Hexation Calculator

What is hexation?

Hexation, denoted , is the sixth operation in the hyper-operation hierarchy. It iterates pentation just as pentation iterates tetration. For positive integers and , means applying pentation times, starting from .

More formally, hexation is defined recursively:

• for

This means each step nests another layer of pentation, creating structures that dwarf even the largest known numbers like Graham's number. For example, is a pentation tower of height 2, which is already beyond comprehension.

The calculator evaluates hexation for small inputs, showing step-by-step construction when possible. For larger values, it falls back to symbolic notation using Knuth's up-arrow notation.

Step-by-step evaluation

When you check "Show iterative breakdown", the calculator displays a table of intermediate values. Each row shows the hexation at that stage, building up to the final result.

For instance, computing :

• Stage 1:
• Stage 2:
• Stage 3: (a pentation of height 4, which is enormous)

As you can see, even small heights quickly produce values that require symbolic representation.

Growth characteristics

Hexation exhibits extreme growth. Each increment in the height parameter multiplies the "size" of the result by an astronomical factor. The growth rate is so rapid that hexation surpasses all functions in the fast-growing hierarchy up to very high ordinal indices.

Compared to tetration and pentation:

• Tetration: grows super-exponentially
• Pentation: grows super-tetrationally
• Hexation: grows super-pentationally

This hierarchy continues indefinitely, with each new operation dwarfing the previous ones.

Applications and context

Hexation and higher hyper-operations are primarily of theoretical interest in mathematics, particularly in:

• Proof theory: Studying the strength of mathematical systems
• Computability theory: Analyzing the limits of computation
• Large number theory: Exploring the structure of enormous numbers

While practical applications are limited due to the immense sizes involved, these operations help mathematicians understand the boundaries of different number systems and the growth rates of functions.