Generator

progression

A generator in the discrete library, called 11 times across 2 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.

progression 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

Two colours avoid a progression up to 8, and no further. The numbers 1 to 8 in the two colours that avoid three equally spaced numbers in one colour, with the number 9 beside them in both colours and the pattern each choice forces.

`show: "monotone"`

A sequence of 3² with no climb and no fall longer than 3. 10 terms plotted in order, each labelled with the longest climb and the longest fall ending at it; the first 9 keep both counters at 3 or below and the last one cannot.

`show: "convex"`

Five points, and the four of them that make a convex quadrilateral. Three sets of five points in general position, with hulls of 5, 4, 3 points; in each, four points forming a convex quadrilateral are ringed and joined.

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.

nothing survives at 9 either ×7
a point set in general position was found for every seed ×1
and all three sizes of hull turned up in the test ×1
and four of them in convex position ×1
and its longest fall is n ×1
and neither colour can be given to the next number without making one ×1
and the extra term's label leaves the n by n square it would have to fit in ×1
and the surviving lengths run right up to it with no gap ×1
between one and four seeded point sets are drawn ×1
every five points in general position hold four in convex position ×1
every term carries a different pair of counters ×1
its longest climb is n ×1
one more term and a climb or a fall of n+1 appears ×1
the claim is tested on between 200 and 20,000 point sets ×1
the colouring drawn contains no such pattern ×1
the exhaustive search runs to between 5 and 14 numbers ×1
the extremal sequence has n² terms ×1
the panels drawn show different hull sizes ×1
the pattern is a progression or a sum ×1
the search reaches a length at which no colouring survives ×1
the search runs to between 5 and 14 numbers ×1
the sequence is built from between 2 and 5 blocks ×1
the view is one the family draws ×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.

Discrete

The sequence that cannot avoid a staircase

Any ten numbers in a row contain four that climb or four that fall. The proof gives every term a pair of counters, notices that no two terms can share a pair, and is finished — with a bound that is exactly right.

Discrete

Three in a row on the number line

Colour the numbers one to eight in two colours and it can be arranged that no three equally spaced numbers agree. Add the ninth and it cannot. The structure being forced is arithmetic rather than graphical, and the proof is a different proof.

The whole library · What the figures prove