The following tricocycloids are all isomorphic to the tricocycloid of the open interval.
The following Python code uses SymPy to check the axioms of a tricocycloid for the positive reals. Note that these follow from the interval tricocycloid in any case.
#!/usr/bin/env python3
from sympy import *
x, y, z = symbols('x y z')
# Tricocycloid of the positive reals
def e(x): return (x / (x+1))
def o(x): return (x / (1-x))
def b(x,y): return (o(e(x)*e(y)))
def w(x,y): return (o(e(x)*(1-e(y))/(1 - e(x)*e(y))))
# Tricocycloid axioms.
simplify( b(x,b(y,z)) - b(b(x,y),z) )
simplify( w(b(x,y),z) - b(w(x,b(y,z)), w(y,z)) )
simplify( w(x,y) - w(w(x,b(y,z)), w(y,z)) )Tags: probability, tricocycloid.