Symmetry and three-dimensional geometry MATH 124

Reflection symmetry Also called line symmetry Appears in early elementary school The reflection line, or line of symmetry, divides the object in two congruent parts. If the object were folded over the reflection line, the two congruent parts would fit each other exactly A shape can have multiple lines of symmetry

Rotation symmetry A shape has rotation symmetry if it can be rotated (turned) around a fixed point until it fits exactly on itself (or the space it previously occupied). The fixed point is called the center of symmetry Every object has 360 degree rotation symmetry, so we don’t consider that one

Translation symmetry A shape has translation symmetry if it can be translated (i.e. slid or shifted) to land on itself (or on the space it previously occupied) Equivalently, an image has translation symmetry if it can be divided by lines into a sequence of identical figures Technically, only infinite objects can have translation symmetry

Symmetry A symmetry of a shape is any movement that fits the shape onto the same set of points it started with

Reflection symmetries in quadrilaterals Zero lines of symmetry: parallelogram, general trapezoid One line of symmetry: kite (diagonal), isosceles trapezoid (line connecting midpoints of bases) Two lines of symmetry: rhombus, rectangle Four lines of symmetry: square

Rotation symmetries in quadrilaterals No rotation symmetry: trapezoids, kites 180 degree rotation symmetry: parallelograms, rhombi, rectangles 90 degree rotation symmetry: squares

Symmetries of regular polygons A regular n-gon has n lines of symmetry. If n is even, then the lines of symmetry connect opposite sides and opposite vertices. If n is odd, then the lines of symmetry connect vertices with opposite sides. A regular n-gon has rotation symmetry of the order n. The smallest angle of rotation possible is 360/n, and any multiple of that angle will also work. For example, a 10-gon can be rotated 36, 72, 108, 144, 180, 216, 252, 288, 324, 360 degrees.

Tessellations A special type of design that employs translation symmetry is called a tessellation.

From Japan

From the Alhambra

M.C. Escher

Tessellation definition A tessellation is the tiling of a plane without gaps or overlaps with a repeating pattern. In geometry, we are interested in regular and semi-regular tessellations.

Regular tessellation

Semi-regular tessellation

Activity You will investigate regular and semi-regular tessellations for homework, but we will get started today. Go to a computer and type “tessellations creator Illuminations.” Work on question 1 with your group and fill out the table as best as you can.

Polyhedron A simple closed surface has exactly one interior, has no holes, and is hollow. The set of all points on a simple closed surface together with all interior points is a solid. A polyhedron (polyhedra is the plural) is a simple closed surface made up of polygonal regions, or faces. The vertices of the polygonal regions are the vertices of the polyhedron, and the sides of each polygonal region are the edges of the polyhedron.

Prism A prism is a polyhedron in which two congruent polygonal faces (bases) lie in parallel planes and the remaining faces are formed by the union of the line segments joining corresponding points in the two bases. The remaining faces are called lateral faces. A prism can be right or oblique. If it is right, then the lateral faces are rectangles, and if it is oblique, then its lateral faces are parallelograms. The lateral faces do not have to all be congruent. Question: If the base of a prism is an n-gon, how many vertices edges faces does the prism have?

Pyramid A pyramid is determined by a polygon and a point not in the plane of the polygon. The pyramid consists of the triangular regions determined by the point and each pair of consecutive vertices of the polygon and the polygonal region determined by the polygon. The polygonal region is the base of the pyramid, and the point is the apex. The faces other than the base are lateral faces. The altitude of the pyramid is the perpendicular line segment from the apex to the plane of the base. A pyramid can also be right or olbique. Question: If the base of a prism is an n-gon, how many vertices edges faces does the prism have?

Euler’s formula All polyhedra satisfy the formula: V + F -2= E.

Nets A two-dimensional representation of a three-dimensional surface is called a net. We will do some practice with cube nets. I will give you a hard copy of the activity, but you can also work online. Search for “Cube nets Illuminations”