Regruntled.com

It’s hard to describe the fascination of the Mandelbrot Set.  It’s something that is “already out there” in the sense that it is implied by some simple equations.  All it takes to explore it is billions of multiplications carried out to 100 decimal places.  It’s a fractal, so at any magnification, it’s somewhat similar.  On the other hand, it never repeats exactly, so it’s always different.  There’s always something that no one has ever seen before because no one has done the multiplications.  You might say that this is a two-dimensional, graphical version of looking at a billion digits of Pi.  A description is here.

Clay Shirky writes in Boing Boing about the traits that make good programmers.  He quotes a study using questions like the one above, given to students on the first day of an introductory programming class:

To write a computer program you have to come to terms with this, to accept that whatever you might want the program to mean, the machine will blindly follow its meaningless rules and come to some meaningless conclusion.

Shirky coins a great phrase, but the study itself is about consistency and rule-following as much as it is about meaninglessness.  The authors themselves introduce meaninglessness by giving  the students a test in a language they don’t understand.  They might as well be giving English speakers a test in Swahili.

It is true that some good programmers have come to programming from mathematics, and that one point of view in the philosophy of mathematics (formalism) holds that mathematics is a game played with meaningless symbols on pieces of paper.  So it is certainly possible to see a connection between programming and meaninglessness.

On the other hand, engineers tend to make good programmers, too, and they’re mainly interested in getting things done in the real world.  If a guy programs a simulation to make sure that a bridge doesn’t fall down, it’s hard to accuse him of meaninglessness.  The shared element is the consistent application of logical rules.

There are a few human languages that are deficient in words for numbers.  One would expect that people who lack words for numbers larger than two would have difficulty performing certain numeric tasks, but recent research suggests that this is not so:

British and Australian researchers assessed 45 indigenous Australian children aged between four and seven years.

They compared those who lived in remote areas and only spoke Warlpiri or Anindilyakawa - two Aboriginal languages with very few number words - with those who lived in Melbourne and spoke English.

There was no difference in numerical ability between the children who spoke languages without number words and the English-speaking children.

Study leader Professor Brian Butterworth, from the Institute of Cognitive Neuroscience at University College London, said two studies in tribes in the Amazon had concluded that words were necessary for exact number tasks but this research showed otherwise.

I don’t know about this.  We seem to have internalized the Sapir-Whorf Hypothesis to the point that learning something new is inseparable from learning a new vocabulary.  On the other hand, crows are said to be able to count up to three, and parrots up to six.  Maybe we have some very simple innate numerical ability, but anything beyond that requires language.

Synchronicity department

After I posted this, I heard about John McCain having more houses than he could count.  I swear that the timing was entirely coincidental.

Daima Taimina, a Latvian mathematician, crochets objects in two-dimensional hyperbolic space.  Such objects  have “extra” area.  An ordinary pot-holder, 12 inches on a side, has one square foot of area.  A hyperbolic pot-holder 12 inches on a side has more than one square foot of area, maybe a lot more, so it has crinkly edges like some kinds of lettuce.  In a two-dimensional spherical space, the pot-holder would have less than one square foot of area (in other words, if you draped the pot-holder over a soccer ball, it would have to have less area in order to lie flat).  Daima provides the extra area by systematically adding extra stitches as she crochets.

Hyperbolic crochet is also helping more women to explore maths. It is part of a new field called “mathematics and the fibre arts” which includes knitting, quilting and weaving as well as crochet.

The Institute for Figuring has a “crocheted coral reef” exhibit at the Southbank Centre in London.

TTB sends a link to a story about a crop circle in Wiltshire, England that seems be a coded representation of the first 10 digits of Pi:

3.141592654

Looks like the math bears are venturing out from the woods.

See also:

• Math humor

A 2-dimensional animation of a 3-dimensional projection of a 4-dimensional convex regular polytope rotating in 2 different directions at the same time.  Don’t try this at home.

Here is the animation, complete with sound effects. What a great way to teach number theory to kids! I even learned three words of Finnish.

Here is a math video so good that it’s fun to watch even if you aren’t interested in the math.

In art, there is the problem of perspective, the realistic representation of three-dimensional objects in two dimensions. In Medieval art, the Bayeaux Tapestry for example, objects are flat and depth is indicated by overlapping. During the Renaissance, artists worked out the rules of perspective by trial and error. In the seventeenth century, the mathematicians formalized the process, beat it to death, and called it projective geometry. And now we have come full circle: a video illustrating one of the basic principles of projective geometry, a video that is itself a work of art.