By George Hart for the Museum of Mathematics What cool things have you made with magnets lately? Anything like the above dodecahedral construction? These geometric sculptures are the creations that resulted from Robert Hodgin playing with thousands of small magnetic spheres and cylinders. I especially like his geometric forms which remind me of Ernst Haeckel’s […]
The problem of tiling a plane has fascinated builders and mathematicians alike since time immemorial. At first glance, the task is straightforward: squares, triangles, hexagons all do the trick producing well known periodic structures. Ditto any number of irregular shapes and combinations of them.
A much trickier question is to ask which shapes can tile a plane in a pattern that does not repeat. In 1962, the mathematician Robert Berger discovered the first set of tiles that did the trick. This set consisted of 20,426 shapes: not an easy set to tile your bathroom with.
With a warm regard for home improvers, Berger later reduced the set to 104 shapes and others have since reduced the number further. Today, the most famous are the Penrose aperiodic tiles, discovered in the early 1970s, which can cover a plane using only two shapes: kites and darts.
The problem of finding a single tile that can do the job is called the einstein problem; nothing to do with the great man but from the German for one– “ein”–and for tile–“stein”. But the search for an einstein has proven fruitless. Until now.
Math Monday: 3D Hilbert Curve from plumbing supplies By George Hart for the Museum of Mathematics The 3D Hilbert curve is an infinitely long path which wiggles around through all regions of a cube, making right angles in a systematic manner. Physical models can be made with additive fabrication technology. The elegant metal version above […]
Mathematical art in the lava By George Hart for the Museum of Mathematics Edmund Harris created this geometric sculpture on a 35 year old lava field in Iceland. It can be understood as a simple form composed of equilateral triangles, but the curved edges where the triangles hinge together soften the geometry, giving it a […]
Balloon polyhedra By George Hart for the Museum of Mathematics For making mathematical models of polyhedra, a convenient and inexpensive material is the long clown balloon. This dodecahedron (made of ten balloons) and icosahedron (made of six balloons) are two examples from a study of Mathematical Balloon Twisting by Erik Demaine, Marty Demaine, and Vi […]
Sierpinski tetrahedron By George Hart for the Museum of Mathematics A classic 3D fractal is the Sierpinski tetrahedron, which is a tetrahedron of tetrahedra of tetrahedra, etc. This fifth-order model is about 8.5 inches along its edges. It is made from nylon by selective laser sintering. If you have access to additive fabrication machines, you […]
MacArthur fellow and MIT Media Lab alumnus Karl Sims brings us this great tutorial on how to build your own complex harmonograph (Wikipedia) for making cool…um…”geometric figures?” I’m looking for a 50-cent mathematician’s word (which may or may not exist) for these periodic spirally figures. Can anybody help me out? [Thanks, David!]
