If Euclid were a taxi driver in New York city, how far could he take you?

In Euclid were a taxi driver in New York city, we would have have what is called the Taxicab geometry where distance between two points is calculated as

d((x1, x2), (y1, y2)) = |x2 - x1| + |y2 - y1|

instead of the Euclidean distance. (The name taxicab geometry came from the real life analogy that, in a city with perpendicular cross streets and lanes, the distance between two stations is the sum of the horizontal distance and the vertical distance - the shortest way a taxicab would have taken you from one station to other.)

It can easily be verified that d is a metric. [d(P, Q) >= 0, d(P,P) = 0 and d(P,Q) = d(Q,P)]. So we can build up our geometry with this metric as the distance function. Lets see where we can get.

Since we have to start with distance, the easiest thing to define is a cirle. If you were to plot all the points that are unit distance away from the origin, you will get a very familiar shape - except that it won't be the 'usual' (Euclidean) cirlce, but a (Eucledean) square centered at the origin whose diagonals are aligned with the axes! To put it loosely, "Circles are squares in taxicab geometry."

Interesting? Lets see what a straight line looks like.

In order to define a straight line, we need two points. Which we readily have. Let us take two points O(0,0) and P(p, q), p, q > 0 as our two points. Now comes the interesting part - how would you define a straight line through two points with just the distance function?

Looking at the properties of the (Euclidean) straight line, we see that for any point X lying on the straight line passing through two distinct O and P, one of the following holds:

(I) d(O,X) + d(X, P) = d(O,P) (for any point 'between' O and P)
(II) |d(O, X) - d(X, P)| = d(O, P) (for any point 'outside' O and P)

This property seems to be a resonable definition of a straight line in Euclidean geometry and it is entirely in terms of distances. Let us carry this over to taxicab geometry and see what happens.

First draw a two perpendiculars through O and P aligned with the axes. They will form a rectangle whose corners are O and P. The horizontal side of this rectangle be p and the vertical side be q.

Consider any point X inside this rectangle. Let its coordinates be (x, y).

Now,
d(O,X) + d(X, P) = |x - 0| + |y - 0| + |p - x| + |q - y|
= p + q
= |p - 0| + |q - 0| = d(O,P)

Thus, according to our definition, any point inside the rectangle lies on the straight line joining P and Q!

Note that the above takes care of only (I) above. What are the set of points that satisfy (II) ?

Also, note that if you try to construct a line between O and X above, you will get an entirely different line! Thus, though two points define a line, any pair of points lying on that line does not define the same line.

Wow! So far we saw that 'a circle is a square' and 'a straight line has two dimensions'. What next?

This is where the problem gets really interesting: How would you define other features of the geometry in terms of distance function alone? What all can you define?

For example, let us see whether we can make a triangle. First of all, you need three distinct points and that's easy. Then you need three line segments connecting the pairs of straight lines. You have that too. Do we have a triangle now?

Not really. A triangle is the set of points that lie inside these three line segments. How do we define 'inside'? That's easy, it is the intersection of some 'above's and 'below's (if you know what I mean). But how do you define 'below' or 'above'?
Or even how do you know two points are on the same side of a given line?

Note that these questions are also equally valid in Euclidean geometry. At some point Euclid made the convenient assumption about existance of a right angle (and all other angles) and proceeded with his geometry. There was no NY city with the perpendicular streets and lanes at his time and he could drive in any direction :-)

A limited world

There is an interesting theorem in mathematics that says that all standard geometrical constructions can be done using just compass alone (as opposed to a compass and a straight edge). Similarly, there is another theorem that says that the same can be achieved using a straight edge alone, provided you have a fixed cirle, drawn in the plane, to start with. (Of course, in either case you'll need the pencil too :-) )

Thus, if the straight-edge was never invented people would still be able to do (Euclidean) geometry - making a compass is, of course, a lot easier than making a straight edge.

Which made me think - how was the first straight edge ever created?

Remember that there were no other measuring instruments to validate the straightness of the edge. Interestingly, this is a corollary to another classic problem in mecahnical engineering - how was the first plane surface made?

I have the (stock) answer to the latter problem, but I will post it in another article.

But the bottom line is, even in a limited world the ingenious mathematician will still be able to survive!