Ἄλογον

The equation

You do not choose the number. You find it.

Ask for a rectangle that, cut in half, gives two rectangles of the same shape. Call the sides 1 and r.

// cut the long side in half
1 × r → two pieces (r/2) × 1

// “same shape” means the same long-to-short ratio
r / 1 = 1 / (r/2)
r = 2 / r
r² = 2
r = √2 = 1.41421356…

There is no other solution. A4 — 210 × 297 mm — is this, rounded to whole millimetres: 297 / 210 = 1.41428. The series starts at A0, which has an area of exactly one square metre.

Fold it

Here it becomes tangible

Two sheets, same height. One in the ratio √2, the other in the golden ratio. Fold them in half, again and again, and look at what remains.

Folds: 0

√2 — your paper

1.4142 start

φ — golden ratio

1.6180 start

√2 gives the same shape at every fold. φ returns every second fold, alternating with 1.2361 — which is, beautifully, exactly 2/φ.

The exact difference from the golden ratio

Two self-similarities, two different operations

The golden ratio has self-similarity too — just under a different operation. That is the whole story:

ratio	operation	what remains
√2 = 1.4142	divide in two	same rectangle
φ = 1.6180	remove a square	same rectangle

A page grid never removes squares. It divides space into equal parts — one column, two, three, four. That is the operation of √2, not of φ. This is why φ, for all its fame, cannot hold a shape in a grid that changes its number of columns.

Screen width: 1280 px · 3 columns each frame: 1.4142

Drag. The count changes; the shape never does. This is exactly what the gallery of works does at every screen width, and it is where the visitor feels the number without naming it.

Why “ἄλογον”

The crisis that broke the Pythagoreans

The square, its diagonal swung down into an arc, and the rectangle that results. This construction is the A series: every sheet of paper you have ever held comes from here.

The Pythagoreans believed that any two magnitudes have a common measure: some length that fits a whole number of times into both. That is what the word sym-metria means. And then their relation can be written as a ratio of whole numbers — a logos.

The diagonal of the square broke it. The proof fits in five lines:

// suppose √2 = p/q in lowest terms
2q² = p²   → p² even → p even
p = 2k   → 2q² = 4k²
q² = 2k²   → q² even → q even

p and q both even — contradiction.

So the side and the diagonal have no common measure at all. Not the millimetre, not the millionth, not any length anyone has imagined or ever will. That is why they were called ἀσύμμετρα — without common measure — and ἄλογα — without ratio, unsayable.

The best fractions come closer and never arrive. 99/70 was the ancient value; it is right to five decimals and wrong in the sixth, forever.

fraction	value	correct digits
3/2	1.5000000	1
7/5	1.4000000	1
17/12	1.4166667	2
41/29	1.4137931	3
99/70	1.4142857	5
577/408	1.4142157	6
665857/470832	1.4142136	11
√2	1.4142136…	—

Why this is your subject

Rational inside irrational

You asked for “symmetry, geometry and logos”. The paper you print on is the one shape that has neither symmetry — in the ancient sense — nor logos.

And here Greek makes a gift no other language can. The mathematical terms for numbers are ῥητός and ἄρρητος: expressible and inexpressible. They are the very same words that mean “spoken” and “unspoken”. It is not a pun; it is the ancient terminology, and it still holds.

This is not a design convention imposed afterwards. It is already measurable in the files: every page the program produces is 1400 × 1980 pixels, that is 0.70707 — and 1/√2 = 0.70711. The contradiction was there before anyone thought of it.

In plain words

Two experiments with paper and a ruler

Everything above fits into two things you can do now, at the table, knowing no mathematics at all.

First: the paper that does not change shape

you need one A4 sheet

fold Take an ordinary sheet of printer paper and fold it in half, bringing one short side onto the other.
look The half sheet has exactly the same shape as the whole. Smaller, but the same — like a photograph of itself.
again Fold again. And again. Always the same shape. You could go on forever, if the paper allowed.

Try it with a square sheet, or a long narrow one: it does not work. The half always comes out a different shape. There is only one shape in the world that does this — and it happens to be the paper we all have at home.

Second: the line that cannot be measured

you need a square and a ruler

draw A square. Measure its side — say, 10 centimetres.
measure Now measure the line from one corner to the opposite one. You will find about 14.1 centimetres. Get a better ruler: 14.14. Better still: 14.142.
search Look for a small piece — a millimetre, half a millimetre, anything — that fits exactly, a whole number of times, into both the side and this line.

You will not find it. And it is not the ruler's fault, nor your eyes: it does not exist. However small you imagine it, it does not fit both. The two lines have no common measure.

The ancient Greeks believed any two lengths have a common measure. When they discovered this is false, it frightened them so much that they named this line ἄρρητη — the one that is not spoken.

Both experiments speak of the same number: 1.4142… In the first it is the shape of the paper. In the second it is the length of the line. It is the same number, and it is never written whole.

I print words on a paper that cannot be spoken.

Twenty-five centuries in nine steps.

All is number.

The measure that does not exist.

A grammar for the unspeakable.

ῥητός and ἄλογος.

The number becomes stationery.

The Republic measures paper.

A0 is one square metre.

Four letters along a line.

Rational text in irrational geometry.