A staircase
Slope
Some staircases feel easy, others leave you breathless. The difference isn't the height. So what is it?
Representation layers
Eight steps, one floor up. Every step is identical to the last: same height, same depth. That repetition isn't an accident — it's exactly what makes a staircase safe to climb.
Switch layers — the scene stays put
What's really going on
Slope isn't a number, it's a ratio — and being a ratio changes everything. Measure it at the first step or the last, across two steps or six, and you get the same answer: 0.58. That constancy is the definition of a straight line. A curve is something whose slope changes from point to point; a line is something whose slope doesn't. It's also why building codes regulate stair ratios — after one step your foot learns where the next one lands and never has to think again.
Analytic form
m = Δy / Δx
Divide the vertical change between two points by the horizontal change. If the answer is the same everywhere, what you have is a straight line.
Same concept, elsewhere
Memorising one example gets you nowhere. You need to recognise the concept independent of its context.
A wheelchair ramp — codes cap the slope at 1/12, which is a statement about ratio, not about law
A road sign reading '8% grade': you drop 8 metres for every 100 you travel
Roof pitch — the minimum ratio that makes rain run off instead of pooling
The rate of a price rise on a chart: vertical change ÷ elapsed time