The Padovan Sequence in Menorca

Photography by Sara Bagur

Photography by Sara Bagur

This sculpture was designed by the students and teachers (Lluís Cervera Pons and Gabriel Seguí de Vidal) of IES Pascual Calbó i Caldes to promote Maths in the population of Menorca. It represents a spiral of equilateral triangles with side lengths which follow the Padovan sequence and the “plastic number”

The plastic number is the unique real solution of the equation x3 = x + 1 which is:

plastic01

This number is approximately equal to ψ=1,324717957244746025960908854…

The other two complex solutions are:

plastic2

ψ is also the limit of the sequence ol ratios P(n+1)/P(n) where P(n) is the Padovan sequence (which is named after the architect Richard Padovan (born in 1935)). The Padovan sequence is the sequence of integers P(n) defined by the initial values P(1)=P(2)=P(3)=1 and the recurrence relation P(n)=P(n-2)+P(n-3). According to Padovan, the plastic number was discovered at the same time by Hans van der Laan.

In the Van der Laan Foundation webpage we can read:

The plastic number

[…] Through experiments with pebbles and then with building materials, he discovered a ratio he called the plastic number. The basis for the plastic number is the relationship between measures belonging to a group of measures. They increase or decrease according to the ratio four to three. The parallel in music is the ratio that relates whole and half notes to each other within an octave. The analogy between the plastic number and music goes even further: in music we can play chords, combinations of tones, with the plastic number we can compose walls and rooms and spaces that are in harmony with each other because they relate to each other as objectively as the tones within a musical key. The plastic number is not a particular measure: it disciplines the relationship between the measures we choose.

Van der Laan expressed his thoughts on the relationship between architecture and nature in this way:

If the body function of the House consists of establishing harmony between the body and its natural environment, the expression of that function will be based on the harmony between the wall that separates and the separate space. It becomes thus registration extension appreciable by the senses, both space-enclosing the separator element. That harmony will depend on the mutual proportions, who speak to the intelligence through the target language of the plastic number and that are established by the rules of the corresponding architectural ordering.

This post has been written by Bara Bagur in the subject Història de les Matemàtiques (History of Mathematics, 2014-15)

Location: The Padovan Sequence sculpture (map)

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