I was just reading through the "What Is Geometry?" article (http://www.cut-the-knot.org/WhatIs/WhatIsGeometry.shtml). The final paragraph of that article says:

"There are many geometries. All of these share some basic elements and properties. Even finite geometries deal with points and lines and universally just a single line may pass through two given points. Thus I believe that a frequently used term "Taxicab Geometry" is a misnomer..."

If I understand correctly, this criterion is violated by spherical geometry, where some pairs of points are joined by an infinite number of lines (as with the meridians all passing through the poles on the earth's surface).

The comment about 'taxicab geometry' might be right, but maybe for a different reason.

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I vaguely remember reading somewhere that a geometry must satisfy:
dr^2 = dx^2 + dy^2 + dz^2 ...and so on for higher dimensions

That definition of a 'geometry' would include spherical geometry and exclude taxicab, but I don't know what other interesting 'geometries' it might exclude.

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By the way, that was an excellent article. I keep forgetting about this site and then stumbling on it again in web searches. Keep up the good work. :)

>If I understand correctly, this criterion is violated by
>spherical geometry, where some pairs of points are joined by
>an infinite number of lines (as with the meridians all
>passing through the poles on the earth's surface).

It's not just that there is an infinity of lines through two antipodes. Any line passing through one also passes through the other. In this sense, the points are indistinguishable. When it comes to defining elliptic geometry, the antipodes are identified and are justly considered as a single point.

As opposed to elliptic geometry, modelled on a sphere, in the commonplace spherical geometry, considerations are usually carried out "in the small," i.e., in the regions that do not include antipodes.

A remark to this effect is certainly in order.

>I vaguely remember reading somewhere that a geometry must
>satisfy:
>dr^2 = dx^2 + dy^2 + dz^2 ...and so on for higher dimensions

There of course no unique and universally accepted definition of geometry. It's rather a field of study that admits a branching, multi-level classification. Yours is probably more relevant to the idea of manifolds and differential geometry.

I would rather admit finite geometries under the same umbrella than make differential geometry into the root of classification.

>By the way, that was an excellent article. I keep forgetting
>about this site and then stumbling on it again in web
>searches.

Thank you for the kind words.