Occurrences of The Dragon Curve in Solution Sets
Having obtained the set of solutions to a differential equation, any linear combination of the solutions will, because of the superposition principle also obey the original equation. In other words, new solutions are obtained by applying a function to the set of existing solutions. This is similar to how an iterated function system produce new points in a set, though not all IFS are linear functions. In a conceptually similar vein, a set of Littlewood polynomials can be arrived at by such iterated applications of a set of functions.
A Littlewood polynomial is a polynomial : where all .
For some |w| < 1 we define the following functions:
Starting at z=0 we can generate all Littlewood polynomials of degree d using these functions iteratively d+1 times. For instance:
It can be seen that for w = (1+i)/2, the above pair of functions is equivalent to the IFS formulation of the Heighway dragon. That is, the Heighway dragon, iterated to a certain iteration, describe the set of all Littlewood polynomials up to a certain degree, evaluated at the point w = (1+i)/2. Indeed, when plotting a sufficiently high number of roots of the Littlewood polynomials, structures similar to the dragon curve appear at points close to these coordinates.
Read more about this topic: Dragon Curve
Famous quotes containing the words occurrences of, occurrences, dragon, curve, solution and/or sets:
“If to be venerated for benevolence, if to be admired for talents, if to be esteemed for patriotism, if to be beloved for philanthropy, can gratify the human mind, you must have the pleasing consolation to know that you have not lived in vain. And I flatter myself that it will not be ranked among the least grateful occurrences of your life to be assured that, so long as I retain my memory, you will be thought on with respect, veneration, and affection by your sincere friend.”
—George Washington (1732–1799)
“If to be venerated for benevolence, if to be admired for talents, if to be esteemed for patriotism, if to be beloved for philanthropy, can gratify the human mind, you must have the pleasing consolation to know that you have not lived in vain. And I flatter myself that it will not be ranked among the least grateful occurrences of your life to be assured that, so long as I retain my memory, you will be thought on with respect, veneration, and affection by your sincere friend.”
—George Washington (1732–1799)
“Come not between the dragon and his wrath.”
—William Shakespeare (1564–1616)
“Nothing ever prepares a couple for having a baby, especially the first one. And even baby number two or three, the surprises and challenges, the cosmic curve balls, keep on coming. We can’t believe how much children change everything—the time we rise and the time we go to bed; the way we fight and the way we get along. Even when, and if, we make love.”
—Susan Lapinski (20th century)
“To the questions of the officiously meddling police Falter replied absently and tersely; but, when he finally grew tired of this pestering, he pointed out that, having accidentally solved “the riddle of the universe,” he had yielded to artful exhortation and shared that solution with his inquisitive interlocutor, whereupon the latter had died of astonishment.”
—Vladimir Nabokov (1899–1977)
“I think middle-age is the best time, if we can escape the fatty degeneration of the conscience which often sets in at about fifty.”
—W.R. (William Ralph)