The joy of joymetry-2

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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.”

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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.

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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.

The Cell City

Welcome to the ancient city called the eukaryotic cell. Through this guided tour I will take you to various tourist spots and give you an overview of what role they play in sustaining this city. Although ancient, it is an ever evolving city equipped with all modernity. All the major places in the cell are in general called the organelles. Let us now enter the city.

THE CELL (modified from enchantedlearning.com): A Roadmap

Cell membrane :The City wall. The city is surrounded on all sides by this highly guarded wall. At the gates are the watch towers, called the channels, made of proteins which check the identity of every molecule entering and exiting the cell. And not everything can pass through it. The cell membrane is mainly made up of lipids and proteins. Some small molecules though can easily pass through the membrane especially if they are small lipid molecules, similar to the ones that constitute the membrane. Plant cells, fungi and certain bacteria are further protected by another stronger structure called the cell wall. Unlike cell membrane they are made of cellulose. Let me zap you to 3nm size and make you lipophilic(the lipid molecules will like you this way and allow you enter the city)

Cytoplasm: Stick together and keep up with me, now that we are entering the hustling and bustling streets of the cell. This is called the cytoplasm. And all those molecules (ATPs, glucose, oligosaccarides, amino acids, RNAs) are rushing around to carry out various activities required for sustaining the cell. Some of them simply diffuse across and other more important ones take the cell’s railroad called the microtubules and the actins.

Nucleus: This is the administrative centre of the cell, where all the major decisions for running the city smoothly are made. Designing, planning and execution of all the laws and strategies are done in accordance to the constitution called the DNA, which contains the complete information to run any given cell city. Only some of the rules are used to run a given city (called the cell type). The nucleus is also surrounded by a wall called nuclear membrane and heavily guarded. The sign post reads RESTRICTED ENTRY. Only the things that the nucleus requires or asks for can enter into it. The sentries here are proteins called porins who check for special ID (tags) on each of the molecules awaiting an entrance. The information in the rule book to be conveyed to the rest of the cell has to be converted to RNA (a language understood by other components in cell) which is then translated to proteins, which are the actual executioners of the information conveyed by the DNA. Apart from DNA, nucleus has one more component called the nucleolus.

Nucleolus: The assembly of the components of an important machine of the cell, the Ribosome, happens here. Why so much care and importance for just this one organelle? Maybe because ribosome is one of the most important machines in the city, which helps in making proteins.

Ribosome: Now, have a look at one of the finest and efficient machine you will ever see- the ribosome. These are responsible for the production of proteins from amino acids (the building blocks of proteins). The mRNAs (messenger RNA, which carry the message/information from DNA) come here and the Ribosome recruits RNA called transfer RNAs (tRNA) which bring in the relevant amino acids to help manufacture proteins (translating the message from DNA). The platform for ribosome to manufacture its products is provided by the endoplasmic reticulum(ER).

Endoplasmic Reticulum (ER): The industrial park. Here is where the protein-making ribosomes are also located. ER is a chain of industries involved in processing and storage of proteins. From the ribosome the proteins move into the ER, where they are helped to assume their final structure (structure of the protein is essential for its function) and some proteins get processed further. During processing various changes like glycosylation (attaching sugar units to protein) is carried out, that are required for the proteins to do their respective jobs. Of course only proteins with tags and the glucose units are allowed inside. Structurally ER is a huge membranous network and hence the name. There are two types of ER: smooth ER and rough ER. Rough ER is so called because it is studded with ribosome on its surface and hence mainly work as processing units and smooth ER (without ribosome) is mainly for storing proteins. Some cells of your body like the muscle cells have a special type of ER called the sarcoplasmic reticulum which acts as reservoir for calcium ion.

Mitochondria: The PowerStation. The functioning of the city requires power supply which is manufactured here inside the mitochondria. Mitochondria have more roles to play in the cell apart from energy generation. You can read more on it here. Some cities like the cells of plants have an additional solar power plant called the chloroplast that uses the sunlight to produce energy (ATP and NADPH) and which in turn is used for producing sugars.


Golgi apparatus: The Postal service. Golgi bodies can process lipids and proteins from ER and sort them to various regions (called “vesicular transport” in case of proteins from ER). Like the post offices, golgi adds address (labels) to the proteins and lipids which are then transported by the cells transportation system comprised of microtubules and actins to their destinations on time.

Lysosomes: The recycling plant. The cell is maintained spic and span by these. They eat up the old organelles, the bacteria that are engulfed by the cell and other cell debris. Lysosomes are loaded with acids(pH 4.5, cell’s pH is 7.5) and enzymes which act as cleaners. Apart from cleaning, they can also help the cell in digesting the ingested food (similar to what your stomach does). There is one more type of recycling plant called the peroxisomes which are very similar to lysosomes. These are the detoxifying units where the toxic peroxides are got rid of.

Centrosomes: Meet the architects of the city. Also called the microtubule organizing centre, it is the region from which microtubules are manufactured for connecting the various regions of the cell. Centrosomes not only help in connecting the regions of cell through the railroads but also look after the maintenance of the structure of the cell. They are also the junction where all railroads converge. Apart from this, centrosomes are vital during cell division (mitosis and meiosis). They form string like structures called mitotic spindles that help in pulling apart and hence distributing the chromosomes (Condensed DNA) equally to the daughter cells.

Vacuoles: Lastly the warehouses, where the food particles are stored. Even engulfed bacteria to be later thrown out are kept segregated here. They also serve as water towers and also play a role in exporting things outside the cell. Their exact usage depends on the type of cell concerned.

That’s it folks, a small tour around the cell city. Hope you enjoyed it. On exiting through the cell membrane you will be zapped back to your original size. Looking forward to meeting you again.