Topics in geometry for teachers

In yesterday's class, we saw a drawing of the (7-point) projective completion of the 4-point affine plane. The drawing in class on 26 September did not include the line that passes through all of the ‘points at ∞’ (the ones labelled E, F, and G), but the one in yesterday's class did include it.

We explored further the motivation behind our adding 3 points to a perfectly good 4-point geometry. The point was that we noticed on 24 September that switching the words ‘point’ and ‘line’ in the axioms of incidence geometry resulted in theorems of incidence geometry, except when there exist parallel lines—so it might be desirable to study geometries without any. The remedy to the fact that our 4-point geometry has parallel lines is simple—namely, to give every collection of such lines a new point at which to intersect. (We think of this point as a ‘point at ∞’. When we draw it, it looks like 2 points, but it can't be really, since distinct lines can be incident with at most 1 point.) If we perform this same procedure on the usual Euclidean (or Cartesian) plane, then we obtain a much larger structure. We drew some pictures to see how this structure might look, and realised that it is just our old interpretation of geometry in terms of great circles on a sphere, but with antipodal points identified. We also saw that the geometry could be realised as the surface obtained by pasting together opposite sides of a Möbius strip in a certain way, but this pasting cannot actually be done in 3-dimensional space, so we didn't physically do it.

In Wednesday's class, we will begin to explore the detailed axiomatic theory of betweenness. The elementary theory of betweenness will be included on Friday's exam.

In today's class, we pointed out that the duality operation discussed in the last class actually should be described as carrying interpretations to interpretations, not necessarily models to models, since Axiom I1 might fail in the dual geometry (it's OK to have parallel lines, but not ‘parallel points’). We formalised the idea that the dual of our 3-point geometry is just the same geometry again by defining the idea of an isomorphism, and gave some examples of isomorphic and non-isomorphic geometries. We defined the notions of projective, affine, and hyperbolic planes in terms of the numbers of parallel lines they have; and observed that, while we have examples of an affine (our 4-point geometry) and a hyperbolic (our 5-point geometry) plane, we do not yet have an example of a projective plane (because our 3-point geometry is “too small”). We thus began our discussion of how to complete an affine plane to obtain a projective plane.

In today's class, we discussed an interpretation (a collection of concrete meanings associated to the primitives) of incidence geometry that is not a model, because the axioms are not satisfied. Specifically, we discussed the setting in which ‘lines’ are great circles on a sphere, and ‘points’ and ‘incident’ have their usual meanings (except that points are considered only on a sphere). In this setting, we have enough points, both overall and per line (axioms I2 and I3); and every two points lie on a line; but some points lie on more than one line, so axiom I1 fails.

We then discussed the somewhat unusual notion of dualisation, in which we build a new interpretation from an old one by switching the primitives ‘point’ and ‘line’ wherever they appear. We observed that doing this changes our three axioms of incidence geometry into three propositions that we have already proved, except that we have to worry about lines being parallel, but there is no such notion for points.

We drew a picture of the dual of the 3-point geometry, and observed that it seems to be just the same geometry again. We will discuss this notion in more detail next time.

In Monday's class, we discussed the notion of a model for a theory (a collection of primitives, in the form of nouns and adjectives, and axioms, in the form of sentences formed from the primitives). To give a model is to define terms that we have previously agreed will be undefined, so it obviously involves having a new collection of primitives in which to define the old ones.

We described the 3-, 4-, and 5-point geometries of last time as very small models of incidence geometry in terms of (a naïve version of) set theory; and observed that, since the abundances of parallel lines in these models are very different, there are some statements about parallel lines—the Euclidean parallel postulate and its negation, for example—that cannot be proven in incidence geometry—not just because we haven't been clever enough, but that it is actually impossible. We say that the Euclidean parallel postulate is independent of the axioms of incidence geometry. We will explore other possible axioms of parallelism, and their consequences, in Wednesday's and succeeding classes.

In today's class, we formalised our proof from last class that, in incidence geometry, any two distinct, non-parallel lines intersect in a unique point. We then stated what are essentially the four other results of incidence geometry, but left their proofs as exercises on the homework. A natural place to look for further results is the study of parallel lines, but we pointed out that it is possible to have models for incidence geometry with no parallel lines; with “just the right number” of parallel lines (in a sense to be made precise on the homework); or with “too many” parallel lines (in a sense that we will discuss on Monday).

In today's class, we completed a ‘paragraph’-style proof that, in incidence geometry, two distinct, non-parallel lines intersect in exactly one point. (This is not the precise statement that was made in class.) We then discussed the advantages and disadvantages of this style of proof, and began to discuss the idea of an “axiomatisation of proof” to allow us to overcome some of the disadvantages. For us, this axiomatisation will come in the form of the structuring of proofs described in logic rule 1 on p. 56 (although we allow one additional sort of statement, a “naming step”, in addition to those described in logic rule 1). We will put this idea into practise in Friday's class.

In Monday's class, we discussed the fact that we would like to be able to specify our geometry sufficiently carefully to avoid bizarre pathologies such as the two-point, one-line geometry discussed in the extra problem for Homework 1. This requires us to specify a lot of axioms—enough to describe all of space! On the other hand, we want to have only a few axioms—because otherwise there's no room left to build on top of them. (To put it differently, there's less elegance in a theory that requires 57 rules to specify it than in one that requires only 5.) One way to do this is to take our idea of decomposing definitions into simpler definitions, and so on all the way down to primitives; and theorems into simpler theorems, and so on down to axioms; and take it a step further by decomposing geometry into “simpler geometries”. We therefore replaced the complicated situation of Euclidean geometry by a much simpler one—incidence geometry—in which we can speak only of points, lines, and incidences among them. We stated the three axioms of this sort of geometry (two of which are designed just to rule out ‘pointless geometries’), observed that it still admits multiple interpretations (which is good!), and started to prove our first theorem.

In today's class, we defined the notions of a ray and an angle. We discovered that it is often possible to define an object simply by saying how to produce it, and how it is related to other objects—for example, that a ray is specified by two points, and that whether a point lies on a ray is a question of between-ness—without ever answering, or trying to answer, the question of what the object we are defining physically ‘is’. The text points out that one way of answering this question is via set theory, but we will often prefer an approach that allows us to work just on the level of objects and their relationships without referring to any particular realisation.

In today's class, we worked on trying to define the non-primitive terms that occur in Euclid's postulates, and discussed some general techniques for working with definitions and theorems. Perhaps the most important is to name everything (while being sure never to use the same name for two different objects). Audrey Lampen also observed that we need a stronger primitive notion than congruence, so we included equality in our list of primitives.

We concluded the class with the following provisional definition of ‘angle’: “An angle is bounded by two non-opposite rays.” This is not far from the definition appearing in the text (on p. 18). We will explore it farther in Friday's class before moving on to Chapter 2. Since we will not spend much time discussing them directly (preferring instead to observe their appearance in the course of proof-building), the student may find it helpful to read Logic Rules 0–12 on pp. 54–66 of the text.

In today's class, we discussed further, but informally, the notions of ‘line’ and ‘line segment’, trying to address the basic question: Is a line segment a line? We eventually settled on the fuzzy idea that a line should “go on forever”, whereas a line segment should “stop”. Since these statements are not phrased in terms of primitives, they cannot be used in mathematical reasoning, so we need something in their place. It turns out that Euclid's Postulate II provides a clever way of saying that line goes on forever without actually having to use any such messy terms as ‘infinity’. We stated this postulate, as well as the other five due to Euclid, and observed that, if one isn't careful, the definitions allow a very small geometry that has only two points and one line! The statement of the postulates involved some as-yet undefined terms, which we will define in Wednesday's class.

In today's class, we went into more detail about the ideas of definitions, primitives (our term for the basic concepts of a field in terms of which others are defined, but that are not themselves defined), axioms, proofs, and theorems, and the analogies between them (for example, that definitions are to primitives as theorems are to axioms). We also tried to come up with a workable set of axioms and primitives for ourselves. Many of our axioms were definitions, introducing the ideas of angles and line segments. The definition of a line segment proposed was “A line segment is the line that exists between two points”, which was later refined to “A line segment is the line that passes through two points”. We closed the class with the observation that such a definition would imply that a line segment is a line. We will investigate this further in Monday's class.

In Wednesday's class, we discussed some of the history and applications of geometry, including a formula due to ancient Egyptians for calculating the area of a circle that turns out, when unravelled, to be approximating π by 3.16. We discussed the necessity of some means of verifying the correctness of geometric rules and statements, and of having a way to correct those that don't survive the verification. The tools to do this are axioms and proofs; we will discuss these in more detail in Friday's class (and through the rest of the course). Students are requested to read Chapter 1 before Friday's class.