41. (Thanks to Doris Schattschneider, Moravian College.) In the text, we discuss some criteria and methods for generating Escher-like patterns that involve just translations or translations and half-turns. A slight variation on one such method allows construction of tilings that feature a tile and its mirror image. We modify a parallelogram so that two reflected tiles are joined and the joined pair tiles by translation. You get to design the shape of the tile and put whatever art you like on it.

Begin with an isosceles triangle and mark the midpoint of each of its sides (it’s best to use graph paper or a dynamic geometry program). Half-turn the triangle about the midpoint of one of its two equal sides to produce a parallelogram.

Modify half of the (odd) third side in any manner—here is where you get to be creative!— joining a vertex to the midpoint of that side. Then reflect that modification in the side and translate it so that it joins the midpoint to the other vertex of that side.

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Now translate the lower half of the modified side to the upper half of the opposite side of the parallelogram, and translate the upper half of the modified side to the lower half of the opposite side of the parallelogram.

Now modify one of the remaining sides of the parallelogram in any way you wish, and then translate that modification to the opposite side of the parallelogram.

Join a midpoint of the diagonal of the parallelogram to a midpoint of the side you have just modified (this line segment will be parallel to the other sides of the parallelogram). Reflect the side just modified in this segment, then translate the reflected modification so that it replaces the diagonal of the parallelogram. Your original parallelogram is now replaced by two tiles that are reflected images of each other. The outline of the joined pair of tiles is a modified par-hexagon that tiles by translations.