by Jackie Anderson and Dan Katz
Problem:

Each of the given images consists of a (darkened) scene from a movie combined with a colored fractal. Identifying the movies will reveal a common bond; each of them features Julia Roberts, who is not visible in any of the scenes.

The common thread of Julia Roberts might suggest that the fractals are (filled) Julia sets, the sets of values where a given polynomial behaves chaotically as the values vary slightly. Specifically, they are examples of the most well-known Julia sets, those that arise from a polynomial fc(z) = z2 + c, where c is a complex number. The c’s that produce connected Julia sets are those in the Mandelbrot set, which is contained in the portion of the complex plane shown in the bottom image.

For each colored arrow, the complex number c corresponding to one of those dots produces the Julia set shown in the corresponding colored image (the set for each point can be viewed using https://www.marksmath.org/visualization/julia_sets/). This identifies ten points in the complex plane. The inequality chain below the bottom image sorts ten variable names, c1 through c10, in increasing order of magnitude (distance from the origin). Comparing the magnitudes of the ten points found above determines which point should be labeled with which variable name.

In each film, the character Julia Roberts plays has a first name with the same number of letters as dots on the matching-color arrow. (The images are alphabetized by these first names.) If the character names are written one letter at a time along the arrow, one of the letters will correspond to a point found above. When these letters are ordered from c1 to c10, they spell the answer to this puzzle, PORTRAYALS.

Here are the movies, characters, extracted letters, point labels and points for each image in the order shown:

Movie Character Letter Label Point Magnitude
Charlotte’s Web CHARLOTTE O c2 (-0.5, 0) 0.500
Mystic Pizza DAISY Y c7 (0.36, -0.38) 0.523
The Pelican Brief DARBY R c3 (-0.36, -0.61) 0.708
Erin Brockovich ERIN R c5 (-1, 0.26) 1.033
Wonder ISABEL S c10 (-0.07, 0.65) 0.654
My Best Friend’s Wedding JULIANNE L c9 (0.02, -0.64) 0.640
Confessions of a Dangerous Mind PATRICIA P c1 (-1.14, -0.25) 1.167
Flatliners RACHEL A c8 (0.28, 0.01) 0.280
Ocean’s Eleven TESS T c4 (-0.11, 0.89) 0.897
Pretty Woman VIVIAN A c6 (-0.67, 0.46) 0.813