Erdős’ age anecdote

Paul Erdős, the most prolific mathematician in history, always made jokes about his age. He said for instance that he is 2.5 billion years old, because in his youth the age of the Earth was said to be 2 billion years old and around 1970, it was known to be 4.5 billion years.

(Source: hyrodium)

This continuous variation of Conway’s Game of Life named SmoothLife is introduced in this arXiv article. It looks astonishingly organic.

(Source: gameoflife-algorithms)

Can two dice be weighted in such a way that the probability of each of the possible outcomes (2 up to 12) is the same? Click here to find out.

Can two dice be weighted in such a way that the probability of each of the possible outcomes (2 up to 12) is the same? Click here to find out.

(Source: mindfuckmath)

Sperner’s lemma
Color the vertices of a triangulated triangle with three colors such that:
each vertex of the main triangle has a different color;
each vertex on an edge of the main triangle is colored with one of the two colors at the end of its edge;
then there exists a small triangle whose vertices are colored with all three different colors. More precisely, there exists an odd number of such triangles.
This result looks playful and innocent but is in fact quite powerful. It is known, for instance, to lead to an easy proof of Brouwer’s fixed point theorem. Its power mainly lies in building bridges between discrete, combinatorial mathematics and continuous mathematics.

Sperner’s lemma

Color the vertices of a triangulated triangle with three colors such that:

  • each vertex of the main triangle has a different color;
  • each vertex on an edge of the main triangle is colored with one of the two colors at the end of its edge;

then there exists a small triangle whose vertices are colored with all three different colors. More precisely, there exists an odd number of such triangles.

This result looks playful and innocent but is in fact quite powerful. It is known, for instance, to lead to an easy proof of Brouwer’s fixed point theorem. Its power mainly lies in building bridges between discrete, combinatorial mathematics and continuous mathematics.

Mathematician Calculates 177147 Ways to Tie a Tie

Earlier this year, Swedish mathematician Mikael Vejedemo-Johansson discovered there are more than 177000 different ways to tie a necktie. His research (article on arXiv) is based upon a formal language describing tie knots. This website randomly generates tie knots according to the classification. Some well-known examples of exotic tie knots:

Trinity tie knot

Trinity knot

Eldredge tie knot

Eldredge

More tie knots can be found here.

It has now been proven beyond doubt that smoking is the major cause of statistics.
Author unknown.

James Grime presents a honeycomb variation on the classical magic square and demonstrates it to be only one—it’s a kind of magic.

(Source: singingbanana)

Playing detective with rolling shutter photos

On some photos of turboprop planes, you can find a strange view of the propellors due to the “rolling shutter” effect of the sensor in the camera.

Now, can one figure out the real number of propellor blades of the plane, and the rate at which the propellor is spinning, by analyzing this kind of photos? Daniel Walsh performs a thorough mathematical analysis.

Glassified is a modified ruler with a transparent display to supplement physical pen strokes on paper with virtual graphics, such as automatic length, angle or area measurements.

Glassified is a modified ruler with a transparent display to supplement physical pen strokes on paper with virtual graphics, such as automatic length, angle or area measurements.

(Source: trigonometry-is-my-bitch, via nobel-mathematician)

Conway’s puzzle M(13)

M(13) is a sliding game proposed by John Conway in 1989, similar to Sam Loyd’s famous 15-puzzle (the classical sliding puzzle) but replacing the role played by the simple alternating group A15 with that of the sporadic simple Mathieu group M12. Its game board is the finite projective plane over the field with three elements.

You can play the puzzle online at Wolfram Demonstrations Projects.


"Design is not the narrow application of formal skills, it is a way of thinking", Chris Pullman

"Design is not the narrow application of formal skills, it is a way of thinking", Chris Pullman

(via geometric-aesthetic)

Borsuk’s conjecture

Karol Borsuk is most famous for the Borsuk-Ulam theorem, discussed in his ground-breaking paper Drei Sätze über die n-dimensionale euklidische Sphäre. The same paper also mentions the following conjecture:

Can every bounded subset of the space Rn be partitioned into (n+1) sets, each of which has a smaller diameter?

Informally, can you cut a n-dimensional shape into (n+1) pieces which are “smaller”, in the sense that the diameter (the maximum distance between two points) decreases? This is an example of a planar shape which fits the conjecture:

Borsuk’s conjecture seems very obvious. Yet, progress was made only slowly: Borsuk himself proved the case n=2 in 1931, Perkal the case n=3 in 1947, Hadwiger for all n but only for smooth convex bodies, Riesling for all centrally symmetric bodies and Dekster for all bodies of revolution. No final proof, however, was found.

The problem was finally solved in 1993 by Jeff Kahn and Gil Kalai. They demonstrated the conjecture of Borsuk actually is false, by constructing a 1325-dimensional counterexample!

The best bound currently known is due to Thomas Jenrich, with n=64.

Math Love by CuteReaper.

Math Love by CuteReaper.

(Source: lifeofmasqueradeus, via themathavenger)

A collection of beautiful mathematics: attractive pictures and fun results