## Angle

An angle is always defined by 2 numbers

Typically those 2 numbers are related to 2 line segments that have a common end point. The coordinate system or the nature of the space in which these line segments are embedded does not matter for the definition of an angle. In the figure below the vector is embedded in a 3-dimensional space but an angle θ requires only 2 numbers for its definition.

The angle θ is defined by

θ = arctan(AC/AE)

In this definition of angle the shape of a triangle is implicit. The vector shown above is the hypotenus of a right triangle and this triangle lies on a flat surface. The sum of the interior angles of a triangle on a flat surface is 180 degrees ($\pi$ radians).

Imagine a triangle drawn on the surface of a sphere.  A spherical surface is positively curved. The sum of the interior angles of a triangle drawn on a positively curved surface is more than 180 degrees.

Now, imagine a triangle drawn on a saddle shaped (negatively curved) surface. The sum of the interior angles of a triangle drawn on a negatively curved surface is less than 180 degrees.

No matter how curved the surface (or space) is an angle can always be defined by 2 intersecting lines of infinitesimal length.

Angle is the best geometrical concept to represent coupling

I mentioned this in  The 3 fundamental variables are coupled pair-wise where I said the pair-wise couplings can be represented by the angles between the edges of the tetrahedron.

In the “New Perspective on Unification” I introduced the concepts of “horizontal attributes” and “vertical attributes.” Horizontal attributes are associated with collectivity and multiplicity. They are attributes of the group behavior. Horizontal attributes are about the cross-sectional universe. Vertical attributes are about individuality, individual histories and individual characteristics. Vertical attributes are about the idiosyncratic behavior. In that article I mentioned that the horizontal and vertical attributes are supposed to be orthogonal. Orthogonality is a measure of independence. But, in reality there is no perfect orthogonality, horizontal and vertical attributes are not independent, they are coupled. The coupling between horizontal and vertical attribute pairs can be represented by an angle as well.

Phase is an angle too

In the image shown above the two curves are essentially the same but there is a time difference between them. This difference in time is called a phase difference or phase shift. The phase difference between the sin(…) and cos(…) is 90 degrees (π/2 radians). In units of cycles the difference is 0.25 cycles.

A phase difference of 90 degrees (π/2 radians) is rather special. It tells us that these sin and cos curves can be represented by a rotating vector on the complex plane.

The r is a complex vector also known as phasor and it is rotating (either clockwise or counter-clockwise) and the Re and Im components are given by

Re: x = r cos(φ)
Im: y = r sin(φ)

where φ = 2π f t

f: number of full rotations per second; t: time in seconds

The angle φ is known as the phase.

Polar angle

The equations of mathematical spirals are given in polar coordinates. The $\displaystyle r$ is the radius and $\displaystyle \theta$ is the angle measured from the positive horizontal axis.

Fermat’s spiral is my favorite. The $\displaystyle a$ in the equation below is a constant.

$\displaystyle r^2 = a^2 \theta$