Turn to Sextant

Tau (τ = 2π ≈ 6.2832) represents one full turn. Its advocates argue that using τ instead of 2π makes formulas cleaner: a quarter-circle is τ/4 instead of π/2, circumference is τr instead of 2πr, and Euler's identity becomes e^(iτ) = 1 (arguably more elegant than e^(iπ) = −1). The Tau Manifesto, published in 2010 by Michael Hartl, sparked a genuine mathematical subculture. Tau Day is June 28 (6.28). The argument has merit but π is so deeply entrenched that adoption remains niche.

Some game engines and shader languages let you specify rotations in turns (0 to 1) rather than degrees (0 to 360) or radians (0 to 2π). Turns map naturally to normalized values — a progress bar from 0.0 to 1.0 directly represents angle completion. The GLSL function fract() wraps any number to the 0–1 range, making turn-based angle arithmetic trivially simple for procedural animations, circular gradients, and clock-face layouts.

Thread pitch is the axial distance a bolt or pipe fitting advances per complete turn. A standard ½-inch NPT pipe thread has 14 threads per inch, so one turn advances it about 1.8 mm into the fitting. Plumbers specify "finger tight plus 2–3 turns" because torque wrenches are impractical in cramped spaces. Spark plug manufacturers use the same approach — "hand tight plus X turns" achieves correct seating force without needing a torque wrench in the field.

Two metrics exist: HTM (half-turn metric) counts any face rotation — 90° or 180° — as one move, while QTM (quarter-turn metric) counts each 90° as one move and each 180° as two. "God's Number" — the maximum moves needed to solve any scramble — is 20 in HTM and 26 in QTM. Computer solvers like Kociemba's algorithm are tuned to HTM because it produces shorter sequences. Human world-record holders now solve random scrambles in under 4 seconds.

Most padlock-style combination locks require you to turn the dial about 3.5 to 4.5 full turns during the opening sequence — multiple full clockwise turns to clear the mechanism, then reverse to the first number, forward to the second, and back to the third. This multi-turn protocol isn't about security (the number of combinations handles that); it's about mechanically engaging and disengaging the internal disc cams in the correct sequence.

A sextant uses two mirrors to superimpose the image of a celestial body onto the horizon. The navigator looks through the eyepiece and sees the horizon directly through a half-silvered mirror, while a second mirror on a movable arm reflects the Sun or star down into the same field of view. You swing the arm until the star appears to sit exactly on the horizon, then read the angle off the graduated arc. The double-reflection design means the arc only needs to span 60° (one sextant) to measure angles up to 120°.

The name refers to the arc of the instrument, not its measurement range. A sextant's arc is one-sixth of a circle (60°), but thanks to the double-reflection principle — where the angle of reflection doubles the arc angle — it can actually measure angles up to 120°. Similarly, an octant (one-eighth of a circle, 45° arc) measures up to 90°. The naming convention describes the physical shape of the tool, not its capability.

Yes, and navies worldwide still require it. The US Naval Academy reintroduced mandatory celestial navigation in 2015 after a decade-long hiatus, citing concerns about GPS vulnerability to jamming, spoofing, and satellite failure. A skilled celestial navigator with a sextant, an accurate clock, and a nautical almanac can determine position to within about 1–2 nautical miles — good enough to make port safely. Several solo round-the-world sailors carry sextants as backup specifically because they have no electronics to fail.

The sextant itself couldn't solve longitude — that required an accurate clock (John Harrison's marine chronometer, completed in 1761). But the sextant was the other half of the solution. A navigator used it to measure the Sun's altitude at local noon to find the exact time of solar noon at their position. Comparing this to Greenwich time on the chronometer gave the time difference, and since Earth rotates 15° per hour, that time difference directly yielded longitude. Sextant + chronometer = position anywhere on Earth.

Sixty degrees is the interior angle of an equilateral triangle — the simplest regular polygon after the square. Honeycomb cells are hexagons (six 120° angles, each the supplement of 60°) because hexagonal packing is the most efficient way to tile a plane. Carbon atoms in graphene and diamond form 60° and 109.5° angles respectively. The 60° angle appears everywhere in nature because it's the geometric consequence of close-packing equal-sized spheres or circles.