plane
A surface is that which has length and breadth only.
The extremities of a surface are lines.
A plane surface is a surface which lies evenly with the straight lines on itself.
—Euclid
A plane is a flat field with two dimensions.
A plane is flat when is contains a line projected by two points on the plane.
The plane is the primary context for the constructions we will create until we venture into 3 dimensions much later.
Euclid’s 5th postulate concerning parallel lines has created consternation amongst mathematicians because it does not hold up for non flat surfaces.
Non-planar surfaces, such as the surface of a sphere, will be defined later.