An Elegant Proof of the Pythagorean Theorem by a Former US President

By Anupum Pant

Abraham Lincoln, the 16th president of the United States, was probably a math whiz. Doesn’t it sound like an extremely rare combination of things a person could possibly be? – a president and a math whiz. And yet, he was not the only one. James Garfield, the 20th president, was very much into mathematics too.

Garfield wasn’t a professional mathematician. He was a president. But, much like Abraham Lincoln, he was very much into geometry! Before he went into politics, he wanted to become a mathematics professor.

While he was a member of the US house of representatives, five years before he was elected president of the US, he came out with a very elegant and unique proof of the Pythagorean theorem (yes, another one of those Pythagorean theorem proofs). Here’s how he did it with the help of a congruent flipped triangle…

Doing it with a piece of paper is really easy…

Pythagorean theorem proofFold a paper and cut 2 exactly same right-angled triangles out of it. Now, put them together as shown in the image here (click the image). Next, write down the area of the trapezium – (a + b) . ½(a + b) – 1

Now write the area of all the 3 triangles and add them. This is what you’d get – 2 x ½ ab + ½ c – 2

Since both these areas are same, just written in a different way, equate them and solve. You’ll end up with the Pythagorean theorem!

a2 + b2 = c2 

Or, simply watch the video to understand better…

A Few Things About Sloths Everybody Should Know

by Anupum Pant

A few days back, on 20th October, Sloth Day was celebrated all around the world. You’d be thinking, what is so good about these strange animals, that makes people have a special day around them. Well, in that case, you need to read this.

What are these creatures?

Sloths are slow animals that make even cows look extremely active. They are so slow that they are almost stationary and algae grows on their hair. Most of their life is spent on trees hanging upside down. They hang on trees to protect themselves from the predators on the ground. Their bodies are so well engineered to stay inverted that the hair on their bodies, is oriented in the opposite direction – growing from stomach to back (This helps them to stay dry by draining water easily). Even dead sloths have been know to retain their grip and remain suspended after death. They come down only around once a week to excrete. They eat, sleep, travel, find partners, mate, give birth and even raise young ones in the canopies.

Although sloths might seem gross, creepy and unseemly, they really aren’t that bad. Sloths are sweet looking [1] [2] [3] animals (especially their babies, they are adorable) who can also swim efficiently and move wisely. We can definitely learn a lot from them.

Their diet is unbelievable

Sloths eat only leaves throughout their lives. They chew leaves slowly like cows to extract whatever nutrients they can. Sloth intestines are also adapted to extract the maximum out of their poor quality food, they are unusually long. They often like to shift to a different kind of leaf after a day or two. This balances their nutrient intake. Humans couldn’t possibly survive on a leafy salad diet for a very long time.

To save energy, sloths drop the temperature of their bodies at night. Even their bodies have more bones than muscles to prevent wastage of energy through muscular movement. After the Orangutan they are the most energy efficient animals.

Other facts about them

Sloths have blunt teeth to chew leaves properly, have large claws to hang on to branches and inverted fur orientation (as also mentioned before). Another interesting thing about them is that they have remained physically un-evolved for a long time because they don’t really have to compete with anyone else for their diet.

Mutualisms

This is where the awesomeness of Sloths come in. Sloths are home to a several kinds of other organism (tiny ones living in their fur). These organism depend on sloths (hosts) for various things and in turn provide an advantage to their hosts. This is called mutualism.

  1. Algae + Sloth – Algae, for instance, uses the long grooves on sloth hair to grow with a secure footing. As a rent for this safe apartment, the algae gives them [sloths] a nice shade of green color to camouflage on trees. This and their still bodies make them virtually impossible to spot with the naked eye. The camouflage protects them from eagles.
  2. Bacteria + Sloth – Apart from the several other bacteria which live inside a sloth to digest the leafy diet, two kinds of Cyanobacteria live on sloth furs too. These bacteria also give sloths a nice gray hue which helps them in the same ways as above.
  3. The Sloth Moth – The Pyralidae Moth also live on Sloths. These feed on the algae which grows on the fur. In return for the good food, moths give them nothing. Yes, nothing. This is called Commensalism.
  4. Others – Similarly, various other organisms like flies, mites and three types of beetles are often found living in a Sloth. Up to 900 beetles have been found on a single Sloth!

There is so much more to write about these amazing little creatures who provide for so many other creatures too. I’ll keep it for the second part that I’ll write some other day. So the next time you see a Sloth crossing the road, carefully pick it up by holding its mid body and gently place it on a tree. Remember to use a glove/cloth.