What does a “plane”mean in geometry?
A plane—in geometry or any other field of mathematics—means a surface such that a
straight line joining any two of its points lies totally in that surface. A plane is consid-
ered to be two-dimensional; when a plane is discussed with higher dimensions, it is
called a hyperplane. Thus, in the majority of mathematical discussions, a plane can be
thought of as a two-dimensional group of points that reach out to infinity in all direc-
tions (for more information about dimensions, see above).
What is a polygon?
The word polygon means “many angles” (Greek polyfor “many”; gonisfor “angle”). It is
a closed figure in a plane made up of line segments (straight with no curves) that inter-
sect only at their vertices (endpoints). In other words, no sides (lines) can touch each
other except at endpoints. A polygon has the same number of sides as it has vertices.
What are some divisions of polygons?
There are two major divisions of polygons: Regular polygonsare convex polygons with
equal sides and length; thus, all sides and angles are congruent (equal). For example,
one of the most famous regular octagons is the stop sign used along roads in the Unit-
ed States: a closed polygon with eight equal sides. The naming of the various polygons
can be challenging, though. For example, a polygon called a regular triangle is also
called an equilateral triangle; another name for the polygon called a regular quadrilat-
eral is a square. Irregular polygonsare those with sides of differing lengths and vari-
able angles. Therefore, unless all the sides of the polygon are of the same length and
all the angles are of the same measure, the polygon is said to be irregular.
But don’t be fooled: The names for the various polygons—such as hexagon,
nonagon, and pentagon, depending on number of sides—don’t just apply to the regu- 179
GEOMETRY AND TRIGONOMETRY
How is the term “surface” used in mathematics?
W
hen most of us think of the word “surface” we often envision our own
world—the thin outer crust of soil and rock we walk and live on. In engi-
neering, the term means the outer part (or the skin with a thickness of zero) of a
body. In science it can apply to a plethora of objects, from geologic structures to
micrometer-sized particles.In mathematics, “surface” also has numerous meanings. The most common
denotes a two-dimensional topological space or three-dimensional Euclidean
space. A surface in mathematics can be complicated and complex, such as in cer-
tain fractals, or extremely simple, such as in a plane.