Generator

lcg

A generator in the computation library, called 6 times across 1 essays. Below: what it draws at its defaults and at each mode an essay asks for, what it checks while drawing, and everywhere it is used.

lcg 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.

At its defaults

256 consecutive pairs from xₙ₊₁ = 137xₙ + 187 mod 256. Consecutive outputs of a linear congruential generator plotted as points of a square, falling on a small family of evenly spaced parallel lines.

`show: "compare"`

Two multipliers, 21 lines against 5. Consecutive output pairs of two linear congruential generators that differ only in their multiplier, one spread over many lines and one collapsed onto few.

`show: "period"`

Which generators visit all 16 values. A table of multipliers and increments with the three conditions that decide whether the generator has the full period, beside the period found by running it out.

`show: "planes"`

1500 triples from a generator, on 15 planes. Consecutive triples of outputs of a linear congruential generator plotted in a cube, seen from an angle that shows them lying on a small number of parallel planes.

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.

the modulus is a whole number between 16 and 65536 ×2
a short vector the modulus annihilates exists ×1
every point of the sequence satisfies the relation exactly ×1
the criterion and the measured period agree ×1
the increment is a whole number between 0 and 1048576 ×1
the modulus is small enough that the arithmetic stays exact ×1
the multiplier is a whole number ×1
the multiplier is a whole number between 2 and 1048576 ×1
the number of points is a whole number between 16 and 4096 ×1
the number of triples is a whole number between 100 and 4000 ×1
the relation involves the middle coordinate, so an edge-on view exists ×1
the relation involves the second coordinate ×1
the relation's normal lies in the screen, so the planes are seen edge on ×1
the triples lie on a small number of planes ×1
the two multipliers give different numbers of lines ×1
the view is one of compare, period, planes ×1
three to eight multiplier-and-increment pairs ×1
two multipliers below the modulus ×1

Where it is called

Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.

Computation

The planes a recurrence cannot leave

One multiplication and one addition, taken modulo a fixed number, produce a sequence that passes for random one value at a time. Taken two or three at a time it does not, and the reason is a whole-number relation that pins every point onto one of a small family of parallel lines.

The whole library · What the figures prove