Defining a Line

The version of Geometry most widely taught in high schools in the United States is an amalgam of the two most basic fields of geometry: Synthetic and analytic. The mixing of these two is done in such a way as to suggest that the fields are complementary, and so the points of differentiation between the two, even when apparently contradictory, are often left to pass without comment. I believe the pedagogical motivation for this is that the average high school mathematics student isn’t particularly interested in this sort of philosophical distinction; High School Geometry is largely seen as a pragmatic field, with little resemblance to what is involved in a “Geometry” course at the graduate level of college (that representing the third major field of geometry, Differential Geometry).

Meanwhile, algebra has continued to gain ground as “the” type of mathematics to be taught at the high school level. My own high school program (in the 1980s) consisted of one year of Algebra, one year of Geometry, one semester of Trigonometry (concurrent with the second semester of Geometry), and two years of advanced math (what my school called Analysis–half algebra and half calculus–and Calculus). Geometry only contained a smattering of algebra; it was largely synthetic, with a high emphasis on proofs, theorems, and axioms. Today’s typical program, dictated by law in Michigan, is two years of Algebra, one year of Geometry, and one additional year of mathematics, the most common being Pre-Calculus (which is best characterized as Algebra 3; in the schools I’ve observed, there is very little I would characterize as calculus in Pre-Calculus). Algebra is also present throughout Geometry, with a downplayed emphasis on theorems and proofs.

In this post, I’m looking at two different approaches to a very basic object in geometry: The line. The synthetic and analytic explanations for what a line is differ significantly. (Note: My comments and examples apply to Euclidean space.)

Synthetic and Analytic Geometry

According to Felix Klein’s “Elementary Mathematics from an Advanced Standpoint/Geometry”, quoted in Wikipedia, “Synthetic geometry is that which studies figures as such, without recourse to formulas, whereas analytic geometry consistently makes use of such formulas as can be written down after the adoption of an appropriate system of coordinates.”

That is to say, all that Euclid stuff, axioms, theorems, undefined objects, drawing and measuring, proof-writing… that’s synthetic geometry. “The midpoint of a segment is the point on the segment that is equidistant from the endpoints” is synthetic geometry.

All the algebra, using the distance formula, using graph paper to lay out objects specifically… that’s analytic geometry. “If \(C\) is the midpoint of \(\overline{AB}\) and \(AC = x + 5\) while \(BC = 2x + 1\), find the value of \(x\)” is analytic geometry (especially when there’s no picture provided).

Much of the time, the two approaches complement each other, and there’s a good deal of overlap between the two. In some cases, though, the different approaches lead to significantly different perspectives.

How can we describe a line? I’m avoiding “define” because, within synthetic geometry, lines are considered an undefined concept. Hilbert identified three basic objects as undefined: The point, the line, and the plane. He then defined all other objects in geometry in terms of these (Wallace and West, Roads to Geometry, 3rd Ed., p. 52).

Axiomatically, a line is generally described as an object that is formed by connecting any two points and extending forever in either direction. Hilbert’s axioms (from Wallace and West, p. 52) are:

  1. Through any two distinct points A, B, there is always a line m.
  2. Through any two distinct points A, B, there is not more than one line m.

In other words, any pair of points in space are on a unique line.

Students come armed into Geometry with the analytic definition, even if they haven’t been explicitly told: A line is all the solutions of an equation \(y = mx + b\), where \(m\) and \(b\) are any real numbers. This definition leaves out a single class of lines, i.e., \(x = a\) (that is, vertical lines). This is of course readily fixable within analytic geometry itself (a line is a set of solutions to either \(y = ax + b\) or \(x = a\)); whether the standard Geometry curriculum addresses this varies.

Superficially, these are different explanations, with different ramifications on how one views geometry in the first place. The axiomatic view does not constrain the object “line” to an arbitrarily absolute coordinate plane. What is relevant to the axiomatic definition is the relative placement of the points. The line (and the points) can be fixed onto an infinite number of coordinate planes, because specific locations aren’t important. What’s important is the relationship of the line to other mathematical objects (including other lines, as well as points and planes) in space.

Meanwhile, the analytic definition fixes the line onto a specific location on a coordinate plane, and defines it in terms of a function. We can use this function to determine whether a given random point on the coordinate plane is on the line, and also what the ordinate of a point is given its abscissa (or vice versa).

As another example of the distinction, how do we define parallel lines?

In analytic geometry: Two distinct non-vertical lines \(l\) and \(m\) are parallel if, given \(l = ax + b\) and \(m = cx + d\), \(a = c\) and \(b \not= d\). More casually, two lines are parallel if they have the same slope. (All vertical lines are parallel.)

In synthetic geometry: Two distinct lines \(l\) and \(m\) are parallel if, for all points \(L\) on \(l\), the distance from \(L\) to \(m\) is constant.

The analytic definition has the benefit of clarity: We don’t need to define (or measure) the distance from a point to a line, we just need to know the equations of the two lines. The synthetic definition has the benefit of abstraction: We can apply it to any two lines without having to have first assigned them onto a coordinate plane.

My concern here is not that either of these approaches is “wrong”, but rather that Geometry texts and teachers don’t point out the distinction enough. Indeed, while teachers are exhorted to couch new information in terms of prior knowledge, Geometry texts tend to start with the synthetic definition in a vacuum and then re-introduce linear algebra, rather than starting with the latter and doing a compare-and-contrast with the Euclid/Hilbert axioms.

Set Theory and Undefined Terms

There is another way to describe a line, which flirts with the infinitesimals of differential geometry and, hence, prepares students for thinking in terms of calculus. This perspective is sometimes hinted at in terms of lines, and there is a lesson on circles as loci in Pearson’s Geometry Common Core (12-6):

  1. A segment \(\overline{AB}\) is the set of points \([A, B]\).
  2. A line \(l\) is the set of points such that \(A, B \in l \Leftrightarrow \overline{AB} \subset l \).

In this manner, we can describe all geometric objects in terms of three concepts: The object “point”, the concept “between” (shown above as \([A, B]\)), and the concept “rotation”.

For instance, a plane is a set of points such that \((A, B, C \in p \wedge C \not\in \overleftrightarrow{AB}) \Leftrightarrow \overleftrightarrow{AB}, \overleftrightarrow{AC} \in p\) (in axiomatic form: If A, B, and C are noncollinear points, then they are on a unique plane). A ray is a subset of a line starting at a specific point on that line. Polygons consist of segments formed between points.

Angles, as well as circles and other curved spaces require the additional concept of “rotation”. The angle \(\angle ABC\) is the amount of needed rotation on point \(B\) from being oriented to \(A\) to being oriented to \(C\). Axiomatically, this is typically defined in terms of rays (\(\angle ABC\) is formed by the intersection of \(\overrightarrow{BA}\) and \(\overrightarrow{BC}\), for instance). But I have begun to wonder if that’s the correct approach, pedagogically. In all of my placements, students have struggled with recognizing angle as something other than a linear measurement: Angle is a distinctly unique measurement, within geometry. By teaching angles in terms of rays, I fear we’re suggesting that \(\angle ABC\) and \(\angle DBC\) (when \(D \in [AB]\) are somehow different, even if we verbally stress that they’re the same. Visually, after all, they ARE different. And we define angles in terms of rays but fairly quickly talk about them in terms of segments… while insisting that rays and segments are not the same thing.

The importance of the angle is that it’s a rotation. Standing in one spot, we can do two kinds of movement: We can walk in a given direction, or we can rotate around the spot. Those are the key measurements in geometry, so why not make them the core objects?

Indeed, with computer languages such as turtle, this is how polygonal shapes are generated. Here is a square generated in turtle:

import turtle

silly = turtle.Turtle()

silly.forward(50)
silly.right(90)     # Rotate clockwise by 90 degrees

silly.forward(50)
silly.right(90)

silly.forward(50)
silly.right(90)

silly.forward(50)
silly.right(90)

turtle.done()

In gist: Start at some point. Go 50 units in one direction. Turn 90 degrees clockwise. Go 50 units, turn 90 degrees, go 50 units, turn 90 degrees, go 50 units, turn 90 degrees. The last turn returns us to our original orientation.

So, here I have a few questions:

  1. Would it make more sense to students to teach geometry in terms of these three concepts (point, between, and rotation), rather than the traditional set (point, line, plane, on, between)?
  2. Are these three concepts really sufficient to define all other geometric terms? (Note that they’d merely be enough to define lines and planes, since lines and planes are sufficient for everything else.)
  3. To what extent should set theory itself be taught in a standard high school Geometry course?

These are questions I’m continuing to think about.

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