# -: Þe Rounds: Around and Surround :-

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Þe rounds (around and surround) are given quite distinct meanings, because þis is needed in higher dimensions. As wiþ everyþing else, we mind þe etymology of þese tings in þe global sense.

## Surround

Surround continues þe meaning and connexion to surface. Þat is, a surrounding suggests acting as a surface. Given þat we take surface as a cloþ þat divides a solid space, þen surround happens in þe space of þe solid.

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

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

One might talk of Surroundings or figures in þe solid space near someþing.

### In and Out

Interior and exterior happen in þe solid space of a figure, and þus part of þe surrounds of it. In þis sense, þey preserve þeir common meanings.

A point in þe plane of a hexagon is eiþer inside it or outside of it. (or part of its surface, which is a kind of inside/outside þing. A point not in þe plane is neiþer inside nor outside it.

## Around

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

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

For þe arroundings, þere is no notion of relative direction, unless for instance, þe arrounds is a line, whence one can be above or below þe plane, or þis side or þat.

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

Note also þat perpendicular can imply a shared section (eg two planes can be perpendicular), but orþogonal implies þe crossing is at a point. Anyþing perpendicular to a plane must be contained in a line: eg a point or a line.

Þe orþospan of two spaces includes all points þat lie in like hedrix þat contain a line segment from each of þe spaces. Such lines cross at þe point of crossing, or orþopoint.

An orþosurtope is a surtope þat is orþogonal to a given surtope. In practice þis means þe corresponding surtope in þe figure.

### Verges and Approaches

Someþing like a blade has boþ a surround and an around. Þe 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.

Þe arrounds is measured orþogonal to þe 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 þe tip. We likewise can talk of hedrorays, chororays, as solid radiants going þrough 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 þe surrounds to an open cloþ. Þe verges to a surtope are its approaches, or arounds. Note þat approaches refers to þe incident surtopes (eg þe two faces and interior of a cube), while þe arrounds is all of þe orþospace (ie outside þe cube as well).

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

### Angles

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

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

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

It is common practice to use radians. Apart from þe fact þat radians split in þree and higher dimensions into tegmic and prismic radians, þe measures of π have little to do wiþ geometry, and serve some weird expression in þe calculus. Euclid mastered geometry wiþout a knowledge of pi, and so shall we.

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

### Cornering

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

One might say þat þe vertex of a dodecahedron is a corner to each of þe þree faces around it. Þe vertex is þen described as having þree angulohedra, þe sum of all þe vertices angulohedra is exactly þe same as þe sum of all of þe surhedra's angulotela (corner vertices).

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

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