Dactylonomy or finger-counting is normally discouraged in school. Which is unfortunate, because it is very interesting how different systems can be used to count up to different numbers. The classical system in Europe counts only up to five on one hand, by identifying the number of raised fingers with the abstract number. This can, however, drastically be improved.

For instance, using the binary system one can count up to 31 on one hand. Once the numbers 1, 2, 4, 8 and 16 are assigned to the fingers, as above, different numbers can be represented by raising or tucking in the fingers. If you want to count ever further, using the same principle on both hands further enhances the system up to 1023! Disadvantage: some dexterousness is required. The number 4 also constitutes some risk of misinterpretation&ldots;

My favorite system is less powerful but also very neat. Use your thumb to count on the three finger bones of each finger. This way, one hand can be used to count up to 12. The other hand is used to display the number of completed 12s, allowing for a total of 144.

(Source: mister-wunderkammer, via imathematicus)

Fractal pizza.

Fractal pizza.

(Source: trollnrage)

A snub cube tessellated with bunnies.

A snub cube tessellated with bunnies.

(Source: cosmicbeachparty, via the-irrationals)

(Source: twocubes)

Physics minus Mathematics

The great probabilist Mark Kac (1914–1984) once gave a lecture at Caltech, with Richard Feynman in the audience. When Kac finished, Feynman stood up and loudly proclaimed, “If all mathematics disappeared, it would set physics back precisely one week.”

To that outrageous comment, Kac shot back with “Yes, precisely the week in which God created the world.”

Fun with Hex

The Hex game was invented by the Danish mathematician Piet Hein, who introduced it in 1942 at the Niels Bohr Institute. It was independently re-invented in 1947 by the mathematician John Nash at Princeton University.

The rules are really simple. Each player has an allocated color, red and blue being conventional. Players take turns placing a stone of their color on a single cell within the overall playing board. The goal is to form a connected path of your stones linking the opposing sides of the board marked by your colors, before your opponent connects his or her sides in a similar fashion (the four corner hexagons each belong to both adjacent sides). The first player to complete his or her connection wins the game.

The game can never end in a tie, a fact proved by Nash: the only way to prevent your opponent from forming a connecting path is to form a path yourself. In other words, Hex is a determined game.

Can you tell which plot above is randomly generated?

Being able to determine if something is “truly” random is not just an investigation carried out by forensic accountants, sociologists, and law enforcement. Rather it is an interesting and complicated mathematical problem. Consider the two plots above. You may look at the on the left and see the clumps, the spacing, and think “That can’t be the random plot.” And yet it is. The plot on the left has been randomly generated, while the plot on the right is a scatter plot of glowworm positions on a ceiling.

So here, the clumps actually help indicate randomness. Try thinking of it in another way: imagine you have two students who were asked to flip a coin 100 times for homework. The first student was diligent and flipped accordingly:

THHHTHTTTTHTTHTTTHHTHTTHT
HHHTHTHHTHTTHHTTTTHTTTHTH
TTHHTTTTTTTTHTHHHHHTHTHTH
THTHTHHHHHTHHTTTTTHTTHHTH

The second student was lazy and decided to make up his flips:

HTTHTTHTHHTTHTHTHTTHHTHTT
HTTHHHTTHTTHTHTHTHHTTHTTH
THTHTHTHHHTTHTHTHTHHTHTTT
HTHHTHTHTHTHHTTHTHTHTTHHT

Now while it might seem strange that the first student has long runs, it fits closer to what one would expect if the flip is random. On the other hand, in the second student’s data, there is less than a 0.1% chance that they wouldn’t get a single run longer than four in a row!

Images and coin flip data were found in this article. It takes a closer look at some of these topics and provides some pretty neat historical background.

(via ryanandmath)

Coffee cup joke

A topologist is drinking coffee from a cup. Suddenly the handle drops off, leaving the topologist in astonishment: the new shape is different, but he can still drink from it! So he keeps on drinking his coffee, until the bottom of the cup drops off. Now he is totally befuddled: the shape is equivalent to the original one, but how can he drink his coffee now?

This square is partitioned into L-shapes, the size of which form the arithmetic progression 1, 2, 3, …, 49. Odd sized L’s are symmetric, while one of the legs of an even sized L is one block longer than the other (in other words, as nearly as symmetrical as possible).

This square is partitioned into L-shapes, the size of which form the arithmetic progression 1, 2, 3, …, 49. Odd sized L’s are symmetric, while one of the legs of an even sized L is one block longer than the other (in other words, as nearly as symmetrical as possible).

(Source: themathkid, via visualizingmath)

You know, people think mathematics is complicated. But mathematics is the simple bit, it’s the stuff we can understand. It’s cats that are complicated. I mean, what is it in those little molecules and stuff that make one cat behave differently than another, or that make a cat? And how do you define a cat? I have no idea.
John Conway

Carl Friedrich Gauss just signed matthen's Tumblr!
Matthen: "I just made this animation tracing out a copy of Gauss's signature he wrote in a book when he was 17. Quite fantastic and elaborate! Perhaps he just wanted to make it difficult for me. Gauss is somewhat of a legend in the history of maths. Reblog this and he’ll be signing your blog too!”
Here is the original signature.

Carl Friedrich Gauss just signed matthen's Tumblr!

Matthen: "I just made this animation tracing out a copy of Gauss's signature he wrote in a book when he was 17. Quite fantastic and elaborate! Perhaps he just wanted to make it difficult for me. Gauss is somewhat of a legend in the history of maths. Reblog this and he’ll be signing your blog too!”

Here is the original signature.

(via lthmath)

This image created by Christopher Culter represents the compact abelian group of 2-adic integers (black points), with selected elements labeled by the corresponding character on the Pontryagin dual group (colored discs).

This image created by Christopher Culter represents the compact abelian group of 2-adic integers (black points), with selected elements labeled by the corresponding character on the Pontryagin dual group (colored discs).

A collection of beautiful mathematics: attractive pictures and fun results