Archimedes’ Screw

In Dresden’s “Old Masters” collection is the famous painting by the Italian painter Domenico Fetti (1588–1623) showing one of the most important mathematicians of antiquity, Archimedes of Syracuse (c. 287 BC-212 BC). Even during the development of the higher calculus in the 16. and in the 17th century, mathematicians relied on the preliminary work of Archimedes.

According to legend, after the conquest of Syracuse by the Romans in the Second Punic War (218–201 BC), Archimedes shouted to the Roman soldier who was about to arrest him as he drew geometric figures in the sand: Noli turbare circulos meos (Do not disturb my circles!), whereupon the latter, enraged, is said to have slain the great scholar with a sword.

A number of mathematical and physical discoveries can be traced back to Archimedes. He described the buoyancy of bodies in liquids and gases, which was later called Archimedes’ principle. He calculated the circular number $\pi$ approximately by means of a 6144 corner (see also the exhibits “What is Pi?”“Proof without Words: The Circular Area“, “My Birthday in Pi“), formulated the law of leverage and designed the “Archimedes’ Grapple” for destroying enemy ships, and he developed the screw or worm pump known as the Archimedean Screw for irrigating fields and arable land. Its crucial component is a helical element called the auger, which can rotate around its central axis and lift water from a low level to a higher level.

In the Maths Adventure Land, there is an Archimedes screw in the εpsilon, the “Adventure Land for Little Ones“, to lift the balls from the lower to the upper level (in the plenulum).

And now … the mathematics:

The Archimedean screw is a special regular screw surface. This is created in the following way by a so-called screwing around an axis, i.e. every point $P_0=(x_0,y_0,z_0)$ changes into a point $P=(x,y,z)$ by a rotation and a shift. If one uses the positively oriented $z$ -axis in a rectangular coordinate system (“Cartesian coordinate system“) as oriented screw axis $a$ and $\varphi$ are the oriented angle of rotation as well as $l$ the oriented sliding length, then with $p=l/2$ :

$\begin{align*}\begin{pmatrix} x\ y\ z\end{pmatrix} =\begin{pmatrix} x_0\cos(\varphi)-y_0\sin(\varphi)\ x_0\sin(\varphi)+y_0\cos(\varphi)\ z_0+p\varphi\end{pmatrix} \quad(1).\end{align*}$

That is, the three equations in (1) describe the rotation by the angle $\varphi$ (around the $z$ -axis) and a shift (in the direction of the $z$ -axis) by the distance $p\cdot\varphi$ .

A screw surface is now created by subjecting not only a point $P$ to a rotation and a shift, but a spatial curve $e$ given by a parameter representation of the following form:

$\begin {pmatrix} x_0\ y_0\ z_0\end {pmatrix} =\begin{pmatrix} x_0(t)\ y_0(t)\ z_0(t)\end {pmatrix} .$

The parameter representation of the resulting screw surface is then

$\begin{align*}\begin {pmatrix} x\ y\ z\end {pmatrix} =\begin {pmatrix} x_0(t)\cos(\varphi)- y_0(t)\sin(\varphi)\ x_0(t)\sin(\varphi)+y_0(t)\cos(\varphi)\ z_0(t)+p\varphi\end {pmatrix} \quad(1) \end{align*}$

with the real parameters $t$ and $\varphi$ .

The Archimedean screw results as a special case of a regular helical surface if one chooses as space curve $e$ the positive real axis, i.e.

$\begin{align*}\begin {pmatrix} x_0\ y_0\ z_0\end {pmatrix} =\begin {pmatrix} t\ 0\ 0\end {pmatrix} \end{align*}$

( $t>0$ ). It has the following parameter representation:

$\begin{align*}\begin {pmatrix} x\ y\ z\end {pmatrix} =\begin {pmatrix} t\cos(\varphi)\ t\sin(\varphi)\ p\varphi\end{pmatrix}.\end{align*}$

( $0\lt t\lt T$ , $0\lt\varphi\lt 2k\pi$ ). Here $T$ is the length of the generating straight line $l$ and $k$ is the number of rotations of $e$ around the $z$ axis (see figure 3 below).

Literature

[1] Klix, W.–D.: Konstruktive Geometrie, Leipzig, 2001.

[2] Schneider, I.: Archimedes. Ingenieur, Naturwissenschaftler und Mathematiker, Darmstadt, 1979.

[3] Wünsch, V.: Differentialgeometrie, Kurven und Flächen, Leipzig, 1997.