Euclidean n-Space \(\mathbb{E}^n\)
The space you know and love. The Euclidean 2-Space is the flat plane where polygons live. The Euclidean 3-space is where polyhedra live.
Set
n copies of the real number line.
Structure
The important thing about Euclidean space is it's representation of the distance between points. When a space has a way of measuring the distance between points, it's referred to as a metric space (same latin root as meter). The Euclidean space is equipped with the "Euclidean metric" which is commonly known as the Pythagorean theorem.
Given two points \(x\) and \(y\) in \(\mathbb{E}^n\), the distance \(d(x,y)\) from \(x\) to \(y\) is calculated as follows: \[ d(x,y)=\sqrt((x_1-y_1)^2 + (x_2-y_2)^2 + ... (x_n-y_n)^2) \]
\(\mathbb{E}^n\) can also be constructed with a list of 5 postulates (rules) that lend themselves nicely to working in the classic geometry style.
- A straight line segment can be drawn joining any two points.
- Any straight line segment can be extended indefinitely in a straight line.
- Given any straight lines segment, a circle can be drawn having the segment as radius and one endpoint as center.
- All Right Angles are congruent.
- If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two Right Angles, then the two lines inevitably must intersect each other on that side if extended far enough.