Towards van der Laan's Plastic Number in the Plane

V. W. de Spinadel ; A. Redondo Buitrago

Journal for geometry and graphics, Tome 13 (2009) no. 2, pp. 163-175

Résumé

In 1960 D. H. van der Laan, architect and member of the Benedictine order, introduced what he calls the "Plastic Number" ψ, as an ideal ratio for a geometric scale of spatial objects. It is the real solution of the cubic equation x³ - x - 1 = 0. This equation may be seen as example of a family of trinomials xⁿ - x - 1 = 0. We define their real positive roots as members of a "Plastic Numbers Family" comprising the well known Golden Mean ϕ, the most prominent member of the Metallic Means Family [see the author, "The family of Metallic Means", Visual Mathematics 1/3 (1999)] and van der Laan's Number ψ. Similar to the occurrence of ϕ in art and nature one can use ψ for defining special 2D- and 3D-objects (rectangles, trapezoids, ellipses, ovals, ovoids, spirals and even 3D-boxes) and look for natural representations of this special number. Laan's Number ψ and the Golden Number ϕ are the only "Morphic Numbers" in the sense of J. Aarts, J. R. Fokkink, G. Kruijtzer ["Morphic Numbers", Nieuw Archief voor Wiskunde 5-2 (2001) 56--58], who define such a number as the common solution of two somehow dual trinomials. We can show that these two numbers are also distinguished by a property of log-spirals. Laan's Number ψ cannot be constructed by using ruler and compass only. We present a planar graphic construction of a segment of length ψ using a dynamical graphics software as well as a computer-independent solution by intersecting a circle with an equilateral hyperbola. This allows to deduce and analyse "Laan-Number figures" like ψ-rectangles with side length ratio 1:ψ and a ψ-pentagons with sides of ratio 1:ψ:ψ²:ψ³:ψ⁴. To this ψ-pentagon we also find a "ψ-Pythagoras Theorem".

Détail

Classification : 51M04, 51M25
Mots-clés : Golden Mean, Plastic Number, Morphic Number, gnomons, spirals