity. An harmonious realization of a (combinatorial) graph is one in which all automorphisms of the graph are induced by isometries of the realization. That is, an harmonious realization is the best-possible way to draw the graph, since it shows all the automorphic structure; this can be helpful for visualization purposes … except when the realizations exist in more dimensions than we can see. Generating harmonious realizations turns out to be a pretty straightforward —though not-well-known— exercise in the eigen-analysis of a graph’s adjacency matrix. (See my post on “spectral” realizations … if only for the pictures.)

Posted 16 July, 2012 by Blue in Harmonious Figures

About the “Hedronometry” Category

By the end of high school, most students should be familiar with the “three-dimensional” Pythagorean theorem (aka, “the distance formula”) that relates the length of the diagonal of a shoebox to the lengths of its edges. (Students should also know that this generalization extends to relate diagonals and edges in “any-dimensional” shoeboxes; always, the square of the diagonal equals the sum of the squares of the edges at a corner.)

However, view students seem to be aware of this result:

Given a “right-corner tetrahedron”, the square of the area of the face opposite the right corner is equal to the sum of the squares of the areas of the other three faces.

Pythagorean Theorem for Right-Corner Tetrahedra

\(X^2+Y^2+Z^2=R^2\)

This result extends to “any-dimensional” space as simply as the distance formula: the square of the hypotenuse is equal to the sum of the squares of the legs. However, unlike the distance formula —which only ever relates one-dimensional lengths— the values being squared here are increasingly “dimensionally-enhanced” aspects of a figure: lengths in 2-d, areas in 3-d, volumes in 4-d, hyper-volumes in 5-d, and so forth.

I was thrilled as a high school junior when I discovered this result, and I was devastated as a college freshman when I discovered that others had beaten me to it by at least a century. Nevertheless, in the years —decades!— since, I’ve worked off and on to find related, completely new, results in the field I call “(Tetra)Hedronometry”, the dimensionally-enhanced trigonometry of tetrahedra. I’ve had a bit of success —in particular, in the realm of non-Euclidean geometry, where (so far as I can tell) absolutely nothing was previously known— and I believe my focus on tetrahedral “pseudo-faces” is a significant contribution to tetrahedral lore. We’ll see.

Posted 16 July, 2012 by Blue in Hedronometry