Dyadic rational — where it appears
Named by 4 essays across one field — each of them below, with the objects they name alongside it.
Seven is not a power of two
Halving a receding depth by diagonals is exact, and every book gives it. Halving reaches a half, a quarter, three eighths — and never a third, however many times it is spent, because no power of two is divisible by three. There is a construction that reaches every whole fraction, it costs three lines rather than a stack of quadrangles, and the extra ingredient is not a measurement.
The circle in the square wants a number
Every manual draws a circle in perspective by inscribing it in a square, crossing the diagonals, and marking four more points about seven tenths of the way out. Seven tenths is right — along the diagonal of the real square. Along the drawn one it is 54 mm off a circle two and a half metres across, and the drawn diagonal is the only diagonal on the paper.
The bays that are not equal
The diagonal repeats a bay exactly and forever, and what it reaches is the whole multiples of that bay and nothing else. An arcade of 1:1.5:2 is 250 mm short of its boundary with no halving, exact after one, and an arcade of 1:1.333:2 is out of reach at every depth whatever. A measuring point lays out all three to 6e-14 px.
What a straightedge reaches on a receding line
Three marks on a receding line fix a coordinate, and everything a straightedge builds from them is a ratio of whole numbers. Three complete quadrangles put 1655 marks on the unit interval with no gap above 0.0038, a third costs two quadrangles where repeated halving never reaches one at all, and the same net built on a line that does not recede arrives at the identical 30 coordinates.
Named alongside it
The objects these essays reach for when they reach for this one.
Cross-ratioHarmonic conjugateStraightedge constructionComplete quadrangleVanishing pointAffine structureHarmonic netIncidenceConditioningConicFixed pointForeshortening