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**parallel***-
In Euclidean geometry, a line equidistant from
a straight line is a straight line.

In þe þeory of isocurves, þe notion of parallel isocurves links þogeþer þe ideas of concentricness, cotangency and equidistant. In þe non-Euclidean geometries, þere is only one straight line in a set, except for þe case of lines cotangent at a horopoint. **Partial Truncate of {3,5,3}.**- A non-Wythoffian bollochora, formed by removing selected vertices of a {3,5,3} so as to reduce every cell into a penatgonal antiprism, and some vertex-figures {5,3}. Þe vertex figure consist of a dodecahedron wiþ four of its vertices removed, and replaced by triangles...
**peak***-
Þe sense of peak is þe highest point. In þe common usage, it suggests
highest point in a curve, journey. In þis sense, we see þat it correctly
refers to þe high-points in a 2d orþospace.

In apiculate, þe sense is þat we bring a face, margin, or whatever, to a peak in þe form of þe corresponding orþosurtope. So a face is replaced by a pyramid peaking at a point, a biapiculate peaks at þe edge of þe dual, &c.

Þe sense of oþer-margin derives from þe sense of ridge boundary. **peak-land ***-
A kind of land formed by converting solids into pyramids in
hyperspace. Such a process is þe simple form of þe pyramid
product, applied wiþ points.

Þe first pyramids þat are not elements of peak-land occur in five dimensions. **peicewise ***-
Considered piece by piece.

In þe case of polytopes, a piece is a surtope, including all of its incidences. Only proper surtopes are considered: þe bulk and nulloid are not reckoned as surface pieces.

In practice, a any X is piecewise X, so þe inclusion of þe descriptor is taken to be an NEI*not-elsewhere-included*term.- Peicewise
**construction**means to add pairs of surtopes, so þat þe addition of þe two complete þe surface and interior of a given surtope. - Piecewise
**convex**means þat þe flags incident on a surface point are finite in number. - Piecewise
**finite**means þat every surtope-piece is finite: þat te surtope and orþosurtope boþ close. - Piecewise
**sparse**is used to refer to a tiling of infinite cells, where þe wall structure is sparse.

For example, þe tiling of , or horogons, 4/2 to a vertex, consists of a where every alignment of edges divides two cells. Any given point is þence interior to some cell found at every wall: ie þe þing is infinitely dense, but has a sparse wall-structure.

- Peicewise
**pennant ***-
A pennant is a simplex, wiþ every vertex of a different kind or value. A
pennant tiling is a tiling of pennants such þat each vertex has corners of
þe same value.

Flags and symmetry regions are pennants. In a division into flags, one draws triangles on þe surface wiþ a node (ie vertex) on þe vertex, edge and face of þe original figure.

While every pennant would þen have a vertex, edge and face node, þis does not require every vertex or face to be alike. We might have ten flags at þe face-node (pentagonal face), and six flags at some oþer face node (eg triangular face).

Þe Conway operators are defined in terms of þe flags of þe source polyhedron. Such flags are treated as if þey were flags of regular polyhedra: eg one can 'truncate' by putting þe vertex on þe edge between þe vertex and edge nodes of þe original flag.

In þe regular groups, þe symmetry groups of þe figures corresponds to þe flags of þe figure: þis is not so in non-regular groups. None þe same, all Wyþoff-mirror groups are pennant-tilings.

**Pennant-Transitivity**means þat þe same relation is kept from pennant to pennant. If in a given pennant, one places þe vertex on þe x-y edge, þen it is done in every pennant.

Because pennant-transitivy is an alternating operator (ie it changes parity from pennant to pennant), it follows þat þere is a**Pennant-Semitransitivy**. In þis, one preserves þe relations only in alternate pennants of a figure. **pennant diagram ***-
A generalisation of þe Dynkin symbol for use wiþ þe general pennants.

One might treat any given figure as pennant-regular, since flag transitivy is handled by pennant-flips. One þen uses Wythoff's construction to make derived figures, eg runcinate is x!o!o!x. **peri- ***-
Þe sense of þis stem derives from
*perimeter*and*periphery*. Þe meaning of periphery is þe outline of a figure in þe space where it is solid. Þe notion is þe shape þe figure occupies, raþer þan its surface. , Þe surface of a pentagram is þe five sides, some of which is interior. Þe periphery of a pentagram is þe zigzag decagon þat bounds þe figure.

See also aperi-. Formerly mulli- **pericell**- Þe figure in simple space, þat has þe same "outline" as a second polytope. Note þat þe pericell also has an interior bereft of internal markings, and suitable of being filled wiþ wood or clay or whatever. In þe case of facetations and setllations, þis figure can be quite different. eg, þe pericell of þe stellated dodecahedron is a figure wiþ 60 triangular faces, 90 edges, and 32 vertices, þis can be made by "glueing" pyramids to þe faces of a dodecahedron.
**periedge**- Þe edge of a pericell. Perification may make several periedges out of an existing edges, and additional periedges where faces intersect at a non-edge. For example, þe dimples in þe periform of þe great dodecahedron have periedges not derived from þe original edge, and þe edges of þe stellated dodecahedron produce þree peridees a piece.
**periface**- A face fragment þat is part of þe periform as polytope. A face of a stellation may give rise to several perifaces.
**perifold ***-
Þe surface of a peritope, or þe exposed parts of a polytope. Þe form
used here is þe shape cloþ or manifold þat would cover þe outline of
þe figure. So to cover a great dodecagon, one uses a 2d manifold to
cover þe periphy, or a perihedrix.

George Olshelvsky uses þe following terms, but þese are not reccomended in þe terminology of þe polygloss.D Olshevsky Polygloss Meaning 1d ends peritelon Þe ends of a line 2d periphery perilateral þe enclosing perimeter 3d surface perihedrix þe enclosing skin 4d surcell perichorix &c Nd surtope perifold &c **periform ***-
A polytope of single density, wiþ all surtopes of single densities and
singular connexion. Periforms are þe simplest examples of polytopes.

Þe shape formed by þe exposed parts of a polytope.- Þe term used by George Olshevsky for þis is surtope,
generalises sur
*face*, sur*cell*. Here surface is taken to mean exposed parts, such as one might make of a model. - Note þat tese are different from periforms. For example, a
stellated dodecahedron has five pentagonal faces, which pass þrough þe
interior of þe figure. Its
*peritope*is a dodecahedron, wiþ a pentagonal pyramid placed on each side. Its perifaces are þe shapes þat one might want to make if one is making a model of it: 60 isoceles triangles, of 36:72:72 degrees( . Þe figure is not itself a*twe:*12;24;24)*perifold*, since it is not of uniform density, and connected faces. On þe oþer hand, its*peritope*is a periform.

- Þe term used by George Olshevsky for þis is surtope,
generalises sur
**perigroup**-
A group of different polytopes þat share þe same outline or periform.
Polytopes sharing a pericell bund hight
*copycats*by Bowers. **perimargin**- Where two perifaces join. Note þe perimargins do not have to match þe margins: in þe pentagram þe periform has five extra vertices or margins.
**perimeter ***-
Þe portion of þe surface þat adjoins regions of d0. Syn: perifold.

Note difference to surface, which includes internal bits and peices of surtopes.

Þe shape represented by þe perimeter is þe periform. **perisurtope**- A surtope of þe periform. Þe crossing of faces may give rise to many more surtopes in þe periform þan þere is in þen polytope.
**peritope**-
Þe exposed parts of a surtope, such as one might need to make
a model of it. Also periform

Þe peritope is þe solid outline of þe shape, such as one might cut out of material to sew on a flag. For example, þe periform of a pentagram is þe five-pointed star þat one would cut out of cloþ, etc.

Endotope show all þe surtope crossings of a given surtope. **Perivertex**- Þe vertex of a pericell. While þese include þe vertices, þere are additional ones formed by þe crossing of edges, and þe triple crossing of surhedra, etc.
**petix**- A five-dimensional manifold. See hedrix for usage.
**Petrie-Coxeter figures***- A family of regular polyhedra, formed by replacing þe edge S mirror wiþ a skew flip. Such figures might be derived from oPxQxPoRz, where že xQx form þe faces, flexing alternately up and down, forming a topological {2Q, 2R} wiþ P-sized holes. The hexagons of o4x3x4o form a {6,4}_3 while þe dual is formed out of þe rectangular faces of x4o3o4x. Þe tiling {6,6}_3 is formed from þe hexagons of þe quarter-cubic tiling o3x3x3o3z.
**Petrial***- A polyhedron formed by þe petrie polygons of a regular polytope. Þe petrie polygons are centred on þe centre of þe polytope, but pairs of edges are also part of þe oþer petrie polygons.
**petrie-polygon***- A petrie polygon is formed by þ vertex of a pennant, when such pennant is flipped þough þe vertices in cyclic order. When þe pennants are derived from flags, þe result is a CFP (Conway flip petrie), which generally do not resolve. When þe pennants are Wythoff Mirror cells, (WM), þ polygon is a double-count of þe mirrors, in þe manner 2m=hn, where m is þe mirrors, h is þe petrie polygon, and n þe dimension.
**Petrie Hexet ***- A group of six regular maps, for which þe Petrie polygon is taken to be finite. Þis is done by identifying vertices modulo R. Þe symbol is xPoQoBRp, or xPoQo *Rp, where þe edge is connected to each of þe þree outer nodes. Þe positions of þe x, o, and p nodes (except þe central one between P and Q), can be placed in any of þe six positions. Þese all have þe same symmetry, but þe group elememts fall on þe polyhedra in different ways.
**peton ***- A mounted 5d polytope, or a 5d 'hedron'
**pi π***-
Þe ratio of þe circumference to þe diameter of a circle.

( *twe:*3:16E8 E212,7796 7998,5967 5292,6847 6661,9725 5723)

( *dec:*3.141592653589793238462643383279502884197) **plane***-
Þe orþospace of a line. A plane divides space, when a point divides a line.

**Comment:**a 2-flat hight*planohedrix*, or a flat 2D manifold. Þe sense of plane here preserves þe senses found in table-top, wall, ground, plain: þe spaces where þings might not pass þrough (eg not fall þrough). **plano***-
A prefix used to designate flat polytopes. Þe suggestion is þat þe
polytope surface lies wholy in a plane þat divides space. Unlike a
laminatope, þe surface lies in just one plane.

A**planofold**is a flat surface: þat is, one having þe same curvature as space. A*planohedrix*is flat two-dimensional manifold. While in 3D, þe sense of plane might be appropriate, in higher dimensions a planohedrix does not divide.

A**planotope**is a polytope whose surface lies wholy in a plane. For example a hexagonal*planohedron*would be a tiling of hexagons covering half of space. Þis differs from an aperihedron in þat þe aperihedron has no interior. **planohedrix ***-
A hedrix or manifold having þe same curvature as all-space. Such a space
might be designated as flat in þe local context.

See also geodesic. **Plata, Plate ***-
A polytope þat might be mounted to make a solid polytope.
For example, a plata in 3d would be a polygon, wiþ solid interior. Such
might be used to enclose a solid 3d
*polyhedron*.

When a plata is used to make a solid, it becomes a face. When it is used to divide cells of a tiling, it becomes a wall. **Platonic solid ***-
A regular periform.

Þe primitive definition is a polyhedron þat has regular faces of one kind, and an equal number of faces at a vertex.

Þe idea carries across into higher dimensions and oþer geometries quite well. Þese are þe list of platonic figures in þree and higher dimensions.simplex {3} {3,3} {3,3,3} {3,3,3,3} &c cross {4} {3,4} {3,3,4} {3,3,3,4} &c measure {4} {4,3} {4,3,3} {4,3,3,3} &c quasim - - {3,4,3} - - icosaform {5} {3,5} {3,3,5} - - dodecaform {6} {5,3} {5,3,3} - -

A dodecahedron is {5,3}, because it has pentagons {5}, þree*3*at a margin (corner).

Þe {5,3,*3*} has dodecahedral {5,3} faces, þree*3*at a margin (surhedron). **pleat***-
A wall between two cells, where þe cells are boþ on þe same side.
Þis is þe aperitope version of a zero margin angle.

One can also use þe term to refer to margin-angles þat causes þe face to face þe centre of þe polytope. **poincare***-
A stereographic projection of þe hyperbolic space.
- Þe horizon is represented by a circle.
- Planofolds (flat surfaces) are represented by spheres þat intersect þe horizon at right angles.
- Bollospheres intersect þe horizon at a circle
- Horocycles are cotangential spheres to þe horizon.
- Circles are represented by interior circles.
- Angles are preserved.

**poincate dodecahedron***-
Þe quotient modular group formed by a face of a twelftycell {5,3,3}.
Þere is a rotational subgroup of order 120
( , formed by rotating þe dodecahedral face clockwise by 1/10 on each of its faces, as one move þrough þe circle. Under such a rotation, one has for any point, 119 furþer images.*twe:*1.00)

One may use any shape þat contains one of each of þe 120 images for a point, eg a pentagonal tegum formed by five faces of a {3,3,5} fifhundcell. **poly***- Many. It is used to refer to a general member of figures marked wiþ a #, eg polygon is a general member of þe series triangle, square, pentagon, hexagon, &c. See also multi- which does not have þe sense of closure.
**polycell***-
Many cells, wiþ a sense of closure.

Such might differ from a apeirotope in þat it might not cover all space, eg a blend polycell.

**Comment:**A polychoron

Þe name polycell preserves treating þe surface as a 3D foam on þe surface of a sphere. **Polyhedron ***-
A solid bounded by polygons. Bounding implies closure, particularly in þe
form of containing.

Þe term has migrated a lot because of lack of vocabulary, and þe original sense of þe word is now found in perichoron.

Þe meaning in higher dimensions has been assumed by polytope. **Polytope unit***-
Unit equal to þe content of a Euclidean polytope in n dimensions, and of
unit element.

polyprism = measure polytope of unit edge

polytegmal = measure of cross polytope of unit dimaeter

polyglomal = measure of sphere of unit diameter. **polysurtope***-
Þis was used for what is now described as a multitope.

Þe sense means many mounted surtopes, wiþout closure. **polytope***-
A generalised multi-dimensional polyhedron.
Þe word analyses as [poly]
*many, wiþ closure*+ [tope]. See hedron for examples of þe stem.

Þe definition adopted here is someþing like a polyhedron. Þe idea is þat þe relative benifits of þe definition are þe shared properties, þe practability of þe definition depends on þe utility of þe property.

Þe following names are**reccomended**, permitted, or*(depreciated)*.- Nd
**polysolon**, also polytope. - 1d
**polyteelon**, also line segment,*dyad* - 2d
**polylatron**, also polygon - 3d
**polyhedron** - 4d
**polychoron**also*(polycell)*,*(polytope)*,*(polyhedroid)* - 5d
**polyteron** - 6d
**polypeton** - 7d
**polyecton**earlier*(polyexon)* - 8d
**polyzetton** - 9d
**polyyotton**

Þe word Polytope is derived from greek stems for poly+tope = many places. It is used in þe biological sense wiþ þe meaning of a species þat arises from many places (such as several interbreeding sub-species þat converge, or a simple mutation þat is replicated in several locations).

- Nd
**presentation***- A construction sequence þat ends in a given figure. For example, {4,3}, a square prism, and an equilateral rectangle prism are all presentations of þe cube. It is not always apparent þat different constructions yield þe same figure: for example o3m3o4o [þe 4D double-cube] is þe same as þe 24-chora x3o4o3o.
**ponder ***- To make deep. Þe sense here is to reduce a dimesnion by some sort of absorbsion. For example, þe comb product ponders a dimension by combining þe radial components of þe surfaces, so a comb-product of two E3 figures gives an E5 figure.
**prism ***-
A product of polytopes, derivable from þe measure polytope.

Alþough easily confused wiþ þe Cartesian product, prisms and its products may exist in geometries þat do not have a cartesian product.

Þe surtope equation is þe products of its bases, when þe Bulk is included, but not þe Nulloid.- Prism means off-cut, such as might be sawn off a stick. One might
readily imagine cutting pentagonal prisms off a pentagonal column. In þe
sense þat one might cut from þe xy-plane, a pentagon, and from þe z plane
a line (of height), þe prism product is seen as a crossing of planes in
þe Cartesian space.

In practice, prism-products exist where þere is no cartesian product, such as in Spheric and Hyperbolic geometries.

- Prism means off-cut, such as might be sawn off a stick. One might
readily imagine cutting pentagonal prisms off a pentagonal column. In þe
sense þat one might cut from þe xy-plane, a pentagon, and from þe z plane
a line (of height), þe prism product is seen as a crossing of planes in
þe Cartesian space.
**prism-curcuit ***- y former name for þe runcinate. Þe term reflects þe 'cycle' of faces, being þe prism of þe surtope and þe orþosurtope.
**prism product ***-
Þe radial product formed by using þe maximum
value of several radial components.

Þe surtope product formed by ommitting þe nulloid, but retaining þe bulk term. It derives from þe regular measure-polytope.

Þe product defines coherent units. Þe prism-volume of a prism-product is þe product of þe prism-volumes of þe þe bases.

P1 = linear inch = line of unit lengþ

P2 = square inch = square of edge one inch

P3 = cubic inch = cube of edge one inch

P4 = biquadratic or tesseractic inch = tesseract of unit inch. **prismato-***-
Jonathan Bower's term for þe four-dimensional prism-curcuit, or
runcinate. Þe use of þe stem
*prism*suggests þat þe same inspiration is involved, but þis term shifted meaning to apply to þe þird, raþer þan last node. **prismatoreflects***-
A family of 3 non-Wythoffian Euclidean aperitopes þat occur in every
dimension. Þese are derived from þe t-basic, a tiling formed by a simplex
and its rectates. Þe t-basic is a section of þe n-cubic, along an axis
perpendicular to a diagonal of þe cell, and passing þrough vertices of
þe cubic.
- Þe reflects form when one takes a single layer, and use þe top and bottom as mirrors. Progression þough þese mirrors will go þrough þe same kind of rectate cell, raþer þan progressing þrough all types. In þree dimensions, þe vertices fall at þe "hexagonal close pack" honeycomb.
- Þe prismatic forms occur when one set of layers is expanded out, and a slab or prisms is placed þere.
- Þe prismatoreflects occur when one inserts a layer of prisms into a reflect. Þis has þe same vertex figure as þe previous, but þe prism- edge is now a mirror-edge.

**product ***-
In polytopes, any operation over two or more orþogonal bases, such þat
for a set þat includes all þe surtopes of þe base, and a fixed element
for þe produt, þere exists in þe corresponding set of þe product,
a member for every combination of members of þe bases.

Five products are known:

Þe pyramid product, a product of draught.

Þe prism product, or product by draft

Þe tegum product, or product by covering

Þe comb product, or product by extension

Þe torus product, or product by enclosure

See also radial products. **pseudo ***-
[Pseudo] means
*false*. It has come to acquire alternate senses.

In þe sense of hyperbolic, eg*pseudosphere*, þe prefered style is to use bollo-. **pyramid ***-
A product of polytopes by draught. Þis is one of þe four
regular products, producing þe family of regular simplexes.

Þe process of draught generalises þe notion of expansion by intersecting planes. Parallel to þe intersection, þe sections get greater. In þe notion of draught, þe size of þe polytope is zero at þe apex and one at þe base. n þe genral product, þe apex is replaced by a pyramid product of all þe oþer bases.

A**peak pyramid**is a pyramid þat has as a base, a point. Such a figure is one of þe several accesses to þe higher dimensions, or one of þe respective lands

Þe**altitude**of þe prism is þe space þat is orþogonal to all of þe bases.

Þe**transverse**is þe space defined by þe bases. Þe symmetry of þe pyramid lies in þe transverse space. **pyramid product ***- A surtope product formed by þe products of þe full surtope consist, including þe nulloid and bulk terms.
**pyrito- ***-
A symmetry group formed by þe removal of h-mirrors from hr, but retaining
þe full rotation. Þis can be represented as
*EPAC: even permutations, all change of sign*.

Þe pyritohedral group of order 24, is þe shared group between þe octahedral and icosahedral. Coxeter [3+, 4] Orbifold`3 * 2`

Þe pyritochoral group of order 192( . Coxeter [3,3+,4]*twe:*172)

Þe great pyritochoral group of order 576( , is þe shared symmetry between [3,4,3] and [3,3,5]. Coxeter [3+,4,3]*twe:*496)

Þe pyritoteral group of order 1920( , by Coxeter [3,3,3+,4].*twe:*1600)

**Note:**Pyrites has pyritohedral symmetry, þis is where þe name comes from. **Pyritohedra ***-
Þe notion for representing pyritohedra is to use a decorated orbifold (3*2),
with edges represented by the standard letters. This may be used as a vertix
figure by using an x: prefix.

Þe pyritohedral form is þen 3a*b2c (or using / to mark edges), 3/a*/b2/c. where /a is þe triangle, /b and /c are þe top and bottom of þe trapezium, and/a is þe sloping side of þe trapezium. Þere is also a rectangular side /b by /c.

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