Always
Counting
Pointing at things until the words run out.
We assume you can count. One, two, three… and a name for each pile. If that is solid, skip ahead. The first real puzzle is a pile too wide to point at one by one.
Assumed
A river, not a school list
Math
Part of Windrose
This is not school chapter order (inte Matematik 1–4). You do not meet a method because a book reached that page. You meet it because someone had a job they could not finish without it.
Four roads: Quantity (how many), Space (place), Structure (pattern), Change. Times is the first time quantity meets space: counting a rectangle.
What we can count — you can finish by pointing and grouping.
The pile we don’t know yet — a number still unnamed
.
Fields & floors
Egypt, and anywhere people laid bricks
Egypt, and anywhere people laid bricks
A rectangle too big to count one-by-one. Tool: (gånger / multiplikation) — How many × Place.
Open
Same floors,
cut up
cut up
Sharing the floor
How many along one side, or how many each. Tool: sharing (division).
Why this comes next: once you can build a floor, the next job is to cut it — tiles per person, or how many along one wall.
Locked
Egypt
c. 1850–1650 BCE
c. 1850–1650 BCE
When it does not come out even
Leftover parts. Tool: leftover parts / fractions (bråk).
Why this comes next: sharing a floor does not always land on a whole tile.
Locked
India
then everywhere
then everywhere
Names for huge counts
So you stop inventing new words. Tool: place value and zero (positionssystem, nolla).
Why this comes next: big floors need short names, not a new word for every size.
To be developed
Egypt “the pile”
later al-Khwārizmī
later al-Khwārizmī
The pile we don’t know yet
A number you do not know yet. Tool: (obekant / ekvation). Old name in some English books: (Egyptian aha — we say pile).
Why this comes next: sometimes the floor is the answer, not the given.
To be developed
Land measure
Thales
Thales
Same shape, bigger
A field or shadow scaled up. Tool: / similar shapes (förhållande, likformighet).
Why this comes next: you already count rectangles; now two rectangles share a shape, not a size.
To be developed
Right corners
Babylon, later named for Pythagoras
Babylon, later named for Pythagoras
The third side you cannot pace
Two lengths at a right angle. Tool: the square-on-the-sides rule (Pythagoras sats).
Why this comes next: similar shapes still need a corner you can trust, and a missing length.
To be developed
Shadows, then sine tables
River bank: pace a baseline, read an angle — a width you cannot tape. Tool: the angle’s ratio (tangens / sinus / trigonometri). On this river it sits after same-shape and the third side. Those pages are not written yet, so this lesson stays Open — a built stop, not a side track.
Open
No right angle
island / span
island / span
Two paced lengths and the angle between — or two angles and one side. Tool: cosine rule / sine rule (cosinussatsen / sinussatsen).
Why this comes next: the right-angle table is not enough once the corner is not square.
Locked
Going around
A point walks the rim of radius 1; height and sideways are sin / cos — past 90° too. Soft angles as rim length (enhetscirkeln, radianer).
Why this comes next: full spins need angles past a right corner.
Locked
Same wave,
now in air
now in air
Spin a radius; plot height — that plot is the wave. Air carries pressure squeeze, not a drawn sine. Tool: sine as the simplest repeating piece (sinusvåg, frekvens, ljud).
Why this comes next: let the circle’s angle keep turning in time — bridge to physics (strings and voice).
Locked
Composing turns
Turn α then β — turns add, sines do not. Tool: angle-addition as composing turns (additionsformler).
Why this comes next: aiming and phase need stacked turns, not lazy sums of heights.
Locked
Newton / Leibniz
rate
rate
A hill or a growing pile — rise per run of the graph. Tool: rate / derivative (derivata).
Why this comes next: the floor was a still rectangle. Now it grows while you watch.
Locked
Newton / Leibniz
total
total
Area under a rate curve = total. Tool: integral (integral).
Why this comes next: flip of steepness — from rate to how much piled up.
Locked
The plane as a move
A pair of numbers as one motion. Tool: complex numbers as rotate+scale, not as a mystery root (komplexa tal).
Why this comes next: you already stretch and turn. This is that move, written as one object.
Locked
Roots of a pile
Where a pile hits zero
Polynomial roots — where the unnamed pile crosses nothing. Listed for completeness; page not written.
Why this comes later: once turn-and-stretch exists, roots in the plane become a job.
To be developed
Why we trust the rule
Proof as trust
Why we trust the rule — not a drill of formal steps for their own sake. Listed; page not written.
To be developed
Reading the table
Reading the table backwards
Which angle has this slope. Tool: inverse of the angle’s ratio (arcsin).
Why this comes next: a table is useless if you can only run it one way.
To be developed
Bookkeeping
Two unnamed piles at once
Keeping two piles honest in the same ledger. Not “matrices as a subject”.
Why this comes next: one unnamed pile. Two is still counting — twice, with rules so they do not collide.
To be developed
Pascal & Fermat
1654
1654
Fair and unfair games
How to split a pot when the match stops early. Tool: chance as counted cases (sannolikhet).
Why this comes next: you can already count rectangles of outcomes. Fairness is a count, not a feeling.
To be developed
Stop. This is as far as we go by hand. Past here you need a specialist’s tools. No topology, no chaos, no group theory, no manifolds. If a later page needs those, it has left this river.