The joy of joymetry-2

Previous

We can prove the theorem very simply, as follows.

Let’s go back to the tilted square sitting on the hypotenuse.

At an instinctive level, this image should make you feel a bit uncomfortable. The square looks potentially unstable, like it might topple or slide down the ramp. And there’s also an unpleasant arbitrariness about which of the four sides of the square gets to touch the triangle.

Guided by these intuitive feelings, let’s add three more copies of the triangle around the square to make a more solid and symmetrical picture:

Let’s recall what we’re trying to prove: that the tilted white square in the picture above (which is just our earlier “large square”— it’s still sitting right there on the hypotenuse of the four triangles along the corners of the bigger square) has the same area as the small and medium squares put together. But where are those other squares? Well, we have to shift some triangles around to find them.

Think of the picture above as literally depicting a puzzle, with four triangular pieces wedged into the corners of a rigid puzzle frame.

In this interpretation, the tilted square is the empty space in the middle of the puzzle. The rest of the area inside the frame is occupied by the puzzle pieces.

Now let’s try moving the pieces around in various ways. Of course, nothing we do can ever change the total amount of empty space inside the frame — it’s always whatever area lies outside the pieces.

The brainstorm, then, is to rearrange the pieces like this:

All of a sudden the empty space has changed into the two shapes we’re looking for — the small square and the medium square. And since the total area of empty space always stays the same, we’ve just proved the Pythagorean theorem!

This proof does far more than convincing; it illuminates. That’s what makes it “elegant.”

Next

The joy of joymetry-1

Is Geometry your favorite math subject in high school?!

Many people, I met over the years, have expressed affection for Geometry. Arithmetic and Algebra? — not many takers there, but geometry, well there is something about it that brings a twinkle to the eye.

I think it’s because geometry kindles the right side of the brain, which appeals to visual thinkers who might otherwise cringe at its cold logic. But other people tell me they loved geometry precisely because it is so logical. The step-by-step reasoning, with each new theorem resting firmly on those already established — that’s the source of satisfaction for many. For those who couldn’t prove certain theorems, the final verse hence proved was a consolation and trick to confuse the examiner (I am told students still follow it.)

My hunch is that people enjoy it because it marries logic and intuition. It feels good to use both halves of our brain isn’t it?

To illustrate the joys of joymetry, let’s revisit the Pythagorean theorem, which you probably remember as a² + b²=c². Part of the goal here is to see why it’s true and appreciate why it matters. Beyond that, by proving the theorem in two different ways, we’ll come to see how one proof can be more “elegant” than another, even though both are correct. =
The Pythagorean theorem is concerned with “right triangles” — meaning those with a right (90-degree) angle at one of the corners. Right triangles are important because they’re what you get if you cut a rectangle in half along its diagonal:

And since rectangles come up often in all sorts of settings, so do right triangles.

They arise, for instance, in surveying. If you’re measuring a rectangular field, you might want to know how far it is from one corner to the diagonally opposite corner. (By the way, this is where geometry started, historically — in problems of land measurement, or measuring the earth: geo = “earth” + metry = “measurement.”)

The Pythagorean theorem tells you how long the diagonal is, compared to the sides of the rectangle. If one side has length a and the other has length b, the theorem says the diagonal has length c, where

For some reason, the diagonal is traditionally called the “hypotenuse,” though I’ve never met anyone who knows why. (Any Latin or Greek scholars there?)

Anyway, here’s how the theorem works. To keep the numbers simple, let’s say a = 3 yards and b = 4 yards. Then to figure out the unknown length c, we add 3² and 4², which, in effect is 9 plus 16. (Keep in mind that all of these quantities are now measured in square yards, since we squared the yards as well as the numbers themselves.) Now, since 9 + 16 = 25, we get c² = 25 square yards, and then take square roots of both sides. This yields c = 5 yards as the length of the hypotenuse.

This way of looking at the Pythagorean theorem makes it seem like a statement about lengths. But traditionally it was viewed as a statement about areas. That becomes clearer when you say it the way they used to say it:

“The square on the hypotenuse is the sum of the squares on the other two sides.” (Math teachers note this)

Notice the word “on.” We’re not speaking of the square “of” the hypotenuse — that’s a excessively modern algebraic concept about multiplying a number (the length of the hypotenuse) by itself C x C. No, we’re literally referring here to a square sitting on the hypotenuse, like this:

Let’s call this the large square, to distinguish it from the small and medium-sized squares we can build on the other two sides:

Then the theorem says that the large square has the same area as the small and medium squares combined.

   Since time immemorial, this marvelous fact has been expressed in a diagram shown below.

Next

Why does a ship float?

Eureka! Eureka! cried Archimedes, jumped out of his bath tub and ran naked. Perhaps he found an answer to a long standing problem in his mind! Now we know his answer as the Archimedes principle.

Principle:

A body wholly or partly immersed in a fluid, undergoes a loss in weight equal to the weight of the fluid it displaces.

(For e.g.)
An aluminium cube of 1ft length weighing 168lb (Fig 1a), when immersed in water (1cubic feet of water =62lbs) undergoes a weight loss equal to the weight of the displaced water , now its weight has apparently decreased to 106lb [168 lb-62lb] (fig 1b)

Principle of floating:


If a body, on being immersed in a fluid would displace a volume of fluid whose weight is greater than that of the body concerned, then that body will float on the fluid. In other words a body floats when it sinks to such a depth that the displaced fluid weighs exactly as much as the floating body. Buoyancy (acting upward) is said to be in equilibrium with the weight of the body.

For e.g.

A 1ft wooden cube (fig 2) weighing about 50lb will float in water, when the submerged part of the cube displaces a volume of water weighting 50lb, counter- balancing the weight of the cube.

What’s the case with the ship?

Apart from floating, a ship must additionally be able to reorient itself after being swung to an inclined position by external force such as wind pressure.
Fig 3a shows the ship in normal position. The weight of this ship acts downward at its centre of gravity S. The counter balancing upward force acts at the centre of buoyancy W, which is the centre of gravity of the displaced volume of water. In normal position (Figure 3a) the points S and W are on the same vertical line. When the ship heels over (Figure 3b & 3c) the centre of gravity of the displaced water shifts to a different position W’.

The upward movement acting here strives to rotate the ship around its centre of gravity S. the intersecting point of the upward force A with the ships axis of symmetry ( vertical dotted line) is called the metacentre M. If the metacentre is located above the centre of gravity S (Fig 3b) the ship will float and return to its normal upright position. On the other hand if the metacentre is below the centre of gravity S (Fig 3c) the ship will capsize when it heels over.

Reference:
An illustrated encyclopedia of technology, Heron books. C.Van Amerongen.