Arithmetic on a dial of 12
clock is one function. Everything below came out of it during this
build, at parameters taken from the essays rather than invented for this page — so a figure
here is the same figure a reader meets in an essay, and if the generator changes, this page
changes with it.
With nothing chosen
Square roots of 2, lifted from one power of 7 to the next
A square root of 2, one 7-adic digit at a time
Newton's method, where near means divisible by 7
How many square roots a number has, modulo each prime power
Multiplication on a dial of 11
What it checks while it draws
Collected by running the family and recording what it asserted, not written here. The count is how many separate times the claim was put to the test while these drawings were made.
- for odd 3 not dividing 1, the count is 0 or 2 at every power ×13
- the cycle through 0 has the length of 2's order modulo 21/gcd(0, 21) ×11
- x² ≡ 2 holds modulo 7^1 ×8
- the residues that 2 leaves in place number gcd(1, 180) ×6
- modulo 8, an odd a has 4 square roots when a ≡ 1 mod 8 and none otherwise ×3
- multiplying by 2 reaches N/gcd(2, 199) of the residues ×3
- the walk reaches m × n / gcd cells on a 3 by 5 grid ×3
- and that length divides 2's order modulo 21 ×2
- 3 to 12 digits ×1
- a dial needs at least three positions ×1
- a dial of 5 to 60 ×1
- a dial of 6 to 400 places ×1
- a multiplier from 2 to N − 1 ×1
- a multiplier sharing no factor with the dial ×1
- a row is a permutation exactly when its multiplier is coprime to the modulus ×1
- a small prime ×1
- an odd prime ×1
- each root reduces to a root one level down ×1
- every cycle of units has the full order ×1
- every Newton step at least doubles the number of correct digits ×1
- every non-zero row is a permutation exactly when the modulus is prime ×1
- every residue has a place on the dial ×1
- every row of the addition table is a permutation ×1
- exactly one digit extends the root, and it is the one the formula gives ×1
- it fills the whole grid exactly when the moduli are coprime ×1
- multiplying `order` times returns to 1 ×1
- the arithmetic and the walk agree ×1
- the dial and the complex circle agree on where each residue sits ×1
- the dial view is one the family draws ×1
- the drawn entry is the arithmetic ×1
- the drawn walk passes the top as often as the division says ×1
- the envelope touches each chord along its own direction ×1
- the first modulus is between 2 and 12 ×1
- the multiplier is not zero ×1
- the orbit visits each residue once before closing ×1
- the order divides m − 1, which is Fermat's little theorem ×1
- the period is the least common multiple ×1
- the second modulus is between 2 and 12 ×1
- the slope is invertible at every step ×1
- the starting digit is a square root of 2 mod 7 ×1
- the walk is between one and 144 steps ×1
- the walk runs at least one full period ×1
- two to eight multipliers ×1
- two to five levels, the top modulus under 20,000 ×1
- two to five primes ×1
- two to six Newton steps ×1
- two to six values of a ×1
- with p odd and not dividing a, every level has as many roots as the first — each lifts in exactly one way ×1
Where it is called
Every figure on this list is drawn by the same rule, so a change to the rule changes all of them at once. That is why the list is published.
A root lifted one digit at a time
On a dial of seven, 3 × 3 is 2. On a dial of forty-nine the square root of 2 must reduce to 3, so there are only seven candidates, and exactly one of them works: 10. On a dial of 343 exactly one lift of 10 works: 108. Each step adds one digit on the left, found by solving a linear equation, and the digits go on for ever — a number …21216213 whose square is 2, in a world where closeness means divisibility by seven.
NumberCounting one rectangle, twice
Whether seven is a square modulo eleven, and whether eleven is a square modulo seven, are two unrelated-looking questions. Their answers are linked, and the link is a rectangle of dots counted along its rows and then along its columns.
DiscreteMultiplying every number on the dial at once
Join every residue on a dial to twice itself and the chords draw a heart-shaped curve with one cusp; join each to three times itself and the curve has two. The picture is the whole multiplication map at once, and it holds three facts: the map splits the dial into cycles whose lengths are orders, those cycles on a dial of 2ⁿ − 1 are the binary necklaces of length n, and the curve is the caustic light draws inside a cup.
NumberNecklaces that prove a theorem
Thread five beads in two colours, thirty-two ways. Two of them are all one colour; the other thirty fall into rings of five. That count, and nothing else, is Fermat's little theorem.
DiscreteNumbers that wrap
A clock does arithmetic. It has finitely many numbers, addition never leaves it, and multiplication behaves entirely differently depending on one property of the size of the dial.
AlgebraThe blocks a subgroup cuts out
Take any part of a group that is closed under composition, and it slices the whole group into blocks of its own size that do not overlap. Everything Lagrange's theorem says is arithmetic about that picture — and whether the blocks can be multiplied is a separate question with a surprising answer.
NumberThe two supplements, and where the eight comes from
The main law relates two odd primes to each other and says nothing about −1 or about 2. Those two are settled separately, by their own counts, and the answers arrive modulo four and modulo eight — which is a clue about where the whole subject is really taking place.
NumberTwo dials at once
Watch one number on two clocks with different faces. If the faces share no factor, every pair of readings occurs exactly once — so two remainders name a number, and a hard calculation can be split into two easy ones.