10/6/2023 0 Comments Fractal dimensionThe fractal dimension of the ternary Cantor set is D H = ln(2)/ln(3) = 0.6309.įig. This clumpiness is an essential feature that distinguishes it from the one-dimensional number line, and it raised important questions about dimensionality. But whereas the real numbers are uniformly distributed, Cantor’s set is “clumped”. It is a striking example of a function that is not equal to the integral of its derivative! Cantor demonstrated that the size of his set is aleph0, which is the cardinality of the real numbers. The set generates a function (The Cantor Staircase) that has a derivative equal to zero almost everywhere, yet whose area integrates to unity. Partially inspired by Weierstrass’ discovery, George Cantor (1845 – 1918) published an example of an unusual ternary set in 1883 in “Grundlagen einer allgemeinen Mannigfaltigkeitslehre” (“Foundations of a General Theory of Aggregates”). It is a fractal with fractal dimension D = 2 + ln(0.5)/ln(5) = 1.5693. This continuous function is nowhere differentiable. Riemann had asked whether the functionįig. Karl Weierstrass (1815 – 1897) was studying convergence properties of infinite power series in 1872 when he began with a problem that Bernhard Riemann had given to his students some years earlier. This blog page presents the history through a set of publications that successively altered how mathematicians thought about curves in spaces, beginning with Karl Weierstrass in 1872. Here is a short history of fractal dimension, partially excerpted from my history of dynamics in Galileo Unbound (Oxford University Press, 2018) pg. From then onward the concept of dimension had to be rebuilt from the ground up, leading ultimately to fractals. Child’s play!īut how do you think of fractional dimensions? What is a fractional dimension? For that matter, what is a dimension? Even the integer dimensions began to unravel when George Cantor showed in 1877 that the line and the plane, which clearly had different “dimensionalities”, both had the same cardinality and could be put into a one-to-one correspondence. Print("Minkowski–Bouligand dimension (computed): ", fractal_dimension(img, 0.And so on to 5 and 6 dimensions and on. Traceback (most recent call last): File "C:\Users\Anup Shahi\Documents\pythonProject\venv\lib\site-packages\IPython\core\interactiveshell.py", line 3418, in run_code exec(code_obj, er_global_ns, er_ns) File "", line 1, in runfile('C:/Users/Anup Shahi/Documents/pythonProject/FD.py', wdir='C:/Users/Anup Shahi/Documents/pythonProject') File "C:\Program Files\JetBrains\P圜harm 2020.3\plugins\python\helpers\pydev_pydev_bundle\pydev_umd.py", line 197, in runfile pydev_imports.execfile(filename, global_vars, local_vars) # execute the script File "C:\Program Files\JetBrains\P圜harm 2020.3\plugins\python\helpers\pydev_pydev_imps_pydev_execfile.py", line 18, in execfile exec(compile(contents+"\n", file, 'exec'), glob, loc) File "C:/Users/Anup Shahi/Documents/pythonProject/FD.py", line 51, in print("Minkowski–Bouligand dimension (computed): ", fractal_dimension(I)) File "C:/Users/Anup Shahi/Documents/pythonProject/FD.py", line 14, in fractal_dimension assert(len(Z.shape) = 2) AssertionError Print("Minkowski–Bouligand dimension (computed): ", fractal_dimension(I))įile "C:/Users/Anup Shahi/Documents/pythonProject/FD.py", line 14, in fractal_dimension Pydev_imports.execfile(filename, global_vars, local_vars) # execute the scriptįile "C:\Program Files\JetBrains\P圜harm 2020.3\plugins\python\helpers\pydev_pydev_imps_pydev_execfile.py", line 18, in execfileĮxec(compile(contents+"\n", file, 'exec'), glob, loc)įile "C:/Users/Anup Shahi/Documents/pythonProject/FD.py", line 51, in ![]() Runfile('C:/Users/Anup Shahi/Documents/pythonProject/FD.py', wdir='C:/Users/Anup Shahi/Documents/pythonProject')įile "C:\Program Files\JetBrains\P圜harm 2020.3\plugins\python\helpers\pydev_pydev_bundle\pydev_umd.py", line 197, in runfile Here what I get after running the programįile "C:\Users\Anup Shahi\Documents\pythonProject\venv\lib\site-packages\IPython\core\interactiveshell.py", line 3418, in run_codeĮxec(code_obj, er_global_ns, er_ns) I used "imageio.imread" but the problem seems to be not resolved. Print( "Haussdorf dimension (theoretical): ", ( np. Print( "Minkowski–Bouligand dimension (computed): ", fractal_dimension( I)) # Fit the successive log(sizes) with log (counts)Ĭoeffs = np. # Actual box counting with decreasing size # Build successive box sizes (from 2**n down to 2**1) # Greatest power of 2 less than or equal to p # We count non-empty (0) and non-full boxes (k*k) ![]() # fractal dimension of a set S in a Euclidean space Rn, or more generally in aĭef fractal_dimension( Z, threshold = 0.9): # Minkowski dimension or box-counting dimension, is a way of determining the # In fractal geometry, the Minkowski–Bouligand dimension, also known as
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