The ease with which a nut is screwed onto a bolt suggests that the thread is uniform along the entire length of the bolt, and the math­e­mat­ical essence of threaded joints is the usage of a curve that can slide on its own. This remark­able curve is called a helix.

The helix can be made by winding a rectan­gular trans­parent sheet with a marked diag­onal on a cylinder. Depending on the length of the sheet and, conse­quently, the angle of the drawn line, the pitch of the helix and the number of turns will vary.

Formally, the helix (cylin­drical) is a line defined by a point which rotates with constant angular velocity around a fixed axis and simul­ta­ne­ously moves along this axis with a constant velocity.

The visual repre­sen­ta­tion and defi­n­i­tion are combined in the para­metric defi­n­i­tion of the helix in a rectan­gular Carte­sian coor­di­nate system: $$ x=a \cos t,\quad y=a \sin t,\quad z=ht. $$ The first two equa­tions show that the projec­tion of a point runs along the base of a straight circular cylinder of radius $a$. The third equa­tion defines the motion along the axis of the cylinder at a constant velocity.

“Good” curves have two basic prop­er­ties in three-dimen­sional space — curva­ture and torsion.

Curva­ture is the speed at which the line curves on a plane and is defined by the radius of the circle whose arc best approx­i­mates the small segment of the curve containing the point in ques­tion. Torsion — the speed at which the curve tends not to be flat, how hard the curve “wants” to leave the plane.

Remark­ably, for fairly smooth curves, curva­ture and torsion completely define its shape.

For the helix, curva­ture and torsion are constant, and the above state­ment implies that only helices do have such a prop­erty!

The constancy of curva­ture and torsion at all points means that the struc­ture of the helix is uniform over all points. As a conse­quence, it follows that a segment of a helix can slide along it in the same way as a segment can slide along a straight line and an arc of a circle can slide along its circum­fer­ence. (A straight line and a circle can be consid­ered as degen­erate, extreme cases of the helix).

Threaded joints, a partic­u­larly bolt or screw, are based on a helix. When screwed in, the thread slides as if on a track.

The helix is the only curve that can slide on its own. And in engi­neering prob­lems, where the pres­ence of such a prop­erty is desir­able or even neces­sary, it is impos­sible to do without helices.

The helix is also the boundary of spiral stair­cases. Climbing up, you are by defi­n­i­tion moving upwards at a constant speed. Both a corkscrew and a fishing auger have the shape of a spiral line, sliding in the mate­rial along the path already taken.

Other etudes in “Wonderful curves”