About
Mathematical curiosity, made executable
An interactive field guide to surprising mathematical behaviour, from the first visible pattern to what is proved, what is only observed, and what remains unknown.
A Mathomaly is
A mathematical phenomenon whose behaviour is more surprising than its rule suggests.
Not an anomaly in the statistical sense. Individual pieces here are called explorations; the brand stays singular, and the plural stays comfortable.
The field is larger than the map.
Is mathematics invented or discovered? Mathomaly treats the question as a productive tension rather than choosing one side. We invent the notation, build the tools and choose where to look, while the structures answer back with consequences we do not control.
Mathematics is a field in more than one sense: a vast, connected terrain, far larger than the part we have mapped. Enter through numbers, shapes, logic or motion and paths that begin far apart can meet at the same truth. A structure uncovered in one corner can surface again in another. It may become useful centuries after it was first understood.
Not every discovery has an obvious use: some become indispensable later, while others are worth finding simply because they are there. Each exploration follows a promising seam by changing the rule, testing its edges, mapping what connects and digging beneath the example until the structure appears.
Mathematics, mined in the mind.
Why it exists
Mathematics is strange because it seems to be both invented and discovered. Once a rule or a set of assumptions is fixed, its consequences are not ours to choose; the structure answers back. Yet those consequences do not arrive already sorted into useful ideas. We decide where to look, what to name, which notation to build, which connections matter and how to make what we find intelligible to someone else.
Imagine an infinite mathematical realm: a field of structures and relations that is already there, but too large for any mind to survey. Finding mathematics within it cannot be passive. We follow one seam, develop the tools that can reach it, test what it contains and make a map that other people can use. The terrain may be discovered, while the route, language and emphasis are created. When the field is inexhaustible, mining it becomes difficult to distinguish from creation.
Mathomaly exists to let people take part in that process. Each exploration begins with a result that seems to be simply there, then gives you ways to change the conditions, map the surrounding terrain and ask which patterns survive. The interaction is an instrument for discovery, while the explanation is a constructed map of what has been proved, what has only been observed and what remains unknown.
This is why the site neither stops at a visual surprise nor demands a formal definition as the price of entry. It begins with direct experience and follows the question towards invariantsDefinition: An invariant is a quantity or property that remains unchanged under a rule., counterexamplesDefinition: A counterexample is one valid case that makes a universal claim false., proofDefinition: A proof is a finite argument showing that a conclusion follows from stated assumptions., generalisation and the unknown. The aim is not to teach a syllabus, but to make questions such as these feel natural to ask: Does this always happen? Which assumptions matter? What changes in another base? Is the pattern real, or an artefact of how it is drawn? Has this been proved, or only checked? What would a counterexample look like?
Principles
Interaction before exposition
The first screen answers “what can I do here?”, not “what is this about?”. You meet the behaviour before you read about it.
Manipulation, not decoration
Motion has to show causality by making clear why one state becomes the next. Animation that exists only to make a page feel lively is hiding something.
Progressive depth
Spark, play, map, understand, proof. You should be able to leave after any layer with something true, and the deepest layer must exist wherever the mathematics warrants it.
Exactness where it is available
IntegerDefinition: An integer is a whole-number value, including zero and negative whole numbers, with no fractional part. and rationalDefinition: A rational number can be written as a fraction of two integers with a non-zero denominator. arithmetic wherever the algorithm permits. A floating-point approximation must never quietly become a mathematical claim.
Make failure interesting
Counterexamples and boundary cases are first-class content. A control that makes the phenomenon stop working usually teaches more than a perfect demonstration.
Never confuse evidence with proof
Every claim states its scope and status. A million successful runs say nothing about an infinite statement.
Who it is for
Several audiences share one product, whose tone should be intelligent and welcoming because rigour does not require opacity.
- Anyone curious
- A satisfying visual surprise in under a minute, with nothing assumed.
- Students
- A bridge from experimenting with something to being able to explain it.
- Teachers
- Reliable demonstrations, shareable states and questions worth asking a class.
- Developers
- Algorithms, state graphsDefinition: A graph is a collection of nodes joined by edges that record relationships or transitions., parameter sweeps and reproducible computation.
- Recreational mathematicians
- Less-common variants, real references, and room to go looking.
- Researchers
- Accurate claim status, primary sources and clearly bounded evidence.
Where to start
The 6174 Attractor is the flagship: a rule you can do in your head, a result that is genuinely startling, and a state spaceDefinition: A state space is the collection of all configurations a rule can occupy. small enough to draw in full, so the proof fits on the page instead of being asserted.
If you would rather know how claims are handled first, read the method.