ideaLive
Fibonacci Spirals
Start with a small adding rule, build growing squares and watch arithmetic become geometry.
Beauty in maths topic
Number and Structure
Number and structure explores exactness, approximation, irrationality and the hidden behaviour of simple rules.
8
live pages
2
prototype tools
4
planned ideas
What this area includes
Fibonacci ratios and limiting behaviour
Irrational directions and lattice points
Structure created by repeated rotations
Links between arithmetic, geometry and proof ideas
Questions to explore
What does it mean for a number to be impossible to write as a fraction?
How can approximation reveal structure?
Why do some ratios create order while others avoid repetition?
Working pages and tools
These are the pages currently available to open from this topic.
ideaLive
Integer Orchard
A thought experiment connecting rational slopes, irrational directions and exact lattice hits.
ideaLive
Rotation Each Time
Repeated turns create seed-packing patterns and reveal why simple fractions make visible arms.
ToolPrototype
Complex Number and Argand Explorer
Explore modulus, argument, multiplication and roots of unity.
ToolPrototype
Classical Ciphers Lab
Explore Caesar and Vigenere ciphers, keys, frequency analysis and why simple ciphers leak structure.
Planned directions
These ideas are not built yet, but they show where this topic could grow next.
ideaPlanned
GCSE Number Sense Trainer
Planned tool for fractions, percentages, ratio, standard form, bounds and estimation.
ideaPlanned
Perfectly Balanced Rhythm
Planned tool for distributing beats as evenly as possible around a cycle, connecting rhythm to modular arithmetic.
ideaPlanned
Tuning and Ratios
Planned investigation of frequency ratios, consonance, equal temperament and why perfect intervals are not always compatible.
ideaPlanned
Modular Arithmetic and Keys
Planned exploration of congruence, inverses, clocks and why modular arithmetic powers cryptography.
Connected topics
themeLive
Beauty in Maths
A portal for connected mathematical explorations: pattern, surprise, structure, nature and emergence.
themeLive
Maths in Music
Rhythm, tuning, symmetry, timing, pattern and musical structure through mathematics.
themeLive
Fractals and Infinite Detail
Self-similarity, recursion, iteration, dimension and patterns that repeat at different scales.
themeLive
Cryptography and Secret Messages
Codes, ciphers, primes, modular arithmetic, public keys and secure communication.
themeLive
Logic, Proof and Paradoxes
Truth, argument, proof, contradiction, paradox and the foundations of mathematical reasoning.
How to use this section
These explorations are particularly good for moving from calculation into mathematical curiosity.