Repetition
What happens if we do the same thing again and again?
Start with 3. Each time, double the number and subtract 1.
Before revealing anything, try to write the next five values on paper. Then reveal the trail and compare it with your prediction.
3?????
The horizontal axis counts how many times the rule has been used. The height shows the value you have reached.
Coordinates for start 3
(0, 3)
On the graph above you may have noticed something special about starting with the number 1.
Starting with 1, we double it to get 2 and subtract 1 to return to 1. This would repeat no matter how many times we tried it.
What happens if we use different rules? What if we multiplied by 3 and subtracted 1? Would this still happen?
For numbers greater than 1 we got further away from 1 each time. Will this always be the case?
Starting with numbers less than 1, our answers became negative. Is this always true?
Tools and visualisations
RepetitionPatternChance
Cellular Automata Explorer
Explore how simple local rules create order, randomness and emergent patterns.
ExplorationTransformationRepresentationRepetition
Complex Number and Argand Diagram Explorer
Explore modulus, argument, multiplication and roots of unity.
ToolChangeRepresentationRepetition
Differential Equation Slope Field Explorer
Explore slope fields, initial conditions and numerical solution curves.
ToolRepetitionPatternChange
Conway's Game of Life Explorer
Run still lifes, oscillators and gliders in a two-dimensional cellular automaton.
ExplorationRepetitionChangePattern
Logistic Map and Chaos Explorer
Iterate a simple recurrence and explore fixed points, cycles and chaos.
ExplorationRepetitionTransformationPattern
Mandelbrot and Julia Set Explorer
Zoom into fractal boundaries created by repeated complex iteration.
ExplorationWhere this appears in school maths
Sequences and functions
Repeated rules connect naturally to term-to-term sequences, function machines and graphs.
Modelling
Population growth, decay and feedback systems often begin with repeated updates.