Friday, February 4, 2011

The taxicab and other whole numbers

Happy birthday Godfrey Harold Hardy, an English mathematician born February 7 1877. Hardy is perhaps best remembered in modern popular culture for the number 1729 that crops up a lot in Matt Groening's Fox Television cartoon Futurama as, for example, the serial number of Bender the bratty robot.

The so-called Taxicab, or Hardy-Ramanujan Number, 1729, is named from a story G. H. Hardy told about a visit fellow mathematical genius Srīnivāsa Aiyangār Rāmānujam he lay ill. "Once, in the taxi from London, Hardy noticed its number, 1729. He must have thought about it a little because he entered the room where Ramanujan lay in bed and, with scarcely a hello, blurted out his disappointment with it. It was, he declared, 'rather a dull number,' adding that he hoped that wasn't a bad omen. 'No, Hardy,' said Ramanujan, 'it is a very interesting number. It is the smallest number expressible as the sum of two cubes in two different ways' ".

Hardy, who once commented "Nothing I have ever done is of the slightest practical use" also noted his protege Rāmānujam remarking "An equation for me has no meaning, unless it represents a thought of God".

Now Professor of Mathematics at Emory University Ken Ono has partly solved a particular puzzle that fascinated Tamil mathematician Ramanujan, by finding fractal relationships in sequences of the so-called Partition values of positive integers.

A Partition of an integer, or whole number, is any combination of other whole numbers that adds up to that number. For example, the number four can be made five ways: from 4, or 3+1, or 2+2, or 2+1+1, or 1+1+1+1. So the Partition number of four is five.

Ono and his colleagues say in their paper i-adic properties of the partition function (pdf) those sequences of numbers are "governed by fractal behavior".

In his book The Fractal Geometry of Nature, Benoit Mandelbrot describes a fractal as "a rough or fragmented geometric shape that can be split into parts, each of which is roughly a reduced-size copy of the whole," a property called self-similarity. A familiar example of fractal self-similarity is a fern frond composed of self-similar but increasingly smaller fronds.

Observations of Fractals in the natural world are thought by some to be evidence of intelligent design.

Hardy, a lifelong atheist, wrote "A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas".

Something to think about.


  1. Do you think Jackson Pollock's art was fractal expressionism?


  2. I think people use the term 'intelligent design' as a way of speaking about God or creationism as opposed to evolutionary theories. There is repetition in 'nature', but I don't necessarily see this as purposeful design, in the sense that a greater being exhibits intent, but rather, that the pattern is or has been useful for propagation in some way. Fractals are interesting and appear to suggest that there is some 'essence', immutable design to the patterns contained within their structure.
    I can see why people would like the notion of 'uniqueness' that emerges out of patterns evident in fractals and DNA, but that doesn't tell me there is a God or an intelligent designer, other than nature, which is the most powerful and unique designer itself.

  3. It might be safe to say that fractal geometry was not a preoccupation for Jackson Pollock. Given the amount of alcohol he drank while he worked, his style is more Foster's expressionism.

  4. and much of that 'Foster's expression was excellent - isn't it amazing what a person can do when they're depressed?!