A trippy journey into fractals.

(Source: magikcentury, via geometric-aesthetic)

Mathematics is the supreme judge; from its decisions there is no appeal.
Tobias Dantzig

(Source: centerofmath)

benny-cheung asked: ay, in your dice post, that solid isn't a pentagonal dipyramid, as the faces are quadrilateral. it's in fact a pentagonal trapezohedron, which is a dual of an antiprism. it is still, however, face-transitive, as antiprisms are vertex-transitive.

Wow, you’re right, thanks for noticing this subtle error!

It’s funny how the Wikipedia page on pentagonal trapezohedra has a section “die”, while the page on pentagonal bipyramids hasn’t :)

Sprouts: a two player game

Sprouts: a pencil-and-paper game with interesting topological, combinatorial and graph theoretical properties, proposed by John Conway and Michael Paterson at Cambridge University in the 1960s. The game is addictive and fun to play, and is not yet solved in general.

  • Draw a line connecting any two dots, making sure the line does not cross any other lines. A line may connect a dot to itself.
  • Add a new dot (the “sprout”) somewhere on the new line.
  • If either of the dots on the ends of the line have three lines connecting it, it must not be used anymore.
  • A dot can also be disabled if both players agree that it is cut off from the rest of the dots, and cannot have any more lines drawn to it.

The winner is the person who draws the last line, leaving the opponent unable to add any more lines using the rules above. The following images show the first few moves in a game of Sprouts.

The regularity of the platonic solids make them especially suited for dice, since the probability that some side shows up is the same for all sides. Unfortunately, the tetrahedron doesn’t have opposite sides, so its labeling should be somewhat special to make it useful. It doesn’t roll as well as the other ones either.
Other semiregular solids can be used as well, such as a pentagonal trapezohedron (top left). The general property which guarantees a fair die is isohedrality or face-transitivity: for any two faces of the solid, there is a symmetry of the entire solid by rotations and reflections, mapping the first face onto the second.
EDIT: thanks to Benny Cheung, who pointed out that the top left die is not a pentagonal dipyramid, but a pentagonal trapezohedron.

The regularity of the platonic solids make them especially suited for dice, since the probability that some side shows up is the same for all sides. Unfortunately, the tetrahedron doesn’t have opposite sides, so its labeling should be somewhat special to make it useful. It doesn’t roll as well as the other ones either.

Other semiregular solids can be used as well, such as a pentagonal trapezohedron (top left). The general property which guarantees a fair die is isohedrality or face-transitivity: for any two faces of the solid, there is a symmetry of the entire solid by rotations and reflections, mapping the first face onto the second.

EDIT: thanks to Benny Cheung, who pointed out that the top left die is not a pentagonal dipyramid, but a pentagonal trapezohedron.

British Mathematical Colloquium

Hi guys, I’m back from London!

The British Mathematical Colloquium was fantastic: lots of interesting talks, workshops and mathematicians. I learned about various nice topics (one especially intriguing was about a connection between latin bitrades, spherical triangulations, groups and codings) and took some great pictures.

London itself was great as well, I bought some nice books and gadgets (such as an icosahedral and dodecahedral die).

Now, it’s time for some maths :)

British Mathematical Colloquium

Tomorrow I depart for London, to attend the BMC 2014 (the British Mathematical Colloquium). There will be workshops and plenary lectures by Michael Atiyah, Endre Szemerédi, Cédric Villani, Don Zagier… and Persi Diaconis will give a public lecture about the life and work of Martin Gardner, so I’m looking forward to it!

Unfortunately, this means that my Tumblr activity will slow down the next few days, but I hope I will learn a lot of new stuff in London to share!


David Cox posted this image today from Jason Davies. It’s an interactive Voronoi diagram of the world’s airports, answering which airport is closest to you. David’s comment: “Instead of having to teach things like perpendicular bisectors and systems of equations, I just wish we could do things like this.”

(Source: mathhombre)

David Cox posted this image today from Jason Davies. It’s an interactive Voronoi diagram of the world’s airports, answering which airport is closest to you. David’s comment: “Instead of having to teach things like perpendicular bisectors and systems of equations, I just wish we could do things like this.

(Source: mathhombre)

For every knot and link, there exist a continuous surface whose boundary is the given knot or link. Such a surface is called a Seifert surface. The image depicts a Seifert surface of the Borromean rings, an intertwined triplet of topological circles with the property that no two of them are actually linked.

For every knot and link, there exist a continuous surface whose boundary is the given knot or link. Such a surface is called a Seifert surface. The image depicts a Seifert surface of the Borromean rings, an intertwined triplet of topological circles with the property that no two of them are actually linked.

How many ways can you arrange a deck of cards? An excellent introduction to the factorial function and its use in combinators, by Yannay Khaikin.

(Source: boballende, via mindfuckmath)

Thirty identical spheres of radius r cover one central sphere of radius R,in such a way that every small sphere touches four others and the big one.Now, there is a very elegant relation the two radii:
R = √5 · r

Thirty identical spheres of radius r cover one central sphere of radius R,
in such a way that every small sphere touches four others and the big one.
Now, there is a very elegant relation the two radii:

R = √5 · r

Seashells: the plainness and beauty of their mathematical description

There exist many strange-looking shell shapes, but with quite a simple mathematical model using only 14 variables, one can describe most of them. Mathematica code included to generate your own shells.

Happy April Fools’ Day!
Sorry for my lack of originality; the original hoax was made by Martin Gardner in his column Mathematical Games in 1975. Gardner claimed the constant on the left is in fact an integer, as conjectured by the Indian math genius Srinivasa Ramanujan. Simon Plouffe hence coined the constant “Ramanujan’s constant”. The hoax was surprisingly well-accepted, but people couldn’t rely on Mathematica those days :)
Of course, Ramanujan’s constant is not an integer. It is irrational (in fact, transcendental) and equals 262537412640768743.999999999999250…
As you can see, the error in the previously stated equality is of an order 10-13, remarkably small compared with the number itself, which is of order 1017!
The number being almost integer is not a mere coincidence as there are deep mathematical objects hiding underneath this seemingly innocuous delectation (modular forms, Heegner numbers, the j-invariant, q-expansions…) but the explanation is far from easy. If you’re interested, you can read about it on Wikipedia.
So, are you guys very disappointed?

Happy April Fools’ Day!

Sorry for my lack of originality; the original hoax was made by Martin Gardner in his column Mathematical Games in 1975. Gardner claimed the constant on the left is in fact an integer, as conjectured by the Indian math genius Srinivasa Ramanujan. Simon Plouffe hence coined the constant “Ramanujan’s constant”. The hoax was surprisingly well-accepted, but people couldn’t rely on Mathematica those days :)

Of course, Ramanujan’s constant is not an integer. It is irrational (in fact, transcendental) and equals 262537412640768743.999999999999250…

As you can see, the error in the previously stated equality is of an order 10-13, remarkably small compared with the number itself, which is of order 1017!

The number being almost integer is not a mere coincidence as there are deep mathematical objects hiding underneath this seemingly innocuous delectation (modular forms, Heegner numbers, the j-invariant, q-expansions…) but the explanation is far from easy. If you’re interested, you can read about it on Wikipedia.

So, are you guys very disappointed?

This number, involving π and the exponential function, evaluates to a perfect integer. It turns out π and e are not that transcendental after all.
Verify if you don’t believe me!
EDIT: just to clear things up, this was a Martin Gardner’s April Fools hoax.

This number, involving π and the exponential function, evaluates to a perfect integer. It turns out π and e are not that transcendental after all.

Verify if you don’t believe me!

EDIT: just to clear things up, this was a Martin Gardner’s April Fools hoax.

(Source: mathispun)

A collection of beautiful mathematics: attractive pictures and fun results