The Tao of τ

Today is 6/28 in Australia. Happy Tau day.

Tau (τ) is a mathematical constant derived from the proportions of a circle, the ratio of a circle's circumference to its radius, approximately 6.28. This is twice the value of the more famous constant Pi (π), approximately 3.14, or the ratio of a circle's diameter to its circumference.

If you like π you will love Tau because it is two π's. But τ doesn't seem to be as popular as π; nobody seems to be memorizing τ to 1000 places. Yet τ has its uses, especially in the solid geometries used in 3D graphics, where circles are less important than spheres, and if you use π you get knee deep in powers of and divisions and multiplications by two.

Physicist Michael Hartl writes in the Tau Manifesto

For millennia, the circle has been considered the most perfect of shapes, and the circle constant captures the geometry of the circle in a single number. Of course, the traditional choice of circle constant is π—but, as mathematician Bob Palais notes in his delightful article “π Is Wrong!”, π is wrong. It’s time to set things right.

The Tau Manifesto, launched on Tau Day 2010, embraces the Tao of τ by doing away with all those inconvenient powers, multiplicands and divisors of two at the heart of quadratic equations and replace them with twice the number of π's.

The Tau of more Pi sounds good to me. But perhaps that is a bit pious. Or 2π -ous!

A short discussion of technique

Julie asked me to write something about how I make the images I make. This is a rather large topic. I guess people would be most interested in how a computer program "makes" an image, and of course any discussion of computer generated imagery tends to focus upon the mechanical process involved. Notwithstanding my own boredom with discussions of software algorithms, and especially hardware setups, I'll give it my best shot.

My background is in real-time computer simulation, what used to be called Virtual Reality, until the term became meaningless with overuse. The two basic elements of this type of computer simulation are form, which is achieved with mathematical models loosely called geometry, and colour, the interaction of simulated light rays with the geometry. The interaction of geometry and light is mediated by a layer of abstraction loosely called a pixel shader. Shaders are applied to regions of geometry to simulate the way light behaves in real life, or not, depending upon the artist and subject.

The image of the violet clouds is a rather large print, an arms-length square that hangs on a wall in my home (cropped here to cope with Blogger's preference for photo-sized images). I love sunrise, and sunsets, and the colours in the image are drawn from many memories of the collision of red and blue light, those colours at opposite ends of light's visible spectrum that produce such a spectacular display when reflected in the world from open water and refracted by clouds.

The image is entirely computer generated from mathematical formulae. I use assistive software called Mojo, developed by Dr. Kenton Musgrave, who worked with Benoit Mandelbrot to produce some celebrated images called Mandelbrots in the 1980s. Mojo has a 400 page user manual, of which around a third expounds a philosophy of fractal geometry.

The software leverages dilation symmetry, a property of self-similarity arising from the application of fractal geometry in mathematics. The image labelled Function Graph Editor [SKY COLOR BLENDS INPUT] gives an example of the way I programmed the software to produce a shader to colour the violet clouds. The "land" and "sea" in the image are programmed the same way.

The software drew the image of the violet clouds in small portions. It took a fast computer a couple of days to calculate the raw data used to create the image in computer memory. The image was then saved on a hard disk drive as a high resolution Truevision (TARGA) RGB image file. Once an image is saved on a computer hard drive it can be printed, resized for use as a graphic on the world wide web, or posted to a photo sharing site as if it were a photograph. Or reused as part of another image.

There are many discussions along the lines of is computer generated imagery art? on the internet. The last I listened to, a rather lengthy harangue from someone I endured at a Monash University seminar around twenty five years ago, was arguing that if you can't see any brush strokes it aint art. Alternatively, a hundred monkeys could turn out the same thing, over time.

Maybe so. But everyone has a medium they are comfortable with, and this is mine.

Pi day

It is 3.14 today. Happy Pi Day.

Pi is a mathematical constant calculated from the ratio of the length of a circle's rim (its circumference) to its breadth (its diameter). It is perhaps the most well known mathematical term.

Over the years the value of Pi has been refined to the approximate value we use today, 3.14159265, Google says.

An ancient (2000BC) document called the Rhind Papyrus gives Pi the value of 3.16045. The fourteenth century South Indian Mathematician Madhava calculated Pi to thirteen decimal places, or 3.1415926535898.

The ratio of a circle's circumference to its diameter can never be an exact fraction, so Pi is both an irrational number, because it can never be calculated to an exact fraction, and a transcendent number. The more you calculate Pi, the more of Pi there is, and, using computers, Pi has been calculated to 5 trillion decimal places. With the rest of infinity to go, there are concerns that machines will never be fast enough to calculate Pi before the universe ends.

Oh, and for the eleet who care about such things, the number 42 is Pi x 13.37. Roughly.

Enjoy Pi Day.

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.