2f:T410,# F-Stops F-Stops defy logic a bit. They're a staple of photographic math, and photographers tend to learn the numbers out of necessity, but if you've ever looked at the numbers and seen that ƒ/1.2 is one stop away from ƒ/0.9 and ƒ/1.8 or one third of a stop away from ƒ/1.1 and ƒ/1.4, something doesn't add up. Looking into the math, f-stops are very systematic; they follow one simple formula: ${tex`\sqrt{2^{AV}}`}, where AV is commonly known as "stops" above or below ƒ/1. The exact output of this, though, is something like ${data.map(d => d3.format(".3")(d.value)).slice(3,9).join(", ")}..., which are certainly not the numbers we're used to. Instead, they're a sort of analogy of the nice round numbers that get engraved on lenses. Round as they are, f-stops that we're accustomed to round not to the nearest tenth, not down, not up. Instead, they seem to round whichever way they please. The table below attempts to look at how each f-stop relates to the aperture value and its output.30:T6ed,

${data.map(d => { const [sign, whole, frac] = d.avLabel; return htl.html``})}

AV	${tex`\textbf{\textit{f(x)}}`}	Value		F-Stop	Delta
${sign}${whole === "0" && !!frac ? "" : whole}${frac}	${tex`\sqrt{2^{${sign === "-" ? sign : ""}${whole === "0" && !!frac ? "" : whole}${frac ?? ''}}}`}	${d.value.toFixed(2)}	${Math.abs(d.av) % 1 === 0 ? "—" : "-"}	ƒ/${d.label}	${d3.format("+.3f")(d.delta)}

31:T5c4,Plot.plot({ marginBottom: 32, height: 800, width: 480, color: { scheme: "Reds" }, x: { inset: 24, ticks: 12, grid: true, domain: [-0.7, 0.7], label: "← Delta between f-stop and common label →", labelAnchor: "center", tickFormat: (d) => (d === 0 ? "0" : d3.format("+")(d)) }, y: { reverse: true, grid: true, ticks: data.map((d) => d.av), tickFormat: (d) => data.find((e) => e.av === d)?.avLabel.join(""), insetBottom: 24, label: "Aperture Value" }, marks: [ Plot.ruleX([0]), Plot.arrow(data, { y1: "av", x1: (d) => 0, y2: "av", x2: "delta", stroke: (d) => d3 .scaleLinear() .domain([0, d.decimalPlaces]) .range(["#aaa", "#ff0000"]) .clamp(true)(Math.abs(d.delta)) }), Plot.dot(data, { y: "av", x: 0, r: 2, stroke: (d) => d3 .scaleLinear() .domain([0, d.decimalPlaces]) .range(["#aaa", "#ff0000"]) .clamp(true)(Math.abs(d.delta)), strokeWidth: 2, fill: "black" }), Plot.text(data, { text: (d) => `ƒ/${d.label}`, x: "delta", y: "av", textAnchor: "start", opacity: (d) => d.delta >= 0, dx: 4 }), Plot.text(data, { text: (d) => `ƒ/${d.label}`, x: "delta", y: "av", textAnchor: "end", opacity: (d) => d.delta < 0, dx: -4 }) ] })32:T744,data = { const roundToNearest = (v, nearest) => Math.floor(v * 10 ** nearest) / 10 ** nearest; const epsilon = 0.01; return Array(43) .fill() .map((el, i) => i / 3 - 1) .map((v, i) => { const av = v; let avLabel = [ Math.sign(av) === 1 || av === 0 ? "+" : "-", Math.floor(Math.abs(av)).toString() ]; const isOneThird = Math.abs((Math.abs(av) % 1) - 1 / 3) < epsilon; const isTwoThird = Math.abs((Math.abs(av) % 1) - 2 / 3) < epsilon; if ((isOneThird || isTwoThird) && avLabel[1] === 0) { avLabel = []; } if (isOneThird) { avLabel.push("⅓"); } if (isTwoThird) { avLabel.push("⅔"); } const fixed = v > 6 ? 0 : 1; const actualValue = Math.sqrt(Math.pow(2, v)); const isRoundedDown = roundedDownAVs.some( (d) => Math.abs(d - v) < 0.0001 ); const label = isRoundedDown ? roundToNearest(actualValue, fixed) : parseFloat(actualValue.toFixed(fixed)); let rounding, roundingSymbol; if (actualValue === label) { rounding = "equal"; roundingSymbol = "="; } else if (isRoundedDown) { rounding = "down"; roundingSymbol = "<"; } else { if (label < actualValue) { rounding = "nearest-down"; roundingSymbol = "≈"; } else { rounding = "nearest-up"; roundingSymbol = "≈"; } } const delta = label - actualValue; return { i, av, avLabel, value: actualValue, reference: reference[i], decimalPlaces: 10 ** (-1 * (label.toString().split(".")[1]?.length || 0)), label, rounding, delta, roundingSymbol, testPassing: reference[i] === label }; }); }33:[] 2d:{"title":"F-Stops","cells":[{"id":0,"value":"$2f","pinned":false,"hidden":false,"mode":"md"},{"id":203,"value":"$30","pinned":false,"hidden":false,"mode":"html"},{"id":539,"value":"$31","pinned":false,"hidden":false,"mode":"ojs"},{"id":535,"value":"### Appendix","pinned":false,"hidden":false,"mode":"md"},{"id":642,"value":"data.map((d) => d.reference).join(\"\\n\")","pinned":true,"hidden":false,"mode":"ojs"},{"id":312,"value":"roundedDownAVs = [2 / 3, 3 + 2 / 3, 5, 9, 12 + 2 / 3, 13]","pinned":false,"hidden":false,"mode":"ojs"},{"id":29,"value":"reference = [\n 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.4, 1.6, 1.8, 2, 2.2, 2.5, 2.8, 3.2, 3.5, 4,\n 4.5, 5.0, 5.6, 6.3, 7.1, 8, 9, 10, 11, 13, 14, 16, 18, 20, 22, 25, 29, 32, 36,\n 40, 45, 51, 57, 64, 72, 80, 90\n]","pinned":false,"hidden":false,"mode":"ojs"},{"id":7,"value":"$32","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W33","stdlib":"1","titling":"auto"}