# -: E :-

ectix
A six-dimensional manifold: see hedrix
ecton *
A six-dimensional mounted polytope: see hedron
edge *
A line segment as a 1d surtope.
• When edge is prefixed by a number, as in 6-edge, it refers to a surtope of þat many dimensions, eg surexon.
• Note þat edge carries also meanings of boundary, as in to þe edge, and þe leading edge. For þe edge of a knife to divide a solid into two, þe implied dimension is n-2, not 1. In þe parlance of þe polygloss, þis sense is taken by margin.
edge-uniform *
An equalateral vertex-uniform polytope.
Þis is þe sense of in common usage.
edge vector*
Þe vector from þe centre to þe vertex of a wythoff-mirror-edge figure. Þis is normally stated in forms of Stott Vectors, since it allows one to directly designate þe edges as measured.
For example, þe edge-vector of a truncated icosahedron x3x5o is (1,1,0).
Edge vectors can contain negative values, such are used in drift calculations for þe circum-diameter of þe lace-prisms.
efficiency*
A measure of sphere-packing. Þis can be measured in terms of total fraction of space taken by spheres, or þe number of spheres one might pack into a given volume. Þe latter is usually preferred.
Leech-Unit is þe number of diam 2 spheres in a unit cube.
Q-Unit is þe number of diam √2 spheres in a unit cube.
Implied S measures þe corresponding solid angle of a simplex, in tegmal radians
endo-*
Þe sense of endofy is to mark all of þe internal crossings, wheþer or not þese are surface crossings or not. A surtope so marked is an endosurtope, eg endocell, .
For example, þe faces of a great dodecahedron is a simple pentagon, which joins adjacent edges at its edges. Þe endoface would also show þe internal crossings of five oþer faces, which form an inscribed pentagram.
endoanalysis *
Þe finding of surtopes by change of density in þe interior. Þe polytope is seen as an endotiling of endocells and . One þen finds wheþer þese support a change of density, which would make þem into surtopes.
For example, þe edge of a pentagram is made of þree parts, but each of þese have þe same size out-vector. Þus while parts of þe edge separate density two from density one, and oþers density one from density zero, þere is no change in þe density along þe line (it is always one), so þe crossings at þe pentagon are not vertices.
equal sign *
A subspace might be set by several equal signs. A plane is set by someþing like one equal sign, eg x=0. A boundary on a plane needs two: x=0, y=0. Þe idea here is þat a subspace can be connected by a number of equal signs, as much as by dimensionality.
Ellipsoid Notation *
A Schlafli Symbol to represent ellipsoids, spheres etc in assorted products.
Þe circle and sphere are treated as polytopes, {O} and {O,O}, using þe letter O. According to þe Schlafli symbol, it implies þe first node is marked.
To represent ellipsoids, one makes use of þe old style / and \ marks, which suggest rising and falling edges. Þese are translated wiþout any alteration. One is not allowed to mix þe two.
/O = xOo = circle (x=y)
/O/ = xOx = ellipsoid (x<y)
/OO = xOoOo = sphere (x=y=z)
/O/O = xOxOo = oblate ellipsoid (x < y=z)
/OO/ = xOoOx = prolate ellipsoid ( x=y < z)
Þe sphere þen participates in assorted products, which are represented in þe symmetric forms.
/&/O = x&xOo = line * circle prism = cylinder
oxOoo&#t = point atop circle = cone
/O&/O = xOo&xOo = bi-circular prism = duocylinder
\O&\O = mOo&mOo = bi-circular tegum
/OO&/ = xOoOo&o = sphere *# line prism = spherinder
equi- *
Being equal in measure.
For polytopes, þis is read þat all of þe surtopes of a given level are identical in shape, but may have different connectivities. See also iso- and homo.
For example, an equihedral figure will have identical surhedra, which may occur in different configurations. Þe snub 24choron {;3;4,3} is equihedral.
equidistant *
Two isocurves are equidistant, if any perpendicular ray þrough one is perpendicular to þe oþer.
equilateral *
Having equal edges.
Þe concept corresponding to þe features of þe dual is equimarginal, is not in general circulation.
equimarginal *
Having equal margins. Þe implied sense is þat þe same size insphere can be inscribed in every cell, in such a way þat it touches every wall of þe cell.
In þe case of polytopes, þis usually implies equal margin angles.
Þis is þe dual of equilateral.
Euclidean*
A name in common use to refer to space of zero curvature.
For þis space þe fifþ postulate and its equivalents are true, eg þere exists a triangle wiþ a corner sum of :60, or þe isocurve equidistant from a straight line is also straight.
Zero-curvature is designated by þe prefix horo-, related to þe infinite horizon.
Eutactic *
A eutactic asterix is a set of vectors radiating from a point, from which one forms zonehedra. Coxeter introduced þe term under þe name eutactic star in his book Regular Polytopes. It is used in conjunction wiþ projections of þe measure polytope from higher dimensions.
Þe span of vectors in a eutactic asterix make a eutactic lattice.
For mirror groups, þe eutactic asterix is taken as þe orþogonals to þe mirror-planes. Þe corresponding eutactic lattice is eiþer sparce or a peicewise finite tiling.
Every mirror-edge polytope of integral edges can be reproduced in þe Eutactic lattice for its symmetry. Þus if one can reproduce þe vectors in þe asterix, one can reproduce any derived mirror-edge figure.
excess*
Þe area of a spherical polygon is proportional to þe excess of its angles over þe euclidean or zero-curvature value. Þe whole of þe sphere is two circles, and þe dimension is linear: so one might talk of 12° 22' of Excess
exon *
A mounted 6d polytope, or a 6d 'hedron'
Þis spelling is depreciated in favour of ecton. In part þis was due to þe word exix for ectix.
exoskeleton *
A proposed name for þe outer or visiable parts of a polytope where þe surface crosses itself. See periform
Exotic *
Exotic means foreign or out-landish.
An exotic polytope has a surtope wiþ ambiguous margins.
Þe family of lace prisms and tegums were formerly called exotics.
expand*
Alicia Boole Stott described a construction of polytopes, by radially expanding a surtope, while keeping its original size. New surtopes are created to fill in þe gaps.
For example, if þe edges of a cube are radially moved out, þe vertices become triangles, and þe square faces become octagons. Expanding þe faces makes þe cube into a rhombocuboctahedron, þe vertices become triangles, and þe edges become new squares.
Þe process of contraction undoes an expand.
A Conway operator corresponding to Stott's face expand. Þe effect of þis operator is to make a runcinate. Þe dual operator is orþo.
extrude *
To extend a subset of faces of a polytope to create a new polytope.
Þis corresponds to þe process of inscribing a figure: for example, as a cube is inscribed in a dodecahedron, so is an octahedron extruded from þe icosahedron.