15 added 6 characters in body edited Feb 11 '17 at 3:13 Martin Rattigan 17399 bronze badges The figure shows a hexagon and triangle tiled by six tiles, which are pairwise congruent. My question is: What is the smallest number of polygonal tiles that will tile both a regular hexagon and an equilateral triangle in such a way that all the edges of the tiles are parallel to an edge of the tiled figure? Small print: I define a polygonal tile as a figure comprised of: A finite set $$P$$ of at least three points. The (straight) lines $$X\pi(X)$$ for each point $$X\in P$$ and some cyclic permutation $$\pi$$ of $$P$$ such that distinct lines do not intersect except at mutual end points. The region contained by the lines. The hexagon, triangle and tiles are assumed to include the vertices, edges and interior. The figures tiled must be the union of exactly one congruent copy of each tile (though the set of tiles itself is allowed to contain congruent figures). No copy of a tile can share an interior point with the copy of any other tile, but may share boundary points with one or more. A congruent copy of a tile may be a "flipped" version (e.g. the pair of cyan tiles in the above diagram are actually mirror images of each other - this is allowed). The figure shows a hexagon and triangle tiled by six tiles, which are pairwise congruent. My question is: What is the smallest number of polygonal tiles that will tile both a regular hexagon and an equilateral triangle in such a way that all the edges of the tiles are parallel to an edge of the tiled figure? Small print: I define a polygonal tile as a figure comprised of: A finite set $$P$$ of at least three points. The (straight) lines $$X\pi(X)$$ for each point $$X\in P$$ and some cyclic permutation $$\pi$$ of $$P$$ such that distinct lines do not intersect except at mutual end points. The region contained by the lines. The hexagon, triangle and tiles are assumed to include the vertices, edges and interior. The figures tiled must be the union of exactly one congruent copy of each tile (though the set of tiles itself is allowed to contain congruent figures). No copy of a tile can share an interior point with the copy of any other tile, but may share boundary points with one or more. A congruent copy of a tile may be a "flipped" version (e.g. the pair of cyan tiles in the above diagram are actually mirror images - this is allowed). The figure shows a hexagon and triangle tiled by six tiles, which are pairwise congruent. My question is: What is the smallest number of polygonal tiles that will tile both a regular hexagon and an equilateral triangle in such a way that all the edges of the tiles are parallel to an edge of the tiled figure? Small print: I define a polygonal tile as a figure comprised of: A finite set $$P$$ of at least three points. The (straight) lines $$X\pi(X)$$ for each point $$X\in P$$ and some cyclic permutation $$\pi$$ of $$P$$ such that distinct lines do not intersect except at mutual end points. The region contained by the lines. The hexagon, triangle and tiles are assumed to include the vertices, edges and interior. The figures tiled must be the union of exactly one congruent copy of each tile (though the set of tiles itself is allowed to contain congruent figures). No copy of a tile can share an interior point with the copy of any other tile, but may share boundary points with one or more. A congruent copy of a tile may be a "flipped" version (e.g. the cyan tiles in the above diagram are actually mirror images of each other - this is allowed). 14 added 1 character in body edited Feb 11 '17 at 1:53 Rubio♦ 33.1k66 gold badges7878 silver badges205205 bronze badges The figure shows a hexagon and triangle tiled by six tiles, which are pairwise congruent. My question is: What is the smallest number of polygonal tiles that will tile both a regular hexagon and an equilateral triangle in such a way that all the edges of the tiles are parallel to an edge of the tiled figure? Small print: I define a polygonal tile as a figure comprised of: A finite set $$P$$ of at least three points. The (straight) lines $$X\pi(X)$$ for each point $$X\in P$$ and some cyclic permutation $$\pi$$ of $$P$$ such that distinct lines do not intersect except at mutual end points. The region contained by the lines. The hexagon, triangle and tiles are assumed to include the vertices, edges and interior. The figures tiled must be the union of exactly one congruent copy of each tile (though the set of tiles itself is allowed to contain congruent figures). No copy of a tile can share an interior point with the copy of any other tile, but may share boundary points with one or more. A congruent copy of a tile may be a "flipped" version (e.g. the pair of cyan tiles in the above digramdiagram are actually mirror images - this is allowed). The figure shows a hexagon and triangle tiled by six tiles, which are pairwise congruent. My question is: What is the smallest number of polygonal tiles that will tile both a regular hexagon and an equilateral triangle in such a way that all the edges of the tiles are parallel to an edge of the tiled figure? Small print: I define a polygonal tile as a figure comprised of: A finite set $$P$$ of at least three points. The (straight) lines $$X\pi(X)$$ for each point $$X\in P$$ and some cyclic permutation $$\pi$$ of $$P$$ such that distinct lines do not intersect except at mutual end points. The region contained by the lines. The hexagon, triangle and tiles are assumed to include the vertices, edges and interior. The figures tiled must be the union of exactly one congruent copy of each tile (though the set of tiles itself is allowed to contain congruent figures). No copy of a tile can share an interior point with the copy of any other tile, but may share boundary points with one or more. A congruent copy of a tile may be a "flipped" version (e.g. the cyan tiles in the above digram are actually mirror images - this is allowed). The figure shows a hexagon and triangle tiled by six tiles, which are pairwise congruent. My question is: What is the smallest number of polygonal tiles that will tile both a regular hexagon and an equilateral triangle in such a way that all the edges of the tiles are parallel to an edge of the tiled figure? Small print: I define a polygonal tile as a figure comprised of: A finite set $$P$$ of at least three points. The (straight) lines $$X\pi(X)$$ for each point $$X\in P$$ and some cyclic permutation $$\pi$$ of $$P$$ such that distinct lines do not intersect except at mutual end points. The region contained by the lines. The hexagon, triangle and tiles are assumed to include the vertices, edges and interior. The figures tiled must be the union of exactly one congruent copy of each tile (though the set of tiles itself is allowed to contain congruent figures). No copy of a tile can share an interior point with the copy of any other tile, but may share boundary points with one or more. A congruent copy of a tile may be a "flipped" version (e.g. the pair of cyan tiles in the above diagram are actually mirror images - this is allowed). 13 deleted 8 characters in body edited Feb 11 '17 at 1:47 Martin Rattigan 17399 bronze badges The figure shows a hexagon and triangle tiled by six tiles, which are pairwise congruent. My question is: What is the smallest number of polygonal tiles that will tile both a regular hexagon and an equilateral triangle in such a way that all the edges of the tiles are parallel to an edge of the tiled figure? Small print: I define a polygonal tile as a figure comprised of: A finite set $$P$$ of at least three points. The (straight) lines $$X\pi(X)$$ for each point $$X\in P$$ and some cyclic permutation $$\pi$$ of $$P$$ such that distinct lines do not intersect except at mutual end points. The region contained by the lines. The hexagon, triangle and tiles are assumed to include the vertices, edges and interior. The figures tiled must be the union of exactly one congruent copy of each tile (though the set of tiles itself is allowed to contain congruent figures). No copy of a tile can share an interior point with the copy of any other tile, but may share boundary points with one or more. A congruent copy of a tile may be a "flipped" version (e.g. the pair of cyan tiles in the above digram are actually mirror images - this is allowed). The figure shows a hexagon and triangle tiled by six tiles, which are pairwise congruent. My question is: What is the smallest number of polygonal tiles that will tile both a regular hexagon and an equilateral triangle in such a way that all the edges of the tiles are parallel to an edge of the tiled figure? Small print: I define a polygonal tile as a figure comprised of: A finite set $$P$$ of at least three points. The (straight) lines $$X\pi(X)$$ for each point $$X\in P$$ and some cyclic permutation $$\pi$$ of $$P$$ such that distinct lines do not intersect except at mutual end points. The region contained by the lines. The hexagon, triangle and tiles are assumed to include the vertices, edges and interior. The figures tiled must be the union of exactly one congruent copy of each tile (though the set of tiles itself is allowed to contain congruent figures). No copy of a tile can share an interior point with the copy of any other tile, but may share boundary points with one or more. A congruent copy of a tile may be a "flipped" version (e.g. the pair of cyan tiles in the above digram are actually mirror images - this is allowed). The figure shows a hexagon and triangle tiled by six tiles, which are pairwise congruent. My question is: What is the smallest number of polygonal tiles that will tile both a regular hexagon and an equilateral triangle in such a way that all the edges of the tiles are parallel to an edge of the tiled figure? Small print: I define a polygonal tile as a figure comprised of: A finite set $$P$$ of at least three points. The (straight) lines $$X\pi(X)$$ for each point $$X\in P$$ and some cyclic permutation $$\pi$$ of $$P$$ such that distinct lines do not intersect except at mutual end points. The region contained by the lines. The hexagon, triangle and tiles are assumed to include the vertices, edges and interior. The figures tiled must be the union of exactly one congruent copy of each tile (though the set of tiles itself is allowed to contain congruent figures). No copy of a tile can share an interior point with the copy of any other tile, but may share boundary points with one or more. A congruent copy of a tile may be a "flipped" version (e.g. the cyan tiles in the above digram are actually mirror images - this is allowed). 12 added 7 characters in body edited Feb 10 '17 at 12:38 Martin Rattigan 17399 bronze badges 11 deleted 33 characters in body edited Feb 9 '17 at 12:02 Martin Rattigan 17399 bronze badges 10 Minor odification of small print for clarity. edited Feb 8 '17 at 12:54 Martin Rattigan 17399 bronze badges 9 added 7 characters in body edited Feb 8 '17 at 12:21 Martin Rattigan 17399 bronze badges 8 Corrected definition of polygonal tile in "small print". My apologies to anyone working with the original edited Feb 8 '17 at 12:04 Martin Rattigan 17399 bronze badges 7 edited tags | link edited Feb 8 '17 at 2:57 JMP 27.4k66 gold badges5454 silver badges117117 bronze badges 6 edited tags | link edited Feb 8 '17 at 2:03 Martin Rattigan 17399 bronze badges 5 edited body edited Feb 8 '17 at 0:15 Martin Rattigan 17399 bronze badges 4 The tiles are not identical. edited Feb 8 '17 at 0:07 elias 9,11633 gold badges2828 silver badges5858 bronze badges 3 The tiles are not identical. edit approved Feb 8 '17 at 0:07 Oliver Ni 67555 silver badges1818 bronze badges 2 added 762 characters in body edited Feb 7 '17 at 23:29 Martin Rattigan 17399 bronze badges 1 asked Feb 7 '17 at 23:23 Martin Rattigan 17399 bronze badges