The history of the fourth dimension: A way of engaging pupils in secondary classrooms Snezana Lawrence To cite this version: Snezana Lawrence. The history of the fourth dimension: A way of engaging pupils in secondary classrooms. CERME 9 - Ninth Congress of the European Society for Research in Mathematics Educa- tion, Charles University in Prague, Faculty of Education; ERME, Feb 2015, Prague, Czech Republic. pp.1846-1852. hal-01288395 HAL Id: hal-01288395 https://hal.archives-ouvertes.fr/hal-01288395 Submitted on 15 Mar 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. The history of the fourth dimension: A way of engaging pupils in secondary classrooms Snezana Lawrence Bath Spa University, Faculty of Education, Bath, UK, [email protected] The National Curriculum in England has, over basis of the problem.3 The organizations such as the past decade, been revised a multitude of times. National Strategies (discontinued 2010) previously Disengagement of pupils was one of the reasons for revi- tried to help teachers devise teaching episodes and sions. In September of 2014, a new curriculum in math- gave suggestions on pedagogy. The new curriculum ematics was introduced, aiming to give greater freedom instead offers an element of autonomy, meaning that to teachers and schools to construct a curriculum and schools and teachers are given freedom to choose and teaching episodes that are engaging and appropriate design the topics and teaching episodes appropriate for their students. This paper investigates how such ep- to their environments. Likewise, the curriculum it- isodes can be constructed through the investigations self lists the skills and knowledge to develop in pupils, based on historical development of mathematical con- but gives no (or minimal) guidance as to the choice cepts and how they could easily link to the new curricu- of topics. lum, offering at the same time greater opportunities for pupils’ engagement. The history of the fourth dimension The choice of topic described in this paper – the his- is one such possible topic, and the paper suggests a way tory of the fourth dimension – arose from two expe- of using it in a secondary classroom. riences: of working with gifted and talented pupils some years ago on the representations of the fourth Keywords: Fourth dimension, engagement, Schläfli, dimension in mathematics (Lawrence, 2012) and the Stringham, Flatland. engagement reported in alternative curricula, such as, for example, Steiner system, which introduces projective geometry and the study of the fourth di- INTRODUCTION mension in the final years of secondary school (Woods et al., 2013). In the decade between 2004 and 2014, mathematics curriculum changed twice, about fifty reports on The paper thus first gives a historical overview of the mathematics education in the country have been topic, and then investigates the possible pedagogi- published, and seven different Secretaries of State cal approaches to develop material based on certain for Education passed through the British Parliament. principles listed. It concludes by showing how an Each of the changes and reports suggested that the unorthodox topic such as this, can nevertheless be state of mathematics education in Britain is trouble- easily linked to the new curriculum and the skills and some; the causes were identified, the evidence was giv- knowledge it aims to develop in pupils. The engage- en (in either a narrative or analytical format), and of ment is expected but not yet empirically proven; this course the suggestions to improve the situation were is proposed as a possible future study, and some ini- recommended.1 The most troublesome of all troubles tial experiments from schools in which the teaching listed was the perceived irrelevance of mathematics pedagogy is trialed are described. and the lack of desire to engage with it.2 HISTORY OF THE FOURTH DIMENSION When exploring the issue of disengagement, teach- ers reported that it was the curriculum that narrows Whilst the concept of the fourth dimension was de- down the topics and the lack of choice to engage with veloped in the nineteenth century, the origins of it different topics from the curriculum that was at the could be traced as far back as the antiquity in the CERME9 (2015) – TWG12 1846 The history of the fourth dimension: A way of engaging pupils in secondary classrooms (Snezana Lawrence) most possibly wide sense of conceptual development. Friedrich, 1834–1882) further simplified this. If one Aristotle for example, discussed it in De Caelo,4 and has a circle and a point outside of it, how can one get Ptolemy denied and disproved it but nevertheless the point into the circle without cutting or crossing mentioned and contemplated upon it (Cajori, 1926, over the circumference? p. 397; Heiberg, 1893, p. 7a, 33). John Wallis, although writing this whilst considering geometric interpre- In 1852, Ludwig Schläfli (1814–1895), a Swiss math- tations of quantities he was developing in the context ematician published a book Theorie der vielfachen of algebra, wrote (Wallis 1685, p. 126): Kontinuität, (Theory of Continuous Manifolds), in which he wrote about the four dimensions. Schläfli A Line drawn into a Line shall make a Plane or looked at Elementa doctrinae solidorum published in Surface; this drawn into a Line, shall make a Solid: 1758, in which Euler described for the first time what But if this Solid be drawn into a Line, or this Plane was to become known as Euler’s characteristic, the into a Plane, what shall it make? A Plano-Plane? expression which conveys the information that in all That is a Monster in Nature, and less possible convex solid bodies the sum of the solid angles and the than a Chimaera or Centaure. For Length, Breadth number of faces is equal to the number of edges add 2. and Thickness, take up the whole of Space. Nor can our Fansie imagine how there should be a Fourth Local Dimension beyond these Three. With the French Revolution, some revolutionary mathematical thinking happened, and Lagrange in particular, spoke of three coordinates to describe the space of three dimensions, introducing time as the Figure 2: Euler’s characteristic first described in Elementa fourth, and denoting it t (Lagrange, 1797, p. 223). doctrinae solidorum, 119 The reader is reminded that this can by no means be We now usually denote Euler’s characteristic by an exhaustive study, but is a sketch of the history of Greek letter chi and describe it for convex polyhedra the fourth dimension and the narrative given is but χ = V − E + F = 2, where V is the number of vertices, E a thread that will later be examined in possible edu- is the number of edges, and F is the number of faces cational setting and application to teaching. in a polyhedron. If we further analyze the formula we notice that we begin from the first variable which Let us then trace further historical development. The counts points (point we earlier took to represent 0th first such opportunity was the example of Möbius dimension); the second variable which numbers the and Zöllner. Möbius (1827) first spoke about an ob- edges in a solid, (representing line, 1st dimension) and ject getting out of a dimension it belonged to in order the third variable, numbering the faces of a solid, (pol- to perform a spatial operation. If one had a crystal, ygon is bound part of a plane, representing the 2nd structured like a left-handed staircase, how would one dimension). get its three-dimensional reflection? Zöllner (Johann Schläfli (1852) showed that this formula is also valid in four dimensions or indeed any higher dimension. We will get there – but let us first look at how he first defined a system which would describe any regular P polytope in any dimension. There is only one polytope in 1st dimension, a line segment, and the Schläfli symbol denoting this is {}. Regular polygons in two dimensions are, for example, triangle {3}, square {4}, pentagon {5}, etc. Figure 1: Zöllner illustrates that in order to perform certain Remembering that he only used these symbols to de- operations in space, objects must exit their current dimension note regular polytopes, we continue. In three dimen- (Zöllner, 1878) sions the five regular polyhedra, Platonic solids, can 1847 The history of the fourth dimension: A way of engaging pupils in secondary classrooms (Snezana Lawrence) be described as {3, 3} – tetrahedron has three-sided The mathematical description of generating the polygons that meet three at each vertex; {4, 3} – cube fourth (and higher dimensions) was first given in an has four-sided polygons three of which meet at each elegant way by William Stringham (Stringham, 1880, vertex; {3, 4} – octahedron has three-sided polygons p. 1): four of which meet at each vertex; {3, 5} – icosahedron has three-sided polygons five of which meet at each A pencil of lines, diverging from a common ver- vertex; {5, 3} – dodecahedron has five-sided polygons, tex in n-dimensional space, forms the edges of three of which meet at each vertex (Schläfli, 1852). an n-fold (short for n-dimensional) angle. There must be at least n of them, for otherwise they would lie in a space of less than n dimensions. If there be just n of them, combined two and two they form 2-fold face boundaries; three and three, they form 3-fold trihedral boundaries, and so on.