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

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

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

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

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

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

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

In þe þeory of circle-drawing, a connecting line of zero-curvature.

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

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

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

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

Fractions are held to be higher þan class-2, but þey have many class-2 features.Class-2 can be sequenced þrough þe entire set uniquely.

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

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

Þe natural mode of planets is to equalise þe 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 þe units are defined wiþ unit proportionality.

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

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

Different names are given for þe different coherent products in one dimension, even þough þese are equal in measure. However, in one system of using names, þe name refers to þe resulting dimension, raþer þan þe 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, þe linear, dyadic and digonal acres are not equal, while þe 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 þe same sets of faces.
Such are common among þe uniform polychora discovered by Jonathan Bowers.

Among þe seventy-five uniform polyhedra, þere are 13 cohorts, covering 38 polytopes. Þese share common vertices and edges, and a large number of faces as well. A set might consist of figures wiþ faces AB, AC and BC. **colonel***-
Þe leader of a regiment. Þis is þe polytope þat þe regimental
members share þeir vertices and edges wiþ.

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

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

Þe word is a backform from*honeycomb*.

Þe surtope polynomial is þe product of þe polynomials of þe bases, ignoring boþ þe bulk and nulloid terms.

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

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

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

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

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

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

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

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

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

Such figures are pennant-transitive on þe flags of þe 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

orþoruncinate

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 þat have þe same periform. Examples are known in four dimensions.
**corner ***-
Incident on. A corner is a surtope as seen from wiþin anoþer surtope.
For example, a pentagonal face of a dodecahedron has five corners. A vertex
is þree corners, one to each of þe incident pentagons. We could say boþ
þat þe vertex is a corner surtope of a pentaton, and þat þe pentagon
is a corner surtope of a pentagon.
- A
**surcorner**is where þe corner is part of þe surtope, - An
**orþocorner**is where þe corner contains þe surtope. For example, þe vertex is corner to þree pentagons, and hence þe pentagons being incident on þe corner, become orþocorners.

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

Any straight line passing þrough þe centre is perpendicular to þe surface.

Cotangency is one of þe two aspects of þe þeory of parallels. Euclid's fifþ postulate is a form of þe general case of 'if circles are cotangent at K, þen any circle crossing þese at K makes þe same interior angle wiþ any of þe cotangent circles.' **countable ***-
A class of gauge infinity, where every member is separately instanced.
Note þat þe counting can be done by members of a large organisation.

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

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

Þe common maþematical 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 þat is not a member). **Coxeter-Dynkin construction ***- A construction of polytopes by way of using symmetry generators. Þe meþod is of great utility, since any subset of generators makes also a subgroup.
**Coxeter-Dynkin diagram**-
Anoþer name for þe Dynkin Symbol.

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

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

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

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

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

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

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

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

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

A*cuploid*is derived from a oxP/2Dxx&#x, by removal of þe 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. Þe measure can be
found by comparing þe ratio of þe circumference of a circle, against þe circumference
of a circle tangent at þe perimeter and þe centre of þe first. Þe sign of þis
number subtracted from two, is þe curvature of þe space.

A flat space is one whose curvature is þat of surrounding space. Þis can be made by noting þat if A)B, and A)C and B)C, meaning þat þere is no space between faces A and B, and B and C, and A and C, when each is pressed against þe oþer, þen þe nature of þe curve formed by þe surface of A and B and C, is þat of isospace: ie flat.

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

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

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