This is the magic hexagon. The numbers add up to 38 along each straight line. It can be called the magic hexagon rather than a magic hexagon because there are no other hexagons numbered 1,2,…,n with this property, no matter how many layers the arrangement has (except for the one which is just one hexagon with 1 written in it, but that’s hardly magical…).
(Source: Mathematical Gems I by Ross Honsberger)

This is the magic hexagon. The numbers add up to 38 along each straight line. It can be called the magic hexagon rather than a magic hexagon because there are no other hexagons numbered 1,2,…,n with this property, no matter how many layers the arrangement has (except for the one which is just one hexagon with 1 written in it, but that’s hardly magical…).

(Source: Mathematical Gems I by Ross Honsberger)

(Source: twocubes, via mathhombre)

Some formulas are plainly beautiful, aren’t they?

(Source: likeaphysicist, via geometricloci)

The Galton board: a cool statistical device illustrating the central limit theorem by the correspondence between the discrete binomial distribution and the continuous normal distribution.

The Galton board: a cool statistical device illustrating the central limit theorem by the correspondence between the discrete binomial distribution and the continuous normal distribution.

(Source: ibmblr)

This chart shows how to solve every quartic polynomial.

Notice it also shows how to solve cubics and quadratics, but it can’t be improved to quintics (polynomials of degree five) or higher, because only polynomials with degree less than five can be solved algebraically in general: this is the Abel-Ruffini theorem. Some specific quintics can be solved, but the method is far more tedious. In 2004 Daniel Lazard wrote out a three-page formula for the roots of a general solvable quintic.

(Source: plus.google.com, via allofmymaths)

(Source: ioanaiuliana21)

Mathematics is like oxygen. If it is there, you do not notice it. If it would not be there, you realize that you cannot do without.
Lex Schrijver
Jeddah is home to the world’s largest outdoor open air art museum. The city abounds with a variety of sculptures, large and small, located throughout the city. This is an enormous sculpture of various drawing tools including a ruler, a compass, and a protractor, and it is one of many around the city that fall in the category of science and engineering. This piece, constructed of steel, is called “Engineers’ Tools” and was designed by Hisham Benjabi and Ali Amin.

Jeddah is home to the world’s largest outdoor open air art museum. The city abounds with a variety of sculptures, large and small, located throughout the city. This is an enormous sculpture of various drawing tools including a ruler, a compass, and a protractor, and it is one of many around the city that fall in the category of science and engineering. This piece, constructed of steel, is called “Engineers’ Tools” and was designed by Hisham Benjabi and Ali Amin.

(Source: ioanaiuliana21)


The brachistochrone
This animation is about one of the most significant problems in the history of mathematics: the brachistochrone challenge.
If a ball is to roll down a ramp which connects two points, what must be the shape of the ramp’s curve be, such that the descent time is a minimum?
Intuition says that it should be a straight line. That would minimize the distance, but the minimum time happens when the ramp curve is the one shown: a cycloid.
Johann Bernoulli posed the problem to the mathematicians of Europe in 1696, and ultimately, several found the solution. However, a new branch of mathematics, calculus of variations, had to be invented to deal with such problems. Today, calculus of variations is vital in quantum mechanics and other fields.

The brachistochrone

This animation is about one of the most significant problems in the history of mathematics: the brachistochrone challenge.

If a ball is to roll down a ramp which connects two points, what must be the shape of the ramp’s curve be, such that the descent time is a minimum?

Intuition says that it should be a straight line. That would minimize the distance, but the minimum time happens when the ramp curve is the one shown: a cycloid.

Johann Bernoulli posed the problem to the mathematicians of Europe in 1696, and ultimately, several found the solution. However, a new branch of mathematics, calculus of variations, had to be invented to deal with such problems. Today, calculus of variations is vital in quantum mechanics and other fields.

(Source: saulofortz, via geometricloci)

(Source: SerketStalker)

I love it how a picture of a cabbage is going viral on Tumblr. A massive 10000+ likes and reblogs, thanks to you all again for the support!

A curved variant of the Sierpiński fractal.

A curved variant of the Sierpiński fractal.

(Source: geometricloci)

Scooping the Loop Snooper

No general procedure for bug checks will do.
Now, I won’t just assert that, I’ll prove it to you.
I will prove that although you might work till you drop,
you cannot tell if computation will stop.

For imagine we have a procedure called P
that for specified input permits you to see
whether specified source code, with all of its faults,
defines a routine that eventually halts.

You feed in your program, with suitable data,
and P gets to work, and a little while later
(in finite compute time) correctly infers
whether infinite looping behavior occurs.

If there will be no looping, then P prints out “Good.”
That means work on this input will halt, as it should.
But if it detects an unstoppable loop,
then P reports “Bad!”—which means you’re in the soup.

Well, the truth is that P cannot possibly be,
because if you wrote it and gave it to me,
I could use it to set up a logical bind
that would shatter your reason and scramble your mind.

Here’s the trick that I’ll use—and it’s simple to do.
I’ll define a procedure, which I will call Q,
that will use P’s predictions of halting success
to stir up a terrible logical mess.

For a specified program, say A, one supplies,
the first step of this program called Q I devise
is to find out from P what’s the right thing to say
of the looping behavior of A run on A.

If P’s answer is “Bad!”, Q will suddenly stop.
But otherwise, Q will go back to the top,
and start off again, looping endlessly back,
till the universe dies and turns frozen and black.

And this program called Q wouldn’t stay on the shelf;
I would ask it to forecast its run on itself.
When it reads its own source code, just what will it do?
What’s the looping behavior of Q run on Q?

If P warns of infinite loops, Q will quit;
yet P is supposed to speak truly of it!
And if Q’s going to quit, then P should say “Good.”
Which makes Q start to loop! (P denied that it would.)

No matter how P might perform, Q will scoop it:
Q uses P’s output to make P look stupid.
Whatever P says, it cannot predict Q:
P is right when it’s wrong, and is false when it’s true!

I’ve created a paradox, neat as can be,
and simply by using your putative P.
When you posited P you stepped into a snare;
Your assumption has led you right into my lair.

So where can this argument possibly go?
I don’t have to tell you; I’m sure you must know.
A reductio: There cannot possibly be
a procedure that acts like the mythical P.

You can never find general mechanical means
for predicting the acts of computing machines;
it’s something that cannot be done. So we users
must find our own bugs. Our computers are losers!

Geoffrey K. Pullum, A proof that the Halting Problem is undecidable

Using the chain rule is like peeling an onion. You have to deal with every layer at a time and if it’s too big you’ll start crying.
Calculus professor (via mathprofessorquotes)
Impossible borromean tribars by Lee Sallows.

Impossible borromean tribars by Lee Sallows.

A collection of beautiful mathematics: attractive pictures and fun results