Interior angle

Angles
The definition of angle follows that of the Polygloss, that is, an angle is that part of space occupied by the figure. Angles are counted around the polytope, that is, in the space that is orthogonal to it.

A wheel spins around its axle, and one dances around the maypole, these actions happen in the place that is orthogonal to the body of the object. The term surround is used to denote forming the edge or limit of the space where the action is solid. One's surroundings are places that limit where one is, and is reachable from where one is.

The angle is denoted as Aa, representing an a-dimensional space where the surtope is a point, and the rest of solid space is made by a Cartesian product with a solid space. So for example, A2 designates the vertex-angle of a polygon, the edge-angle of a polyhedron, and so forth. It is the margin-angle, or fraction of space where that the polytope occupies at the margin.


 * A0: This represents the interior of a polytope, and so the value is always 1.


 * A1: This represents a point on the interior of a face (or facet), the angle is always 1/2 or 0:60.00.00


 * A2: This is the angle formed between two adjacent faces, or the angle around a surtope of N-2 dimensions.  It can be thought of as a point at the centre of a circle, with two radiating arms.  The angle is that between the arms.  To bring A2 to solid, one must take the cartesian product of this and a solid space of n-2 dimensions.  It can be measured in degrees or radians.


 * A3: The A3 is the third-angle, where the product is between a vertex-in-a-sphere and the solid section of a surtope of n-3 dimensions.


 * An: The final angle is the portion occupied by the vertex-verge (that is, the vertex and its aroundings),

Method of calculation

 * A0:  This is always 1.  It does not matter if the area is denser than unity.
 * A1:  This is always 1/2.  The 'angle of a line' is over the end of it, from occupied to empty, so it is 1/2.
 * A2:  This is derived from the supplement of the face normals, where it is not easy to calculate the angle between faces.
 * A3:  Angles in A3 are given by the spherical excess or spherical defect.  Hyperbolic tilings are comprised of polygons that add to more than the full circle, the excess here is a measure of how much space the vertex would occupy.
 * A4+:  No algorithm is at hand for angles A4 and higher.  Instead, we resort to disecting lattices and other euclidean tilings.

Radians and Degrees
Little use is made of radians and their solid measures, since polytopes are integeral and radians are not. The normal angle used is the circle.

The circle of 360° is used for A2, and for measures of A3, the angles of excess, where a sphere is 720° E are used.

For all angles, the unit is the whole sphere, divided into base 120, and preceeded by a colon, thus 0:30 is a right-angle, 0:15 is an octant, and so forth. This is the unit i normally do hand calculations in.

The full circle is coherent to that the product of angles is the angle of the product. Thus the octagon (3/8) times a line (1/2) gives a octagon prism (3/16).

Solid Sextant
The solid sextent is the vertex-angle of a simplex in each dimension. It is formed by radii and chords being equal. Since this measures the simplex vertex-angle, and from this, we might calculate the cross-polytope solid angle, some time is spent looking at this feature.