[math-fun] New way to visualise complex functions
Tim Large suggested an idea of how to visualise functions from the complex plane to the extended complex plane. Some preliminary results are shown on the webpage below: http://cp4space.wordpress.com/2013/01/27/visualising-complex-functions/ I should be able to make this into a Wolfram Demonstration. Sincerely, Adam P. Goucher
The general idea is a very good one (and has been thought of independently by a number of people.) But using a world map (in particular the one Adam showed in his lovely images) is not optimal, because so many colors in different parts of the initial globe are identical or almost so. Using a face may be better but also has the disadvantage of requiring us to distinguish the left half from the right half. At least we know meromorphic functions preserve orientation, so this shouldn't be extremely hard, but still requires a disambiguation that may not be particularly helpful. (Though for certain functions having a real period equal to the width of the face, it could be especially helpful.) But in general, an asymmetrical familiar picture may be most helpful. One common device is to use a square with left->right showing one gradient, and bottom->top showing another. For example, left->right might show the colors between white and red, and bottom->top might show dots of increasing radius. Or similarly there could be one gradient showing angle (perhaps the circular "spectrum" of saturated colors) and another one (e.g., saturation) to show increasing radius. These abstract gradients aren't necessarily viscerally more helpful than a familiar picture of something, but they do aid in quickly identifying real and imaginary components, or radius and angle, of the function values. --Dan On 2013-01-27, at 9:47 AM, Marc LeBrun wrote:
="Adam P. Goucher" <apgoucher@gmx.com> Tim Large suggested an idea
Nice! You might also try, instead of a world map, using a human face as the base image, making expressions of expressions, so to speak.
I have committed a few of these conformal mappings, using immersive 360°x180° panoramas as source. http://www.flickr.com/photos/sbprzd/sets/72157594172266668/ A map is certainly an immediately recognizable object, and the shape and location of continents is clear in our minds. Using other images offers more artistic liberty - at the expense of clarity in showing the behaviour of a complex function, which is what Adam is trying to do. Best, Seb On 27 January 2013 21:53, Dan Asimov <dasimov@earthlink.net> wrote:
The general idea is a very good one (and has been thought of independently by a number of people.)
But using a world map (in particular the one Adam showed in his lovely images) is not optimal, because so many colors in different parts of the initial globe are identical or almost so.
Using a face may be better but also has the disadvantage of requiring us to distinguish the left half from the right half. At least we know meromorphic functions preserve orientation, so this shouldn't be extremely hard, but still requires a disambiguation that may not be particularly helpful. (Though for certain functions having a real period equal to the width of the face, it could be especially helpful.)
But in general, an asymmetrical familiar picture may be most helpful.
One common device is to use a square with left->right showing one gradient, and bottom->top showing another. For example, left->right might show the colors between white and red, and bottom->top might show dots of increasing radius. Or similarly there could be one gradient showing angle (perhaps the circular "spectrum" of saturated colors) and another one (e.g., saturation) to show increasing radius.
These abstract gradients aren't necessarily viscerally more helpful than a familiar picture of something, but they do aid in quickly identifying real and imaginary components, or radius and angle, of the function values.
--Dan
On 2013-01-27, at 9:47 AM, Marc LeBrun wrote:
="Adam P. Goucher" <apgoucher@gmx.com> Tim Large suggested an idea
Nice! You might also try, instead of a world map, using a human face as the base image, making expressions of expressions, so to speak.
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Speaking about distortions: this map by Giacomo Faiella (updated version) is a palindromic planisphere (will you find the "negative" Italy?) Best, É. http://www.cetteadressecomportecinquantesignes.com/FAIELLA.pdf
Tim Large suggested an idea [...]
I've used Postscript-rendered text to show conformal mappings, e.g.: http://home.pipeline.com/~hbaker1/sigplannotices/sigcol07.pdf The actual Postscript code to do this is in the Postscript version of the paper itself(!) : http://home.pipeline.com/~hbaker1/sigplannotices/sigcol07.ps.gz At 07:29 AM 1/27/2013, Adam P. Goucher wrote:
Tim Large suggested an idea of how to visualise functions from the complex plane to the extended complex plane. Some preliminary results are shown on the webpage below:
http://cp4space.wordpress.com/2013/01/27/visualising-complex-functions/
I should be able to make this into a Wolfram Demonstration.
Sincerely,
Adam P. Goucher
participants (6)
-
Adam P. Goucher -
Dan Asimov -
Eric Angelini -
Henry Baker -
Marc LeBrun -
Seb Perez-D