Plastic number

Abstract: In this article we will construct plastic number in a heuristic way, explaining its relation to human perception in three-dimensional space through architectural style of Dom Hans van der Laan. Furthermore, it will be shown that the plastic number and the golden ratio special cases of more general definition. Finally, we will explain how van der Laan’s discovery relates to perception in pitch space, how to define and tune Padovan intervals and, subsequently, how to construct chromatic scale temperament using the plastic number.

Abstract: In 1960 D.H. van der Laan, architect and member of the Benedic- tine order, introduced what he calls the "Plastic Number" , 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, n = 2,3,.... Considering the real positive roots of these equations we define these roots as members of a "Plastic Numbers Family" (PNF) com- prising the well known Golden Mean � = 1,618..., the most prominent member of the Metallic Means Family (12) and van der Laan's Number = 1,324... Similar to the occurrence ofin art and nature one can use 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 and the Golden Numberare the only "Morphic Numbers" in the sense of Aarts et al. (1), who define such a number as the common solution of two somehow dual trinomials. We can show that these two numbers are also dis- tinguished by a property of log-spirals. Laan's Number cannot be constructed by using ruler and compass only. We present a planar graphic construction of a segment of length 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 -rectangles with side length ratio 1 : and a -pentagons with sides of ratio 1 : : 2 : 3 : 4 . To

Abstract: In the present text, we present some possibilities to formalize the mathematical content and a historical context, referring to a numerical sequence of linear and recurrent form, known as Sequence of Padovan or Cordonnier. Throughout the text some definitions are discussed, the matrix approach and the relation of this sequence with the plastic number. The explicit exploration of the possible paths used to formalize the explored mathematical subject, comes with an epistemological character, still conserving the exploratory intention of these numbers and always taking care of the mathematical rigor approached

Abstract: Between 1920 and 1991, the Dutch Benedictine monk and architect Dom Hans van der Laan (1904–91) developed his own proportional system based on the ratio 3:4, or the irrational number 1.3247. . ., which he called the plastic number. According to him, this ratio directly grew from discernment, the human ability to differentiate sizes, and as such would be an improvement over the golden ratio. To put his theories to the test, he developed an architectural language, which can best be described as elementary architecture. His oeuvre — four convents and a house — is published on an international scale. His buildings have become pilgrimage sites for practicing architects and institutions that want to study and experience his spaces. His 1977 book Architectonic Space: Fifteen Lessons on the Disposition of the Human Habitat , translated into English, French, German and Italian, still inspires architects today, as does his biography, Modern Primitive , written by the architect Richard Padovan in 1994. But beyond the inspiration of his writings and realisations, the actual application of the plastic number in Van der Laan’s designs is unclear. Moreover, Van der Laan’s theories seem to be directed towards one goal only: to present the plastic number as the only possible means by which eminent architecture can be achieved, making them a target for suspicion and critique. To understand and evaluate Van der Laan’s application of the plastic number, this paper approaches it as a practical design tool. It analyses its genealogy and defines its key concepts. From that framework, Van der Laan’s architectonic space is interpreted as a design methodology that combines antique tectonic theories reminiscent of writers from Plato to Vitruvius with more recent atectonic approaches towards space through experience and movement.