Proof Without Words - Pi

In mathematics there is a concept known as ‘Conformal Mapping’ which allows you convert a given shape to a completely different one by making a transformation.

In the joukowski transform you take all the points on a circle and apply the following transform:

And the resulting transformed points resemble an aerofoil shape. Pretty cool huh ?

** Conformal mappings are a really cool topic in complex analysis but also equally extensive. If you want to know more about them click here

Blackest is the new black: Scientists have developed a material so dark that you can't see it...

Puritans, Goths, avant-garde artists, hell-raising poets and fashion icon Coco Chanel all saw something special in it. Now black, that most enigmatic of colours, has become even darker and more mysterious.

A British company has produced a “strange, alien” material so black that it absorbs all but 0.035 per cent of visual light, setting a new world record. To stare at the “super black” coating made of carbon nanotubes – each 10,000 times thinner than a human hair – is an odd experience. It is so dark that the human eye cannot understand what it is seeing. Shapes and contours are lost, leaving nothing but an apparent abyss.

Thought this was important to post.

Crazy Looking Bamboo Tower Creates 25 Gallons of Drinking Water Per Day From Thin air

This crazy looking tower creates 25 gallons of drinking water per day from thin air. It’s basically an atmospheric water collector which gathers dew from the air.

“…The 9 m tall bamboo framework has a special fabric hanging inside capable to collect potable water from the air by condensation…”

It’s called the WarkaWater:

“…The name ‘WarkaWater’ comes from the Warka Tree, a giant wild fig tree native to Ethiopia, traditionally used for public gatherings and school education. The Warka Tree is an archetype of the Biennale theme ‘Common Ground’…”

The simple and practical, yet elegant design powers out ahead of any of the commercial atmospheric water generators on the market which cost thousands more to build than this.

This is a wonderful water generation idea, that’s inexpensive, and actually beautiful to look at.

Water is life, and being designed after a tree. This is a real Tree of Life.

It’s easy:  Form a triangle, then a hexagon, then a bicycle wheel

This is the general mechanism on how a spider spins its web (talk about engineering, right?). When an insect gets caught in its web, the vibrations caused by the insect is felt by the spider which then rushes to engulf its prey.

Now here is the trippy part ; This is the effect of drugs on the pattern of the web.

Hope you are having a great week. Have a good one!

* Spider spinning a web (video) (if you find a better full video let us know)

** Spiders on drugs -  NASA article ; Video

How did the Greeks know ?

Greeks had a strong geometric approach towards problems and as a result their methods are very intuitive.

In this post, we will look at the Method of exhaustion formulated by Archimedes that stands out as a milestone in the history of mathematics

Method of Exhaustion - Archimedes

                                                       Source

In order to find the bounds of pi, Archimedes came up with a remarkably elegant ‘algorithm’, which is as follows:

Lower bound

Inscribe a n-sided polygon in a circle —> Measure its perimeter(p) —> Measure its diameter(d) —> pi_min = p/d —-> Repeat with  n+1 sides.

Upper bound

Circumscribe a n-sided polygon in a circle —> Measure its perimeter(p) —> Measure its diameter(d) —> pi_max = p/d —-> Repeat with  n+1 sides.

And by following this procedure one could obtain the upper and lower bounds of pi !

Heres an animation made on geogebra for a circle of diameter 1. Watch how the lower and upper bounds vary.

Archimedes did this for a 96 sided polygon and found the value of pi  to be between 3.14103 and 3.1427. This is a good enough approximation for most of the calculations that we do even today!

Happy Holidays !

Handy Tool Definitions

I found some handy definitions of tools on edn.com for those of you who are unclear about them. So, I put together this reference guide. 

http://ow.ly/VyRt30dOlwi 

But that’s not all it can do. Microsoft and NASA teamed up to “bring” you, yes you, to Mars.

Follow @the-future-now

There's an app for that

A “master list” of apps that are essential to student life.

Flashcards

Quizlet - Flashcard app that has pre downloaded flashcards and you can make your own flashcards. Available on multiple platforms

Duolingo - Language flashcard app

Flashcards+ - Another flashcard app

Math

Luna Calculator - calculator app ($2.99 on iOS)

Wolfram Alpha - online math solver

Symbollab - online math solver

Productivity & Planning

Forest - Grows a tree for every thirty minutes you are off your phone. If you touch your phone, the tree dies. ($0.99 on iOS)

FocusNow - Free version of Forest for iOS. Instead of trees, it grows fruits and vegetables. 

30/30 - Great time management app

Any.Do - Great app for creating to-do lists, organize tasks, syncs with other devices. You can also change the priorities on certain tasks. 

Studious - Great app on Android that reminds you when homework is due, when exams are and even silences your phone during class. 

StayFocused - A chrome extension that you can customize to limit the time on websites to stay productive

Self Control - Allows you to block your own access to distracting websites for a certain amount of time. Useful for people who get distracted on the computer (Also known as Cold Turkey for Windows).

My Study Life - Student planner app

myHomework Student Planner - another Student planner app

To-do List - extensive to do list, like an online bullet journal

Learning

Itunes U - For any apple device users, a app that gives you access to free educational courses 

TED - The whole catalog of TED talks at your fingertip. 

Wikipedia - Free encyclopedia that can be used for general references

Khan Academy - study videos (absolutely amazing app/website)

Music 

Coffitivity - coffee shop soundtrack

8tracks - great playlists, especially study playlists

Wake Up Calls

Alarmy - good alarm that will force you out of bed

Sleep Cycle - measures your sleep cycle

Math Alarm - alarm clock that forces you to do math to turn the alarm off, get’s your brain going 

Class Lectures and Note Taking

Dragon Dictation - takes notes for lectures (useful for classes where professors talk faster than you can type)

XMind - mind mapping tool

Notability - another app for recording lectures. ($2.99 on iOS)

These are just a few study apps that can be useful for studying. I will add more (or make a new list) later as I learn about them; feel free to message me to add some to this list. 

Alloys: Steel

According to Dictionary.com, steel is “any of various modified forms of iron, artificially produced, having a carbon content less than that of pig iron and more than that of wrought iron, and having qualities of hardness, elasticity, and strength varying according to composition and heat treatment: generally categorized as having a high, medium, or low-carbon content”. 

Perhaps the most well known alloy around, as well as one of the most common materials in the world, steel is essentially iron with a small percentage of carbon (and, on occasion, one or more other elements). Not enough carbon and you’re stuck with wrought iron, too much carbon and you get cast iron. The graph above is a binary iron-carbon phase diagram that goes from zero percent carbon to about 6.5 percent, illustrating the various phases that can form.

Steel has been known about since ancient times, some pieces dating back to 1800 BC, but it was the invention of the Bessemer process during the industrial revolution that really popularized the alloy. (Technically, similar methods had been used before, particularly in China and Japan, but Henry Bessemer invented the modern method, industrializing it and obtaining a patent in 1856.)

Mainly used in construction, the alloy has been used for almost every possible application: from office furniture to steel wool, from bulldozers to washing machines, and from wires to watches, the possibilities are pretty much endless. Steel is also one of the world’s most-recycled materials, able to be used more than once, with a recycling rate of over 60% globally.

The addition of carbon allows the steel to be stronger than the iron it’s made from. Adding nickel and manganese increases its tensile strength, chromium increases hardness and melting temperature, and vanadium also increases hardness while making it less prone to metal fatigue. Stainless steel has at least eleven percent chromium, whereas Hadfield steel (which resists wearing) contains twelve to fourteen percent manganese. Check out these links for more information on the effects of adding certain elements.  

Sources: 1 (top images), 2 (bottom images)