• The shape of the Wankel rotor

    I’ve written a number of posts about curvilinear triangles that are not the Reuleaux triangle, including MIT’s Kresge Auditorium, triforce string art, valve covers, a patio table, and the logo of Whale Cove, Nunavut. I’ve long intended to write about another obvious topic in this theme, the curved-triangle rotor of the Wankel engine, but was finally pushed into doing so by seeing that two recent popular mathematics books, How Round Is Your Circle? (2008) and Icons of Mathematics (2011) repeat the falsehood that Wankel rotors are Reuleaux triangles. They are not.

  • Linkage

    • The five bridges puzzle (). Sort of like the bridges of Königsberg, but stochastic. A cute puzzle with a connection to percolation theory and connection games.
  • Sorting with integer offsets

    Here’s a cute exercise for the next time you’re teaching radix sorting in an algorithms class:

  • Subpract

    I’ve written here before about subtraction games, two-player games in which the players remove tokens from a pile of tokens, the number of removed tokens is required to belong to a designated subtraction set, and the goal is to make the last move. For instance, subtract a square, a game I studied at FUN 2018, is of this type, with the subtraction set being the square numbers.

  • Linkage

    • Graduata data structures online (), finally done and graded. Warning: dry voice-over-slides videos, and some mistakes, because I didn’t have time to put together anything more sophisticated or edit more carefully.
  • Infinite threshold graphs, four different ways

    One of the difficulties of extending results from finite graphs to infinite ones is that it is not always obvious how to extend the definitions. A single class of finite graphs may correspond, in the infinite graph world, to several different natural classes of infinite graphs. One of the ways this can happen is through orderings: if a class of graphs has a natural ordering on its vertices (say, through a construction in which graphs in this class are built up by adding one vertex at a time) then we might get several classes of infinite graphs with different ways of restricting or not restricting this vertex ordering.

  • Linkage

  • Linkage

    • Famous people’s bookshelves, visible as they teleconference from home (, via). The story calls this inadvertent, but that seems dubious. Much of my teleconferencing has books behind me, deliberately, mostly as a clean background in a part of the house where it’s convenient to sit and lighting is good, but also because very few views in my house avoid books. On the other hand, at least one colleague has substituted a fake background from a library…
  • The inbox of a triangle

    In affine geometry, the minimum-area ellipse surrounding a given triangle and the maximum-area ellipse within it (its two Steiner ellipses) are concentric and similar. This can be seen easily by performing an affine transformation to an equilateral triangle, observing that in this case these ellipses are concentric circles (the circumcircle and incircle), and that the extreme ellipses of the transformed shape are the transforms of the extreme ellipses of the original shape. In Cartesian geometry, where the Cartesian coordinates can be independently linearly transformed or swapped, something similar turns out to happen. In this case, the minimum-area axis-parallel rectangle surrounding a given triangle (its bounding box) and the maximum-area rectangle within it (let’s call it the inbox) are always similar, though not concentric.

  • Hanoi vs Sierpiński

    The Hanoi graphs and Sierpiński graphs both look like the Sierpiński triangle, and have a very similar recursive construction from triples of smaller graphs of the same type, but they are not quite the same graphs as each other. The Sierpiński graphs (left, below) are the graphs of the vertices and boundary edges of partially-constructed Sierpiński triangles; they can also be formed from three smaller Sierpiński graphs by identifying pairs of extreme vertices (the vertices of degree two at the three corners of the triangular layout). The Hanoi graphs (right, below) are the state spaces of the tower of Hanoi puzzle, in which rings of different size are moved one at a time between three pegs, only allowing moves that keep the rings sorted on each peg. They also have a construction from three smaller Hanoi graphs, but where the pairs of extreme vertices are connected by an edge rather than identified.

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