Lee Mac Posted November 27, 2009 Share Posted November 27, 2009 Following with the Mathematical trend to my threads... here is another extremely interesting branch of Mathematics. Relating to my other Chaos Theory thread, here are a few attractors that you can create in ACAD. The attractors are created by solving various Non-Linear Differential Equations numerically, (using Euler's method in this case). I have modelled three attractors for you to experiment with: The Lorenz Attractor: This stems from a simplification of the equations used to model Weather systems, and is the general attractor that most people think of when they think about Chaos Theory... as Edward Lorenz was the pioneer of Chaos theory. The Rössler Attractor: The Duffing's Attractor: This attractor is obtained when numerically solving the non-linear ordinary differential equation that is Duffing's Equation. The equation models a damped oscillator, such as a weighted non-uniform spring. To solve the equation numerically, I have reduced the second order differential equation to two first order equations, and then used Euler's method to generate the attractor. Various Attractors can be generated by varying the input parameters (a,b and c in the code). (defun c:lorenz (/ iLim i h a b c x0 y0 z0 x y z) (setq iLim 10000 i -1 h 0.01 a 10. b 28. c (/ 8. 3.) x0 0.1 y0 0. z0 0.) (entmake '((0 . "POLYLINE") (70 . )) (while (< (setq i (1+ i)) iLim) (setq x (+ x0 (* h a (- y0 x0))) y (+ y0 (* h (- (* x0 (- b z0)) y0))) z (+ z0 (* h (- (* x0 y0) (* c z0)))) x0 x y0 y z0 z) (entmake (list '(0 . "VERTEX") '(70 . 32) (cons 10 (list x y z))))) (entmake '((0 . "SEQEND"))) (princ)) (defun c:rossler (/ iLim i h a b c x0 y0 z0 x y z) (setq iLim 10000 i -1 h 0.01 a 0.2 b 0.2 c 5.7 x0 0.1 y0 0. z0 0.) (entmake '((0 . "POLYLINE") (70 . )) (while (< (setq i (1+ i)) iLim) (setq x (+ x0 (* h (- (- y0) z0))) y (+ y0 (* h (+ x0 (* a y0)))) z (+ z0 (* h (+ b (* z0 (- x0 c))))) x0 x y0 y z0 z) (entmake (list '(0 . "VERTEX") '(70 . 32) (cons 10 (list x y z))))) (entmake '((0 . "SEQEND"))) (princ)) (defun c:duffings (/ iLim i h a b x0 y0 z0 x y z) (setq iLim 10000 i -1 h 0.04 a 0.2 b 0.3 x0 0. y0 0. z0 0.) (entmake '((0 . "POLYLINE") (70 . )) (while (< (setq i (1+ i)) iLim) (setq x (+ x0 (* h y0)) y (+ y0 (* h (+ (- x0 (* x0 x0 x0) (* a y0)) (* b (cos z0))))) z (+ z0 h) x0 x y0 y z0 z) (entmake (list '(0 . "VERTEX") '(70 . 32) (cons 10 (list x y z))))) (entmake '((0 . "SEQEND"))) (princ)) I hope you enjoy this little exploration into this fascinating area of mathematics, and, of course, if you have any questions - please ask. Lee Quote Link to comment Share on other sites More sharing options...
CAD USER Posted February 11, 2016 Share Posted February 11, 2016 Amazing.... Quote Link to comment Share on other sites More sharing options...
BIGAL Posted February 11, 2016 Share Posted February 11, 2016 Cool ! oh maybe it was the 1/2 empty bottle of scotch Quote Link to comment Share on other sites More sharing options...
Happy Hobbit Posted February 11, 2016 Share Posted February 11, 2016 Incredible...How...?...What..?..Why..? Good intellectual exercise Lee Quote Link to comment Share on other sites More sharing options...
Lee Mac Posted February 11, 2016 Author Share Posted February 11, 2016 Thanks all This thread is now quite old - a better write-up may be found on my site here. This is just one of a few Mathematical Endeavours: Attractors Fractals Iterated Function Systems Koch Snowflake Logistic Map Sierpinski Triangle Enjoy! Lee Quote Link to comment Share on other sites More sharing options...
tzframpton Posted February 11, 2016 Share Posted February 11, 2016 Lee, you should also add the Fibonacci Sequence in there. Quote Link to comment Share on other sites More sharing options...
Lee Mac Posted February 12, 2016 Author Share Posted February 12, 2016 Lee, you should also add the Fibonacci Sequence in there. Thanks for the suggestion Tannar - I'll see what I can do! Quote Link to comment Share on other sites More sharing options...
Lee Mac Posted March 8, 2016 Author Share Posted March 8, 2016 Going with the theme of this thread, I was recently inspired to write the following new addition to my site: Sierpinski Triangle Quote Link to comment Share on other sites More sharing options...
tzframpton Posted March 8, 2016 Share Posted March 8, 2016 The 3D one is spectacular!! Man I love seeing what math can do. Quote Link to comment Share on other sites More sharing options...
Lee Mac Posted March 8, 2016 Author Share Posted March 8, 2016 The 3D one is spectacular!! Man I love seeing what math can do. Thanks Tannar! As you can tell, I'm also fascinated by fractal geometry Quote Link to comment Share on other sites More sharing options...
TheCADnoob Posted March 9, 2016 Share Posted March 9, 2016 These things are great! Quote Link to comment Share on other sites More sharing options...
Lee Mac Posted March 9, 2016 Author Share Posted March 9, 2016 These things are great! Thanks, I'm glad you like them! Quote Link to comment Share on other sites More sharing options...
TheCADnoob Posted March 9, 2016 Share Posted March 9, 2016 Id love to eventually get to the point where i would be able to create things like these. Quote Link to comment Share on other sites More sharing options...
Lee Mac Posted March 9, 2016 Author Share Posted March 9, 2016 Practice, practice, practice... Quote Link to comment Share on other sites More sharing options...
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