College Intermediate Algebra

http://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/int_alg_tut3_sets.htm

The Human Skeleton

Matrix Rotation Analysis

In linear algebra, a rotation matrix is a matrix that is used to perform a rotation in Euclidean space. For example the matrix

rotates points in the xy-Cartesian plane counterclockwise through an angle θ about the origin of the Cartesian coordinate system. To perform the rotation using a rotation matrix R, the position of each point must be represented by a column vector v, containing the coordinates of the point. A rotated vector is obtained by using the matrix multiplication Rv. Since matrix multiplication has no effect on the zero vector (i.e., on the coordinates of the origin), rotation matrices can only be used to describe rotations about the origin of the coordinate system.
Rotation matrices provide a simple algebraic description of such rotations, and are used extensively for computations in geometry, physics, and computer graphics. In two-dimensional space, a rotation can be simply described by an angle θ of rotation, but it can also be represented by the four entries of a rotation matrix with two rows and two columns. In three-dimensional space, every rotation can be interpreted as a rotation by a given angle about a single fixed axis of rotation (see Euler's rotation theorem), and hence it can be simply described by an angle and a vector with three entries. However, it can also be represented by the nine entries of a rotation matrix with three rows and three columns. The notion of rotation is not commonly used in dimensions higher than 3; there is a notion of a rotational displacement, which can be represented by a matrix, but not associated single axis or angle.
Rotation matrices are square matrices, with real entries. More specifically they can be characterized as orthogonal matrices with determinant 1:

.

The set of all such matrices of size n forms a (generally not commutative) group, known as the special orthogonal group SO(n).

Excel Programming web site

http://office.microsoft.com/en-us/excel-help/overview-of-formulas-HP010081865.aspx

Overview of formulas.

The Koch Snowflake

The area inside the Koch snowflake can be described as the union of infinitely many equilateral triangles (see figure). Each side of the green triangle is exactly 1/3 the size of a side of the large blue triangle, and therefore has exactly 1/9 the area. Similarly, each yellow triangle has 1/9 the area of a green triangle, and so forth. Taking the blue triangle as a unit of area, the total area of the snowflake is

The first term of this series represents the area of the blue triangle, the second term the total area of the three green triangles, the third term the total area of the twelve yellow triangles, and so forth. Excluding the initial 1, this series is geometric with constant ratio r = 4/9. The first term of the geometric series is a = 3(1/9) = 1/3, so the sum is

Thus the Koch snowflake has 8/5 of the area of the base triangle.

http://en.wikipedia.org/wiki/Geometric_series

Trigonometry Functions

Interval Mathematics