Plane
A plane is a flat, two-dimensional surface that extends infinitely in all directions, with no thickness.
Formula
ax + by + cz = d \text{ (equation of a plane in 3D)}
Definition
A plane is a perfectly flat surface that extends forever in all directions, with length and width but no thickness at all, like an endless, perfectly flat tabletop. It is uniquely determined by three non-collinear points, or by a line and a point not on that line, and is named by three points (plane $ABC$) or a single capital letter. In $\mathbb{R}^3$ it is an affine subspace of dimension $2$, expressed as $\{P + su + tv : s, t \in \mathbb{R}\}$ for a point $P$ and two linearly independent vectors $u, v$, or equivalently as the solution set of $ax + by + cz = d$, where $(a, b, c)$ is the normal vector.
Example
A sheet of paper is like a small piece of a plane, and the floor of a classroom represents one too, except a true plane extends past the walls forever in every direction. Points $A(0,0,0)$, $B(1,0,0)$, and $C(0,1,0)$ determine the $xy$-plane in 3D space, and any two intersecting lines lie in exactly one plane, while parallel planes never intersect. The plane through points $A, B, C$ has normal $n = (B - A) \times (C - A)$, equation $n \cdot (r - A) = 0$, and the distance from point $Q$ to the plane is $|n \cdot (Q - A)| / |n|$.
Key Insight
A plane is two-dimensional, with two directions to move but no up-down thickness, which is what separates it from solid 3D objects; two distinct planes either are parallel or intersect in exactly one line, a property used in architecture and engineering. In projective geometry, a plane is dual to a point, and the duality between points and planes, where any theorem about points and lines has a dual theorem about planes and lines, is a deep structural symmetry that differential geometry extends further through tangent planes on curved surfaces.