![Plastic number square spiral](https://upload.wikimedia.org/wikipedia/commons/d/d2/Plastic_number_square_spiral.svg)
In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the plastic number (also known as the plastic constant, the plastic ratio, the minimal Pisot number, the platin number,
Siegel
Siegel (also Segal or Segel), is a German and Ashkenazi Jewish surname. it can be traced to 11th century Bavaria and was used by people who made wax seals for or sealed official documents (each such male being described as a ''Siegelbeamter''). ...
's number or, in French, ) is a
mathematical constant
A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. Cons ...
which is the unique real solution of the
cubic equation
In algebra, a cubic equation in one variable is an equation of the form
:ax^3+bx^2+cx+d=0
in which is nonzero.
The solutions of this equation are called roots of the cubic function defined by the left-hand side of the equation. If all of the ...
:
It has the exact value
:
Its decimal expansion begins with .
Properties
Recurrences
The powers of the plastic number satisfy the third-order linear recurrence relation for . Hence it is the limiting ratio of successive terms of any (non-zero) integer sequence satisfying this recurrence such as the
Padovan sequence
In number theory, the Padovan sequence is the sequence of integers ''P''(''n'') defined. by the initial values
:P(0)=P(1)=P(2)=1,
and the recurrence relation
:P(n)=P(n-2)+P(n-3).
The first few values of ''P''(''n'') are
:1, 1, 1, 2, 2, 3, 4, 5 ...
(also known as the Cordonnier numbers), the
Perrin number
In mathematics, the Perrin numbers are defined by the recurrence relation
: for ,
with initial values
:.
The sequence of Perrin numbers starts with
: 3, 0, 2, 3, 2, 5, 5, 7, 10, 12, 17, 22, 29, 39, ...
The number of different maxima ...
s and the
Van der Laan numbers
A van is a type of road vehicle used for transporting goods or people. Depending on the type of van, it can be bigger or smaller than a pickup truck and SUV, and bigger than a common car. There is some varying in the scope of the word across th ...
, and bears relationships to these sequences akin to the relationships of the
golden ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0,
where the Greek letter phi ( ...
to the second-order
Fibonacci
Fibonacci (; also , ; – ), also known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano ('Leonardo the Traveller from Pisa'), was an Italian mathematician from the Republic of Pisa, considered to be "the most talented Western ...
and
Lucas
Lucas or LUCAS may refer to:
People
* Lucas (surname)
* Lucas (given name)
Arts and entertainment
* Luca Family Singers, also known as "lucas ligner en torsk"
* ''Lucas'' (album) (2007), an album by Skeletons and the Kings of All Cities
* ''L ...
numbers, akin to the relationships between the
silver ratio
In mathematics, two quantities are in the silver ratio (or silver mean) if the ratio of the smaller of those two quantities to the larger quantity is the same as the ratio of the larger quantity to the sum of the smaller quantity and twice t ...
and the
Pell number
In mathematics, the Pell numbers are an infinite sequence of integers, known since ancient times, that comprise the denominators of the closest rational approximations to the square root of 2. This sequence of approximations begins , , , , an ...
s.
The plastic number satisfies the
nested radical
In algebra, a nested radical is a radical expression (one containing a square root sign, cube root sign, etc.) that contains (nests) another radical expression. Examples include
:\sqrt,
which arises in discussing the regular pentagon, and more co ...
recurrence
:
Number theory
Because the plastic number has the
minimal polynomial it is also a solution of the polynomial equation for every polynomial that is a multiple of but not for any other polynomials with integer coefficients. Since the
discriminant
In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the origi ...
of its minimal polynomial is −23, its
splitting field
In abstract algebra, a splitting field of a polynomial with coefficients in a field is the smallest field extension of that field over which the polynomial ''splits'', i.e., decomposes into linear factors.
Definition
A splitting field of a poly ...
over rationals is
This field is also a
Hilbert class field In algebraic number theory, the Hilbert class field ''E'' of a number field ''K'' is the maximal abelian unramified extension of ''K''. Its degree over ''K'' equals the class number of ''K'' and the Galois group of ''E'' over ''K'' is canonicall ...
of
As such, it can be expressed
[ in terms of the ]Dedekind eta function
In mathematics, the Dedekind eta function, named after Richard Dedekind, is a modular form of weight 1/2 and is a function defined on the upper half-plane of complex numbers, where the imaginary part is positive. It also occurs in bosonic string t ...
with argument ,
:
and root of unity
In mathematics, a root of unity, occasionally called a Abraham de Moivre, de Moivre number, is any complex number that yields 1 when exponentiation, raised to some positive integer power . Roots of unity are used in many branches of mathematic ...
. Similarly, for the supergolden ratio
In mathematics, two quantities are in the supergolden ratio if the quotient of the larger number divided by the smaller one is equal to
:\psi = \frac
which is the only real solution to the equation x^3 = x^2+1. It can also be represented using ...
with argument ,
:
Also, the plastic number is the smallest Pisot–Vijayaraghavan number
In mathematics, a Pisot–Vijayaraghavan number, also called simply a Pisot number or a PV number, is a real algebraic integer greater than 1, all of whose Galois conjugates are less than 1 in absolute value. These numbers were discovered by Axel ...
. Its algebraic conjugate
In mathematics, in particular field theory, the conjugate elements or algebraic conjugates of an algebraic element , over a field extension , are the roots of the minimal polynomial of over . Conjugate elements are commonly called conju ...
s are
:
of absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
≈ 0.868837 . This value is also because the product of the three roots of the minimal polynomial is 1.
Trigonometry
The plastic number can be written using the hyperbolic cosine
In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the u ...
() and its inverse:
:
(See Cubic function#Trigonometric (and hyperbolic) method.)
Geometry
There are precisely three ways of partitioning a square into three similar rectangles:
#The trivial solution given by three congruent rectangles with aspect ratio 3:1.
#The solution in which two of the three rectangles are congruent and the third one has twice the side length of the other two, where the rectangles have aspect ratio 3:2.
#The solution in which the three rectangles are mutually non congruent (all of different sizes) and where they have aspect ratio ''ρ''2. The ratios of the linear sizes of the three rectangles are: ''ρ'' (large:medium); ''ρ''2 (medium:small); and ''ρ''3 (large:small). The internal, long edge of the largest rectangle (the square's fault line) divides two of the square's four edges into two segments each that stand to one another in the ratio ''ρ.'' The internal, coincident short edge of the medium rectangle and long edge of the small rectangle divides one of the square's other, two edges into two segments that stand to one another in the ratio ''ρ''4.
The fact that a rectangle of aspect ratio ''ρ''2 can be used for dissections of a square into similar rectangles is equivalent to an algebraic property of the number ''ρ''2 related to the Routh–Hurwitz theorem
In mathematics, the Routh–Hurwitz theorem gives a test to determine whether all roots of a given polynomial lie in the left half-plane. Polynomials with this property are called Hurwitz stable polynomials. The Routh-Hurwitz theorem is importan ...
: all of its conjugates have positive real part.
The intersections of the circle ''x''2 + ''y''2 = 1 with the curve ''x''3 = ''y''2 occur at Cartesian coordinates (''ρ''−1,''ρ''−3/2) and (''ρ''−1,-''ρ''−3/2).
History and names
Dutch architect and Benedictine monk
, image = Medalla San Benito.PNG
, caption = Design on the obverse side of the Saint Benedict Medal
, abbreviation = OSB
, formation =
, motto = (English: 'Pray and Work')
, found ...
Dom Hans van der Laan
Dom Hans van der Laan (29 December 1904 – 19 August 1991) was a Dutch Benedictine monk and architect.
He was a leading figure in the Bossche School. His theories on numerical ratios in architecture, in particular regarding the plastic number, ...
gave the name ''plastic number'' ( nl, het plastische getal) to this number in 1928. In 1924, four years prior to van der Laan's christening of the number's name, French engineer had already discovered the number and referred to it as ''the radiant number'' (french: le nombre radiant). Unlike the names of the golden ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0,
where the Greek letter phi ( ...
and silver ratio
In mathematics, two quantities are in the silver ratio (or silver mean) if the ratio of the smaller of those two quantities to the larger quantity is the same as the ratio of the larger quantity to the sum of the smaller quantity and twice t ...
, the word plastic was not intended by van der Laan to refer to a specific substance, but rather in its adjectival sense, meaning something that can be given a three-dimensional shape. This, according to Richard Padovan
Richard Padovan (born 1935) is an architect, author, translator and lecturer.
In the 1950s he studied at the Architectural Association School of Architecture; he has practised architecture in several European countries, and taught at the Universit ...
, is because the characteristic ratios of the number, and , relate to the limits of human perception in relating one physical size to another. Van der Laan designed the 1967 St. Benedictusberg Abbey
St. Benedictusberg Abbey, also Mamelis Abbey, is a Benedictine monastery established in 1922 in Mamelis, a Hamlet (place), hamlet which administratively falls within Vaals, Netherlands. It is a rijksmonument.
Since 1951 St. Benedictusberg has belo ...
church to these plastic number proportions.
The plastic number is also sometimes called the silver number, a name given to it by Midhat J. Gazalé
Midhat Gazalé (22 July 1929 – 8 June 2009) was an international telecommunications and space consultant and a visiting Professor of Telecommunications and Computer Management at the University of Paris IX. He served as President of AT&T-Fra ...
and subsequently used by Martin Gardner
Martin Gardner (October 21, 1914May 22, 2010) was an American popular mathematics and popular science writer with interests also encompassing scientific skepticism, micromagic, philosophy, religion, and literatureespecially the writings of Lewis ...
, but that name is more commonly used for the silver ratio
In mathematics, two quantities are in the silver ratio (or silver mean) if the ratio of the smaller of those two quantities to the larger quantity is the same as the ratio of the larger quantity to the sum of the smaller quantity and twice t ...
one of the ratios from the family of metallic means
The metallic means (also ratios or constants) of the successive natural numbers are the continued fractions:
n + \cfrac
= ;n,n,n,n,\dots= \frac.
The golden ratio (1.618...) is the metallic mean between 1 and 2, while the silver ratio (2.414. ...
first described by Vera W. de Spinadel in 1998.
Martin Gardner
Martin Gardner (October 21, 1914May 22, 2010) was an American popular mathematics and popular science writer with interests also encompassing scientific skepticism, micromagic, philosophy, religion, and literatureespecially the writings of Lewis ...
has suggested referring to as "high phi", and Donald Knuth
Donald Ervin Knuth ( ; born January 10, 1938) is an American computer scientist, mathematician, and professor emeritus at Stanford University. He is the 1974 recipient of the ACM Turing Award, informally considered the Nobel Prize of computer sc ...
created a special typographic mark for this name, a variant of the Greek letter phi
Phi (; uppercase Φ, lowercase φ or ϕ; grc, ϕεῖ ''pheî'' ; Modern Greek: ''fi'' ) is the 21st letter of the Greek alphabet.
In Archaic and Classical Greek (c. 9th century BC to 4th century BC), it represented an aspirated voicele ...
("φ") with its central circle raised, resembling the Georgian letter pari ("Ⴔ").
See also
*Snub icosidodecadodecahedron
In geometry, the snub icosidodecadodecahedron is a nonconvex uniform polyhedron, indexed as U46. It has 104 faces (80 triangles, 12 pentagons, and 12 pentagrams), 180 edges, and 60 vertices. As the name indicates, it belongs to the family of snu ...
*Supergolden ratio
In mathematics, two quantities are in the supergolden ratio if the quotient of the larger number divided by the smaller one is equal to
:\psi = \frac
which is the only real solution to the equation x^3 = x^2+1. It can also be represented using ...
Notes
References
*.
*.
*.
*.
External links
Tales of a Neglected Number
by Ian Stewart
Plastic rectangle and Padovan sequence
at Tartapelago by Giorgio Pietrocola
*
{{Authority control
Euclidean plane geometry
Cubic irrational numbers
Mathematical constants
History of geometry
Visual arts theory
Composition in visual art