Archimedes screw

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

Figure 1: Archimedes of Syracuse

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 wanted 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 approximately calculated the circular number \pi 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 the lever and designed the “Grapple of Archimedes” for the destruction of enemy ships, and he developed the screw or worm pump known as the Archimedean screw for the irrigation of fields and fields. Its crucial component is a helical element called the screw, which can rotate about its central axis and lift water from a low level to a higher level.

Figure 2: The screw pump according to Archimedes

In the MATHEMATICS ADVENTURE LAND there is an Archimedean screw in the εpsilon, the “ADVENTURE LAND for little ones”, to lift (in the plenulum) the spheres from the lower to the upper level.


And now … the mathematics of it:

The Archimedean screw is a special regular screw surface. It 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 if \varphi are the oriented rotation angle and l the oriented shift length, then with p=l/2:

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

That is, the three equations in (1) describe rotation by the angle \varphi (about 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

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

with 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.

    \[\begin{pmatrix} x_0\ y_0\ z_0\end{pmatrix}=\begin{pmatrix} t\ 0\ 0\end{pmatrix}\]

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

    \[\begin{pmatrix} x\ y\ z\end{pmatrix}=\begin{pmatrix} t\cos(\varphi)\ t\sin(\varphi)\ p\varphi\end{pmatrix}.\]

(0\lt t\lt T, 0\lt\varphi\lt 2k\pi). Where 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).

Figure 3: The Archimedean helical surface

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.