Tilings (Tessellations) and Angles

Tilings cover a plane with identicle shaped pieces that do not overlap or leave any blank spaces. Tiling can be done with one, two, or any infinate number of shapes.

Example

 

There are two types of Tiling.

1. Regular: a pattern created by repeating a regular polygon. There are only three regular tessellations. They are triangles, squares, and hexagons. Only the initial shape is used and adds up to 360 degrees at the vertex. Basically the shapes fit tother with one another all the way around each other perfectly. (Shape ONLY)

 

2. Semi-Regular: made of two or more regular polygons and the pattern at every vertex has to be the same. This requires more than one shape to make the pattern because not all shapes can perfectly surround their own shapes without any gaps. A semi-regular tiling uses more than one regular polygon in order to add up to 360 degrees at the vertex. There are only eight semi-regular tessellations. (Shape+ different shape, etc.)

Angles

It is often easiest to understand an angle first before trying to understand them when it comes to shapes like a right triangular prism or right rectangular prism.

There are three kinds of angles: acute, obtuse, and reflex. An (A)acute angle is an angle in which its measure is less than 90 degrees. A (B)obtuse angle has a measure of more than 90 degrees but less than 180 degrees. A (C)reflex angle has a measure of more than 180 degrees but is less than 360 degrees. (note: the initial side of an angle doesn't move but the terminal side does)

(A)

(B)

(C)












Example(s) of if you know one angle, how to find the other.

Transformations Concerning Math Shapes

This week I learned of four types of transformations of shapes (geometric figures). They are translation, rotation, reflection and glide reflection.



Translation refers to a transformation of a graph or geometric figure is "picked up" and is carried to another location, without a change to the size or orientation of the figure.

Rotations refers to a transformation where a plane figure (e.g. figure on a graph or near a center of rotation) is rotated around a fixed center point. Basically, one point on a plane (which is the origin point) does not move, but everything else on that plane moves around that point by a given angle.

Reflection refers to a transformation where a shape (geometric figure) is reflected across a line (called the axis of reflection), which creates a mirror image.

Glide Reflection refers to a transformation that is a combination of both reflection and translation.

Understanding Complex Shapes

There are many steps to understanding and identifying complex shapes. The first step is differentiating between the different types of shapes there can be in one family. For example, the polygon has three different classes. The first is the equilateral polygon in which all sides are congruent. The second class, the equiangular polygon, has angles which are all congruent. And the third is the regular polygon which is characterized by being a convex polygon that is both equiangular and equilateral.

Another step is understanding the different angles and measures in shapes. For example, in polygons there are three types of angles. The first is the interior angle (A) which is the inner part of the angle, the second is the exterior angle (B) which lies on the outside of the shape. The last type of angle is the central angle (C) which is measured by a vertex at the circles center creating adjacent line segments.

Angles in a Polygon

The third step to understanding complex shapes is being able to calculate their angles. There is a formula that allows you to do this with the different types of angles in polygons. In the formulas n=number of angles:

-Interior angle (A)

•     (n-2)x180/n

-Exterior angle (B)

•    360/n

-Central angle (C)

•    360/n

Hexagon Example:

(A)  (6-2)x180/6 = 120 degrees

(B)  (360/6) = 60 degrees

(C)  (360/6) = 60 degrees