It is known if we use two convex polygons with equal sides we can cover the plane periodically in few ways. One new way to cover the plane periodically is if we use rhombuses and octagons of equal integer sides. My question is Is it possible for the rhombuses to have diagonals which are integer numbers and the octagons to have angles integer numbers of degrees in this type of tiling?
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Sure. Let the octagons be regular (so with angles 135 degrees) with sides of length $\sqrt2$, then the rhombi are squares with diagonals of length 2.
EDIT: Let's show that this is the only solution, up to similarity. In fact, up to similarity, it's the only solution with rational diagonals and rational angles.
Each vertex in the tiling is the meeting point of two octagons and one rhombus. Since the octagons have rational angles, the rhombi must also have rational angles.
The diagonals of the rhombi cut them into (four congruent) right triangles. These triangles have rational angles (since the diagonals bisect the angles of a rhombus) and rational legs (since the diagonals bisect each other; "leg" means side other than hypotenuse), so we have a rational angle whose tangent is a rational number. But it's well-known that the only such angles are zero and 45 degrees. Zero is irrelevant here, and 45 degrees gives the solution with squares.
answered Jul 1, 2012 at 2:58
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$\begingroup$ @Gerry.The rhombi can not be squares $\endgroup$ Jul 1, 2012 at 3:30
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4$\begingroup$ Up to your old tricks, eh, Vassilis? Post a question, and as soon as it gets an answer, change the question? Is there really any point in answering any of your questions, then? $\endgroup$ Jul 1, 2012 at 4:11
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$\begingroup$ Anyway, your problem reduces to finding a rational angle with a rational tangent, and it's known that only zero and 45 degrees qualify, so all you'll get is what I already gave. $\endgroup$ Jul 1, 2012 at 4:16
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$\begingroup$ No tricks. What this new tiling is all about is the following: Is there any Pythagorean triplet which makes the angles of the triangle integer number of degrees? It is very peculiar tiling of the plane due to the fact that the rhomboid tiles can get infinitely skinny if we keep bisecting one of the diagonals of the rhombus, keeping the sides of the rhombuses and octagons the same length. $\endgroup$ Jul 2, 2012 at 2:56
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2$\begingroup$ As I already said, there is no right triangle with rational legs and rational angles, other than the isosceles; this trivially implies there is no Pythagorean triplet such that the angles of the triangle have integer numbers of degrees. $\endgroup$ Jul 2, 2012 at 4:13