= Thread start
A \b[thread start] is one continuous helical ridge of a <screw thread>. A single-start thread has one such ridge; a multiple-start thread has several interleaved ridges. The number of starts controls how <screw lead> and <thread pitch> differ:
$$
L=np,
$$
where $n$ is the number of starts, $p$ the axial spacing between adjacent thread crests and $L$ the advance of one ridge in a complete turn.<Selecting the Optimal Screw Technology by Tolomatic>[Screw Terminology, printed pp5–6, single/double-start diagrams and lead versus pitch]{ref}
An ideal cylindrical construction explains the relation. At reference radius $R$, represent the $j$th start by the <helix>
$$
\mathbf r_j(\theta)=
\left(R\cos\theta,\ R\sin\theta,\ \frac{L\theta}{2\pi}+jp\right),
\qquad j=0,\ldots,n-1.
$$
At one fixed angular position, successive complete turns of the $j$th start occur at axial offsets $kL+jp$. When $L=np$, these offsets are $p(nk+j)$. As $k$ ranges over the integers and $j$ over the $n$ starts, they give every integer multiple of $p$. Adjacent crests can therefore be spaced by $p$ even though following one continuous ridge for a turn advances by $np$.
For example, a three-start thread with $p=2\,\mathrm{mm}$ has $L=6\,\mathrm{mm}$. At a chosen angular position, crests at axial coordinates $0,2,4,6\,\mathrm{mm}$ do not identify four starts: the crests at 0 and $6\,\mathrm{mm}$ can be successive turns of the same start. Following the three ridges gives advances $0\to6$, $2\to8$ and $4\to10\,\mathrm{mm}$ over one turn. The unwrapped geometry in <screw thread> displays this construction.
With the screw restrained axially and its matching <nut (hardware)> guided against rotation, a quarter-turn in this example advances the nut by $L/4=1.5\,\mathrm{mm}$. A single-start thread with the same pitch advances only $0.5\,\mathrm{mm}$. Counting starts is therefore essential when predicting a <lead screw>'s motion from its pitch.
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