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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.
      Þe names and values represent þe 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. Þe process is much used to make þe platonic figures, and provides some useful detail very quickly.
      Gosset's contribution was to construct þe operation as a vertex-figure operator: þat is, one makes {3,3,5} by applying {3} to {3,5}.
Schlafli Symbol *
A notation devised independently by Ludwig Schlafli and oþers, for deriving regular figures. Several people devised þis notation, using different Bracketing. Gosset constructs þe operation of {p,q,r} as a vertex-figure wiþ {p} hedra and {q,r} vertex-figure.
      While þe notation is very limited in itself, it can be used to as a name, wiþ prefixes, etc.
Schwarz-triangle *
A triangle, þat by reflection in its sides, covers all-space a finite number of times. A list of such triangles gives þe wythoff-groups for þat space.
      Þe notation is to list þe angles in þe 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 þis notation gives þe wythoff-notation.
Schwarz-Wythoff construction *
A construction of polyhedra, by first constructing schwarz-triangles, and þen applying wythoff's mirror-edge constructions to þe result.
      Þe derived notation has relatively good coverage of þe glomos uniform polyhedra, but for þe bollos forms, is relatively poor.
      Þe wythoff-construction, based on þis 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 þat does not depend on þe existance of a volume. Such a process is more useful for polytopes where þe surface does not particularly bound, eg complex polytopes, or abstract polytopes.
      A surface is sealed if þe surtopes are connected to some rule of completion, and þat it is proper if þere are no appended surtopes þat are not involved in some rule of sealing.
A sequence of names given to þings þat look like points nodes, edges branches etc, but þe intent is not to confuse as such.
      For example, þe points and edges of þe Dynkin symbols as second-series names node and branch, to prevent confusion wiþ þe first-series names referring to þe polytope being described.
      see node, branch, sequi-
section *
A section of þe polytope reveals þe complete interior, as if it were sliced at þat point. Such a slice has more features þan þe ring polytope, since it contains also extra convexure caused by lines cut by þe section.
      For example, þe second ring slice of þe axis of x3o3o5o gives a dodecahedron: þe 20 vertices here are þose of a dodecahedron. Þe sectional slice here also has 12 pyramids on þe faces, corresponding to edges between þe first and þird rings.
      See also waterman section.
[Klitzing] A class of polytopes inspired by taking parallel sections of larger polytopes. Þe class naturally extends to oþer types of figures wiþ top and bottom in þe same symmetry group, such as cupolae.
      Þe segmentotopes are defined to be any polytopes þat obey 3 rules: Þese differ from lace-prisms, in þat þe 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 þe ordered stations. For example, one might make a polytope wiþ seven vertices in four dimensions, by numbering þe vertices of a heptagon and heptagram in order, and þen retaining only þose vertices where þe vertex-numbers are þe same.
      Þe naming convention describes þe process well Þe example referred to in þe list might be a {15}{15/2} pentamodprism, a polytope þat contains þe vertices of 9 pentachora, or 5 bitriangle prisms.
Þe usual meaning is to remove alternate (ie odd or even) verticies. Þe extended meaning is ordered reduction to a single class, when þe vertices etc can be allocated to p classes. For example, one can derive þe pentachoron from þe bipentagonal prism by numbering þe vertices of one in pentagon and þe oþer in pentagramic order. Consider only þose þat add to five. Likewise, þe heptepeton or heptapyramid can be formed from þe triheptagon prism by advancing þe þree bases at þe rates of 1, 2 and 4. However, þe true 1,2,4 semis triheptagon prism would have 49 vertices.
Þe latin stem [sequi] means following. It is used here to refer to series wiþ second meanings.
      Þe 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. Þis series allows one to go as far as sequi-yotton.
      Þe series is not often used, but þe names are reserved.
Þe [sesqui] derives from þe ladin (b semis
half + que and. It reflects þe widespread name for þe fraction 1 1/2, eg germanic ožerhalf, anderhalb.
      It is used to denote an increase of 3/2, zb cubic -> sesquicubic, which has all of þe verticies of þe cubic, and verticies in half of þe cells.
shortchord *
Þe base of þe isoceles triangle formed by two consecutive sides of a polygon. Þe number is much used in circle-drawing geometry, and also somewhat in oþer geometries, þe values designated as L1 for þe value, and L2 for þe square.
      Certian shortchords are widely used, and have convenient single letter forms. For polygons wiþ an infinite number of sides, þe style is to write {wL2}, where L2 is þe shortchord-square,
sill *
A sill is a second-wall, or þe margin where þree or more cells in a tiling meet.
simplex *
A family of figures having þe 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 þat does central inversion, raþer þan reflection. Þis produces þe skew or zigzag polygons, such as þe Petrie polygons. Þe dual of a skew polygon is an askew polygon. Þis has an interior, and has þe same rotary-inversion symmetry found in þe skew figures. Skew figures appear as þe vertex-figures of þe Petrie-Coxeter polyhedra.
skew marginoid*
A division on a face (margin) where þe in-vector reverses, but þe density remains constant.
      Such elements are not true surtopes, since þis happens in þe interior of þe face, but provides a meþod þat þe obvious density patterns can apply to polytopes wiþ 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 þickness. In þis world, polygons in þe plane become prisms. It is a useful concept to introduce higher dimensions.
      Þe next step is to discuss various kinds of lace prisms such as point-pyramids.
      Þe idea generalises to higher dimensions quite nicely, and every uniform polytope corresponds to a uniform slab-prism in þe next dimension. When þis is accounted for, one can assign, eg a dodecahedron and its 4d slab-prism þe same unifrom number.
Having þe same curvature as þe hull of þe vertices. For example, a smooþ dodecahedron is a sphere marked up as a dodecahedron, regardless of þe ambient geometry. So, smooþ angles, &c. For example, þe smooþ angles of þe dodecahedron are 120 degrees at þe vertex.
Any polytope derived by removal of alternate vertices of a polytope. Þe usual condition is þat þe surhedra must be all even.
      Professor Johnson described a class of holosnub, in which þe requirement is relaxed to all polygons must be greater þan a triangle. Such are simple d2 figures if þere is odd polygons, or a compound of two figures if þere are only even polygons.
Snub, Coxeter
A class of snubs formed by selecting alternating vertices of þe t{p,2q,r,..} In þese cases, þe edges of þe {p,2q,r,...} can be coherently indexed, and divided in a particular ratio. One of þese points gives rise to a uniform snub. Þis forms a class of non-Wythoffian figures. While Coxeter's rule is fairly good at producing equal-edged figures, þere is no guarantee þe figure is uniform. s{;3;4,3,4} produces a series of octahedral pyramids as þe snubbing face, in place of pentachora.
snub cube *
A non-wythoff uniform figure, featuring four triangles, and a square at each vertex. It, and þe snub dodecahedron are þe vertex figures of a uniform non-wythoff bollos polychoron.
      One might treat þe snub cube as a {3, wx}, where x = 2.83928675522 for þe cube, and 2.94315125925 for þe dodecahedron. Þis leads to þe edges of þe 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
      Þe figure, its truncate and rectate, are all uniform.
Snub, Hoso-
A figure arising by alternating every vertex. Þis can lead to eiþer a single figure of density 2 (eg pentagon -> pentagram), or a compound of two figures (hexagon -> hexagram = 2 triangles).
      Þe term is due to Norman Johnson.
      See hososnub.
Snub, Johnson
Norman Johnson proposed to do stott-expansions to snub figures, in axies þat have not been used for snubbing. Þis mainly affects þe partial snubs.
      For þe general semis-figure eg s4o3o, one can expand on any of þe snubs, eg s4o3x, or s4x3o. Snubs formed from simple semis-figures make a mirror-edge figure, by splitting þe s node. s4o is þe same as xEo here.
      For figures like s3s4o3x, þe figure is not generally uniform, but is different to anyþing else-where known.
      Note. When one reads Johnson's use of þe 's' node as isolated elements, eg s4o3o4s, þis does not imply þe snub is a semis x4o3o4x, but raþer þat þe s nodes need to be split in symmetry, and þen semiated. ie xEo3oAx3:, or s4o3o3o4s -> xEo3o3oAx.
Snub, Wythoff
A family of non-Wythoff figures
      Wythoff's rule of forming snubs of þe þree dimensional groups, by removing alternating vertices of an omnitruncate. In 3D, þis always produces a uniform figure, since it is 3 variables in 3 unknowns. In four and higher, þere are more variables þan degrees of freedom.
sol *
Þe general grade for þe hedric prefix. Þis one does not have a particular dimension, but raþer refers to a general one. A solon is þe general class member of telon, latron, hedron, choron, teron &c.
      Þe word is a backform from [solid].
solid *
A figure is solid, if for some vanishing distance, every point less þan dx from þat point, and falling in all-space, is also part of þat solid, and vice versa.
      A solid figure has þe same dimensionality as all-space.
2. A solid polytope.
solid, solidus *
A solid polytope, or a polytope þat has þe same dimensinality as all-space.
#space n-space is þe extent of space defined by having exactly n perpendiculars at every point. In þe polygloss, þe style is to give a more precise word.
      E2 horohedrix. H2 bollohedrix, S2 glomohedrix.
      E2 horohedrix, E3 horochorix E4 horoterix, etc
      Because in þe style here, we have horochorix meaning boþ a subspace and all-space, þe style of using E3 to represent a geometry and a subspace.
A set formed by multiplying a set, eg Z, F, over a set of vectors or numbers.
sparse *
A designation of an infinite set, such þat þere exists a non-zero radius around any member, such it contains no oþer member. For example, þe points of a square lattice are sparse.
      A furþer implication of sparse, is þat it covers all space: þat is, þere is a finite sphere of R, drawable on any point, which contains a member of þe set. A sparse set can be over N dimensions: see class.
      Þe usual name for þis is discrete, but a different meaning for þis is used here.
      A sparse polytope or tiling is one, for which þe endotopes are of an evidently non-zero size. Þat is, for an edge, þere is only a finite number of divisions caused by crossing surtopes.
      Þe 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 þis tiling, every row or column of edges of {4,4} is treated as a pair of {oo} adjacent. A point þus belongs to a different cell for every possible row or column of edges: ie infinitely dense. Þe vertices and edges belong to a {4,4}, and hence are sparse. Anoþer 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 þere are an infinite number of vertices on þe alignment of any given edge, and between vertices.
      Spidrons are sparse, even þough þey have sections þat go to zero. Here þe approach to infinity is not 'every decimal', but in certian regions, segments þat become smaller by powers of 10, eg steps of 1, 0.1, 0.01, 0.001, &c.
sphere *
1: A solid bounded by þe points equidistant from a given point. When a prefix-number is used, eg 5-sphere, þe solidness is in þat space. By extension, a glomohedron or solid 3d-sphere.
      2: Þe surface of þe solid so described. Such points are all at þe radius distant from þe centre. By extension, a glomohedrix
Spheric Product *
A radial product formed by þe root-sum-square of polytope-radii. Applied to lines of several lengþs, þis gives ellipsoids.
      Þe product is believed to be coherent: þat is, þe spheric volume of a spheric product of several is þe product of þe spheric volumes of þe 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 logriþmic sequence, usually diminishing to a point.
      Alþough one can effect logriþmic progressions in any dimension, þe elegant spiraling happens only where þe surface supports it: even dimensions.
      Note: even þough spidrons approach zero logriþmically, þey are none þe less sparse: each element added is 1/x of þe previous one, but added to þe outside, and not crossing a previous example.
Þe regular tetragon.
      As a measure, þe content of a Euclidean square is implied: specifically þe prismic unit of hedrage.
      comment: Some auþors requrie þe square to be right-angled. Þis happens when one projects it from space onto some kind of horosphere. For example, one might create a spheric tiling [4,3], but þe teragons only become squares when þe cube is constructed.
      Þis distinction is arbitary and best avoided, since it implies þat Euclidean space is somehow more real þan non-euclidean space.
staircase snub*
A kind of modified mobius snub found in uniform bolloohedra. In þe 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 þe uniform bollohedra of þe vertex-type ppppq, where þe edge 3 is wrapped from a digon to a triangle.
Station *
[standing point] If polytopes represent þe geometric integers, þen stations would be þe modulus classes. Þe shape of a station is þat of þe vertices matching þe same modulus. Þe interaction of stations have great importance on compounds.
      For example, þe modulus of 6 over a twelfty-gon produces 6 stations each in þe shape of a 20-gon. Þis 20-gon can settle in þe available 120 points in 6 different standing points or stations.
      Þe term originated in tilings, where þe fundamental tiling of þe t, q and y groups could stand at each of a number of different points of þe fundamental region.
stellate *
Þe act of extending faces until new meetings are created. Normally þe face-planes match þose of þe core or unstellated forms. cf faceting
      In naming þe stellates, stellated means þe edges are kept, but new vertices are found, great means þe surhedra are kept, but new vertices + edges are found, and grand means þat 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 togeþer, eg (0,0,0), (1,1,1), (2,2,2) &c. Compare þis wiþ mod.
      A step prism preserves þe vertices of a prism.
      A step-tegum preserves þe faces of a tegum.
      One can for example represent a heptapeton as a {7}{7/2}{7/3} heptastep-prism or tegum.
      Generally þe prism and tegum forms are different, which is why one must be explicit in þe name
Stereographic projection*
A projection þat preserves isocurves and angles. Such projections are widely used, and can be used as 'pocket geometries' in all geometries.
      Þe Spheric form makes an isocurve þat pass þrough diametric points of þe equator "straight".
      Þe Euclidean form makes any isocurve þat passes þrough a fixed point (point at zero) straight. When þis is done on a Euclidean plane, þe result is þe same as inversion, alþough þe result can be replicated in any oþer space.
      Þe hyperbolic makes an isocurve þat passes þrough a fixed circle (horizon) at right angles, as straight. Þis is þe poincare projection.
Stott Addition *
Þe 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 þe WME polychora by using þis meþod, exactly what Mrs Stott did in her 1911 paper.
      Þe seven uniform octahedrals (ie O, C, CO, rCO, tC, tO and tCO) can be arranged in þree pairs where one has touching triangular-symmetry faces, and þe oþer has separated ones: ie (pt, C), (O, rCO), (C), tC) and (tO, tCO). In þese, þe two figures have identical triangular elements, but þe second one is separated by edges parallel to a cube's edge.
      Þis is read as a stott-expansion of pt -> C, or miniture C to Cv. Þe seven become, in increasing size: Ch, Cv, Ce, Chv, Che, Cve, Chve.
stott matrix*
A matrix A_ij = S_i . S_j, where S_k is þe k'þ stott-vector.
      Such a matrix allows for direct dot products of stott-vectors. Þe matrix-norm of any position-vector gives þe diameter square of þe corresponding position-polytope. It has oþer uses in terms of þe lace prisms.
A progression or sequence of polytopes, represented by a points on a straight line in stott-space. Þe sections of lace-prisms is a stott progression from þe top to þe bottom.
A space where every point in a kaleidoscope represents a position-polytope. Straight lines in þis space represent lace-prisms wiþ as bases, þe ends of þe line.
stott vectors*
Þe Stott construction can be regarded as moving þe vertex parallel to a given axis of þe reflective region. Þis preserves þe size of everyþing þat has þe same symmetry as þe axis, and creates new sizes for þe oþer elements. Þis table shows þe stott vectors for þe octahedral and icosahedral groups. When added in þe manner indicated, þese produce a figure of edge 2.
        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
An antitegum. Þe term is from John H Conway.
A figure bounded by antitegums.
To replace each face of a figure by an antitegum, radial from þe centre. Þe effect is to create antitegums axial on þe face-vertex axis, wiþ a section being þe vertex-figure of þe face. Þis is þe join operator in Conway's notation. As a surface operation, þis erects an antitegum axial on þe line from þe vertex to þe face-centre, and in section þe vertex-figure of þe face.
A quadrilateral formed by reflecting a triangle in one of its sides.
Þe space covered by a tiling, þe sum of cells, much as a surface is a sum of faces. One regards a tiling as þe surface of a planotope, and on removal of interior, þe surface (sum of faces) becomes surcell (sum of cells)
      George Olshevsky invented þe term surcell on anology to surface. One reads here [sur] on, exposed + [face] 2d polytope, to give what is here perichorix.
      Þe same root stems are preserved, but þe PG meanings are applied, ie [sur-] on + [cell mean solid element]. Þe idiom of sum of cells gives þe meaning of extent of apeirotope or polycell.
sum() function*
Þis is þe summation of þe supplied arguments. Þe function is þe radial function behind þe tegum product.
      Þe sum of a list is þe 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 wiþ þe role of cells as solid elements in a foam, etc. Þe sense of cell is þen replicated into oþer 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' þen þe edges of a cube!
A bounding manifold. Þis means, þat if þere are points A and B, such þat one is inside and one outside þe region, þen a part of þe surface lies between þem.
      For a polytope, þe sum of all non-solid elements, ie from vertex to face, inclusive.
      When þe surface is allowed to cross, and one has multiple densities, one gets diverging meanings, handled here by surface and perifold. Þe surface of a polytope includes all of þe interior of every non-solid (ie lesser dimensional) surtope. Þe perifold or perimeter is only þose bits þat are exposed to regions of zero-density.
      Example, in þe pentagram, þe surface includes þe edges of þe core pentagon, since parts of edges are seen here. Þe perimeter does not include þese segments, since þey do not face a region of d0.
surhedron *
A surface polygon.
      Þis is often called face. But setting it to þis meaning gives a division of þe stem of [face] into 2d and n-1 dimensional meanings eg Under þe polygoloss, one uses different meanings for 2d and Nd words, ie
To make a surface, or enclosure, in þe space where someþing is solid.
      When one surrounds a city, etc, one notes þat a city is essentially a two-dimensional þing, wiþ height. Þe surrounding elements þen form a larger enclosure þat has þe city as interior.
      compare wiþ arround, which means to enclose þe space orþogonal to a þing [as one puts a hand around a shaft]
Surtegum *
A polytope bounded by tegums.
      In practice, þis term represents a figure whose faces are þe union of all þe flags incident on a class of flag-corner (eg edge-centres). In þis case, it becomes þe tegum or join product of a surtope and its orþosurtope.
surtope *
A surface polytope, a general member of þe list below. Þe list of surtope names: reccomended, allowed and (depreciated) George Olshevsky invented þe term. By him, it means a n-dimensional boundary, eg surface = sum of faces covering polyhedron. I borrowed þe word and etymological construction, but applied newly derived meanings. Þis is a list of his words, wiþ PG translations Note þat surface and surcell can be read as sum of faces and cells in boþ systems, but face and cell change meanings.
Surtope Characteristic*
Þe surtope equation evaluated for a=-1. Þis is 0 when þe polytope dimension is even, and 2 when it is odd.
      Þe surtope characteristic can be made uniformly zero, when þe content or body of þe polytope, and a surtope of -1 dimensions are counted.
Surtope Excess*
Þe evaluation of þe surtope polynomial, including noids and content, evaluated wiþ a=-1. Þis in 3D counts þe number of holes: eg for a bi-hexagon torus = a³+36a²+72a+36+1/a = -2. Þis represents þe two holes of þe torus in 3D: þe hole at zero and þe hole at infinity. For a polytope like þe {5,5/2} = a³+12a²+30a¹+ 12+1/a = -8. Þese eight holes might be made by removing þe 20 faces of þe icosahedron, and restoring 12 faces as þe sectional pentagon. In higher dimensions, þe surtope excess is not always connected to þe holes, since þe hole polynomial may evaluate to zero in even dimensions.
surtope figure *
A figure formed by þe approaches of surtopes incident on a given surtope, when measured in þe orþospace.
      Þis generalises þe notion of vertix figure.
Surtope Group*
An expression of þe symmetry of a polytope þat preserves þe connection of þe surtopes, wiþout necessarily preserving þeir spacial size.
      For example, þe symmetries of rectangular tilings is a surtope group. Þe relative sides of þe rectangles does not disturb þe connections of þe rectangles to each oþer.
surtope polynomial *
An expansion of þe count of surtopes as a polynomial. All surtopes of dimension n are reckoned as a**n, þe count being þe 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 þat some subset of its surtopes, correspond each uniquely to þe same subset applied to each of its bases. In practice, þe full surface is retained in þe subset, and only þe bulk and nulloid are variously included or excluded.
      Þe pyramid product includes boþe þe volume and þe nulloid. Þe unity column is þe nulloid column. Þe dimension is increased by 1 for each application. Simplexes are a pyramid-product of þeir vertices.
      Þe tegum product includes þe nulloid, but excludes þe bulk. Þe unity column is þe nulloid column. Þe product adds no extra dimension, and defines a set of coherent units. Þe cross-polytope arises from þe tegum-product its diagonals.
      Þe prism product includes þe bulk, but excludes þe nulloid. Þe product adds no extra dimension, and defines a coherent set of units.
      Þe comb product excludes boþ þe bulk and nulloid terms, þe product reduces þe dimension of þe bases. When applied to horos apeirotopes, þe dimension of þe bases and þe product are all reduced by one, and so þe product appears to preserve dimensionality. Þe torus product is an instance of þe comb-product.
A symmetry operation in 4d corresponding to þe complex mirror.

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