Mathematical recreations and essays . squaretiles can be easily constructed by the repeated use of the fourelementary analla^gmatic arrangements given in the angular * Monsieur A. Hermann has proposed an analogous theorem for polygonscovering the surface of a sphere. t On this, see the second edition of the French translation of this work,Paris, 1908, vol. ii, pp. 26—37. X See Mathematical Questions from the Educational Times, London, vol. x,1868, pp. 74—76 ; vol. lvi, 1892, pp. 97—99. The results are closely connectedwith theorems in the theory of equations. B. R. 5 66 GEOMETRICAL RECREATIONS
Mathematical recreations and essays . squaretiles can be easily constructed by the repeated use of the fourelementary analla^gmatic arrangements given in the angular * Monsieur A. Hermann has proposed an analogous theorem for polygonscovering the surface of a sphere. t On this, see the second edition of the French translation of this work,Paris, 1908, vol. ii, pp. 26—37. X See Mathematical Questions from the Educational Times, London, vol. x,1868, pp. 74—76 ; vol. lvi, 1892, pp. 97—99. The results are closely connectedwith theorems in the theory of equations. B. R. 5 66 GEOMETRICAL RECREATIONS [CH. IV spaces of the accompanying diagram. In these fundamentalforms A represents one colour and B the other colour. Thediamond-shaped figure in the middle of the diagram representsan anallagmatic pavement of 256 tiles which is symmetricalabout its diagonals. In half the rows and half the columnseach line has 10 white tiles and 6 black tiles, and in theremaining rows and columns each line has 6 white tiles and A B B B B A B B. B B A B B B B A An Anallagmatic Isochromatic Pavement. 10 black tiles. Such an arrangement, where the differencebetween the number of white and black tiles used in each lineis constant, and equal to Vm, is called isochromatic. If m isodd or oddly even, it is impossible to construct anallagmaticboards which are isochromatic. Interesting problems can also be proposed when the tiles aretriangular, whether equilateral or isosceles right-angled. Twoequal isosceles right-angled tiles of different colours can be CH. IV] GEOMETRICAL RECREATIONS 67 put together so as to make a square tile as shown in themargin We can arrange four such tiles inno less than 256 different ways, making 64distinct designs. With the use of more tilesthe number of possible designs increases withstartling rapidity*. I content myself withgiving two illustrations of designs of pave-ments constructed with sixty-four such tiles, all exactly alike.
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