Proportion functions in three dimensions

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1989

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The paper presents a functional equation approach to the construction and characterization of proportion functions on three-dimensional boxes, extending some classical considerations of plane geometry which were motivated by architectural problems. Let D : = (0, ∞) and I : = [1, ∞). A function f: D3 →I will be called normalized if f(x, x, x) = 1 for all x > 0 and symmetric if f(x1, x2, x3) =f(xσ(1), xσ(2), xσ(3)) for all x1, x2, x3 > 0 and for any permutation σ of the set {1, 2, 3}. A proportion function in three dimensions is a three-place function f from D3 into I which is normalized, symmetric and satisfies a condition of the form {Mathematical expression} for all mappings α:D3 →D3 belonging to a fixed set B of bijections of D3. Two boxes of sides x, y, z and ξ,ηz with the common edge z are homothetic iff {ξ, η} = {zy/x, z2/x}. This motivates to characterize functions f from D3 into I which are normalized, symmetric and satisfy {Mathematical expression} Also the equation {Mathematical expression} (case of two boxes with a common face) in place of the previous one is important in this context. All the corresponding proportion functions (replace α in the definition of a proportion function by the functions in the functional equations above) are determined. © 1989 Birkhäuser Verlag.

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