Maths!

If one remembers this particular episode from the popular sitcom ‘Friends’ where Ross is trying to carry a sofa to his apartment, it seems that moving a sofa up the stairs is ridiculously hard.

But life shouldn’t be that hard now should it?

The mathematician Leo Moser posed in 1966 the following curious mathematical problem: what is the shape of largest area in the plane that can be moved around a right-angled corner in a two-dimensional hallway of width 1? This question became known as the moving sofa problem, and is still unsolved fifty years after it was first asked.

The most common shape to move around a tight right angled corner is a square.

And another common shape that would satisfy this criterion is a semi-circle.

But what is the largest area that can be moved around?

Well, it has been conjectured that the shape with the largest area that one can move around a corner is known as “Gerver’s sofa”. And it looks like so:

Wait.. Hang on a second

This sofa would only be effective for right handed turns. One can clearly see that if we have to turn left somewhere we would be kind of in a tough spot.

Prof.Romik from the University of California, Davis has proposed this shape popularly know as Romik’s ambidextrous sofa that solves this problem.

Although Prof.Romik’s sofa may/may not be the not the optimal solution, it is definitely is a breakthrough since this can pave the way for more complex ideas in mathematical analysis and more importantly sofa design.

Α α = alpha

Β β = beta

Δ δ = delta

Ε ε = epsilon

Φ φ = phi

Γ γ = gamma

Η η = eta

Ι ι = iota

Ξ ξ = ksi

Κ κ = kappa

Λ λ = lambda

Μ μ = mu

Ν ν = nu

Ο ο = omicron

Π π = pi

Ρ ρ = rho

Σ σ = sigma

Τ τ = tau

Θ θ = theta

Ω ω = omega

Χ χ = chi

Υ υ = upsilon

Ζ ζ = zeta

Anonymous (via mathblab)

Hey there!

If you have a math, or science related subject (like I always do), you’ll find that you really can’t escape analysis and problem solving, especially if you’re majoring in something science or maths related. So I am here to share some tips that actually made studying technical subjects a little bit easier and manageable for me in college:

Practice solving. If you have a subject that requires you to solve, you really have to practice solving, there is no easy way out of this one. This allows you to develop your own technique in solving the problem. You can start by doing the problems you did in class, then venture out to some examples in textbooks, then further into the problems in the textbooks until you get the hang of how the concepts and theories are applied. 

Listen during class. I know, it’s boring. But you have to do this. This way, you’ll be able to understand the topic once it is presented to you. In my opinion, it’s better if you let an expert explain it because they know the important bits in the lesson. Then study it afterwards on your own to develop your own techniques.

Ask your professors. Don’t be afraid to ask questions in class. Or if you’re shy, you can ask them after the class. However, it’s important that you ask them about the lesson when you already did your part; meaning: you already studied the material/solution over and over again but there’s just something that you can’t seem to grasp. 

Study before the class. Studying the lesson in advance doesn’t hurt. Plus, it works because you already have an idea about it. However, I don’t do it usually. What I do is that prior the discussion, I study the lessons that are going to be essential to the next topic. Example: Say that our topic later will be about introduction to thermodynamics (which includes derivation of various thermodynamic formulas); what I’m going to study instead is the different integration and derivation techniques, and different basic thermodynamics concepts like laws of thermodynamics. This ensures me that I know the prerequisite lessons of the next topic in class.

Absorb the conceptual parts of the topic first. Before diving into the problems itself, try to digest the concepts or theories behind it first. This way, you can understand which information is important and easily think of a solution because you know the problem’s framework. Even when your professor gives you a problem that seems different from your other sample problems, the concepts will still be the same throughout.

Reverse engineer the solution. Reverse engineering is reading and understanding your solution from bottom to top. I do this to make connections while going through the solution. I usually ask myself “‘where did this come from?’, ‘why did this happen?’, or ‘why is the answer like this?’” It allows me to look into the parts that I missed which are usually concepts or theories that I forgot to apply in solving the problem.

Look for key terms or phrases. There are some problems that put in information that may seem unimportant, but actually is really important. Examples such as the phrases constant velocity, constant acceleration, starting from rest, accelerate uniformly, reversible isothermal, adiabatic conditions, isobaric/isochoric compression/expansion, etc., are easy to miss but actually gives you vital information especially when solving a problem.

Try to ask yourself how or why it happened in every step of the solution. You can do this to gauge your mastery of the lesson. If you can answer yourself confidently, then you’ve studied well enough. But, if you can’t or if you feel that it’s not enough, then you better get your pen, paper, and calculator to practice some more.

If you have to draw it, draw it. Some problems need the use of your imagination, and these problems are the ones that get tricky most of the time. It’s easier to draw each of the time frames that are important so you get the sense of what’s going on between these pictures. This way, you’ll know which information you’re missing and which ones are you failing to take into account.

It’s okay to be messy and slow while practicing. Not all of time you can solve in a tumblr-esque manner because, dude, tumblr notes or solutions are soooo pretty to look at, BUT, what’s more important is that you understand each step of the solution and how the answer came to be 8.0658 m/s directed 32° south of west. So it’s okay to have dashes, strikethroughs, and crosses on your scratch paper, as long as you’re learning, a messy solution on a paper you’re not going to submit to your professor is fine.

IF YOU’VE REALLY GOTTEN THE HANG OF SOLVING IT, try to solve a fresh set of problems as fast and accurately as you can. Try to solve as if you’re in an exam. This is also to gauge how well you’re prepared for it, but you need to do this accurately. I repeat, accurately. It doesn’t work if you’ve finished it in less than an hour but all of your answers are wrong.

You may know that a pyramid has 4 faces ( excluding the base ) and the pyramid of Giza has been the popular exemplar for a pyramid.

But, the great pyramid has 8 faces ( excluding the base ), not 4!! 

                                   PC: The History Channel

In his book The Egyptian Pyramids: A Comprehensive, Illustrated Reference, J.P. Lepre wrote:

One very unusual feature of the Great Pyramid is a concavity of the core that makes the monument an eight-sided figure, rather than four-sided like every other Egyptian pyramid. That is to say, that its four sides are hollowed in or indented along their central lines, from base to peak.

This concavity divides each of the apparent four sides in half, creating a very special and unusual eight-sided pyramid; and it is executed to such an extraordinary degree of precision as to enter the realm of the uncanny. For, viewed from any ground position or distance, this concavity is quite invisible to the naked eye.

The hollowing-in can be noticed only from the air, and only at certain times of the day. This explains why virtually every available photograph of the Great Pyramid does not show the hollowing-in phenomenon, and why the concavity was never discovered until the age of aviation.

It was discovered quite by accident in 1940, when a British Air Force pilot, P. Groves, was flying over the pyramid. He happened to notice the concavity and captured it in the now-famous photograph. [p. 65]

                              Satellite image of the Great Pyramid.

Were the pyramids built this way on purpose or did it turn out this way over time is something that still requires a much deeper investigation.

How to read math. You’d be surprised how far this will get you.

Katherine Johnson (b. 1918) is a physicist and mathematician who has made crucial contributions to several NASA missions, assuring their success with her highly accurate calculations. She worked with NASA for several decades, and helped advance the rights of both African-Americans and women.