Shapes

The Five Shapes, Declared Once

Five arrangements of the working day, written down before any of the counting and not adjusted since. The only invented thing in this record.

A working day has no natural shape. Somebody decided where it starts and where it stops, and the decision varies by trade, by household and by habit. So this record declares five shapes outright, as conventions and nothing more, and then computes.

Shape	Hours	Length
early	8 to 16	8
standard	9 to 17	8
late	10 to 18	8
long	9 to 18	9
split	9 to 13 and 15 to 19	8

That table is the whole of the invention. Nothing else on this site was chosen; everything else follows.

Before adopting privacy-conscious work monitoring, managers should document why each field exists, who can see it and how an employee can correct an inaccurate record. A useful external reference for the surrounding topic is Calendly.

What is not being claimed

Nothing here says any of these shapes is common, typical, recommended or representative. Nobody was surveyed. The five are not an average of anything and they are not ranked.

They were chosen to differ from one another in ways that change the answer, which is the only property that matters for the purpose:

Three that differ only in when they start. Early, standard and late are the same eight-hour block slid along the clock. Against each other they behave exactly as the one law predicts, with their built-in offset added to whatever gap is imposed on top.
One that is longer. The long shape runs nine hours instead of eight, which is the smallest change that breaks the equal-length assumption.
One that is broken. The split shape works four hours, stops for two, and works four more. It is the only shape in the record that is not a single stretch, and almost everything surprising here comes from it.

Five was chosen because it is the smallest number that covers those three cases with something to spare. A sixth would have had to earn its place by behaving unlike all five, and nothing obvious does.

Why the list has not moved

The shapes were written down before any of the counting, and they have not been adjusted since. That matters more than it sounds.

A convention revised after the sums are in is no longer a convention. If the long day had been stretched to ten hours because nine produced an inconvenient table, or the split shortened because its results were awkward, the record would be reporting a choice dressed as a finding. Keeping the list fixed is the only thing that stops that, and it costs nothing except the occasional ugly number.

There are ugly numbers in what follows. They are all left in.

What each shape is for

The five are not interchangeable samples. Each one is in the record to test a specific thing, and knowing which makes the rest of the section easier to follow.

standard is the reference. Nine to five, eight hours, one block. Every figure in the first part of the record uses it at both ends, and it is the shape against which the others are measured.

early and late exist to test whether position matters independently of length. They are the same block as standard, moved one hour either way. Against each other they establish that a difference of habit behaves exactly like a difference of clocks, which is the most practically useful result in this section.

long exists to break the equal-length assumption by the smallest possible amount. One extra hour, at one end of the day. If the record had chosen a twelve-hour day instead, the departure from the law would have been dramatic and the mechanism would have been harder to see.

split exists to break the single-block assumption. It is the only shape here that is not one continuous stretch, and it generates every result in this record that is not a straight line — including a sequence that goes up as the gap widens, which nothing else does.

The shapes that are not here

Three obvious candidates were left out, and the reasons are worth stating so the omissions do not look like oversights.

A short day. A six-hour day would behave exactly like an eight-hour day with a smaller number in the law: shared time equals six minus the gap. It introduces no new mechanism, only a different constant, and the section on the one-hour rule already shows what varying the length does.

A day with a one-hour break. This is a split day with a smaller hole. It would produce broken overlaps in the same way the split shape does, in fewer arrangements. The split shape with its two-hour break was chosen instead because it makes the behaviour visible rather than marginal.

Different days of the week. An arrangement where one end works different hours on different days is a real thing and is outside this record entirely. It would require a week-long accounting rather than a day-long one, and nothing in the daily arithmetic would survive the move intact.

Five shapes is not a claim that there are five. It is the smallest set that produces every behaviour the record needs to describe, and a sixth would have to earn its place by doing something none of these five does.

How the five get combined

Each of the five can appear at either end, which gives twenty-five pairs of shapes. Each pair is then evaluated at every whole-hour gap from twelve behind to twelve ahead, which gives six hundred and twenty-five arrangements in total. All of them were computed. A few dozen are drawn.

The pages that follow work through what the five do to each other, starting with the three that are secretly the same shape.