In the standard full Mandelbrot construction, which number domains are used…
2012
In the standard full Mandelbrot construction, which number domains are used for the iterated state xn and the parameter z in the recurrence
xn+1 = xn2 + z ?
Answer: B. Both xₙ and z range over the complex-number domain ℂ. — ConceptThe Mandelbrot set is defined in the complex parameter plane ℂ. For each parameter z in ℂ, its orbit starts at zero and is generated by repeated…
- A.
Both xₙ and z are restricted to the real-number domain ℝ.
- B.
Both xₙ and z range over the complex-number domain ℂ.
- C.
xₙ is restricted to ℝ, while z ranges over ℂ.
- D.
xₙ ranges over ℂ, while z is restricted to ℝ.
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Show answer & explanation
Correct answer: B
Concept
The Mandelbrot set is defined in the complex parameter plane ℂ. For each parameter z in ℂ, its orbit starts at zero and is generated by repeated squaring and addition.
The orbit is therefore a sequence in ℂ; the real axis ℝ is only a special slice of that plane, not the domain of the full construction.
Application
Write the parameter as z = c + di, where c and d are real.
Initialize the orbit with x0 = 0 + 0i.
The first iterate is x1 = x02 + z = c + di, so the orbit can immediately have both real and imaginary components.
Because ℂ is closed under squaring and addition, every later xn is also a complex value.
Cross-check and contrast
The domain labels in the choices describe these distinct assignments:
ℝ × ℝ confines both variables to the real-axis slice.
ℝ × ℂ restricts the state while allowing a complex parameter.
ℂ × ℝ allows a complex state but restricts the parameter to the real-axis slice.
Thus the full construction uses xn ∈ ℂ and z ∈ ℂ: both are complex-valued.