What are the different approaches of tetration?

[Daniel]

Great question, in a sense answering that question is central to the Tetration Forum. I have asked that very question two or three times in the fifteen year history of the Forum. But at the moment understanding tetration is more about understanding the work of a number of individuals than the idea that anyone is sitting on a grand synthesis of all tetration techniques.

The following is a progression through the different levels of tetration research that I have seen a number of people make.

• Numerical simulation
  
• Algorithmic
  
• Complex tetration - power series, Schroeder's and Abel's equations, based on fixed points
  • Aldrovandi, Helms - matrices
    
  • Geisler - combinatorics
    
• Dynamics - Kneser, Ecalle
  
• Real tetration - Trappmann, Kouznetsov, Paulsen, plus a number of members from this Forum.