A problem in Linear Algebra by Jim Hefferson:(adsbygoogle = window.adsbygoogle || []).push({});

Euclid describes a plane as \a surface which lies evenly with the straight lines

on itself". Commentators (e.g., Heron) have interpreted this to mean \(A plane

surface is) such that, if a straight line pass through two points on it, the line

coincides wholly with it at every spot, all ways". (Translations from [Heath], pp.

171-172.) Do planes, as described in this section, have that property? Does this

description adequately define planes?

The answer is ambiguous:

Euclid no doubt is picturing a plane inside of R^3. Observe, however, that both R^1 and R^2 also satisfy that definition.

So what about R^4 and beyond? Do planes in R^4 (and beyond) have the property stated above? Is this property a definition of planes in R^4 (and beyond)? Is it even a definition of planes in R^3?

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# Plane in higher dimensions

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