# -: The Rounds: Around and Surround :-

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The rounds (around and surround) are given quite distinct meanings, because this is needed in higher dimensions. As with everything else, we mind the etymology of these tings in the global sense.

## Surround

Surround continues the meaning and connexion to surface. That is, a surrounding suggests acting as a surface. Given that we take surface as a cloth that divides a solid space, then surround happens in the space of the solid.

When one surrounds something, one treats it as living in some meaningful space, eg a plane, and the surrounds happen in the plane.

A surface or periphery surrounds a figure, regardless of its relative dimension. For example, the four edges surround a square, even if it should be in three or eighteen dimensions. This is because the solid space of the square is the hedrix it falls in, and correspondingly the four lines do divide the hedrix.

One might talk of Surroundings or figures in the solid space near something.

### In and Out

Interior and exterior happen in the solid space of a figure, and thus part of the surrounds of it. In this sense, they preserve their common meanings.

A point in the plane of a hexagon is either inside it or outside of it. (or part of its surface, which is a kind of inside/outside thing. A point not in the plane is neither inside nor outside it.

## Around

Around happens in a space where the figure's solid space cuts at a point.

From the aroundings one can draw a line through any point of the figure in the arounded space. Thus, we do not have to grace ourselves in crossing the surface to make our way to the interior of a polygon from the arrounds.

For the arroundings, there is no notion of relative direction, unless for instance, the arrounds is a line, whence one can be above or below the plane, or this side or that.

Ortho- is taken to mean a space notionally at right angles to a figure. In practice, we use ortho- to mean any point-crossing space, and right ortho- to infer additional perpendiculary.

Note also that perpendicular can imply a shared section (eg two planes can be perpendicular), but orthogonal implies the crossing is at a point. Anything perpendicular to a plane must be contained in a line: eg a point or a line.

The orthospan of two spaces includes all points that lie in like hedrix that contain a line segment from each of the spaces. Such lines cross at the point of crossing, or orthopoint.

An orthosurtope is a surtope that is orthogonal to a given surtope. In practice this means the corresponding surtope in the figure.

### Verges and Approaches

Something like a blade has both a surround and an around. The surround of a blade is its tip, measured as a fabric. A pin has a telic tip, while a knife has a latric tip. An iron, of course has a hedric tip.

The arrounds is measured orthogonal to the tip. So a pin has a choric section, a knife has a hedric section, and an iron has a latric section.

A latric section can be considered as a ray radiating from the tip. We likewise can talk of hedrorays, chororays, as solid radiants going through a solid hedrix or chorix. A cube-corner forms a chororay, or point in 3D, while an cube-edge forms a hedroray, or 2D ray.

A generalised blade is a verge, or the surrounds to an open cloth. The verges to a surtope are its approaches, or arounds. Note that approaches refers to the incident surtopes (eg the two faces and interior of a cube), while the arrounds is all of the orthospace (ie outside the cube as well).

A pin becomes a telic chororay. A knife becomes a latrix hedroray, and the iron becomes a hedric chororay, ie [tip-cloth + section-ray]

### Angles

Angles refer to the fraction of the right orthospace occupied by the figure. Regardless of the dimension, two faces meet at a margin, and the corresponding angle is a hedroray, or circle-wedge.

We call this the margin-angle, although one sees 'dihedral angle' in this usage. All angles are considered as the fraction of space taken by the ray, measured against all-space.

For practical purposes, one might divide these angles into powers of 120, and express these accordingly in that base.

It is common practice to use radians. Apart from the fact that radians split in three and higher dimensions into tegmic and prismic radians, the measures of π have little to do with geometry, and serve some weird expression in the calculus. Euclid mastered geometry without a knowledge of pi, and so shall we.

I had found the solid angle of the polytope hight {3,3,5/2}, long before i had come to understand that 2π² cubical radians made the glomochorix (or surface of the 4-sphere. One measures radians by the area of surface the angle cuts with a sphere in euclidean space. In non-euclidean space, this is dodgy to say the least.

### Cornering

For the polygloss, cornering is a relation of mounting, where one surtope completely contains another. It is a bi-directional relation: if a point corners on a polygon, the polygon corners on a point.

One might say that the vertex of a dodecahedron is a corner to each of the three faces around it. The vertex is then described as having three angulohedra, the sum of all the vertices angulohedra is exactly the same as the sum of all of the surhedra's angulotela (corner vertices).

Such might be demonstrated by drawing lines from each surhedra to its vertices. In effect, we count these lines as incidences, starting at one class of surtope, and ending at another.

A direct incidence happens between a surtope and its faces. The diagram of all direct incidences forms a Hasse diagram, or incidence antitegum, because the shape formed by the surtopes and direct incidences form the vertices and edges of the antitegum.

One makes an incidence matrix, by listing surtopes as rows, and angulotopes as columns. Where a row crosses a column, one notes how many times that angulotope is incident on a surtope. One can do individual or type incidence matrices, based on every surtope separately, or grouped by types.

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