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The dissection of equilateral triangles into equilateral triangles

  • W. T. Tutte (a1)

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In a previous joint paper (‘The dissection of rectangles into squares’, by R. L. Brooks, C. A. B. Smith, A. H. Stone and W. T. Tutte, Duke Math. J. 7 (1940), 312–40), hereafter referred to as (A) for brevity, it was shown that it is possible to dissect a square into smaller unequal squares in an infinite number of ways. The basis of the theory was the association with any rectangle or square dissected into squares of an electrical network obeying Kirchhoff's laws. The present paper is concerned with the similar problem of dissecting a figure into equilateral triangles. We make use of an analogue of the electrical network in which the ‘currents’ obey laws similar to but not identical with those of Kirchhoff. As a generalization of topological duality in the sphere we find that these networks occur in triplets of ‘trial networks’ N1, N2, N3. We find that it is impossible to dissect a triangle into unequal equilateral triangles but that a dissection is possible into triangles and rhombuses so that no two of these figures have equal sides. Most of the theorems of paper (A) are special cases of those proved below.

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In the latter case and are distinct because, since n > 1, E r is not the whole of δ.

For rs, c rs is thus minus the number of elements of T with bases on and vertices on (except that for the purposes of this enumeration the element having a vertex at W σ is deemed to have it on S σ).

Aitken, A. C., Determinants and matrices (Edinburgh, 1939), pp. 6971.

For definitions of the terms of combinatorial topology used here, reference may be made to Seifert, and Threlfall, , Lehrbuch der Topologie (Leipzig and Berlin, 1934), to Alexandroff, and Hopf, , Topologie (Berlin, 1935), or to Lefschetz, Algebraic Topology, American Math. Soc. Colloquium publications, vol. 27. Here we use the results of Chapter V, § 3 of the second of these works.

In Figs. 1–3, any pair of elements having a common side is represented as a rhombus. The numbers represent the lengths of the sides of the containing polygons, or of the dissected figure.

(A) § (3·3).

The 2-cell is that one incident with the given edge of D p and not of colour σ.

Seifert and Threlfall, p. 87 (Example 3).

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