John Horton Conway
John Horton Conway (1937–2020) was a renowned English mathematician known for his work in group theory, number-theory, and combinatorial game theory. He is perhaps most famous for inventing the Game of Life, a cellular automaton that exhibits complex emergent behavior from simple rules.
Key Contributions
- Game of Life: A zero-player game that demonstrates how simple rules can lead to complexity and emergence.
- Surreal Numbers: Developed a system of numbers that includes all real numbers as well as infinite and infinitesimal quantities.
- Conway’s Base 13 Function: A pathological function that maps every real interval to the entire set of real numbers, challenging standard intuitions about continuity and function behavior.
- Look-and-Say Sequence: A sequence of integers where each term is derived from the description of the previous term.
Conway’s Base 13 Function
This concept illustrates a function that is discontinuous everywhere yet takes on every possible real value in any given interval. It was designed to provide a counterexample to the idea that a function must be “well-behaved” or continuous to be meaningful.
- Definition: The function uses base-13 representation of real numbers, utilizing digits 0-9 and symbols for positive/negative infinity and decimal points to encode ASCII characters.
- Behavior: If the base-13 expansion of an input contains a specific pattern resembling an ASCII-encoded equation (e.g.,
+123.), the function outputs the result of that equation; otherwise, it outputs 0. - Implication: This demonstrates that a function can be bounded nowhere and still be mathematically valid.
- Resource: See Conway’s Base 13 Function: Explaining its Unique Mathematical Definition for a detailed explanation.
References
- Conway’s Base 13 Function: Explaining its Unique Mathematical Definition https://www.youtube.com/watch?v=zsBaApyf_Ns