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Mathematical Impressions: The Surprising Menger Sponge Slice | Simons Foundation
The Menger Sponge, a well-studied fractal, was first described in the 1920s. The fractal is cube-like, yet its cross section is quite surprising. What happens when it is sliced on a diagonal plane?
Superb explanation and beautiful animations.
“How many mathematical logicians does it take to replace a lightbulb??
None: They can’t do it, but they can prove that it can be done.
How many numerical analysts does it take to replace a lightbulb??
3.9967: (after six iterations).
How many classical geometers does it take to replace a lightbulb??
None: You can’t do it with a straight edge and a compass.
How many constructivist mathematicians does it take to replace a lightbulb??
None: They do not believe in infinitesimal rotations.
How many simulationists does it take to replace a lightbulb??
Infinity: Each one builds a fully validated model, but the light actually never goes on.
How many topologists does it take to screw in a lightbulb??
Just one. But what will you do with the doughnut?
How many analysts does it take to screw in a lightbulb??
Three: One to prove existence, one to prove uniqueness and one to derive a nonconstructive algorithm to do it.
How many Bourbakists does it take to replace a lightbulb: ?
Changing a lightbulb is a special case of a more general theorem concerning the maintain and repair of an electrical system. To establish upper and lower bounds for the number of personnel required, we must determine whether the sufficient conditions of Lemma 2.1 (Availability of personnel) and those of Corollary 2.3.55 (Motivation of personnel) apply. Iff these conditions are met, we derive the result by an application of the theorems in Section 3.1123. The resulting upper bound is, of course, a result in an abstract measure space, in the weak-* topology.”
"Another picture from our wonderful blackboard artist Ryan Budney in Victoria, Canada. He’s taken a 4-dimensional sphere and removed a knotted 2-dimensional sphere (i.e. the surface of a beach ball), and tried to triangulate the resulting space. Triangulation means breaking up a complicated space into simpler pieces – triangles! – and then explaining how these pieces fit together. The triangulation here has precisely one vertex (a 0-dimensional triangle), one edge (a 1-dimensional triangle), 4 triangles (i.e. your usual garden-variety triangles), 5 tetrahedra (3D triangles) and the two pentachora (4-dimensional triangles). The bottom picture shows how the pieces are glued together. The top picture is completely unrelated.
The knotted 2-dimensional sphere is called a Cappell-Shaneson knot because it is sitting inside a ‘homotopy 4-sphere’. That is, the space the knot is sitting in looks like a normal sphere from a distance but might turn out not to be upon closer inspection. If it isn’t, then it would provide a counterexample to the Poincaré Conjecture, which is still open in 4 dimensions (but was famously solved for 3 dimensions by Grigori Perelman in 2003).” Via What’s On My Blackboard?