Image: SAM DERBYSHIRE
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Listen to this PodcastPolynomials, the meat and potatoes of high-school algebra, are foundational to many aspects of quantitative science. But it would take a particularly enthusiastic math teacher to think of these trusty workhorses as beautiful.
As with so many phenomena, however, what is simple and straightforward in a single serving becomes intricately detailed—beautiful, even—in the collective.
On December 5 John Baez, a mathematical physicist at the University of California, Riverside, posted a collection of images of polynomial roots by Dan Christensen, a mathematician at the University of Western Ontario, and Sam Derbyshire, an undergraduate student at the University of Warwick in England.
Polynomials are mathematical expressions that in their prototypical form can be described by the sum or product of one or more variables raised to various powers. As a single-variable example, take x2 - x - 2. This expression is a second-degree polynomial, or a quadratic, meaning that the variable (x) is raised to the second power in the term with the largest exponent (x2).
A root of such a polynomial is a value for x such that the expression is equal to zero. In the quadratic above, the roots are 2 and –1. That is to say, plug either of those numbers in for x and the polynomial will be equal to zero. (These roots can be found by using the famous quadratic formula.) But some roots are more complex. Take the quadratic polynomial x2 + 1. Such an expression is only equal to zero when x2 is equal to –1, but on its face this seems impossible. After all, a positive number times a positive number is positive, and a negative number times a negative number is positive as well. So what number, multiplied by itself, could be negative?
Imaginary numbers were, well, imagined into existence to fit the bill. Based on the number i, the square root of –1, imaginary numbers are unusual in that they do not represent a tangible physical quantity. (You cannot have i dollars—at least, not if you wish to pay your bills.) Polynomial roots can be either real or imaginary—that is, they may or may not have an imaginary component.
What Christensen and Derbyshire did was plot the roots of entire families of single-variable polynomials, imposing constraints on the polynomials' degrees and coefficients. (Coefficients are the multipliers of the variable terms—in the polynomial 4x - 2, the coefficients are 4 and –2, respectively.) For example, Christensen plotted the roots of every polynomial whose degree is six or less and whose coefficients are integers between –4 and 4.
The horizontal axis in Christensen's and Derbyshire's plots is the real numbers; the vertical axis is the imaginary numbers. So a real root, such as –1, would fall on the horizontal axis; a purely imaginary root such as 2i would fall on the vertical axis. The rest of the imaginary numbers—those with both real and imaginary components—fill out the quadrants of the graph. For instance, the imaginary number 3 - 2i would be represented by the point aligning with 3 on the horizontal (real) axis and –2 on the vertical (imaginary) axis.
What happens when these families of roots are plotted en masse? Intricate and intriguing patterns emerge that should appeal even to the most math-averse. Take a look at Christensen's and Derbyshire's images to see for yourself.
Slide Show: Polynomial Plot
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23 Comments
Add CommentPolynomials solved by Newton's method produce even more complex patterns:
Reply | Report Abuse | Link to thishttp://en.wikipedia.org/wiki/Newton_fractal
While these plots, in general, are pretty, they are the basis for finding the broadcast pattern for radio (and TV) towers. Complex numbers may seem imaginary, but the signal you receive on your radio, TV, or wireless device are real.
Reply | Report Abuse | Link to thisI wonder if the structure of the atom or the Universe is not hidden in such formulae. As if the GOD unleashed a formula and things then took their own inevitable course. It will be great if like God Particle, we can also unravel God formula.
Reply | Report Abuse | Link to thisI wonder if the structure of atom or the universe is hidden in some such formulae or the formula. AS if the God unleashed such a formula and things then took their inevitable course from there on. It would be interesting if like God particle, the God formula too can be uncovered (in terms of our maths).
Reply | Report Abuse | Link to thisI retired from teaching Mathematics two years ago. I WANT TO GO BACK!
Reply | Report Abuse | Link to thisIt is very likely but try to consider reading the Quran it is very scientific it contains alot of recently establish theories about space and planets is mentioned in the Quran which was revealed 14 centuries ago. The 19th coeficient partern may reveal something to do with our solar system or universe or multiverse.
Reply | Report Abuse | Link to thisI have no idea what on earth this article is talking about, but dang, "the sharply defined figures looming from the mist" sure are pretty.
Reply | Report Abuse | Link to thisThe esoteric meaning of the word GOD is Geometry of Divinity. The Golden Ratio is found throughout nature and is most beautifully displayed in the spiral shell of the Nautilus.
Reply | Report Abuse | Link to this"In the 1960s and '70s Swiss engineer and medical doctor Hans Jenny performed experiments using sound frequencies on various materials such as water, sand, dust, liquid plastic and milk.
Dr Jenny placed the material on a metal plate attached to a crystal oscillator which was controlled by a frequency generator capable of producing a wide range of vibrations. He then filmed and photographed what happened on the plate.
He coined the term Cymatics (the study of wave-form phenomena), which is derived from the Greek 'kyma' meaning 'wave', and 'ta kymatica' meaning 'matters pertaining to waves'."
You can see Dr Jenny and his device along with some pictures of the resulting phenomena at this link http://www.unitedearth.com.au/sound.html including the Sri Yantra (composed of 9 ascending and descending isosceles triangles) that formed when the OM was intoned.
You can look at a current You Tube video of the same type of experiments based on Dr Jenny's work at this link http://curezone.org/forums/fm.asp?i=862325.
Dr. Emoto did experiments with the effects on water of spoken and written words, sounds, and music. He would then freeze samples of the water and observe the crystalline structures that formed. You can see some of the results at these links http://enwaterment.com/ and http://www.life-enthusiast.com/twilight/research_emoto.htm.
All of Creation is formed of God's Light, Energy and Consciousness and has Innate Intelligence. We were created in the image and likeness of God and within our Heart of hearts God placed an Unfed Threefold Flame of Power (blue, Father), Wisdom (yellow, Son) and Love (pink, Holy Spirit) that bursts forth from a Cube of White Fire (Mother). This Threefold Flame is our connection point to our divinity and it is the portal to the kingdom of God that Jesus said was within us. This is the access point to the Higher Mind, the Christ Mind that St Paul admonished us to put on. This is the source of the "still small voice of conscience". For more, visit my blog at http://carltonnewman.blogspot.com.
We don't need to wax mystical to appreciate the beauty of mathematical images (sometimes called geometry) but it is a very old and natural human response. From the beginning, mathematicians have "seen" the beauty of nature through the medium of mathematical symbols and geometrical patterns. Plato was the first philosopher, whose works we posses, to build a philosophy around mathematics. His philosophy is, essentially, an ecstatic, mystical song of thanks for the mind's capacity to discover patterns which exist in nature and which are not simply inventions. We rationalists celebrate that which seems to be a "gift from God" but which is really simply our ability to discover and show others these deep and beautiful patterns that are the heart of nature. Our happiness is so great, sometimes, that we can't stop ourselves from singing and shouting with words like God, truth and beauty .... when the simple word "mathematics" does not seem adequate ...
Reply | Report Abuse | Link to thisSeeing as we are all being very esoteric I shall offer up some thoughts. It is clear that the universe has a superlative regularity both at the level of the galaxy and the natural world on this planet, thus dear Paul Davies has us all heading towards the "God" factor. What this mapping of the maths shows though is that the universe gets a bit lost at the square root of -1 and goes into a bit of a tizz, seemingly losing data around it like a black hole, a bit like the Hollywood robot "this does not compute". Maybe when trying to equate the quantum levels with the macro we should be trying to find a theory of everything that involves a theory of non-compliance rather than unification. Thus the universe in its majesty makes it all work like clockwork by having a system of things that don't compute, such as in brutal nature of the natural world, or as at the level of genes, or the edge of the universe, and quantum factors. It isn't a perfect universe, just a clever one. Can anyone work that bit out, the bit that looks and has a design which appears as though it isn't computing because it won't, so that is how the rest of it DOES work elegantly? Does that help- "God" as a trickster? Better call it the theory of implausibility or universal non-compliance rather than unified theory. We are looking in the wrong direction for the answers perhaps.
Reply | Report Abuse | Link to thisThe graphics are lovely but I would LOVE to know the "Simple Polynomial" that each is a plot of and what graphing method was used.
Reply | Report Abuse | Link to thisscohn,
Reply | Report Abuse | Link to thisThese images are not plot of a single polynomial. Also the polynomials here are calculated using complex number arithmetic formulas; not usual real number math.
See here: http://en.wikipedia.org/wiki/Complex_numbers
I think there are multiple known methods for solving complex number based polynomials. You would need to write a computer program to generate all possible polynomials, and solve all of them and plot all roots (each x,y) to replicate these images.
I, at least, did not mention God in any religious sense. It only stands for an unimaginable indescribable root of everything, for want of a more appropriate word in my limited vocabulary. I was just curious if with some basic building blocks (in the language of our physics), their unique values and their organization by means of some such basic formulae, (in the language of our mathematics) the world as we know as of now can be plausibly explained and then extrapolated to what it will be. I humbly suggest that we appreciate that we will be like one of those seven blind men exploring elephant and also accept that we may fall into the "king has no clothes" syndrome while not challenging the prevailing wisdom. As they say, what we like is not necessarily good, what we believe in is not necessarily true and that all questions are open. Havel or is it Gide , has said that befriend those seeking the truth, not the ones who have found it.
Reply | Report Abuse | Link to thisDear cc_ctc,
Reply | Report Abuse | Link to thisNice to see a sober [? - no need for a literally zero blood-alcohol level ;)], realistic, non-religious, hyper-realistic perspective; one offering a realistically messy, smeared-out, (not pedantic but prudent and plausible) philosophical position) on What Is going on.
Thinking polynomial roots may conceal some secret code is not the smart thing to do. Obviously, beauty, love, and justice are hardly understandable through mere numbers. There should be a superset to science.
Reply | Report Abuse | Link to thiswell,
Reply | Report Abuse | Link to thiswhere are the fractals
hiding super dimensions -
that last slide does look pretty confused/ing -
Reply | Report Abuse | Link to thisin contrast to the others
all that amorphous cloudy black fog back from the edge within the border,
who said that?
Willington at 04:59 AM on 12/29/09 WROTE.....
Reply | Report Abuse | Link to thisI retired from teaching Mathematics two years ago. I WANT TO GO BACK!
Willington...i say "GO BACK YOUNG MAN!!!!!"...my Father began working as a consultant when he was 72...went on do many great and wonderful things. Happiest years of his life...great thoughts...extraordinary friendships and time to contemplate the universe in the company of the young.
It was a fascinating thing to see...just like these images!
Patterns are nice, but which equations yield true non-patterns? True spontaneity or chance patterns?
Reply | Report Abuse | Link to thisConcerning Fractals-
Reply | Report Abuse | Link to thisSpecific geometry may help understand phenomenons beyond our natural intuition. Still, such mathematical tools are merely functional, not charged with ultimate significance.
That's way more interesting than when I had to find poles and zeros for control systems using Laplace transforms and a slide rule. The detail is probably more due to the floating point representation than the polynomial itself, but my 12" slide rule only had 3 sig figs.
Reply | Report Abuse | Link to thisDoes anyone think that there is a similarity here with FRACTALS or CHAOS?
Reply | Report Abuse | Link to thisI like pi
Reply | Report Abuse | Link to this