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Journal for Geometry and Graphics 13 (2009), No. 2, 163--175
Copyright Heldermann Verlag 2009

Towards van der Laan's Plastic Number in the Plane

Vera W. de Spinadel
Laboratorio de Matemática y Diseño, Universidad de Buenos Aires, Argentina

Antonia Redondo Buitrago
Dep. de Matemática, I.E.S. Bachiller Sabuco, Avenida de España 9, 02002 Albacete, Spain

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:ψ234. To this ψ-pentagon we also find a "ψ-Pythagoras Theorem".

Keywords: Golden Mean, Plastic Number, Morphic Number, gnomons, spirals.

MSC: 51M04; 51M25

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