Monday, 18 October 2010

i = imaginary

Mathematics began when humans needed a system to count objects. We could use our fingers to count 1,2,3,4,... apples, for example. Our numbering system is based on multiples of ten, most likely because we humans have ten fingers (including thumbs). We call these numbers the set of natural numbers. It was the ancient Egyptians who developed a set of distinct hieroglyphic symbols that could be used to represent natural numbers (up to one million).

This was great but we also needed to do things like divide two apples among three people. Thus fractions were invented and we could calculate how give each person ⅔ of an apple.

This set of numbers was satisfying to ancient Greek philosophers and mathematicians. They argued that all measurable quantities could be expressed as a fraction. Pythagoras showed this was wrong when he proved that the length of the diagonal across a square with unit lengths was equal to the square root of 2.

The square root of 2 cannot be described as any fraction of integers. In fact, using fractions, we can only write down an approximation of √2 such as 1.4142, the actual value simply cannot be written. Hippasus of Metapontum, proved that this was true.

Unfortunately, the Pythagoreans could not accept that any measurable number should be so irrational and thus, according to one legend, they sentenced Hippasus to death by drowning.

Negative numbers began to appear at around 100 BC in China, and much later in the West (around 300 AD in Greece). Negative numbers were also fiercely resisted, although they eventually proved useful to numerically represent things such as quantities of debt.

Taken together, we call this the set of Real numbers. They are real in the sense that each of them can be physically plotted on a horizontal line (above) with zero in the middle, negative real numbers extending to the left and positive numbers to the right.

These numbers give us a powerful tool for performing many calculations, and can represent any measurable, or countable quantity. But there was more trouble on the horizon.

i = impossible

In the 1530's Italian mathematician, Girolamo Cardano, had figured out a formula for solving quadratic equations. In certain cases, however, his formula gave answers with components that were impossible to calculate because they included such values as the square root of -1.

It had long been established that two equal numbers, when multiplied together, always produce a positive number. Thus only positive numbers have a square root. How then could √-1 be considered a number if it is not real. Cardano simply reasoned that his formulas did not always work properly and rejected the cases where the answer included √-1.

i = imaginary

Cardano's contemporary, Rafael Bombelli of Bologna, realized that if you just ignored the fact that √-1 is impossible and represent it with a symbol (say i), then you could work your way through Cardano's rejected formulas, because each instance of i was matched by an instance of -i. By matching them up you can cancel them out (i.e. i - i = 0), and you are left with an answer that is both rational and correct.

Bombelli went on to develop the theory of complex numbers, the set of all numbers that could be represented by adding a Real number, and some Imaginary number (a multiple of √-1). These are now written using the formula a + bi, where a and b are real numbers and i = √-1.

In the centuries following Bombelli, mathematicians gradually came to accept imaginary numbers (kicking and screaming in some cases, but at least not drowning anybody) as a serious and useful mathematical tool.

In the beginning of the 19th century, three mathematicians, Carl Friedrich Gauss, Jean-Robert Argand, and Caspar Wessel came up with a clever way to physically represent complex numbers, the Argand Diagram.

Since imaginary numbers do not belong on the horizontal line representing Real numbers, these three independently came up with the idea of plotting imaginary numbers on a vertical line that intersects the real number line at 0.

Now, instead of representing numbers on a one dimensional line, we had a new set of numbers that could be plotted on a 2-dimensional grid. This geometric representation of complex numbers proved to be both insightful and useful. Simply manipulating the geometry of numbers represented in this plane is the analogue of performing arithmetic operations on those numbers.

For example, if you want to multiply a number on this diagram by i, you simply rotate the number on the diagram by 90º.

Above is a geometric representation for the multiplication 
of two complex numbers (A x B) = X.

Einstein and the 4th Dimension

Among other things, complex numbers are useful in General and Special Relativity (I'm walking on thin ice here by the way). The 4-dimensional field equations used by Einstein to represent gravitational bodies in space-time can be represented and manipulated using complex number algebra where the 4th dimension, time, is represented by the imaginary component.

Moving right along from mathematics, to contemporary architecture: In the three dimensional world we live in, how might we represent the geometry of, say, a 4-dimensional cube?

One approach is to use the same idea of projection that we use to draw a 3-dimensional cube on a 2-dimensional sheet of paper. This beautiful animation (below) demonstrates a rotating 4-dimensional cube projected onto 3-dimensions.

The Danish architect, Johann Otto von Spreckelsen, understood this when he designed the wonderful La Grande Arche de la Défense building in Paris in 1985.

While considering these projections, their connection to complex number theory and Argand diagrams, I decided to design a 4-dimensional projection where the inner cube is rotated by 90º along the vertical axis. The resulting shape is shown below.

Taking this a step further, I added a square to the center of the inner cube and rotated that a further 90º. This modified cube is shown below.

That is interesting, but I want to develop some practical Cubist forms that I can use for such things as a supporting column. The blue shape shown below uses the same geometry as the cube above, but it makes a solid of that shape's hollow core, I then cut away the exterior edges...

That's a pretty cool shape whose geometry has meaning, historically, philosophically and mathematically. I think it has a lot of potential and so I plan to be using this in some future projects. As John of Salisbury wrote in 1159, "we are like dwarfs on the shoulders of giants".

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