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)
“And then at last our bliss
Full and perfect is,
But now begins; for from this happy day
The old Dragon underground,
In straiter limits bound,
Not half so far casts his usurped sway,
And, wroth to see his kingdom fail,
Swinges the scaly horror of his folded tail.”
—John Milton (1608–1674)
“The years-heired feature that can
In curve and voice and eye
Despise the human span
Of durance—that is I;
The eternal thing in man,
That heeds no call to die.”
—Thomas Hardy (1840–1928)
“What is history? Its beginning is that of the centuries of systematic work devoted to the solution of the enigma of death, so that death itself may eventually be overcome. That is why people write symphonies, and why they discover mathematical infinity and electromagnetic waves.”
—Boris Pasternak (1890–1960)
“It is time to be old,
To take in sail:—
The god of bounds,
Who sets to seas a shore,
Came to me in his fatal rounds,
And said: ‘No more!’”
—Ralph Waldo Emerson (1803–1882)