1. The Hilbert curve is a curve which completely covers the square. We can obtain it by folding a U-shaped curve recursively, as shown.

    # vimeo.com/199059577 Uploaded 2,772 Plays 1 Comment
  2. Take three numbers in the complex plane, z₁, z₂, z₃ and consider the Julia set for the Newton iteration process for solving (z - z₁) (z - z₂) (z - z₃) = 0. If we move z₁, z₂, z₃ about in the plane, we get a changing Julia set.

    # vimeo.com/199059402 Uploaded 220 Plays 0 Comments
  3. The Julia set for the Newton iteration for solving z³ - 1 = 0. Here we zoom in by a factor of one million to watch the self-similar endlessly repating structure of the set.

    # vimeo.com/199059077 Uploaded 206 Plays 0 Comments
  4. # vimeo.com/117221911 Uploaded 349 Plays 0 Comments
  5. Given a pair of numbers (a,b), let us say that an (a,b)-polynomial is a polynomial in which every coefficient is either a or b. There are 2^(n+1) such polynomials of degree n, provided a and b are not equal.

    For any given pair (a,b) we can draw all the zeroes of all the (a,b)-polynomials of degree at most 11, where zeroes of lower degrees are represented by larger circles. Let us call this an (a,b)-image.

    The animation shows (a,b)-images as the pair (a,b) moves in a line segment from (1,1) to (-1,1). In other words, a changes from 1 to -1, while b stays fixed.

    # vimeo.com/109100676 Uploaded 289 Plays 0 Comments

Visual math

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