By Sherman Stein, Sandor Szabó
Usually questions about tiling house or a polygon result in questions pertaining to algebra. for example, tiling by way of cubes increases questions on finite abelian teams. Tiling through triangles of equivalent parts quickly consists of Sperner's lemma from topology and valuations from algebra. the 1st six chapters of Algebra and Tiling shape a self-contained therapy of those issues, starting with Minkowski's conjecture approximately lattice tiling of Euclidean house by way of unit cubes, and concluding with Laczkowicz's fresh paintings on tiling by means of comparable triangles. The concluding bankruptcy offers a simplified model of Rédei's theorem on finite abelian teams. Algebra and Tiling is offered to undergraduate arithmetic majors, as lots of the instruments essential to learn the publication are present in general top point algebra classes, yet lecturers, researchers mathematicians will locate the ebook both attractive.
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Extra info for Algebra and Tiling: Homomorphisms in the Service of Geometry
0) is in 43 Cubical Clusters L. ) Now ( z , 1, 0, . . , 0) = X 1 * 1 + · • · + Χ 2 η ί η , 2 2 where at least o n e of t h e x , 1 < i < r, is not 0 since t h e vector ( z , 1 , 0 , . . , 0) is moved in our construction. T h u s 2 i 2 2 2 = X 2 1 0 1 1 "I 1" X n O n l , 2 where at least o n e of t h e x i , 1 < i < r, is not 0. , 6 so that bi is in - α ; ι + Q and 2 r 22 + 1- X2rb φ 0. Η r But before we make such a choice, we must k e e p in mind that there may be other vectors in L of t h e form (z, 1 , 0 , .
If we delete either one of the assumptions, it is false. For instance, the cluster consisting of the two unit squares whose centers are (0,0) a n d (2,0) Q-lattice tiles the plane but does not Z-lattice tile it. Also the cluster consisting of the four unit squares corresponding to (0,0), (1,0), (1,1) and (0,1) lattice tiles the plane in such a way that not all the vectors have integer coordinates. Exercise 1. Check that the two clusters mentioned tile as claimed. After proving these theorems we turn o u r attention to the problem of filling a box with bricks.
T h e cube with vertices ( ± 1 , ± 1 , ± 1 ) contains a nonzero vector of t h e lattice spanned by the rows of B. ) 1^31^1 + 632^2 + 633^31 < 1 have an integer solution other than the trivial one, ( 0 , 0 , 0 ) . Now let t > 1 b e a real n u m b e r a n d α a n d 6 be real numbers. As a special 18 ALGEBRA AND TILING case of (1) t h e inequalities I i i i + 0x2 — |0xi + tx2 < 1 atxz\ 6ix | < 1 — 3 |0x + 0ΐ2 + ( l / i ) x | < 1 2 x 3 have a nontrivial integer solution. T h u s there are integers xi, x , and £ 3 , with X3 > 0, such that 2 ο xi 1 X2 X3 X3^' X3 <—,, t , X3< Χ ί 2 3 and consequently xi X3 Exercise 20.