I was reminded of Weierstrass' elliptic function by John Cook's recent post about elliptic functions:
(where 
are the points of a lattice in the complex plane).
are the points of a lattice in the complex plane).
The technique of adding up one term for each lattice point is really neat, and it's extra great that the function turns out to have such amazing properties.
Since John's post showed what pretty pictures come from using phase portraits to visual complex functions, I decided to make a video to show me what the partial sums that converge to the Weierstrass function. I'm adding a bunch of terms starting from the origin going outward, but you could add the terms in any order.
The picture on the left is a zoomed in version showing two periods in each direction. You can see the function getting more periodic as we add more terms.
The code is here. The code is O(n^2) (where n is the number of terms I want images for) when it really should be O(n), since I'm in effect asking for a lot of repeated computation by making each image separately. Sorry!
The Weierstrass Elliptic Function is an interesting example of how mathematical concepts can be explored computationally and visually. The use of lattice points and partial sums provides a clear way to observe how the function gradually approaches its characteristic periodic behavior.
ReplyDeleteThe visualization of the partial sums is particularly useful because complex mathematical functions can often be difficult to understand from equations alone. Computational visualization allows these mathematical patterns to be represented graphically and examined more intuitively. This makes the topic relevant to Data Visualization Course.
The article also demonstrates how programming can be used to experiment with mathematical functions and generate visual representations. Python is widely used for numerical computation and visualization, making Python Online Course a relevant resource for learners interested in implementing similar mathematical experiments.
ReplyDeleteStudents interested in taking computational approaches further can explore Machine Learning Projects For Final Year, where mathematical and computational techniques are applied to practical problems.
ReplyDeleteSimilarly, advanced computational models often involve mathematical functions, numerical methods, and visual analysis. Learners exploring these areas can also look at Deep Learning Projects For Final Year.
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