Sir Petrie has noticed that the relation between the height and the base (ratio 3/2 = 6/4) conceal the Pythagorean Triangle (the first “Pythagorean Triplet”) https://en.wikipedia.org/wiki/Fine-structure_constant#Numerological_explanations_and_multiverse_theory 137
"The Elder Wand," he said, and he drew a straight vertical line on the parchment. "The Resurrection Stone," he said, and he added a circle on top of the line. "The Cloak of Invisibility," he finished, enclosing both line and circle in a triangle, to make the symbol that so intrigued Hermione. "Together," he said, "the Deathly Hallows."
Note that this infinite process is not dependent upon the starting shape being a triangle—it is just clearer that way. The first few steps starting, for example, from a square also tend towards a Sierpinski triangle. Michael Barnsley used an image of a fish to illustrate this in his paper "V-variable fractals and superfractals."