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Balanced binary arrays iii: the hexagonal grid

Part of: Designs and configurations Design of experiments

Published online by Cambridge University Press:  09 April 2009

Sheila Oates-Williams  and
Anne Penfold Street
Sheila Oates-Williams
Affiliation:
Departments of Mathematics University of QueenslandSt. Lucia, Queensland 4067, Australia
Anne Penfold Street
Affiliation:
Departments of Mathematics University of QueenslandSt. Lucia, Queensland 4067, Australia
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Abstract

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We consider the following problem arising in agricultural statistics. Suppose that a large number of plants are set out on a regular grid, which may be triangular, square or hexagonal, and that among these plants, half are to be given one and half the other of two possible treatments. For the sake of statistical balance, we require also that, if one plant in every k plants has i of its immediate neighbours receiving the same treatment as itself, then k is constant over all possible values of i. For square and triangular grids, there exist balanced arrays of finite period in each direction, but for the hexagonal grid, we show that no such balanced array can exist. Several related questions are discussed.

MSC classification

Secondary: 05B30: Other designs, configurations 05B45: Tessellation and tiling problems 62K99: None of the above, but in this section
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 28 , Issue 4 , December 1979 , pp. 479 - 498
Copyright
Copyright © Australian Mathematical Society 1979

References

Referneces

Grünbaüm, Branko and Shephard, Geoffrey C. (1977–1978), ‘Tilings by regular polygons’, Math. Mag. 50, 227–247; 51, 205–206.CrossRefGoogle Scholar
Macdonald, Sheila Oates and Street, Anne Penfold (1979), ‘Balanced binary arrays II: the triangular grid’, Ars Combinatoria (to appear).Google Scholar
Street, Anne Penfold and Macdonald, Sheila Oates (1979), ‘Balanced binary arrays I: the square grid’, Comb. Maths. Vl, Proc. Sixth Aust. Conf. on Comb. Maths (Armidale, 1978) (Springer-Verlag Lecture Notes in Maths., edited by Horadam, A. F. and Wallis, W. D.) (to appear).Google Scholar