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AlonsoMcLaren
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A problem in Linear Algebra by Jim Hefferson:
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 de fine 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?
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 de fine 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?