Proportion functions in three dimensions

Summary

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.

LetD : = (0, ∞) andI : = [1, ∞). A functionf: D ³ →I will be called normalized iff(x, x, x) = 1 for allx > 0 and symmetric iff(x ₁,x ₂,x ₃) =f(x _σ(1),x _σ(2),x _σ(3)) for allx ₁,x ₂,x ₃ > 0 and for any permutation σ of the set {1, 2, 3}. A proportion function in three dimensions is a three-place functionf fromD ³ intoI which is normalized, symmetric and satisfies a condition of the form

$$f(x,y,z) = f(\alpha (x,y,z))forallx,y,z > 0,$$

for all mappings α:D ³ →D ³ belonging to a fixed setB of bijections ofD ³.

Two boxes of sidesx, y, z and ξ,ηz with the common edgez are homothetic iff{ξ, η} = {zy/x, z ²/x}. This motivates to characterize functionsf fromD ³ intoI which are normalized, symmetric and satisfy

$$f(x,y,z) = f\left( {z,\frac{{zy}}{x},\frac{{z^2 }}{x}} \right)forallx,y,z > 0.$$

Also the equation

$$f(x,y,\sqrt {xy} ) = f\left( {\frac{{y^2 }}{{\sqrt {xy} }},\sqrt {xy} ,y} \right)forallx,y,z > 0$$

(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.