Three Teens Knotted Into the Menger Sponge

Knots are intricate tangles of intertwined lines that may appear complex at first glance but have profound applications, from DNA repair to bow-tying to managing money at the quantum level (according to some mathematicians). Malors Espinosa was therefore determined to give his students knotting-related problems that would both challenge and excite their minds – including these questions about knots by Malors Espinosa himself!

The objective was to demonstrate that every knot could be embedded within a fractal known as the Menger sponge, consisting of two identical, connected sponge-like faces. But first Joshua Broden, Noah Nazareth and Niko Voth needed to figure out a way to stretch a knot into three dimensions without accidentally creating holes; they did this using an arc presentation diagram which provided detailed information about where strands pass in front and behind one another.

To create an arc presentation, one begins with a set of points on a grid and applies rules to convert these points into horizontal and vertical segments that cross paths with one another; wherever these segments cross, connect them by lines – this creates the iconic knot image. However, when trying to place such knots onto Menger sponges this method no longer works: there should be holes in their material where directly opposite faces meet in coordinate space; thus making an arc presentation unsuitable for display.