Euclidean geometry

The ancient Greeks managed to prove quite a lot about mathematics in the guise of geometry. Euclid systematised much of this in his Elements (of geometry). This book started from five common notions and five postulates – collectively known as Euclid's axioms – and proceeded to prove a broad collection of results.

Parallel

His fifth postulate, which it is easy to believe even Euclid didn't like much (in the Elements, he avoided using it for as long as possible), stands out as the only axiom that is not intuitively obvious. Ultimately, it has been found to be the one that distinguishes the geometry of a flat surface from that of a curved surface (or smooth manifold).

Euclid's fifth postulate concerns a pair of lines which cross a third: if the internal angles they make with that third on a given side sum to less than half a turn, he postulates that – if extended indefinitely – they must meet on that same side of the given line. This postulate is commonly referred to as the parallel postulate because of its implications for parallel lines: it could equally be regarded as specifying the equivalence of any pair among the following notions of parallelness. Two lines may be deemed parallel if

If any two of these are equivalent, then so are the rest. It should be noted that further notions of parallelism are possible, that may be identified with one or more of the above, either individually or as a consequence of the equivalence of some pair of them. For example, two lines deemed parallel by at least some subset of the above satisfy:

With any of these notions of parallelism, Euclid's fifth postulate becomes equivalent to for any line L and point P not on L, there is exactly one line parallel to L which passes through P. There are many other axioms to which Euclid's fifth postulate is equivalent. One of the nicest is the assertion that any triangle can be scaled arbitrarily without changing either its angles or the ratios among the lengths of its sides. Euclid's fifth has also been shown equivalent to Pythagoras's theorem.

Lengths

We can make sense of straight lines as shortest possible routes between points, if only by thinking in terms of pulling a piece of string taut between those points; and we can identify intervals between points on straight lines as line segments. We can then consider placing a rigid physical object (e.g. a ruler) along a line segment with marks on the object that line up with the ends of the segment and, by moving the object around and comparing its markings with the end-points of other segments, establishing an equivalence among line segments that we may think of as equality of length. From that we can come to a notion of an addition of lengths which, in turn, gives us a notion of scaling lengths by whole numbers.

We might take a different view of rigid movement, at least for vertical displacements, if gravity were stronger: unless the rigid object is neutrally buoyant in the surrounding medium, when rotated to a vertical alignment it would stretch or compress, depending on whether supported from below or above, enough to matter. So Jovian geometers might treat vertical displacements as special, where we mostly don't – but even they would likely still apply the same reasoning as us to horizontal displacements, which is where geometry (or joveometry) comes from originally.

Equality in detail

We can express moving an object around, within the two-dimensional Euclidean plane, in terms of rotations and translations. As it happens, each of these can be expressed in terms of reflections: any translation can be achieved by reflecting in two parallel mirrors separated by half the translation displacement; any rotation by a pair of reflections in mirrors that meet at the centre of rotation making an angle equal to half the rotation angle. By doing our geometry on sheets of thin paper using ink dark enough to soak through the paper, we can fairly easily see that reflection – realised as flipping the paper over and looking at its other side – also preserves lengths; so we could satisfy ourselves that translation and rotation do so as a consequence of reflections doing so.

When we use a pair of compasses (with their hinge screw tight enough), we expect its sharp end and the end of the marker (e.g. a pencil) in its other arm to stay a fixed distance apart as we turn the whole and mark out (an arc of) a circle centred on wherever we've pinned the sharp end; so all points marked are the same distance from the centre, illustrating our intuition that rotating a line segment (from the centre to a point on the circle) about one of its end-points preserves length. We can place a straight-edge (e.g. a ruler) along a straight line and mark the points on the ruler that coincide with the end points of a segment, then move the ruler elsewhere; if we keep the ruler parallel to the line, this illustrates translation preserving length between the marked points; otherwise, we illustrate some combination of translation and rotation preserving length. For any line segment and its reflection in a given mirror, we can show how to use rotations and translations to move either onto the other, thereby showing that reflections do indeed preserve lengths; so we can obtain this property of reflections from that of rotations and translations, just as we can do the converse.

Either way, we thus obtain a notion of equality of lengths of line segments, which can be expressed in terms of reflections or in terms of rotations and translations, or in terms of all three. Combination of rotaitons, reflections and translations always preserve length. A transformation that preserves lengths is described as an isometry. Later, I hope I'll remember to come back and show that any isometry is necessarily expressible as a translation, a rotation and an optional reflection (the translation may be through no distance, the rotation through no angle, so in fact all three parts are optional; but I have to expressly make the reflection optional because there's no reflection that has no effect, like fatuous translation and rotation), at least given some variant on the parallel postulate: but for that I'll need things I haven't yet introduced.

Ordering and length

So we can identify an equivalence among line segments characterised by there being an isometry that maps the whole of one segment onto the whole of the other, notably thus moving the end-points of one onto the end-points of the other. We call that equivalence equality of length.

By translating one line-segment to where it has one end-point coincident with an end-point of another, then rotating the first line-segment's image until it lies along the same straight line as the other, with its other end-point on the same side as the other's other end-point, we can establish a segment equivalent to the first that overlaps the second, with an end-point in common; we can then ask where the other end point of each is in relation to the other. If the other end-points coincide, that's exactly the situation that says the two line segments we started with are equivalent – they have equal length. Otherwise, as we move along the line from their shared end-point, we have to come to the other end-point of one of them first; we then say that this line segment is shorter than the other, which is longer than it.

By sliding the shorter one around inside the longer one, flipping either end-to-end and otherwise applying isometries we can establish that these relations between them – whether they have equal length or, if not, which is shorter than the other – aren't dependent on how we lined them up with each other, so do provide us with a well-defined relation among line segments. We can show that one line segment being shorter than a second that's shorter than a third does indeed imply the first is shorter than the third; and every line segment has equal length with itself; so we end up establishing is shorter than and is longer than as ordering relations among line segments, with equality of length as an equivalence, such that for any given pair of line segments exactly one of these three relations relates the first of the pair to the second. Furthermore, if one line segment is shorter than another, then any line segment of equal length to the first is shorter than any line segment of equal length to the second.

Given equality of length as an equivalence, orthodoxy would then identify a length as an equivalence class of line segments; I (almost equivalently) formally specify a length to be a line segment understood modulo the equivalence. Either way, we then obtain an ordering among lengths for which any length is either shorter than (exclusive) or longer than any distinct length.

Adding lengths

In establishing the ordering of lengths, I showed how we can move any line segment, representative of a length, onto the straight line that another segment lies on: I did so for the moved segment overlapping the other with an end in common, but a barely-distinguishable construction would let me have the moved segment not overlap the other except for having an end in common; the two segments are then said to abut one another; and they are contiguous along a common straight line. By considering the segment between their other ends, ignoring the two ends at which they abut, we obtain a line segment that we can think of as a sum of the moved original segment with the other segment. If either segment is empty, i.e. its two end-points coincide, this sum of segments is the other segment. Otherwise, the sum is necessarily longer than either of the segments added to make it.

By considering the result of moving around such a sum of segments and looking at the images, within the moved sum, of the two parts that we added to make it, we can establish that adding a moved image of a first segment to a second segment produces a sum segment that has equal length with each of

The first then establishes that the addition is commutative (a.k.a. abelian); the latter that it respects length equivalence and thus reduces to a closed binary operator on lengths. By considering three lengths and adding one to the result of adding the other two, then moving this sum around and comparing it with the result of adding that one to one of the other two, then adding the result of that to the last, we can establish associativity and thus that our binary operator is a combinator.

Next, given a first line segment, suppose two others each, when added to the first, gives the same sum; within that sum, one portion is taken up by the first line segment, the other by the images of the others; thus the others can be moved onto that common portion of the sum, so each has length equal to that of this portion and thus the two others have equal length. Thus, when adding lengths, if two lengths add to another to give equal answer, the two lengths were actually equal, which means our addition of lengths is cancellable. This makes it, in fact, an addition in a formal sense that I've analysed in detail, with lots of interesting and useful consequences. In particular, by repeatedly adding a segment to itself, we can scale lengths by naturals; I'll refer to the results as whole number multiples of the length so scaled; and to any length as a whole number fraction of its postive natural scalings.

All empty segments clearly have equal length and adding this length to any other segment gives the length of that segment, so an empty segment, considered as a length, is an additive identity; we name this length zero and denote it 0, in the usual way for an addition. It is easy enough to show that 0 is shorter than any other length. The addition isn't complete because there's nothing you can add to a longer length to get a shorter length; any sum of lengths is at least as long as the two lengths added and is longer than either unless one of them is the length of empty segments. However, one can always subtract one length from another unless the former is longer than the latter, which lets us recover the longer/shorter ordering from the addition. Given its familiar relationship to addition, it is natural to denote is shorter than by < and is longer than by > in the usual way, just as for the natural numbers.

Highest Common Factor

A great deal of ancient Greek geometry served as a surrogate for what we now discuss in terms of whole number arithmetic. The ancient Greeks were particularly interested in which lengths were rationally commensurate; this arises when there is some length unit for which each of the lengths is a whole number multiple of that unit. If I have two lengths and there is some such unit, of which each is a whole-number multiple, then any whole-number fraction of that unit will also suffice. It's thus interesting to find the largest unit for which each of a pair of lengths is a whole-number multiple of that unit.

If two lengths are both whole number multiples of some unit then the difference between those lengths is also a whole number multiple of the same unit; so mark off on the longer length as many copies as you can of the shorter one and now look for a common unit between the shorter one and the remainder of the longer one. If there was no remainder, then the shorter length is a unit we can use – and, obviously, the longest such unit, since nothing longer can fit within the shorter length, much less do so repeatedly. Otherwise, the remainder is, by construction (we removed as many copies of the shorter as we could), shorter than the formerly shorter length, so the formerly shorter length is now the longer and the former remainder is the shorter; as we are still looking for a largest unit that subdivides each a whole number of times, we can now repeat the procedure, until we get no remainder and our shorter length is the unit we were looking for.

When applied to whole numbers, this process is known as Euclid's algorithm for computing highest common factors. For example, take the numbers 217 and 231; subtracting the smaller from the larger we get 14 and now consider 14 and 217; we get 14×15 = 210, so dividing 217 by 14 we get remainder 7 and now consider 7 and 14; since 7 divides 14 we get remainder 0 and 7 was our highest common factor of 217 and 231.

When applied to arbitrary lengths, this algorithm isn't necessarily guaranteed to terminate: if the lengths aren't rationally commensurate, you'll always be left with some remainder and the lengths you're considering shall always be getting smaller and smaller. If the ratios of the lengths ever repeat, you can use this (with the scalings used to subtract multiples of the shorter from the longer, at each step) to obtain a continued fraction representation of the ratio of the original lengths.


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