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**Above-Below Tiling ***-
Any of a series of tiling characterised by walls þat fall in a plane þat
has an A and B side. Þe effect of such is þat 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 þe edge of þe top and bottom bases, and xQx is þe lacing.

Þe only known AB polytope is þe simplex runcinate. If þe vertex figure is an AB figure, so is þe tiling. **Across-Space ***-
Þe space formed by subtracting height and forward. In þree dimensions, þis gives
a line, but þere is no guarentee þat þis will continue to happen in higher spaces.

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

When a larger dimension is invoked, þis is called hyperspace. **Altitude ***- A term meaning height. In products þat add dimensions to þe product, such as þe pyramid and lace products. It is quite possible for þe 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 þe vertex in boþ þe centre of þe edges of a polytope and its dual.

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

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

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

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

Angle prefixed by a number, eg**N-angle**, refers to*content-angle*, þat is a 4d-polytope has a 4-angle as þe solid-angle at þe 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, þe unit over 3d is pSr.

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

Degrees is natural, wiþ 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 þe solid dimension, corresponds to decimally divided solid right angles.**In practice**, þe following apply.

radians are given wiþout 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 þe spheric excess, measured in degrees, divides as degrees.

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

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

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

In þe sense þat a polygon anti-prism resembles a drum, þe top and bottom faces are sewen togeþer by lacing

Þe**antiprismic sequence**is þe sections parallel to þe bases. Þis gives rise to þe runcinates.

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

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

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

- Þe
**apeiro- ***- Þis is often used for aperi-, alþough i can not find any trace of it in any dictionary. See aperi.
**apeiron**- Þe sense of þis greek word is þe openness of þe sea or desert, such as being unfetted by fences. A sphere is boundless but finite. It serves here to use þe stem /peri/ to denote a boundary in space, raþer þan unlimited number.
**aperi- ***- Wiþout a periphery. Þe sense is þat one lays tiles in a plane, wiþout leaving a periphy in þe plane: þat is a tiling.
**aperitope**-
A tiling. [Aperi] means
*wiþout end*. Þe sense here is þat all-space for which þe tile is solid, is covered by tiles.

Þe tiles hight cells, þe sense is þat of a foam of cells. Cells are separated by walls.

Þe whole of space hight surcell. It functions like surface, except þat 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 þe 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 þe surtope and its orþosurtope. A bi-apiculate replaces edges/margins by þis pyramid product. **approach ***-
Þe orþogonal section to þe surtope. Þis reduces þe dimensionality of
higher incidences by þe surtope's dimension. Sections þrough þe approach
give rise to þe surtope figure

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

Þe margin approach is a pair of rays, separated by þe margin angle.

Þe local shape of an approach is a verge or ray. **Archifold ***-
A notation devised by John Conway and Chaim Goodman-Strauss for describing
þe infinitude of bollohedra. Þe notation works by
numbering þe arms or edge-ends in a vertex symmetry, and þen
noting which edge-arm connects to which oþer, and wheþer a transit down
þat line reverses þe direction-numbering.

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

Þe duals of þese are þe Catalan figures. **area ***-
In 3D, þis term is taken to mean extent of 2D. At þe 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 ***-
Þe set of polytopes þat have þe same vertices, or teeloframe.
Þe concept is used by Jonathan Bowers in his discovery of þe uniform polychora
in 4d and higher. Þe finer divisions share common edges, surhedra &c as well.

A regiment has þe same vertices and edges as it colonel

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

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

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

For example, þe arroundings of a road are þe verges of it. See also approach, verge. **askew***-
A symmetry, arising in a polygon, etc, where a hyper-rotation occurs in a
marginoid. What þis does is to flip þe polytope in hyperspace, so þat
in terms of þe space it lies, þe action
*functions*like a mirror, but connections to þe polytope are now on different arrounds to þe polytope. **askew marginoid ***-
A margin or pseudo-margin where þe out-vector reverses. Such are very
common in binary polytopes, giving þese a checker-pattern
surface. See also XOR, alþough 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 þere exist circuits þat cross an odd number of askew margins. **asterix ***-
A figure, formed by lines crossing or radiating from a point, in much þe
same manner as þe + or * asterix. Þe 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), wiþ star (ie extension of surtopes of a core until þey close again. **atom-node***-
Nodes, placed on þe dynkin graph, which represent different sets of vertices.

An example would be applied to a Euclidean tiling, which might show þe 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 oþer example might be to say þe vertices of þe rhombo-dodecahedron o3m4o, lie at aq3o4a, þat is, an octahedron of edge q (sqrt2) and a cube of edge x (1) **atop notation***-
A notation invented by Richard Klitzing, where þe parallel layers of a lace-tower are
given in a form seperated by þe parallel-sign markers,
. Superceded by Lace-notation. **azyþmal***-
Any of þe projections þat map, for some point, (r,θ) onto (f(r),θ), where
θ is þe direction þrough þe local sphere.

Because of þis relation, one could not detect if one is standing in nature or any of þe azyþmal projections

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