The plastic constant, sometimes also called le nombre radiant, the minimal Pisot number, plastic number, plastic ratio, platin number, Siegel's number, or silver
number, is the limiting ratio of the successive terms of the Padovan
sequence or Perrin sequence. It is denoted
using 
or 
and given by
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
(OEIS A060006), where 
denotes a polynomial
root. It is therefore an algebraic number
of degree 3.
It is also given by
 |
(4)
|
where
![r=-1/5[-j(tau_0)]^(1/3),](/images/equations/PlasticConstant/NumberedEquation2.svg) |
(5)
|
where 
is the 
-function
and the half-period ratio is equal to 
.
The plastic constant 
was originally studied in 1924 by Gérard Cordonnier when he was 17. In his
later correspondence with Dom Hans van der Laan, he described applications to architecture,
using the name "radiant number." In 1958, Cordonnier gave a lecture tour
that illustrated the use of the constant in many existing buildings and monuments
(C. Mannu, pers comm., Mar. 11, 2006).
satisfies the algebraic identities
 |
(6)
|
and
 |
(7)
|
and is therefore is one of the numbers 
for which there exist natural
numbers 
and 
such that 
and 
.
It was proven by Aarts et al. (2001) that 
and the golden ratio 
are in fact the only such numbers.
The identity 
leads to the beautiful nested radical identity
![P=RadicalBox[{1, +, RadicalBox[{1, +, RadicalBox[{1, +, ...}, 3]}, 3]}, 3].](/images/equations/PlasticConstant/NumberedEquation5.svg) |
(8)
|
The plastic constant is also connected with the ring of integers 
of the number field 
since it the real root of the Weber
function for the smallest negative discriminant with class
number 3, namely 
.
In particular,
 |  |  |
(9)
|
 |  |  |
(10)
|
 |  | ![-[(eta(tau))/(sqrt(2)eta(2tau))]^(24)](/images/equations/PlasticConstant/Inline37.svg) |
(11)
|
 |  |  |
(12)
|
(OEIS A116397), where 
is the Dedekind
eta function.
The plastic constant is also the smallest Pisot number.
The plastic constant satisfies the near-identity
 |
(13)
|
where the difference is 
.
Surprisingly, the plastic constant is connected to the metric properties of the snub icosidodecadodecahedron. It is also involved in the definition of maverick graphs.
The plastic constant satisfies 
, where 
is the golden ratio, so
has a reciprocal proportion triangle.
,
, 
, 
." Mar. 7, 2019.