A special Pisot number is a Pisot number 
for which 
is also a Pisot
number. Smyth (1999, p. 1) proved that there are exactly 11 special Pisot
numbers.
The map 
is an involution,
since 
for 
.
Its unique fixed point in this interval
is 2. The other ten special Pisot numbers form five pairs 
, listed below (Smyth 1999, p. 1). In each
row, 
is the real root greater than 1 of the stated minimal
polynomial.
| degree |  | constant | OEIS | minimal polynomial |  | constant | OEIS |
| 3 | 1.3247179572... | plastic constant  | A060006 |  | 4.0795956234... |  | |
| 4 | 1.3802775690... | superplastic
ratio  | A086106 |  | 3.6296581267... |  | |
| 3 | 1.4655712318... | supergolden ratio  | A092526 |  | 3.1478990357... |  | |
| 2 | 1.6180339887... | golden
ratio  | A001622 |  | 2.6180339887... |  | A104457 |
| 3 | 1.7548776662... | square
of the plastic constant  | A109134 |  | 2.3247179572... |  |
All ten paired values are algebraic units, while 2 is not (Smyth 1999, p. 1).
The first five self-summing ratios, with 
, consist of 2 and the first
four values of 
in the table. These are the only self-summing ratios
that are Pisot numbers. The fifth value of 
is not a self-summing
ratio.
The contracting algebraic conjugates of these ten units define tent tiles associated with asymmetric tent maps. The two parameters of polynomial degree 2 yield one-dimensional tiles, the six of degree 3 yield planar tiles, and the two of degree 4 yield three-dimensional tiles (Scheicher et al. 2026).
