18 November 2020

Ellipsograph of Archimedes as a simple LEGO construction

A joint work with Sara Jodlbauer

School curriculum related to elementary geometry usually focuses on "compass and ruler" constructions where the traditional Euclidean construction steps are allowed to generate a figure. While it is possible to construct any point of an ellipse that is given by its parameters, school curriculum seldom covers any methods how to efficiently draw an arc or the full ellipse by home-made mechanical constructions.

To draw an ellipse on your own, several options exist. One of them, the gardener method is mentioned in many textbooks, and it is based on the locus definition, namely:

Given two fixed points \(F_{1}\), \(F_{2}\) called the foci and a distance \(2a\) which is greater than the distance between the foci, the ellipse is the set of points \(P\) such that the sum of the distances \(|PF_{1}|\), \(|PF_{2}|\) is equal to \(2a\): \[E=\{P\in \mathbb {R} ^{2}:\,|PF_{2}|+|PF_{1}|=2a\}.\]

By using this definition, the gardener puts down two stakes and loops a piece of rope around them. Using a stick, he pulls the loop taut and marks the points around a curve.

In this article we consider a different method, namely the ellipsograph (or trammel) of Archimedes that consists of two shuttles which are confined to perpendicular channels or rails and a rod which is attached to the shuttles by pivots at fixed positions along the rod.

We remark that such constructions are well-known and very popular as commercial toys. Also as LEGO constructions there are several options. A recent YouTube video contributed by Brixe63 shows a complicated mechanism that is definitely based on the same idea, but consists of several LEGO bricks. Another approach (among several others) can be found on Rebrickable, contributed by JKBrickWorks, it uses 47 parts. In our contribution we focus on minimizing the number of parts, and, at the same time, we focus on drawing the ellipse, not just the building up the motion.

The figure above shows how the ellipsograph is built. Easy sliding is assured by using two flat tiles for the shuttles. Their collision-free arrival in the channels is solved by using \(1\times4\) flat tiles. However, their somewhat long size has some drawback, namely, that the shuttles require a bigger distance to each other.

We use 24 LEGO parts to build the construction and a G2 type pen refill. We highlight that this construction well harmonizes with a set of other LEGO bricks, described at a GitHub page; it allows the students to draw a high variety of other algebraic curves. On the other hand, static balancing of our contribution is an important part: a counterweight is required to ensure stable motion.

Our contribution can draw almost 100% of a quite big ellipse that fits on an A4 sheet of paper. A very small part of the arc cannot be drawn because the shuttles get too close in that part of the movement. As future work we address solving this problem, too.

Finally, we point the reader to LEGO's official computer aided design tool, LEGO Digital Designer. We used this program to visualize our concept in a digital way. The LXF file that contains our work is available here. Also the building instructions can be checked out.

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See also a filtered list of the entries on topics GeoGebra, technical developments or internal references in the Bible.

Zoltán Kovács
Linzer Zentrum für Mathematik Didaktik
Johannes Kepler Universität
Altenberger Strasse 54
A-4040 Linz