§1The helix at work
Wrap an inclined plane around a cylinder and it becomes a helix; give the helix a cross-section and it becomes a screw thread — a machine that converts rotation into axial motion and torque into axial force.
Everything a thread does follows from that one geometric idea. Turn the screw one revolution and it advances by its lead, so a small torque acting through a long circumferential path becomes a large force acting through a short axial one — the mechanical advantage of the inclined plane, continuously applied. The whole fasteners section already used the result: the torque-and-tension page turned wrist-torque into tens of kilonewtons of preload through exactly this mechanism. What this section adds is the thread itself as a subject — its precise geometry (§2–3), the friction arithmetic that decides whether it holds or runs back (§4), the different cross-sections evolved for fastening, moving and sealing (§5), and, in the pages that follow, each system in detail and the ways threads are measured and made. One machine, many dialects; this page is the grammar.
Contents§2The anatomy
A thread is described by three diameters, one angle and one length — and once the form is fixed, a single dimension, the pitch, generates everything else.
The major diameter d is the largest, over the crests of an external thread — the number in the name, the 10 of M10. The minor diameter is the smallest, at the roots, and is what remains to carry tension. Between them lies the most important and least visible of the three: the pitch diameter d2, the diameter at which the thread ridge and the groove are exactly equal in width — the surface on which the flanks actually bear, the diameter every fit is toleranced on, and the one the measuring page goes to such lengths to reach. The flank angle — 60° included for the metric and Unified worlds — sets the shape of the vee, and the pitch p is the axial distance from one thread to the next. From p alone, the whole profile follows: the sharp-vee height is H = 0.866 p (the hero’s dashed construction), and the working crests and roots are that triangle truncated by fixed fractions of H — arithmetic the calculating-thread-dimensions page works through constant by constant. Name the form and the pitch, and the thread is fully determined; that determinism is what makes threads interchangeable at all.
Contents§3Pitch, lead and starts
Pitch is the spacing of the threads; lead is how far the screw advances per turn. On the ordinary single-start thread they are equal — and the distinction only bites, hard, when they are not.
An M10×1.5 has d2 = 10 − 0.6495 × 1.5 = 9.026 mm. Single-start, its lead equals its pitch, and the helix climbs at λ = atan(1.5 ÷ π × 9.026) = 3.03° — a gradient of about one in nineteen, gentler than a wheelchair ramp. Cut the same 1.5 mm profile as a two-start thread — two parallel ridges interleaved — and the pitch is unchanged but the lead doubles to 3.0 mm, the screw advances twice as far per turn, and λ steepens to 6.04°. Multi-start threads buy speed of advance without coarsening the tooth: camera focus rings, bottle caps that open in half a turn, and fast-acting valve spindles all use them. The price is paid in §4’s arithmetic — a steeper helix is a less self-locking, more back-drivable one — and in manufacture, since the starts must be indexed around the blank with perfect equality, which is precisely the problem the indexing page at the end of this section exists to solve.
§4Self-locking and efficiency
A thread holds its load without unwinding when its helix is shallower than its friction — and working the numbers shows every fastening thread sits deep inside that condition, at the deliberate cost of efficiency.
The vee shape wedges the nut onto the flanks, so the effective friction is the true coefficient divided by the cosine of the flank half-angle: at μ = 0.15 on a 60° thread, μ′ = 0.15 ÷ cos 30° = 0.173, giving a friction angle ρ′ = 9.83°. The M10’s lead angle of 3.03° lies far below it — the load’s attempt to unwind the thread is defeated by friction with a three-to-one angular margin, so an axially loaded M10 cannot back-drive, ever; vibration loosening (the machine screws page) works by momentarily relieving the friction, not by overcoming it. The same numbers price the security: as a torque-to-force machine, η = tan 3.03° ÷ tan(3.03° + 9.83°) = 23.2% — roughly three-quarters of every newton-metre applied to a bolt is spent on friction. Fig. 1 draws the whole curve: efficiency would peak near 71% at a 40° helix, but a fastener wants no part of that hill. It is engineered to be a bad machine, because a bad machine run forward is an excellent lock run backward — the single trade that separates the fastening threads of §5 from the power threads.
§5The families
Every thread form is a different answer to the same question — how much of the flank angle to trade away — and the family table below is the map of the whole section.
| Family | Included angle | Form | Built for |
|---|---|---|---|
| ISO metric | 60° | vee, flat crests, rounded root | fastening — the world default |
| Unified (UNC/UNF) | 60° | vee, inch pitches | fastening — North America and aerospace |
| Whitworth (BSW/BSF) | 55° | vee, fully radiused crest and root | the 1841 original; survives in BSP sealing |
| Acme / Tr | 29° / 30° | trapezoidal, depth p/2 | power transmission — lead screws, jacks |
| Buttress | 7° + 45° | asymmetric | heavy one-direction load — presses, vices |
| Square | 0° | square rib | the efficiency ideal — rarely cut today |
| Read the angle column as a dial. Wide flanks (60°) wedge, centre well, forgive sloppy fits and lock hard — fastening virtues. Narrow flanks approach the square thread’s efficiency and low bursting force — power virtues — at the cost of the wedging that keeps a fastener tight. The Acme sits deliberately in between, and the buttress cheats by being square on one flank only. Each family has its own page ahead. | |||
§6Hands, history and the section ahead
Two conventions and one timeline complete the grammar: which way the helix winds, and how a century of chaos became two surviving systems.
A thread is right-hand unless marked otherwise — it advances when turned clockwise, the way intuition and every tool expects — and left-hand threads (designated LH) are reserved for places where rotation would unscrew a right-hand one: the left pedal of a bicycle, one end of a turnbuckle, grinder spindle nuts. The history is short and consequential. Before 1841 every workshop cut its own threads and nothing interchanged; Whitworth’s 55° standard was the first national system, Sellers’ 60° flat-crested form (1864) became the American answer, the wartime agony of incompatible British and American hardware produced the Unified inch compromise in 1948, and the ISO metric thread then swept the field to become the default for the planet — leaving the Unified system entrenched in North American and aerospace practice and Whitworth’s form alive chiefly inside pipe fittings. That is exactly the itinerary of the coming pages: the Unified system, the arithmetic that generates any thread’s dimensions, the metric system, then the power forms (Acme, buttress), the 55° legacy, pipe threads and the sealing problem, the miscellany of special forms, and finally how threads are measured and made — tapped, cut, rolled, ground, milled and indexed.
Contents§7Quick reference
The working core of the page on one card rack.
Anatomy
major · pitch d2 · minor
H = 0.866 p from p alone
Lead
L = n p · tan λ = L/(π d2)
M10: 3.03° · two-start 6.04°
Self-locking
λ < ρ′ (μ′ = μ/cos 30°)
3.03° ≪ 9.83° — cannot back-drive
Efficiency
M10 as a machine: 23.2%
a bad machine is a good lock
Families
60° · 55° · 29/30° · 7°/45°
the angle is the dial