asymmetrical and exist in two distinct forms related by mirror-reflection. Dissection of the sphinx and some other asymmetric rep-tiles requires use of both the original shape and its mirror-image. Rep-tiles and polyforms Some rep-tiles are based on polyforms like polyiamonds and polyominoes, or shapes created by laying equilateral triangles and squares edge-to-edge. Squares If a polyomino is rectifiable, that is, able to tile a rectangle, then it will also be a rep-tile, because the rectangle will have an integer side length ratio and will thus tile a square. This can be seen in the octominoes, which are created from eight squares. Two copies of some octominoes will tile a square; therefore these octominoes are also rep-16 rep-tiles. Four copies of some nonominoes and nonakings will tile a square, therefore these polyforms are also rep-36 rep-tiles. Equilateral triangles Similarly, if a polyiamond tiles an equilateral triangle, it will also be a rep-tile. Right triangles A right triangle is a triangle containing one right angle of 90°. Two particular forms of right triangle have attracted the attention of rep-tile researchers, the 45°-90°-45° triangle and the 30°-60°-90° triangle. 45°-90°-45° triangles Polyforms based on isosceles right triangles, with sides in the ratio 1 : 1 : , are known as polyabolos. An infinite number of them are rep-tiles. Indeed, the simplest of all rep-tiles is a single isosceles right triangle. It is rep-2 when divided by a single line bisecting the right angle to the hypotenuse. Rep-2 rep-tiles are also rep-2n and the rep-4,8,16+ triangles yield further rep-tiles. These are found by discarding half of the sub-copies and permutating the remainder until they are mirror-symmetrical within a right triangle. In other words, two copies will tile a right triangle. One of these new rep-tiles is reminiscent of the fish formed from three equilateral triangles. 30°-60°-90° triangles Polyforms based on 30°-60°-90° right triangles, with sides in the ratio 1 : : 2, are known as polydrafters. Some are identical to polyminoes and polyiamonds, others are distinct. Multiple and variant rep-tilings Many of the common rep-tiles are rep- for all positive integer values of . In particular this is true for three trapezoids including the one formed from three equilateral triangles, for three axis-parallel hexagons (the L-tromino, L-tetromino, and P-pentomino), and the sphinx hexiamond. In addition, many rep-tiles, particularly those with higher rep-n, can be self-tiled in different ways. For example, the rep-9 L-tetramino has at