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
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 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".
Keywords: Golden Mean, Plastic Number, Morphic Number, gnomons, spirals.
MSC: 51M04; 51M25
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