Teaching Multidimensional Spaces and Non-Euclidean Geometry by Analogies: Limits in Conceiving and Explaining Ideas

The use of analogies to explain difficult subjects has always been reckoned as an expedient of great utility. Analogical reasoning from Y to X can lead to correct conclusions, but not all implications that can be correctly drawn from Y can be applied to X. In a learning context, the use of analogies without the corresponding warning about differences between the subjects will produce invalid knowledge and may represent a barrier to later learning of the correct concept. This paper analyses the use of analogies to further the understanding of non-Euclidian geometry and multidimensional space, addressing some famous comparisons that were suggested in the late 19th century. The present work discusses specific historical cases involving Charles Hinton, Johann Zöllner, Hermann von Helmholtz and Henri Poincaré and the use of analogies in explaining the new geometries, and analyses how the careful use of analogies can become a helpful means in the learning process.