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**cantic***-
Norman Johnson's stem for the second marked node, so also
*cantellated*,*cantetruncate*. **#-cantellated***-
A process of deriving a figure, by taking the mid-points of the edge of a
#-rectate. In four and higher dimensions, this produces a set
of distinct uniform figures.

The Dynkin symbol for a cantellated figure marks every node connected to the rectate node, but leaves the rectate node unmarked. **cantitruncate***-
The result of truncating the #-rectate of a figure. In four
and higher dimensions, this produces a uniform figure.

The Dynkin symbol for a cantellated figure marks the rectate node and all nodes directly connected to it. **captian***- The notional unit, to which all units of a company share all vertices, edges, and surhedra to. The unit is one of the army terms.
**cartesian***-
A product in Euclidean space, relying on parallel straight lines.

One can implement the prism and comb products through the Cartesian product, but the prism and comb products give rise to different surtope polynomials. Note that the comb and prism products exist in non-Euclidean geometries, where the cartesian product does not. **catalan***-
A class of margin-uniform figures, which are not platonic,
tegums, or antitegums. In practice, these are defined
as the duals of the Archimedean figures.

An inclusive or outer sense is also known: it corresponds to the duals of uniform figures. The term used for this sense is margin-uniform. **cavity***-
An interior region of a shape, that is disconnected from the outer surface.

Such become proper holes in , Treating cavities and dyads as proper holes gives a rich insight into the nature of holes at a lesser dimension. **cell ***-
A solid surtope. The sense is that a cell is taken to be as a bubble in
a foam covering .
- The sense of surchoron or 3d surtope comes from regarding the surface of a 4d polytope as a 3d foam.

**chord***-
The straight line drawn between two points on a curve.

In the theory of circle-drawing, a connecting line of zero-curvature.

A shortchord is the base of a triangle formed by two consecutive edges of a polygon. **CHEVN letters ***-
A code formed by the last five dimensions of surtope. The letters
derive from sur
**C**horon, sur**H**edron,**E**dge,**V**ertex,**N**ulloid. The normal style is to write the highest dimensions to the left. **chorix ***-
A 3-manifold or cloth of three dimensions. One might cut solid chora
or polyhedra from the fabric of a chorix. See hedrix

Note that it is piecewise 3d, and can contain things like loops and wormholes in it. **choron ***- A mounted 3d polytope, or a 3d 'hedron'
**circle***- The locus of points equidistant from a real point.
**circle-drawing ***-
An approach to non-Euclidean geometry by treating all circles as Euclidean.

The geometry is inspired by the sphere, where the surface is Spheric, and internal chords might be taken in terms of some Euclidean space. **circum***- The sense of circum is the same as that of surround, normally by means of touching the most distant points.
**class ***-
A number system of class N is a discrete set that resolves co-metrically and
ly over N dimensions. The usual examples are the Zp deriving from
the span of chords of a polygon, which lead to the N solutions of J(2p). Some
other systems have been given class numbers.

The set by closing '/n' (Bn) is always 'class 2'.

Fractions are held to be higher than class-2, but they have many class-2 features.Class-2 can be sequenced through the entire set uniquely.

**Clifford ***-
A name allocated to non-planar equidistants in S3, or the rotation of a glomocloron
(4-sphere), such that perpendicular rotations are at the same speed. The effect of
this is that every point moves on a great circle around the centre of the sphere.

One can show clifford parallels, by noting that every even dimension corresponds to a Complex-Euclidean of half as many dimensions, and then introduce a time-scalar wt, when multiplied by every point, causes it to go in a great circle around the centre.

The natural mode of planets is to equalise the energy in each mode of rotation, is to tend towards clifford rotations.

Clifford rotations come in left and right parity, corresponding to left and right product of quarterions.

See also Lissajous, Wheel rotations. **coherent ***-
A series of measures, where the units are defined with unit proportionality.

For example, the representation of angle as a fraction of space is coherent to the function of orthoexpansion: the vertex of an octagon and the edge of an octahedral prism both have an angle of 0:45 = 3/8.

The volumes measured in the prism, tegum and spheric scales are coherent to the respective products: the x-volume of an x-product is the product of the x-volumes of the bases.

Different names are given for the different coherent products in one dimension, even though these are equal in measure. However, in one system of using names, the name refers to the resulting dimension, rather than the scale (ie cubic X is X as a 3-volume, regardless of what X is, eg one might convert cubic acres into cubic gallons). Accordingly, the linear, dyadic and digonal acres are not equal, while the square, rhombic and circular acres are equal (to one acre).

See prism product, tegum product spheric product. **cohort***-
A set of separate polytopes, derived by blending on the same sets of faces.
Such are common among the uniform polychora discovered by Jonathan Bowers.

Among the seventy-five uniform polyhedra, there are 13 cohorts, covering 38 polytopes. These share common vertices and edges, and a large number of faces as well. A set might consist of figures with faces AB, AC and BC. **colonel***-
The leader of a regiment. This is the polytope that the regimental
members share their vertices and edges with.

See also army, latroframe **comb ***-
A product derivable from the regular tiling of measure polytopes.

In Euclidean space, this is the Cartesian product applied to tilings, but it also applies in spaces where the cartesian product does not exist, such as hyperbolic space.

The word is a backform from*honeycomb*.

The surtope polynomial is the product of the polynomials of the bases, ignoring both the bulk and nulloid terms.

The comb product of polytopes is the cartesian products of their surfaces. For example, the comb product of 2 pentagons gives a connected sheet in 4d of 25 squares. This is also hight hotel. **comb product ***-
A surtope product of two figures, excluding both the
nulloid and bulk. It reduces dimension by every application.

The comb-product of tilings gives a tiling, but because it is really only meaningful to take cartesian products in horic space, the comb product is noted there. The comb product of a horogon gives rise to the infinite family of cubics in every dimension. **Company***- A set of polytopes having the same set of vertices, edges and surhedra. The figure that these are notionally shared with hight captian.
**complex polytope***-
A class of polytope derived by relaxing the dyadic rule, and allowing more
than two N-surtopes to be mutually incident on the same N-1 and N+1 surtope.

In practice, the margin figure becomes a complex multiplication, rather than a reflection. **compound***-
A polytope for which one can not access all faces by traversing the margins.

While such figures have a unity of purpose, the surface is now falls into separate parts, giving the impression that it is a composite of separate figures. **composite***- A number of separate polytopes considered together. Normally, these are solid in the same subspace, but have no 'unity of being'.
**concentric ***-
Having the same centre.

When horopoints and bollopoints are taken into consideration, this gives rise to an equidistant curve situation.

Concentricness is one of the two aspects of parallelism. **#-content***-
The measure of extent of a #-manifold, made by one or more figures.

One might derive a specific term for n-content from the n-manifold as,

2-content hedrage, 3-content*chorage*&c. **Conway-Kepler rule***- A rule that says in polyhedra, that expand = ambo ambo. In higher dimensions this does not give a runcinate but a cantellate. In 3d, it happens that the cantellate is the same as the runcinate.
**Conway Operators***-
A series of surface operators, particularly for polyhedra.

One treats the flags of the source polytope as if it were a Wythoff mirror-group, and constructs the wythoff mirror-edge and mirror-margin figures accordingly.

Such figures are pennant-transitive on the flags of the source figure.V F D name polygloss x4o3o - - d dual - o4o3x t

k- truncate

kistruncate

apiculatex4o3o

m4o3oa

j

pambo

join

propellerrectate

surtegmate

-o4x3o

o4m3oe

o- expand

orthoruncinate

strombiatex4o3x

m4o3mb

m- bevel

meta- x4x3x

m4m3ms

g

rsnub

gyro

reflect- s4s3s

**Conway-Thusrston ***-
John Conway implemented şe Orbifold notation to describe William Thurston's list of
groups on şe plane. Şis consists of şe following devices.
- numbers, being a cone or rotation only
- * An asterix, representing a chain of mirrors
- numbers, represting şe angles pi/x between şe mirrors.
- * Furşer disjoint mirror-chains
- x miracle, representing a mirror-less reflection
- o wander, representing a slideless non-rotation.

- <> Şe mirror runs along şe edge, to make it a mirror-margin.
- [] Şe edge runs to a reflected form of şe vertex (ge mirror-edge).
- () Şe edge runs to a rotated form of şe vertex (eg skew-edge)

**copycat***- Jonathan Bower's term for a pair of figures that have the same periform. Examples are known in four dimensions.
**corner ***-
Incident on. A corner is a surtope as seen from within another surtope.
For example, a pentagonal face of a dodecahedron has five corners. A vertex
is three corners, one to each of the incident pentagons. We could say both
that the vertex is a corner surtope of a pentaton, and that the pentagon
is a corner surtope of a pentagon.
- A
**surcorner**is where the corner is part of the surtope, - An
**orthocorner**is where the corner contains the surtope. For example, the vertex is corner to three pentagons, and hence the pentagons being incident on the corner, become orthocorners.

- A
**cotangent ***-
Two isocurves are cotangential, if the line drawn from the centres of two
isocurves cross the surface at the same point.

Any straight line passing through the centre is perpendicular to the surface.

Cotangency is one of the two aspects of the theory of parallels. Euclid's fifth postulate is a form of the general case of 'if circles are cotangent at K, then any circle crossing these at K makes the same interior angle with any of the cotangent circles.' **countable ***-
A class of gauge infinity, where every member is separately instanced.
Note that the counting can be done by members of a large organisation.

While one might not readily count to a million, it is not inconcievable that a large organisation, such as a tax office, can instance a million separate files.

A smaller gauge-infinity might be the permutated infininty, that is, the size of a set where all permutations have been instanced. For our example above, a set of a million records corresponds to the complete instancing of all six-digit numbers, and so six would be the permutation infinity.

The common mathematical rule of countable is one for which an integeral value might be allocated. Such a number is catalogued here as discrete infinities (ie one for which one can between any two members, find a instance that is not a member). **Coxeter-Dynkin construction ***- A construction of polytopes by way of using symmetry generators. The method is of great utility, since any subset of generators makes also a subgroup.
**Coxeter-Dynkin diagram**-
Another name for the Dynkin Symbol.

Coxeter derived the symbol as a means of describing reflective groups. It was after he read Wythoff's 1912 reconstructions of Stott's construction that he realised that polytopes could also be symmetry motifs.

E B Dynkin independently discovered the graph later, in relation to families of Lie groups. **crind product**- A coherent radiant product, based on the rss() or root-sum-square product. The crind power of a unit line, gives rises to unit-spheres, both of nature and of measure.
**cross-polytope***-
Coxeter's name for the family of regular polytopes formed by the tegum product.

In practice, the cross polytope family may be used as a basis of coherent units for the cross and pyramid products, where the diagonal of the cross is taken as a unit.

Cross-polytopes are the first polytope of bouyland, and one might describe the regular form as a regular poly-bouy tegum. **cube***-
The name for the tri-slab prism, or measure polyhedron.

In Non-euclidean geometry, the cube shape is pressed out of alignment with the measure: that is, a cube is no longer the measure. One must read that the cube defines a measure of chorix equal to the content of a zero-curvature or horospace cube. **cubic***-
The adjective describing both cubes and the tiling of cubes, four at a margin.

When used as a noun, it refers to the tiling of measure-polytopes in all dimensions.

*Semicubic*refers to the symmetry of alternate vertices or cells of the cubic.

*Half-cubic*refers to the symmetry of alternate vertices of the measure polytope

*Quarter Cubic*refers to the symmetry of alternate vertices and cells of the cubic. **cupola***-
In three dimensions, this refers to the lace prism oxPxx&#x. In higher
dimensions, it refers to lace-prisms, where the marked nodes of one base
are a subset of the bases of the other.

A*cuploid*is derived from a oxP/2Dxx&#x, by removal of the doubly- wound xP/2Dx, and reconnecting squares to triangles as appropriate. **curvature ***-
Curvature is an intrinsic measure of space, which equates to 1/R^2. The measure can be
found by comparing the ratio of the circumference of a circle, against the circumference
of a circle tangent at the perimeter and the centre of the first. The sign of this
number subtracted from two, is the curvature of the space.

A flat space is one whose curvature is that of surrounding space. This can be made by noting that if A)B, and A)C and B)C, meaning that there is no space between faces A and B, and B and C, and A and C, when each is pressed against the other, then the nature of the curve formed by the surface of A and B and C, is that of isospace: ie flat.

In Euclidean geometry, the flat surface is the same as the special surface of zero curvature. In non-euclidean geometry, the flat space is a special case of an equidistant, such as a line of lattitude, and the zero-curvature is a special kind of curve. **CZn, CZZ***-
The set CZn is the complex cyclotomic numbers, defined by the span of 1^(1/2n).
This intersects the reals in the set Zn, the span of chords of a {N}-gon.

Te set CZZ is the union of all CZn. No proper fraction is a member of the set CZZ.

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