So, no prefixes...Do planes in higher dimensions satisfy Euclid's definition?

[AlonsoMcLaren]

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?

 

[chiro]

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?

Hey AlonsoMcLaren.

Just to clarify I'm interpreting the quote to mean that you have a flat surface (plane) and a line and if the line passes through any two points then the line runs through infinitely many points on the plane.

If this is the case, the answer should be used for any plane of any dimension, although the line will only run through a subset of points even though the number of points is infinite.

The way to do this formally is to show that the line is perpendicular to the normal vector and that the two points satisfy the plane equation. Showing that the direction vector of the line is perpendicular to the normal vector with two solutions to the plane equation being the two points can be expanded to show that the line crosses infinitly many points on the plane and in effect lies on the plane.

For the n-dimensional construction of a plane, use the relationship n dot (r - r0) = 0 wher dot is the dot product (Assuming cartesian/euclidean space), r is an arbitrary point to satisfy plane-equation and r0 is a specific known point on the plane. When this is 0 for a given r, then r lies on the plane.