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

In the theory of isocurves, the notion of parallel isocurves links thogether the ideas of concentricness, cotangency and equidistant. In the non-Euclidean geometries, there is only one straight line in a set, except for the 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}. The vertex figure consist of a dodecahedron with four of its vertices removed, and replaced by triangles...
**peak***-
The sense of peak is the highest point. In the common usage, it suggests
highest point in a curve, journey. In this sense, we see that it correctly
refers to the high-points in a 2d orthospace.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

The shape represented by the perimeter is the periform. **perisurtope**- A surtope of the periform. The crossing of faces may give rise to many more surtopes in the periform than there is in then polytope.
**peritope**-
The exposed parts of a surtope, such as one might need to make
a model of it. Also periform

The peritope is the solid outline of the shape, such as one might cut out of material to sew on a flag. For example, the periform of a pentagram is the five-pointed star that one would cut out of cloth, etc.

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

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

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

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

A**planofold**is a flat surface: that is, one having the same curvature as space. A*planohedrix*is flat two-dimensional manifold. While in 3D, the 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. This differs from an aperihedron in that the aperihedron has no interior. **planohedrix ***-
A hedrix or manifold having the same curvature as all-space. Such a space
might be designated as flat in the local context.

See also geodesic. **Plata, Plate ***-
A polytope that might be mounted to make a solid polytope.
For example, a plata in 3d would be a polygon, with 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.

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

The idea carries across into higher dimensions and other geometries quite well. These are the list of platonic figures in three 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}, three*3*at a margin (corner).

The {5,3,*3*} has dodecahedral {5,3} faces, three*3*at a margin (surhedron). **pleat***-
A wall between two cells, where the cells are both on the same side.
This is the aperitope version of a zero margin angle.

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

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

One may use any shape that contains one of each of the 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 with a #, eg polygon is a general member of the series triangle, square, pentagon, hexagon, &c. See also multi- which does not have the sense of closure.
**polycell***-
Many cells, with a sense of closure.

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

**Comment:**A polychoron

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

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

The meaning in higher dimensions has been assumed by polytope. **Polytope unit***-
Unit equal to the 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***-
This was used for what is now described as a multitope.

The sense means many mounted surtopes, without closure. **polytope***-
A generalised multi-dimensional polyhedron.
The word analyses as [poly]
*many, with closure*+ [tope]. See hedron for examples of the stem.

The definition adopted here is something like a polyhedron. The idea is that the relative benifits of the definition are the shared properties, the practability of the definition depends on the utility of the property.

The 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**

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

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

Although easily confused with the Cartesian product, prisms and its products may exist in geometries that do not have a cartesian product.

The surtope equation is the products of its bases, when the Bulk is included, but not the 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 the
sense that one might cut from the xy-plane, a pentagon, and from the z plane
a line (of height), the prism product is seen as a crossing of planes in
the Cartesian space.

In practice, prism-products exist where there 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 the
sense that one might cut from the xy-plane, a pentagon, and from the z plane
a line (of height), the prism product is seen as a crossing of planes in
the Cartesian space.
**prism-curcuit ***- y former name for the runcinate. The term reflects the 'cycle' of faces, being the prism of the surtope and the orthosurtope.
**prism product ***-
The radial product formed by using the maximum
value of several radial components.

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

The product defines coherent units. The prism-volume of a prism-product is the product of the prism-volumes of the the bases.

P1 = linear inch = line of unit length

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 the four-dimensional prism-curcuit, or
runcinate. The use of the stem
*prism*suggests that the same inspiration is involved, but this term shifted meaning to apply to the third, rather than last node. **prismatoreflects***-
A family of 3 non-Wythoffian Euclidean aperitopes that occur in every
dimension. These are derived from the t-basic, a tiling formed by a simplex
and its rectates. The t-basic is a section of the n-cubic, along an axis
perpendicular to a diagonal of the cell, and passing through vertices of
the cubic.
- The reflects form when one takes a single layer, and use the top and bottom as mirrors. Progression though these mirrors will go through the same kind of rectate cell, rather than progressing through all types. In three dimensions, the vertices fall at the "hexagonal close pack" honeycomb.
- The prismatic forms occur when one set of layers is expanded out, and a slab or prisms is placed there.
- The prismatoreflects occur when one inserts a layer of prisms into a reflect. This has the same vertex figure as the previous, but the prism- edge is now a mirror-edge.

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

Five products are known:

The pyramid product, a product of draught.

The prism product, or product by draft

The tegum product, or product by covering

The comb product, or product by extension

The torus product, or product by enclosure

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

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

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

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

The**altitude**of the prism is the space that is orthogonal to all of the bases.

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

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

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

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

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

**Note:**Pyrites has pyritohedral symmetry, this is where the name comes from. **Pyritohedra ***-
The 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.

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

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