John Conway initially thought that his cellular automata game of Life could not lead to an unlimited number of active cells starting from a finite number of cells.

Computer experimentation by Bill Gosper led to the discovery in 1970 of the Gosper Glider Gun, a finite collection of cells which leads to a recurring state which continues to emit "spaceships" or "bullets" which continue out into the periphery infinitely. I can't find a link to the specifics of the discovery beyond the remark that Gosper won a 50$US reward from Conway for making this finding.

The glider gun was also used in constructions involving the manipulations of the "bullets" in such a way as to form a computational apparatus, ultimately proving the Turing-completeness of Conway's Life. I do know that a lot of random configuration discovery was done for the Life cellular automaton and for other CA by creating random patterns, then masking the lattice with a pattern which blanked out the peripheral cell values and left the random pattern alone in the center of the screen, and then allowed the CA rules to run. Escapees such as gliders would continue their motion into the blanked areas and be easily discerned in a simulation.

The first glider-type pattern was discovered by tracing the evolution of the finite starting patterns.

David Eppstein also appears to have written a program that helps in finding these spaceship types of pattern, which should also count as mathematical discoveries and exploration via computer programs.

:gfind A program by David Eppstein which uses de Bruijn graphs to search for new spaceships. It was with gfind that Eppstein found the weekender, and Paul Tooke later used it to find the dragon. It is available at http://www.ics.uci.edu/~eppstein/ca/gfind.c (C source code only).

John Conway initially thought that his cellular automata game of Life could not lead to an unlimited number of active cells starting from a finite number of cells.

Computer experimentation by Bill Gosper led to the discovery in 1970 of the Gosper Glider Gun, a finite collection of cells which leads to a recurring state which continues to emit "spaceships" or "bullets" which continue out into the periphery infinitely. I can't find a link to the specifics of the discovery beyond the remark that Gosper won a 50$US reward from Conway for making this finding.

The glider gun was also used in constructions involving the manipulations of the "bullets" in such a way as to form a computational apparatus, ultimately proving the Turing-completeness of Conway's Life. I do know that a lot of random configuration discovery was done for the Life cellular automaton and for other CA by creating random patterns, then masking the lattice with a pattern which blanked out the peripheral cell values and left the random pattern alone in the center of the screen, and then allowed the CA rules to run. Escapees such as gliders would continue their motion into the blanked areas and be easily discerned in a simulation.

The first glider-type pattern was discovered by tracing the evolution of the finite starting patterns.

David Eppstein also appears to have written a program that helps in finding these spaceship types of pattern, which should also count as mathematical discoveries and exploration via computer programs.

:gfind A program by David Eppstein which uses de Bruijn graphs to search for new spaceships. It was with gfind that Eppstein found the weekender, and Paul Tooke later used it to find the dragon. It is available at http://www.ics.uci.edu/~eppstein/ca/gfind.c (C source code only).

John Conway initially thought that his cellular automata game of Life could not lead to an unlimited number of active cells starting from a finite number of cells.

Computer experimentation by Bill Gosper led to the discovery in 1970 of the Gosper Glider Gun, a finite collection of cells which leads to a recurring state which continues to emit "spaceships" or "bullets" which continue out into the periphery infinitely. I can't find a link to the specifics of the discovery beyond the remark that Gosper won a 50$US reward from Conway for making this finding.

I do know that a lot of random configuration discovery was done for the Life cellular automaton and for other CA by creating random patterns, then masking the lattice with a pattern which blanked out the peripheral cell values and left the random pattern alone in the center of the screen, and then allowed the CA rules to run. Escapees such as gliders would continue their motion into the blanked areas and be easily discerned in a simulation.

The first glider-type pattern was discovered by tracing the evolution of the finite starting patterns.

David Eppstein also appears to have written a program that helps in finding these spaceship types of pattern, which should also count as mathematical discoveries and exploration via computer programs.

:gfind A program by David Eppstein which uses de Bruijn graphs to search for new spaceships. It was with gfind that Eppstein found the weekender, and Paul Tooke later used it to find the dragon. It is available at http://www.ics.uci.edu/~eppstein/ca/gfind.c (C source code only).

John Conway initially thought that his cellular automata game of Life could not lead to an unlimited number of active cells starting from a finite number of cells.

Computer experimentation by Bill Gosper led to the discovery in 1970 of the Gosper Glider Gun, a finite collection of cells which leads to a recurring state which continues to emit "spaceships" or "bullets" which continue out into the periphery infinitely. I can't find a link to the specifics of the discovery beyond the remark that Gosper won a 50$US reward from Conway for making this finding.

The glider gun was also used in constructions involving the manipulations of the "bullets" in such a way as to form a computational apparatus, ultimately proving the Turing-completeness of Conway's Life. I do know that a lot of random configuration discovery was done for the Life cellular automaton and for other CA by creating random patterns, then masking the lattice with a pattern which blanked out the peripheral cell values and left the random pattern alone in the center of the screen, and then allowed the CA rules to run. Escapees such as gliders would continue their motion into the blanked areas and be easily discerned in a simulation.

The first glider-type pattern was discovered by tracing the evolution of the finite starting patterns.

David Eppstein also appears to have written a program that helps in finding these spaceship types of pattern, which should also count as mathematical discoveries and exploration via computer programs.

:gfind A program by David Eppstein which uses de Bruijn graphs to search for new spaceships. It was with gfind that Eppstein found the weekender, and Paul Tooke later used it to find the dragon. It is available at http://www.ics.uci.edu/~eppstein/ca/gfind.c (C source code only).