Polynomial

“If any philosopher had been asked for a definition of infinity, he might have produced some unintelligible rigmarole, but he would certainly not have been able to give a definition that had any meaning at all.” Bertrand Russell

In this brief essay on the ‘infinite,’ I do not want to ramble on about uncertainties or truths. My aim here is not to lecture. Rather it is to encourage… So instead of joining the dots together in an obvious proclamation of basis, I am happy to quote certain others’ works that have more pertinently and eloquently touched aspects of the ‘infinite’ over the years, with a hope that the reader’s mind will naturally settle on the splendor lying behind the complex and distracting facades of catechism.

1. William Blake

William Blake (28 November 1757 – 12 August 1827) was an English poet, painter, and printmaker. Largely unrecognized during his lifetime, Blake is now considered a seminal figure in the history of both poetry and the visual arts of the Romantic Age. His prophetic poetry has been said to form “what is in proportion to its merits the least read body of poetry in the English language”. His visual artistry has led one modern critic to proclaim him “far and away the greatest artist Britain has ever produced”. Although he only once journeyed farther than a day’s walk outside London during his lifetime, he produced a diverse and symbolically rich corpus, which embraced ‘imagination’ as “the body of God”, or “Human existence itself”.

In one of his most insightful poems “The Auguries of Innocence”, he states:

“To see a world in a grain of sand,
And a heaven in a wild flower,
Hold infinity in the palm of your hand,
And eternity in an hour.”

For the full poem, please visit: http://www.artofeurope.com/blake/bla3.htm

What could such majestic tapestry mean? Well… Perhaps to the bovine logician, or the unenlightened literary reader, this idea might fall short of the lofty missive prescribed by Blake’s godly eye. But thankfully, with today’s scientific awareness of all things great and small, this notion may be better ‘understood’ with only a lax dedication towards ‘knowing’ the world around oneself better.

2. Niels Fabian Helge von Koch

Niels Fabian Helge von Koch (January 25, 1870 – March 11, 1924) was a Swedish mathematician who gave his name to the famous fractal known as the Koch snowflake, one of the earliest fractal curves to be described.

He was born into a family of Swedish nobility. His grandfather, Nils Samuel von Koch (1801–1881), was the Attorney-General of Sweden. His father, Richert Vogt von Koch (1838–1913) was a Lietenant-Colonel in the Royal Horse Guards of Sweden.

Von Koch wrote several papers on number theory . One of his results was a 1901 theorem proving that the Riemann hypothesis is equivalent to a strengthened form of the prime number theorem.

He described the Koch curve in a 1904 paper entitled “On a continuous curve without tangents constructible from elementary geometry” (original French title: “Sur une courbe continue sans tangente, obtenue par une construction géométrique élémentaire”).

The Koch snowflake (or Koch star) is a mathematical curve and one of the earliest fractal curves to have been described. (Actually Koch described what is now known as the Koch curve, which is the same as the now popular snowflake, except it starts with a line segment instead of an equilateral triangle. Three Koch curves form the snowflake.)

The Koch curve is a special case of the Cesaro curve where:

,

which is in turn a special case of the de Rham curve.

One can imagine that it was created by starting with a line segment, then recursively altering each line segment as follows:

1. divide the line segment into three segments of equal length.
2. draw an equilateral triangle that has the middle segment from step 1 as its base and points outward.
3. remove the line segment that is the base of the triangle from step 2.

The Koch snowflake/star is generated using the same recursive process but starting with an equilateral triangle rather than a line segment. After doing this once for the Koch snowflake, the result is a shape similar to the Star of David.

The Koch curve is the limit approached as the above steps are followed over and over again.

The Koch curve has an infinite length because each time the steps above are performed on each line segment of the figure there are four times as many line segments, the length of each being one-third the length of the segments in the previous stage. Hence the total length increases by one third and thus the length at step n will be (4/3)ⁿ: the fractal dimension is log 4/log 3 ≈ 1.26, greater than the dimension of a line (dimension 1) but less than Peano’s space-filling curve.

Ever smaller and smaller… As one zooms into the Koch curve, steady self-similarity is exuded infinitly:

3. Karl Menger

In mathematics, the Menger sponge is a fractal curve. It is the universal curve, in that it has topological dimension one, and any other curve (more precisely: any compact metric space of topological dimension 1) is homeomorphic to some subset of it. It is sometimes called the Menger-Sierpinski sponge or the Sierpinski sponge. It is a three-dimensional extension of the Cantor Set and Sierpinski Carpet. It was first described by Austrian mathematician Karl Menger in 1926 while exploring the concept of topological dimension.

Each face of the Menger sponge is a Sierpinski cerpet; furthermore, any intersection of the Menger sponge with a diagonal or medium of the initial cube M₀ is a Cantor set.

The Menger sponge is a closed set i.e. it contains its own boundary (unlike the Mandelbrot set); since it is also bounded, the Heine-Borel theorem implies that it is compact. Furthermore, the Menger sponge is uncountable and has Lebesque measure 0.

The topological dimension of the Menger sponge is one, the same as any curve. Menger showed, in the 1926 construction, that the sponge is a universal curve, in that any possible one-dimensional curve is homeomorphic to a subset of the Menger sponge, where here a curve means any compact metric shape of Lebesgue covering dimension one; this includes trees and graphs with an arbitrary countable number of edges, vertices and closed loops, connected in arbitrary ways.

In a similar way, the Sierpinski cerpet is a universal curve for all curves that can be drawn on the two-dimensional plane. The Menger sponge constructed in three dimensions extends this idea to graphs that are not planar, and might be embedded in any number of dimensions. Thus any geometry of quantum loop gravity can be embedded in a Menger sponge.

Interestingly, the Menger sponge simultaneously exhibits an infinite surface area and encloses zero volume. This idea of the infinite held within the finite is perhaps not such a revelation as it might initially seem…

A ‘simpler’ more visual way to understand the complexity of Menger’s idea can be seen in the follow animation:

http://www.pure-mirage.com/html/MillersMengerSpongeFastPlay.htm

4. Benoît B. Mandelbrot

Benoît B. Mandelbrot (born 20 November 1924) is a French mathematician, best known as the father of fractal geometry. He is Sterling Professor of Mathematical Sciences, Emeritus at Yale University; IBM Fellow Emeritus at the Thomas J. Watson Research Cente; and Battelle Fellow at the Pacific Northwest National Laboratory. He was born in Poland. His family moved to France when he was a child, and he was educated in France.

In mathematics, the Mandelbrot set, named after Mandelbrot himself, is a set of points in the complex plane, the boundary of which forms a fractal. Mathematically, the Mandelbrot set can be defined as the set of complex values of c for which the orbit of 0 under iteration of the complex quadratic polynomial z_n+1 = z_n² + c remains bounded. That is, a complex number, c, is in the Mandelbrot set if, when starting with z₀=0 and applying the iteration repeatedly, the absolute value of z_n never exceeds a certain number (that number depends on c) however large n gets.

In other words… Part of the charm of the set is that it springs from such a simple equation: z² + c. The terms z and c are complex numbers, which consist of an imaginary number (a multiple of the square root of –1) combined with a real number. One begins by assigning a fixed value to c, letting z = 0 and calculating the output. One then repeatedly recalculates, or iterates, the equation, substituting each new output for z. Some values of c, when plugged into this iterative function, produce outputs that swiftly soar toward infinity. Other values of c produce outputs that eternally skitter about within a certain boundary. This latter group of c‘s, or complex numbers, constitutes the Mandelbrot set.

When plotted on a graph consisting of all complex numbers, the members of the set cluster into a distinctive shape. From afar, it is not much to look at: it has been likened to a tumor-ridden heart, a beetle, a badly burned chicken and a warty figure eight on its side.

A closer look reveals that the borders of the set do not form crisp lines but seem to shimmer like flames. Repeated magnification of the borders plunges one into a bottomless phantasmagoria of baroque imagery. Some forms, such as the basic heartlike shape, keep recurring but always with subtle differences.

The Mandelbrot set shows more intricate detail the closer one looks or magnifies the image, usually called “zooming in”. The following example of an image sequence zooming to a selected c value gives an impression of the infinite richness of different geometrical structures, and explains some of their typical rules.

The magnification of the last image relative to the first one is about 10,000,000,000 to 1. Relating to an ordinary computer monitor, it represents a section of a Mandelbrot set with a diameter of 4 million kilometres. Its border would show an inconceivable number of different fractal structures…

And here I will leave you with a quotation…

“Pure mathematics is, in its way, the poetry of logical ideas.”  Albert Einstein