
Journal for Geometry and Graphics 13 (2009), No. 2, 163175 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 vspinadel@fibertel.com.ar Antonia Redondo Buitrago Dep. de Matemática, I.E.S. Bachiller Sabuco, Avenida de España 9, 02002 Albacete, Spain aredondo@sabuco.com 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 x^{3}  x  1 = 0. This equation may be seen as example of a family of trinomials x^{n}  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 3Dobjects (rectangles, trapezoids, ellipses, ovals, ovoids, spirals and even 3Dboxes) 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 52 (2001) 5658], 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 logspirals. 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 computerindependent solution by intersecting a circle with an equilateral hyperbola. This allows to deduce and analyse "LaanNumber figures" like ψrectangles with side length ratio 1:ψ and a ψpentagons with sides of ratio 1:ψ:ψ^{2}:ψ^{3}:ψ^{4}. To this ψpentagon we also find a "ψPythagoras Theorem". Keywords: Golden Mean, Plastic Number, Morphic Number, gnomons, spirals. MSC: 51M04; 51M25 [ Fulltextpdf (350 KB)] for subscribers only. 