Plane Geometry | Research & Encyclopedia Articles

Plane Geometry

Plane geometry is that subsection of geometry that deals with figures in a two-dimensional plane. Euclidean geometry, sometimes called parabolic geometry, is divided into two subsections: plane geometry, geometry dealing with figures in a plane, and solid geometry, geometry dealing with solid in three-dimensional space. Plane geometry is sometimes called two-dimensional Euclidean geometry and deals with figures such as circles, lines, polygons and the like. Plane geometry is concerned with the study of figures in two-dimensional Euclidean space, which is usually denoted R², also known as the Euclidean plane. This branch of Euclidean geometry is also focused on studying the properties of flat surfaces.

Plane geometry is a branch of Euclidean geometry, developed by Greek mathematician Euclid in the 4th century B.C., that is governed by Euclid's five postulates as laid out in his work The Elements. In the early 20th century mathematicians recognized that Euclid's postulates were incomplete in that concepts such as between, inside, and outside were not made precise. In 1902 David Hilbert developed a modern axiom system removing the incompleteness of Euclid's system. Later, American mathematician George Birkhoff laid out an axiom system for plane geometry in his work, "A set of postulates for plane geometry, based on scale and protractor", published in 1923. Sanders MacLane formulated a modified version of Birkoff's axiom system that was published in 1959. Birkoff's system consists of five axioms that are similar in function to Euclid's five postulates:

• 1) Existence axiom
• 2) Ruler axiom (introduces the ruler function)
• 3) Protractor axiom (introduces the protractor function)
• 4) Betweenness axiom
• 5) Similarity axiom

The difference between Birkoff's axiom system and Euclid's postulates is that Birkoff's system uses real numbers and their properties. Euclid's The Elements never mentioned distance and angular measure but instead used concepts such as congruence of segments and angles. In the early 1960s Birkoff's axioms were again modified by the School Mathematics Study Group providing a new standard for teaching high school geometry.

There are six basic assumptions used in plane geometry:

• 1) Only one line can be drawn through two distinct points.
• 2) Two straight lines can intersect at only one point.
• 3) The length of the line segment joining two points is the shortest distance between those two points.
• 4) A geometric figure can be moves without altering its size or shape.
• 5) A point divides a line into two infinite subsets of points of the line.
• 6) A straight line divides a plane into three subsets of points: two half planes and the line itself.

There are also four types of symmetry used in plane geometry: rotation, translation, reflection, and glide reflection. Rotation in plane geometry means to turn a figure around. A rotation has a center and an angle associated with it. Translation in plane geometry means to move a figure without rotating or reflecting the figure. A translation has a direction and a distance associated with it. Reflection means to produce a mirror image of a figure and every reflection has a mirror line, a line between the figure before and after reflection. Glide reflection is a reflection of a figure combines with a translation along the direction of the mirror line. The six assumptions along with the four types of symmetry gives mathematicians a precise way of approaching plane geometry.