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A unit of measuring solid angle in four dimensions. A 4-sphere divides into 120 s-units, An s-unit is 120 f-units.
      The names and values represent the symmetry regions of 4s [3,3,3] and 4f [3,3,5] respectively.
Schlafli-Gosset construction *
A construction by successive application of an polygonal operator. The process is much used to make the platonic figures, and provides some useful detail very quickly.
      Gosset's contribution was to construct the operation as a vertex-figure operator: that is, one makes {3,3,5} by applying {3} to {3,5}.
Schlafli Symbol *
A notation devised independently by Ludwig Schlafli and others, for deriving regular figures. Several people devised this notation, using different Bracketing. Gosset constructs the operation of {p,q,r} as a vertex-figure with {p} hedra and {q,r} vertex-figure.
      While the notation is very limited in itself, it can be used to as a name, with prefixes, etc.
Schwarz-triangle *
A triangle, that by reflection in its sides, covers all-space a finite number of times. A list of such triangles gives the wythoff-groups for that space.
      The notation is to list the angles in the form of π/p, for assorted p, eg 2 3 5 means a triangle of corners :30, :20, :12, or in degrees; 90, 60, 36.
      Adding a decoration to this notation gives the wythoff-notation.
Schwarz-Wythoff construction *
A construction of polyhedra, by first constructing schwarz-triangles, and then applying wythoff's mirror-edge constructions to the result.
      The derived notation has relatively good coverage of the glomos uniform polyhedra, but for the bollos forms, is relatively poor.
      The wythoff-construction, based on this form, has wide currency, as a result of a 1954 monograph written by Coxeter, Longeut-Higgens and Miller.
sealed surface *
A model of polytope closure that does not depend on the existance of a volume. Such a process is more useful for polytopes where the surface does not particularly bound, eg complex polytopes, or abstract polytopes.
      A surface is sealed if the surtopes are connected to some rule of completion, and that it is proper if there are no appended surtopes that are not involved in some rule of sealing.
A sequence of names given to things that look like points nodes, edges branches etc, but the intent is not to confuse as such.
      For example, the points and edges of the Dynkin symbols as second-series names node and branch, to prevent confusion with the first-series names referring to the polytope being described.
      see node, branch, sequi-
section *
A section of the polytope reveals the complete interior, as if it were sliced at that point. Such a slice has more features than the ring polytope, since it contains also extra convexure caused by lines cut by the section.
      For example, the second ring slice of the axis of x3o3o5o gives a dodecahedron: the 20 vertices here are those of a dodecahedron. The sectional slice here also has 12 pyramids on the faces, corresponding to edges between the first and third rings.
      See also waterman section.
[Klitzing] A class of polytopes inspired by taking parallel sections of larger polytopes. The class naturally extends to other types of figures with top and bottom in the same symmetry group, such as cupolae.
      The segmentotopes are defined to be any polytopes that obey 3 rules: These differ from lace-prisms, in that the edges must be uniform. An lace prisms on elliptic bases and uniform edges is a segmentatope, but lace prisms can have unequal edges.
A process of reducing a cartesian product of several station-systems, according to some modulus property of the ordered stations. For example, one might make a polytope with seven vertices in four dimensions, by numbering the vertices of a heptagon and heptagram in order, and then retaining only those vertices where the vertex-numbers are the same.
      The naming convention describes the process well The example referred to in the list might be a {15}{15/2} pentamodprism, a polytope that contains the vertices of 9 pentachora, or 5 bitriangle prisms.
The usual meaning is to remove alternate (ie odd or even) verticies. The extended meaning is ordered reduction to a single class, when the vertices etc can be allocated to p classes. For example, one can derive the pentachoron from the bipentagonal prism by numbering the vertices of one in pentagon and the other in pentagramic order. Consider only those that add to five. Likewise, the heptepeton or heptapyramid can be formed from the triheptagon prism by advancing the three bases at the rates of 1, 2 and 4. However, the true 1,2,4 semis triheptagon prism would have 49 vertices.
The latin stem [sequi] means following. It is used here to refer to series with second meanings.
      The meaning here is a prefix for second-series surtopes. One might describe a node as a sequi-teelon, and a branch as a sequi-lateron. This series allows one to go as far as sequi-yotton.
      The series is not often used, but the names are reserved.
The [sesqui] derives from the ladin semis half + que and. It reflects the widespread name for the fraction 1 1/2, eg germanic otherhalf, anderhalb.
      It is used to denote an increase of 3/2, zb cubic -> sesquicubic, which has all of the verticies of the cubic, and verticies in half of the cells.
shortchord *
The base of the isoceles triangle formed by two consecutive sides of a polygon. The number is much used in circle-drawing geometry, and also somewhat in other geometries, the values designated as L1 for the value, and L2 for the square.
      Certian shortchords are widely used, and have convenient single letter forms. For polygons with an infinite number of sides, the style is to write {wL2}, where L2 is the shortchord-square,
sill *
A sill is a second-wall, or the margin where three or more cells in a tiling meet.
simplex *
A family of figures having the minimum requirements for a given dimension: n+1 vertices.
surtope -> ~Peta ~Tera ~Chora ~Hedra Edge Vertex Nulloid -
CHEVN -> P T C H E V N Schlaffli
dimension -> 5 4 3 2 1 0 -1 Symbol
triangle - - - 1 3 3 1 {3}
tetrahedron - - 1 4 6 4 1 {3,3}
pentachoron - 1 5 10 10 5 1 {3,3,3}
hexateron 1 6 15 20 15 6 1 {3,3,3,3}
A symmetry operation that does central inversion, rather than reflection. This produces the skew or zigzag polygons, such as the Petrie polygons. The dual of a skew polygon is an askew polygon. This has an interior, and has the same rotary-inversion symmetry found in the skew figures. Skew figures appear as the vertex-figures of the Petrie-Coxeter polyhedra.
skew marginoid*
A division on a face (margin) where the in-vector reverses, but the density remains constant.
      Such elements are not true surtopes, since this happens in the interior of the face, but provides a method that the obvious density patterns can apply to polytopes with a non-orientable surface can still have an interior.
slab-land *
In 3d one can replace a plane by a layer or slab of given thickness. In this world, polygons in the plane become prisms. It is a useful concept to introduce higher dimensions.
      The next step is to discuss various kinds of lace prisms such as point-pyramids.
      The idea generalises to higher dimensions quite nicely, and every uniform polytope corresponds to a uniform slab-prism in the next dimension. When this is accounted for, one can assign, eg a dodecahedron and its 4d slab-prism the same unifrom number.
Having the same curvature as the hull of the vertices. For example, a smooth dodecahedron is a sphere marked up as a dodecahedron, regardless of the ambient geometry. So, smooth angles, &c. For example, the smooth angles of the dodecahedron are 120 degrees at the vertex.
Any polytope derived by removal of alternate vertices of a polytope. The usual condition is that the surhedra must be all even.
      Professor Johnson described a class of holosnub, in which the requirement is relaxed to all polygons must be greater than a triangle. Such are simple d2 figures if there is odd polygons, or a compound of two figures if there are only even polygons.
Snub, Coxeter
A class of snubs formed by selecting alternating vertices of the t{p,2q,r,..} In these cases, the edges of the {p,2q,r,...} can be coherently indexed, and divided in a particular ratio. One of these points gives rise to a uniform snub. This forms a class of non-Wythoffian figures. While Coxeter's rule is fairly good at producing equal-edged figures, there is no guarantee the figure is uniform. s{;3;4,3,4} produces a series of octahedral pyramids as the snubbing face, in place of pentachora.
snub cube *
A non-wythoff uniform figure, featuring four triangles, and a square at each vertex. It, and the snub dodecahedron are the vertex figures of a uniform non-wythoff bollos polychoron.
      One might treat the snub cube as a {3, wx}, where x = 2.83928675522 for the cube, and 2.94315125925 for the dodecahedron. This leads to the edges of the bollos {3,3,wx}, as:
      {3,3,wq} e2 = 3.222 262 523 29, e2(r) = 0.446 156 936 68, e2(t)=0.208 634 614 10
      {3,3,wf} e2 = 14.590 539 153 7, e2(r) = 0.784 483 679 43, e2(t)=0.381 952 222 52
      The figure, its truncate and rectate, are all uniform.
Snub, Hoso-
A figure arising by alternating every vertex. This can lead to either a single figure of density 2 (eg pentagon -> pentagram), or a compound of two figures (hexagon -> hexagram = 2 triangles).
      The term is due to Norman Johnson.
      See hososnub.
Snub, Johnson
Norman Johnson proposed to do stott-expansions to snub figures, in axies that have not been used for snubbing. This mainly affects the partial snubs.
      For the general semis-figure eg s4o3o, one can expand on any of the snubs, eg s4o3x, or s4x3o. Snubs formed from simple semis-figures make a mirror-edge figure, by splitting the s node. s4o is the same as xEo here.
      For figures like s3s4o3x, the figure is not generally uniform, but is different to anything else-where known.
      Note. When one reads Johnson's use of the 's' node as isolated elements, eg s4o3o4s, this does not imply the snub is a semis x4o3o4x, but rather that the s nodes need to be split in symmetry, and then semiated. ie xEo3oAx3:, or s4o3o3o4s -> xEo3o3oAx.
Snub, Wythoff
A family of non-Wythoff figures
      Wythoff's rule of forming snubs of the three dimensional groups, by removing alternating vertices of an omnitruncate. In 3D, this always produces a uniform figure, since it is 3 variables in 3 unknowns. In four and higher, there are more variables than degrees of freedom.
-strophic *
A multiple covering of a polytopic, as in bistrophic, twice-covered. In the sense of a spool-star, viz PpDdRd.
sol *
The general grade for the hedric prefix. This one does not have a particular dimension, but rather refers to a general one. A solon is the general class member of telon, latron, hedron, choron, teron &c.
      The word is a backform from [solid].
solid *
A figure is solid, if for some vanishing distance, every point less than dx from that point, and falling in all-space, is also part of that solid, and vice versa.
      A solid figure has the same dimensionality as all-space.
2. A solid polytope.
solid, solidus *
A solid polytope, or a polytope that has the same dimensinality as all-space.
#space n-space is the extent of space defined by having exactly n perpendiculars at every point. In the polygloss, the style is to give a more precise word.
      E2 horohedrix. H2 bollohedrix, S2 glomohedrix.
      E2 horohedrix, E3 horochorix E4 horoterix, etc
      Because in the style here, we have horochorix meaning both a subspace and all-space, the style of using E3 to represent a geometry and a subspace.
The span over a finite set, is the sum of members of that set multiplied by integers. For example, Z5 is the span of chords of a pentagon, is numbers of the form Y+øZ, where ø is 1.61803398875, and Y and Z are integers.
sparse *
A designation of an infinite set, such that there exists a non-zero radius around any member, such it contains no other member. For example, the points of a square lattice are sparse.
      A further implication of sparse, is that it covers all space: that is, there is a finite sphere of R, drawable on any point, which contains a member of the set. A sparse set can be over N dimensions: see class.
      The usual name for this is discrete, but a different meaning for this is used here.
      A sparse polytope or tiling is one, for which the endotopes are of an evidently non-zero size. That is, for an edge, there is only a finite number of divisions caused by crossing surtopes.
      The quantum polytopes are sparse by nature.
      A tiling is peicewise-sparse if it has a sparse framework, and infinite cells. An example is {oo, 4/2}. In this tiling, every row or column of edges of {4,4} is treated as a pair of {oo} adjacent. A point thus belongs to a different cell for every possible row or column of edges: ie infinitely dense. The vertices and edges belong to a {4,4}, and hence are sparse. Another example is '3,5,3,5/2', whose edges belong to a {3,3,3,5}.
      A polytope like {5/2,4} or {8,8/3} are not sparse, since there are an infinite number of vertices on the alignment of any given edge, and between vertices.
      Spidrons are sparse, even though they have sections that go to zero. Here the approach to infinity is not 'every decimal', but in certian regions, segments that become smaller by powers of 10, eg steps of 1, 0.1, 0.01, 0.001, &c.
sphere *
1: A solid bounded by the points equidistant from a given point. When a prefix-number is used, eg 5-sphere, the solidness is in that space. By extension, a glomohedron or solid 3d-sphere.
      2: The surface of the solid so described. Such points are all at the radius distant from the centre. By extension, a glomohedrix
Spheric Product *
A radial product formed by the root-sum-square of polytope-radii. Applied to lines of several lengths, this gives ellipsoids.
      The product is believed to be coherent: that is, the spheric volume of a spheric product of several is the product of the spheric volumes of the bases.
      S1 = dyadic inch = inch of unit diameter
      S2 = circular inch = hedrage (area) of circle of unit diameter
      S3 = spheric inch = chorage (volume) of sphere of unit diameter
      S4 = glomic inch = terage of glome (4-sohere) of unit diameter.
      See also prism product, tegum product
spidron *
A class of polytope featuring progressions of surtopes in logrithmic sequence, usually diminishing to a point.
      Although one can effect logrithmic progressions in any dimension, the elegant spiraling happens only where the surface supports it: even dimensions.
      Note: even though spidrons approach zero logrithmically, they are none the less sparse: each element added is 1/x of the previous one, but added to the outside, and not crossing a previous example.
The regular tetragon.
      As a measure, the content of a Euclidean square is implied: specifically the prismic unit of hedrage.
      comment: Some authors requrie the square to be right-angled. This happens when one projects it from space onto some kind of horosphere. For example, one might create a spheric tiling [4,3], but the teragons only become squares when the cube is constructed.
      This distinction is arbitary and best avoided, since it implies that Euclidean space is somehow more real than non-euclidean space.
staircase snub*
A kind of modified mobius snub found in uniform bolloohedra. In the diagrams below, each different number represents a different edge class.

    x 4 x 2 x 4 x 2 x      o 2 x 3 x 2 o 1 o
    3   1   3   1   3      2   1   2   3   2
    x 2 x 4 x 2 x 4 x      x 3 x 2 o 1 o 2 x
    1   3   1   3   1      1   2   3   2   1
    x 4 x 2 x 4 x 2 x      x 2 o 1 o 2 x 3 x
    3   1   3   1   3      2   3   2   1   2
    x 2 x 4 x 2 x 4 x      o 1 o 2 x 3 x 2 o

      Normal snub           Staircase snub

An example was presented in Marek Čtrnack's enumeration of the uniform bollohedra of the vertex-type ppppq, where the edge 3 is wrapped from a digon to a triangle.
Station *
[standing point] If polytopes represent the geometric integers, then stations would be the modulus classes. The shape of a station is that of the vertices matching the same modulus. The interaction of stations have great importance on compounds.
      For example, the modulus of 6 over a twelfty-gon produces 6 stations each in the shape of a 20-gon. This 20-gon can settle in the available 120 points in 6 different standing points or stations.
      The term originated in tilings, where the fundamental tiling of the t, q and y groups could stand at each of a number of different points of the fundamental region.
stellate *
The act of extending faces until new meetings are created. Normally the face-planes match those of the core or unstellated forms. cf faceting
      In naming the stellates, stellated means the edges are kept, but new vertices are found, great means the surhedra are kept, but new vertices + edges are found, and grand means that surchora are kept but all below are new.
step-prism, step-tegum *
Stepping is a form of semiation over concurrent modulo classes, wherein every modulo ring is stepped together, eg (0,0,0), (1,1,1), (2,2,2) &c. Compare this with mod.
      A step prism preserves the vertices of a prism.
      A step-tegum preserves the faces of a tegum.
      One can for example represent a heptapeton as a {7}{7/2}{7/3} heptastep-prism or tegum.
      Generally the prism and tegum forms are different, which is why one must be explicit in the name
Stereographic projection*
A projection that preserves isocurves and angles. Such projections are widely used, and can be used as 'pocket geometries' in all geometries.
      The Spheric form makes an isocurve that pass through diametric points of the equator "straight".
      The Euclidean form makes any isocurve that passes through a fixed point (point at zero) straight. When this is done on a Euclidean plane, the result is the same as inversion, although the result can be replicated in any other space.
      The hyperbolic makes an isocurve that passes through a fixed circle (horizon) at right angles, as straight. This is the poincare projection.
Stott Addition *
The addition of stott-vectors. One sees stott-addition appear in stott progressions.
Stott Construction*
A construction of polytopes by expansion, or insertion of edges of new kinds. One can derive all of the WME polychora by using this method, exactly what Mrs Stott did in her 1911 paper.
      The seven uniform octahedrals (ie O, C, CO, rCO, tC, tO and tCO) can be arranged in three pairs where one has touching triangular-symmetry faces, and the other has separated ones: ie (pt, C), (O, rCO), (C), tC) and (tO, tCO). In these, the two figures have identical triangular elements, but the second one is separated by edges parallel to a cube's edge.
      This is read as a stott-expansion of pt -> C, or miniture C to Cv. The seven become, in increasing size: Ch, Cv, Ce, Chv, Che, Cve, Chve.
Stott-Schläfli notation*
The notation used by mathematicians, who mainly concentrate on the overall symmetry group, rather than the polytopes constructed. It involves putting a Stott operator t_ in front of a Schläfli symbol . Replaced by an inline system.
stott matrix*
A matrix A_ij = S_i . S_j, where S_k is the k'th stott-vector.
      Such a matrix allows for direct dot products of stott-vectors. The matrix-norm of any position-vector gives the diameter square of the corresponding position-polytope. It has other uses in terms of the lace prisms.
A progression or sequence of polytopes, represented by a points on a straight line in stott-space. The sections of lace-prisms is a stott progression from the top to the bottom.
A space where every point in a kaleidoscope represents a position-polytope. Straight lines in this space represent lace-prisms with as bases, the ends of the line.
stott vectors*
The stott construction can be thought of as representing the kaleidoscope as an oblique coordinate system, and the vertices of the figure being derived by being a measured distance from each of the walls.
      The construction is increase one coordinate at a time until the desired position is reached, amounts to drawing different levels of surtope from a zero-sized polytope. Such causes a plane to move outwards from the centre, the contents of that plane not disturbed, but the space between the planes are drawn into prisms and other shapes by the action.
        V         E         H           Vectors making uniform fig

      1.0.0     0.1.1     r2.r2.r2       v     octa    icosa
                                         e     CO      ID
      f.1.0    2f.0.0     f2. 0. 1       h     cube    dodeca
      f.1.0    f2.1.f     f2. 0. 1      v+e    t oct   t ico
      f.1.0    f2.1.f      f. f. f      e+h    t cub   t dod
      f.1.0    f.f2.1      f. f. f      v+e    srCO    srID
      f.1.0    f.f2.1      1. 0. f2    v+e+g   grCO    grID

      One can construct a stott-matrix from the primary coordinates (vectors with one '1' coordinate and the rest 0), by dotting these into a matrix, so for example, the cell (a,b) = a·b = Sab (Sab is the scalar product of a, b in the older style).
An antitegum. The term is from John H Conway. Specifically, it can be presented as the intersection of lace cones of figures in the dual position, where both cones might point in the same direction.
A figure bounded by antitegums.
To replace each face of a figure by an antitegum, radial from the centre. The effect is to create antitegums axial on the face-vertex axis, with a section being the vertex-figure of the face. This is the join operator in Conway's notation. As a surface operation, this erects an antitegum axial on the line from the vertex to the face-centre, and in section the vertex-figure of the face.
A quadrilateral formed by reflecting a triangle in one of its sides.
The space covered by a tiling, the sum of cells, much as a surface is a sum of faces. One regards a tiling as the surface of a planotope, and on removal of interior, the surface (sum of faces) becomes surcell (sum of cells)
      George Olshevsky invented the term surcell on anology to surface. One reads here [sur] on, exposed + [face] 2d polytope, to give what is here perichorix.
      The same root stems are preserved, but the PG meanings are applied, ie [sur-] on + [cell mean solid element]. The idiom of sum of cells gives the meaning of extent of apeirotope or polycell.
sum() function*
This is the summation of the supplied arguments. The function is the radial function behind the tegum product.
      The sum of a list is the numeric sum, eg sum(3,5,7) = 15.
surchoron *
A surchoron is a 3d element part of a larger polytope's surface.
      It is variously called cell, by analogy with the role of cells as solid elements in a foam, etc. The sense of cell is then replicated into other words, as cellule.
It is best to use cell as element of tiling, solid surtope, and use surchoron for 3d surtope. In 6D, 3d elements are no more 'cellule' then the edges of a cube!
A bounding manifold. This means, that if there are points A and B, such that one is inside and one outside the region, then a part of the surface lies between them.
      For a polytope, the sum of all non-solid elements, ie from vertex to face, inclusive.
      When the surface is allowed to cross, and one has multiple densities, one gets diverging meanings, handled here by surface and perifold. The surface of a polytope includes all of the interior of every non-solid (ie lesser dimensional) surtope. The perifold or perimeter is only those bits that are exposed to regions of zero-density.
      Example, in the pentagram, the surface includes the edges of the core pentagon, since parts of edges are seen here. The perimeter does not include these segments, since they do not face a region of d0.
surhedron *
A surface polygon.
      This is often called face. But setting it to this meaning gives a division of the stem of [face] into 2d and n-1 dimensional meanings eg Under the polygoloss, one uses different meanings for 2d and Nd words, ie
To make a surface, or enclosure, in the space where something is solid.
      When one surrounds a city, etc, one notes that a city is essentially a two-dimensional thing, with height. The surrounding elements then form a larger enclosure that has the city as interior.
      compare with arround, which means to enclose the space orthogonal to a thing [as one puts a hand around a shaft]
Surtegum *
A polytope bounded by tegums.
      In practice, this term represents a figure whose faces are the union of all the flags incident on a class of flag-corner (eg edge-centres). In this case, it becomes the tegum or join product of a surtope and its orthosurtope.
surtope *
A surface polytope, a general member of the list below. The list of surtope names: reccomended, allowed and (depreciated) George Olshevsky invented the term. By him, it means a n-dimensional boundary, eg surface = sum of faces covering polyhedron. I borrowed the word and etymological construction, but applied newly derived meanings. This is a list of his words, with PG translations Note that surface and surcell can be read as sum of faces and cells in both systems, but face and cell change meanings.
Surtope Characteristic*
The surtope equation evaluated for a=-1. This is 0 when the polytope dimension is even, and 2 when it is odd.
      The surtope characteristic can be made uniformly zero, when the content or body of the polytope, and a surtope of -1 dimensions are counted.
Surtope Excess*
The evaluation of the surtope polynomial, including noids and content, evaluated with a=-1. This in 3D counts the number of holes: eg for a bi-hexagon torus = a³+36a²+72a+36+1/a = -2. This represents the two holes of the torus in 3D: the hole at zero and the hole at infinity. For a polytope like the {5,5/2} = a³+12a²+30a¹+ 12+1/a = -8. These eight holes might be made by removing the 20 faces of the icosahedron, and restoring 12 faces as the sectional pentagon. In higher dimensions, the surtope excess is not always connected to the holes, since the hole polynomial may evaluate to zero in even dimensions.
surtope figure *
A figure formed by the approaches of surtopes incident on a given surtope, when measured in the orthospace.
      This generalises the notion of vertix figure.
Surtope Group*
An expression of the symmetry of a polytope that preserves the connection of the surtopes, without necessarily preserving their spacial size.
      For example, the symmetries of rectangular tilings is a surtope group. The relative sides of the rectangles does not disturb the connections of the rectangles to each other.
surtope polynomial *
An expansion of the count of surtopes as a polynomial. All surtopes of dimension n are reckoned as a**n, the count being the multiplier. eg eight polygons is written as 8a².
      For example, a cuboctahedron has 14 surhedra, 24 edges and 12 vertices, so its surtope equation becomes 14a² + 24a¹ + 12a°.
surtope product *
A polytope formed such that some subset of its surtopes, correspond each uniquely to the same subset applied to each of its bases. In practice, the full surface is retained in the subset, and only the bulk and nulloid are variously included or excluded.
      The pyramid product includes bothe the volume and the nulloid. The unity column is the nulloid column. The dimension is increased by 1 for each application. Simplexes are a pyramid-product of their vertices.
      The tegum product includes the nulloid, but excludes the bulk. The unity column is the nulloid column. The product adds no extra dimension, and defines a set of coherent units. The cross-polytope arises from the tegum-product its diagonals.
      The prism product includes the bulk, but excludes the nulloid. The product adds no extra dimension, and defines a coherent set of units.
      The comb product excludes both the bulk and nulloid terms, the product reduces the dimension of the bases. When applied to horos apeirotopes, the dimension of the bases and the product are all reduced by one, and so the product appears to preserve dimensionality. The torus product is an instance of the comb-product.
A symmetry operation in 4d corresponding to the complex mirror.

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