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Two Approaches to Geometrography Journal for Geometry and Graphics Volume 14 (2010), No. 1, 15–28. Two Approaches to Geometrography Jorma K. Merikoski1, Timo Tossavainen2 1Department of Mathematics and Statistics, University of Tampere FI-33014 University of Tampere, Finland email: jorma.merikoski@uta.fi 2School of Applied Educational Science and Teacher Education, Univ. of Eastern Finland P.O. Box 86, FI-57101 Savonlinna, Finland email: timo.tossavainen@uef.fi Abstract. Geometrography studies the complexity of ruler-and-compass con- structions. According to our knowledge, so far the only remarkable work in this field is by E.´ Lemoine. We survey his method and present another method based on statistical modelling. We compare these methods by studying the complexity of certain constructions of a perpendicular to a line and a regular pentagon. A purpose of this paper is to show that geometrography has potential to enrich geometry and graphics education in school. The latter, statistical, method also provides tools for a more advanced analysis of error propagation through geomet- ric transformation. Key Words: geometric constructions, complexity, statistical modelling MSC 2010: 51M15 1. Introduction Ruler-and-compass constructions, which only less than a century ago constituted a core of mathematical education, are today almost fully ignored in teaching geometry. Indeed, nowa- days even in upper secondary school, geometry education merely builds on students’ intuitive and informal knowledge of their environment and space rather than on strict deductive rea- soning. The focus is preferably on applications than on the logical structure of geometric knowledge. Also most ICT based dynamic geometry environments are designed to support teaching and learning problem solving skills related to, for example, measuring, rather than general deductive thinking. For a more detailed discussion, see, e.g., [15]. As an example of this global and rapid development, we mention that in Finland, the PISA winning country, only a few decades ago upper secondary school’s matriculation examination regularly contained ruler-and-compass constructions. For example, in 1951 students were asked to draw a circle sector whose arc is 72◦ and area is equal to the area of a given circle. ISSN 1433-8157/$ 2.50 c 2010 Heldermann Verlag 16 J.K. Merikoski, T. Tossavainen: Two Approaches to Geometrography Today Finnish students are introduced to, if any, only a few trivial constructions like angle bisection etc. There are several obvious reasons why ruler-and-compass constructions do not deserve so much emphasis in mathematics education today. On the other hand, there are also grounds why these constructions, in our opinion, warrant a little more attention than they nowadays are paid to. For example, studying the three classical unsolvable Greek construction problems still invokes even amateurs’ interest toward mathematics in general. Also the value of these constructions to the fine arts is timelessly consistent. There are, however, also more mod- ern and concrete possibilities to increase the relevance of ruler-and-compass constructions to mathematics education — and eventually for all mathematical sciences. For instance, computer scientists have already noticed that [2, p. 85] “. the algorithmic problems occurring in geometric reasoning have also an enormous scientific appeal [to com- puter science]. In the past few years, geometric reasoning problems have provoked a whole spectrum of new algorithmic techniques.” In other words, any suitable approach to study, for example, the complexity of algorithms in the context of geometry would arguably enrich mathematics teaching in school. Further, in school, statistics is often introduced only as an application of mathematics and the fact that both of these autonomous disciplines promote each other is not usually noticed. We believe that any relevant way to diversify students’ knowledge in statistics is welcome, too. Last but not least, the consistent problem of math- ematics education in universities is related to the training of prospective researchers: how to find reasonable ways to introduce students with the process of doing research in mathematics already at the level of basic studies? As some recent articles, e.g., [6] and [17] verify, it is still possible to achieve new results relying only on undergraduate mathematics. We see that participation into yielding such results is authentic and, hence, a motivating context for such activities. Therefore, we want to speak for surveying topics in the intersections of different areas of mathematical sciences – such as geometric constructions and statistics – since new, but only moderately challenging, observations are most probably found in these junctions. We shall show that geometrography, i.e., studying the complexity of ruler-and-compass constructions, can answer the above demands. Especially, the new approach to geomet- rography to-be-introduced below provides in upper secondary school and in undergraduate mathematics education — particularly in mathematics teacher education — a natural setting to discuss many modern mathematical phenomena including algorithmic thinking and pro- gram design, in a classical geometric context. Indeed, this approach requires that geometric constructions are described precisely in an algorithmic form for computer software; the syntax and method of this description are yet simple and easy to learn. Further, the approach enables learners to study at a general level relations between abstract mathematical structures and their models, and how statistics can support mathematical reasoning and problem solving. Finally, it also provides a complementary approach to graphics education. We shall first bring up the almost forgotten work of E.´ Lemoine [14] which seems to be so far the only comprehensive treatise in this field. We will survey his approach in Section 2. In Section 3, we will present an alternative, statistical, approach due to S. Mustonen [18]. Its main idea is that, in practice, the ruler and compass are never placed exactly right. The errors so arising are modelled statistically. The measure of the final error, defined in a certain reasonable way, is computed using simulation experiments. It turns out that the result expresses, besides the inaccuracy, also the complexity of the construction. We will illustrate both approaches and compare them by studying first a construction of a perpendicular to a line in Section 4. Second, we will consider three constructions of a regular J.K. Merikoski, T. Tossavainen: Two Approaches to Geometrography 17 pentagon in Section 5. The results of this section motivate us to study correlations of certain complexity measures for the constructions of the regular pentagon. We will do it in Section 6. The rise of CAD and the arithmetic of floating point numbers behind it have vitalized research on error propagation of geometric constructions in ICT environments, e.g., [11, 21, 23, 24, 25]. In Section 7, we will shortly compare Mustonen’s approach and that of Hu and Wallner [11]. 2. Lemoine’s geometrography A trivial way (see, e.g., [13]) to define the complexity of a ruler-and-compass construction is simply to count how many (straight) lines and circles must be drawn. So, if l lines and c circles are required, then the complexity is l + c. But these operations are geometric, not arithmetic, and as such incommensurable. Experience makes us to think that drawing a circle is more complicated than drawing a line, yet also the opposite can be argued, see [5]. Anyway, they are here considered equally complex and, if necessary elsewhere, this matter may be taken into account by defining also the “symbol” lL + cC where L and C are indeterminates expressing relative complexities of drawing a line and a circle, respectively. This definition, nevertheless, ignores the fact that single operations with the same instru- ment may have different complexities. For example, to draw a line through two given points is obviously more complicated than to do it through one given point. Further, to draw a circle with a given center and radius is clearly more complex than to do it with a given center only. Therefore a deeper analysis is needed and that has been done by Lemoine [14] (see also [3, 4, 7, 12, 20]). He distinguishes the following basic operations. L1. Place the ruler through a given point. L2. Draw a line. C1. Place one leg of the compass on a given point. C2. Place one leg of the compass on an indeterminate point of a given line. C3. Draw a circle. If the numbers of these operations in a construction are respectively l1, l2, c1, c2, c3, then the complexity of the construction is defined by l1 + l2 + c1 + c2 + c3. Actually Lemoine calls it simplicity (and so do also the other references above) but we find “complexity” more appropriate. Since the summands are again incommensurable, it is more precise to define (as Lemoine does) the symbol of the construction by l1L1 + l2L2 + c1C1 + c2C2 + c3C3. Here the indeterminates L1, L2, C1, C2, C3 express relative complexities of the respective operations. For brevity, we denote the symbol by the 5-tuple (l1, l2,c1,c2,c3). These basic operations can be trivially partitioned into two sets according to the instru- ment but there is also another natural partition. In L1, C1 and C2 an instrument is placed, while in L2 and C3 it is used. We may plausibly think that the only factor effecting on accuracy is how exactly the instrument is placed; once it is done, then “arbitrarily thin” and (subject to the placing) “arbitrarily exact” lines and circles can be drawn. This leads us to define the inexactness of the construction by l1 + c1 + c2. Actually Lemoine calls it exacti- tude (and so do also [4, 7, 12], while [3] and [20] do not mention this concept) but we find “inexactness” more appropriate. It turns out that in practice there is no essential difference between complexity and inexactness. We will discuss this in Section 6. Lemoine’s geometrography is not widely known today, neither did it get much attention even in his time.
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