The name
The name is a Sinhala word. Its root is saṅgata: coherent, gone together. Things that are saṅgata fit each other. They agree. Nothing in them is out of place.
The word also had a precise job in the old commentaries. A scholar could not open a new section without first stating its saṅgati: the thread that tied it to the section before, and the reason it had the right to come next. A section with no thread had no place in the text.
That rule is the whole of this project. Every page states its thread to the page before. Every page makes a promise to the page after.
A formal system you have already heard of
Sanskrit is a language. It is also a formal system, and the oldest one we have in full.
About twenty-four centuries ago, Pāṇini wrote a grammar of Sanskrit in close to four thousand short rules called sūtras. A sūtra is a few words long. Rules refer to other rules. A later rule uses what an earlier rule defined. Some rules are rules about rules: they say which rule wins when two apply. From a few hundred roots and these rules, every correct word of the language can be built, and a wrong one can be shown to be wrong, step by step.
Computer scientists met this grammar again in the twentieth century. The notation they use to define programming languages has the same shape, and some of them named it after both its inventors: the Pāṇini-Backus form.
Here is what Sanskrit shows. A complex idea is not a large thing. It is a small set of parts and an explicit rule for joining them. What makes an idea hard is not its size. It is a missing rule.
What Sangathika formalises
Most courses treat advanced mathematics as a summit. You climb through the foundations, and the advanced ideas wait at the top for the few who arrive.
Sangathika treats mathematics as a spine. There is no summit. Each idea is a bone shaped to hold the next. What you can carry at the top is only what is aligned below.
Most lessons mix several ideas in one chapter.
Here every page is a sūtra: one idea, stated in one sentence of fewer than twenty words. The rest of the page is the commentary that builds it, in six layers, in a fixed order.
Most teaching asks you to read, then do.
Here you act first. You drag the arrow, set the dial, move the towns. Then you predict what will happen. Only then do you read why.
Most definitions are words to memorise.
Here a definition is code. The page that introduces the dot product adds a function called dot to the leaf's library. The next page calls it, and you can read the call. By the end of a leaf you hold a small grammar you built one rule at a time, and every rule in it runs.
Most courses change their examples with every chapter.
Here the same few objects return: a 3-4-5 triangle, two towns and the road between them, one dial, fifteen students and their marks. They are the roots of the grammar. Each time you meet one, it means more than last time.
Most textbooks keep pure and applied mathematics in separate rooms.
Here an application hangs from the exact bone it needs. A cipher hangs from the inverse. A search engine's ranking hangs from an eigenvector. You reach the application the moment its spine is strong enough.
Most difficulty is hidden until you fail.
Here every challenge declares its level before you open it: Seed, Branch or Canopy. You choose your climb.
Advanced mathematics is not far away. It is what a strong spine can carry.
How to read a page
Act: drag, set or step the object on the first plate before you read anything. Predict: commit to an answer before the plate shows it. Read: the six layers, in order, each adding one clause to the page's one sentence. Check: six quick questions, then the challenges, each with its level declared.
The cast
Four objects that return across the leaves. Each leaf uses at least one. No object is ever redefined; each return adds one new reading.
Three levels of challenge
For the teacher
Sequencing, principles and how to contribute
Every lesson page carries a teacher note: why the page sits where it does, the misconceptions it targets, and why each challenge is hard. Every leaf page states the sequencing of its whole leaf. The seven principles behind every page are: need before name, act before read, predict before reveal, one idea per page, misconception first, same objects with new eyes, and spaced retrieval.
To propose a page or report an error, see how to contribute.
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