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**Above-Below Tiling ***-
Any of a series of tiling characterised by walls that fall in a plane that
has an A and B side. The effect of such is that all surtopes can be
designated by a series of A's and B's.

Such often have for a vertex-figure, a simplex-antiprism, and a dynkin symbol xPo[3o]Px or xPo[3o]PoQz, where xPo is the edge of the top and bottom bases, and xQx is the lacing.

The only known AB polytope is the simplex runcinate. If the vertex figure is an AB figure, so is the tiling. **Across-Space ***-
The space formed by subtracting height and forward. In three dimensions, this gives
a line, but there is no guarentee that this will continue to happen in higher spaces.

In four dimensions, the across space is two-dimensional, there is no more any way of setting a set of axies, then there is of laying clocks face-up on the floor, that the twelve-position ought point a given direction. **All-Space ***-
The totality of space under consideration. For example, the allspace of
two-dimensions is the Euclidean plane.

When a larger dimension is invoked, this is called hyperspace. **Altitude ***- A term meaning height. In products that add dimensions to the product, such as the pyramid and lace products. It is quite possible for the total dimension of a polytope to be altitude.
**ambiate ***-
John Conway's term for rectate. As
**ambo**, it is also one of his operators.

In 3d it corresponds to placing the vertex in both the centre of the edges of a polytope and its dual.

In higher dimensions #-ambiate means placing the vertex in the centre of the #-surtopes: so a bi-ambiate means the vertices fall in the centres of the surhedra.

Ambi means*both*, the sense here is that the centres of the edges of a polyhedron correspond to those of its dual: the process pf ambiation corresponds to setting the vertex at the centres of both kind of edges. In higher dimensions, the sense of both is less obvious. **angle ***-
Derived from the Latin for
*corner*, or a measure in relation to its content.

**surface-angle**is measured in relation to the surface of the sphere. An arc is held to be a length, and the surface of the sphere is measured in superficial units of arc. Natural, tegmal and degrees follow this form.

**content-angle**is measured in relation to the fraction of the sphere interior. While functionally not distinct to surface-angle, it has a different dimension: an arc is held to be a pie-slice rather than an arc. The tegmal, twelfty and metric scales might be treated in this way.

Angle prefixed by a number, eg**N-angle**, refers to*content-angle*, that is a 4d-polytope has a 4-angle as the solid-angle at the vertex.- - Natural tegmal Degrees twelfty metric circle C2 2π rad 2 π 360° 100° 400 grad sphere C3 4π sr 8 π t2r 720° E 100° 8000 - C4 2π² p3r 12 π² t3r 64800 1 0000 160000 - C5 8π²/3 p4r 64 π² t4r 86400 1 0000 32E5 - C6 π³ p5r 120 π³ t5r 5832000 100.0000 64E6 - C7 16π³/15 p6r 768 π³ t6r 6220800 100.0000 128E7

Natural is prismatic S-radians, the unit over 3d is pSr.

Tegmal is tegmatic S-radians, the unit being tSr: 1 pSr = S! tSr

Degrees is natural, with π=180. Divided 60-wise or decimally

Twelfty is solid space divided twelftywise, multiplied by 120 for each 2 dimensions

Metric is 20 raised to the solid dimension, corresponds to decimally divided solid right angles.**In practice**, the following apply.

radians are given without unit.

Degrees are deg \ 60 minutes \ 60 seconds \ decimally. or deg \ decimals.

Twelfty C2 \( deg \*twe:*100)( min \*twe:*100)( sec \ twelftywise.*twe:*100)

For C3, Astronomers use square degree = π/180 degE.

For C3, degreeE is the spheric excess, measured in degrees, divides as degrees.

For C4, the usual style is C4 \ 120 s \ 120 f. s and f correspond to the angle of the symmetries of {3,3,3} and {3,3,5} respectively.

For C5 and higher, there is no established unit: the twelfty-scale is used. **Angluotope ***-
A corner surtope. For example, a single vertex of a dodecahedron is the
corners of three different pentagons. In this sense, we see the corner
relation gives rise to the same vertex being part of three different
surtopes.

The relation of incident on is also allowed. It is correct to refer to a the three pentagons as*angluohedra*incident on a vertex. **Angluotope Matrix ***- A matrix formed, by writing in each column, the name of a kind of surtope, and each row, each kind of angluotope. Where a row crosses a column, one writes the number of incident angluotopes on a surtope. Where the angluotope and surtope are the same, one writes the count or proportional count of the named surtope.
**anticomb ***- A tiling of antiprisms, with an antitegmal vertex. Such things arise from the real projection of the complex polygons. Like combs, anticombs are self-dual.
**antiprism ***-
A lace prism formed on duals. The top and base are
connected by a series of faces, being the pyramid product of
a and its orthotope in the dual.

In the sense that a polygon anti-prism resembles a drum, the top and bottom faces are sewen together by lacing

The**antiprismic sequence**is the sections parallel to the bases. This gives rise to the runcinates.

The tegum product of antiprisms on P,Q,... is an antiprism on the pyramid product of P,Q,... **antitegum ***-
A polytope made from the intersection of point-pyramids of duals, the
apex of each being in the centre of the base of the other.

Every surtope of an antitegum is an antitegum, formed by a surtope and its dual.

The prism product on antitegums on P,Q... is an antitegum on the pyramid product of P,Q,...- The
**antitegmal sequence**are the slices perpendicular to the main axis. These represent the truncate by descent of faces. - The verticies of an antitegum on P correspond to the surtopes of P, including the nulloid and content. The edges correspond to direct incidences, and because of this, one might construct an incidence antitegum, where two surtopes are incidence on each other, if they lie at ends of an antitegmal axis. According to Dr Richard Klitzing and Prof Norman Johnson, this figure corresponds to the "Hasse diagram" for the polytope P.
- The
**antitegmal cluster**is the result of replacing faces of a polytope by antitegums. What happens to the surface is that it gets replaced by antitegums.

- The
**apeiro- ***- This is often used for aperi-, although i can not find any trace of it in any dictionary. See aperi.
**apeiron**- The sense of this greek word is the openness of the sea or desert, such as being unfetted by fences. A sphere is boundless but finite. It serves here to use the stem /peri/ to denote a boundary in space, rather than unlimited number.
**aperi- ***- Without a periphery. The sense is that one lays tiles in a plane, without leaving a periphy in the plane: that is a tiling.
**aperitope**-
A tiling. [Aperi] means
*without end*. The sense here is that all-space for which the tile is solid, is covered by tiles.

The tiles hight cells, the sense is that of a foam of cells. Cells are separated by walls.

The whole of space hight surcell. It functions like surface, except that it does not divided.

An aperigon is simply a line marked into equal segments. Where one specifically requires a*Euclidean*line, one should use horogon. **apiculate ***-
To raise to a peak. In polytope terms, it means to set a pyramid on the face
of: for example, an apiculated dodecahedron is made by attaching pyramids to
its faces, to give sixty faces.

In higher dimensions, apiculation amounts to replacing a surtope by a pyramid of the surtope and its orthosurtope. A bi-apiculate replaces edges/margins by this pyramid product. **approach ***-
The orthogonal section to the surtope. This reduces the dimensionality of
higher incidences by the surtope's dimension. Sections through the approach
give rise to the surtope figure

The face approach is a line, passing normal to the surface. In the simple case it sets in and out, but when the surface is allowed to cross, the notion of out-vectors &c come to play.

The margin approach is a pair of rays, separated by the margin angle.

The local shape of an approach is a verge or ray. **Archifold ***-
A notation devised by John Conway and Chaim Goodman-Strauss for describing
the infinitude of bollohedra. The notation works by
numbering the arms or edge-ends in a vertex symmetry, and then
noting which edge-arm connects to which other, and whether a transit down
that line reverses the direction-numbering.

The term combines*Archi*medean and orbi*fold*. **Archimedean Figures ***-
The Edge-Uniform figures, not being platonic,
prisms or antiprisms.

The duals of these are the Catalan figures. **area ***-
In 3D, this term is taken to mean extent of 2D. At the moment it is best
avoided, or used for extent of dividing space.

For extent of 2D space, use hedrage.

One might use surface content for facing-extent. **army ***-
The set of polytopes that have the same vertices, or teeloframe.
The concept is used by Jonathan Bowers in his discovery of the uniform polychora
in 4d and higher. The finer divisions share common edges, surhedra &c as well.

A regiment has the same vertices and edges as it colonel

A company has the same vertices, edges, and surhedra as its captian

The dual concept is a navy, the set of polytopes that have the same face, margin, &c as its leader **arm ***- An end of an edge, incident on a vertex in a polyhedron. The term occurs in archifolds. See also sill.
**around ***-
The term is used in the sense of in the space orthogonal to.

One might wind cotton around a spool, or dance around the maypole, in the sense that one is not in the alignment of the spool, or the maypole. For the sense of enclosing the solid space, see surround. **aroundings**-
The arroundings are the elements that are parallel to or equidistant
from something.

For example, the arroundings of a road are the verges of it. See also approach, verge. **askew***-
A symmetry, arising in a polygon, etc, where a hyper-rotation occurs in a
marginoid. What this does is to flip the polytope in hyperspace, so that
in terms of the space it lies, the action
*functions*like a mirror, but connections to the polytope are now on different arrounds to the polytope. **askew marginoid ***-
A margin or pseudo-margin where the out-vector reverses. Such are very
common in binary polytopes, giving these a checker-pattern
surface. See also XOR, although askew margins are not restricted
to binary polytopes.

Askew margins are how non-orientable surfaces might be made to contain a volume: A surface is non-orientable, if there exist circuits that cross an odd number of askew margins. **asterix ***-
A figure, formed by lines crossing or radiating from a point, in much the
same manner as the + or * asterix. The coordinate system of a set of
vectors form an asterix.

Examples of asterix are {4/2}, {6/3} and {8/4}, all of which occur in symmetry groups (eg {P,P,4/2:}, {6,6/2,6/3} and {8,8,8/4}.

See also eutactic asterix.

**Note:**One should not confuse asterix (essentially lines crossing at a point), with star (ie extension of surtopes of a core until they close again. **atom-node***-
Nodes, placed on the dynkin graph, which represent different sets of vertices.

An example would be applied to a Euclidean tiling, which might show the locations positions of different atoms in a salt, such as o4o3(Na)A(Cl), which places atoms Na, Cl at alternate vertices of a cubic.

n other example might be to say the vertices of the rhombo-dodecahedron o3m4o, lie at aq3o4a, that is, an octahedron of edge q (sqrt2) and a cube of edge x (1) **atop notation***-
A notation invented by Richard Klitzing, where the parallel layers of a lace-tower are
given in a form seperated by the parallel-sign markers,
. Superceded by Lace-notation. **azythmal***-
Any of the projections that map, for some point, (r,θ) onto (f(r),θ), where
θ is the direction through the local sphere.

Because of this relation, one could not detect if one is standing in nature or any of the azythmal projections

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