A trion is obtained by taking an equilateral triangle, dividing the edges into n segments of equal length, and cutting from the center of the triangle to two subdivision points on different edges. This will give a particular quadrilateral. If you divide the edges into just 3 segments, there are three different trions, which fit nicely into a single triangle:
This is the single puzzle piece from a previous post. We have also seen this mechanism (explained to me by Alan Schoen) to produce what I called hexons. Today we will look what trions we get when we divide the triangle sides into four segments.
There are six such trions, which fit nicely into two triangles. They can, as we did with the hexons, also be arranged in groups of six around a (former) triangle vertex, to create hexagonal pillows, i.e. hexagons whose edges can remain straight or possess inward or outward kinks.
There are too many of those for my taste, but there is only one (not counting its mirror) that uses each trion exactly once, namely the one to the right. In the spirit of perfect solitairity, this makes an engaging single puzzle piece.
Can you extend the tiling above so that it tiles the plane? Using it as is gets a bit dizzying (but notice the triangle pattern on the left), so I have replaced it by a simpler version that contains all the essentials.
Two hexagons match along an edge if either both sides have no arrow, or you can keep following the arrow, as in the example. Below are two simple examples of periodic tilings:
There is much more one can do with this piece, but for now let’s end with a homework puzzle: Can you fill the board below so that everything matches?
The Inner Frame 