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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 that many dimensions, eg surexon.
• Note that edge carries also meanings of boundary, as in to the edge, and the leading edge. For the edge of a knife to divide a solid into two, the implied dimension is n-2, not 1. In the parlance of the polygloss, this sense is taken by margin.

edge-uniform *

An equalateral vertex-uniform polytope.
      This is the sense of in common usage.

edge vector*

The vector from the centre to the vertex of a wythoff-mirror-edge figure. This is normally stated in forms of Stott Vectors, since it allows one to directly designate the edges as measured.
      For example, the edge-vector of a truncated icosahedron x3x5o is (1,1,0).
      Edge vectors can contain negative values, such are used in drift calculations for the circum-diameter of the lace-prisms.

efficiency*

A measure of sphere-packing. This can be measured in terms of total fraction of space taken by spheres, or the number of spheres one might pack into a given volume. The latter is usually preferred.
      Leech-Unit is the number of diam 2 spheres in a unit cube.
      Q-Unit is the number of diam √2 spheres in a unit cube.
      Implied S measures the corresponding solid angle of a simplex, in tegmal radians

endo-*

The sense of endofy is to mark all of the internal crossings, whether or not these are surface crossings or not. A surtope so marked is an endosurtope, eg endocell, .
      For example, the faces of a great dodecahedron is a simple pentagon, which joins adjacent edges at its edges. The endoface would also show the internal crossings of five other faces, which form an inscribed pentagram.

endoanalysis *

The finding of surtopes by change of density in the interior. The polytope is seen as an endotiling of endocells and . One then finds whether these support a change of density, which would make them into surtopes.
      For example, the edge of a pentagram is made of three parts, but each of these have the same size out-vector. Thus while parts of the edge separate density two from density one, and others density one from density zero, there is no change in the density along the line (it is always one), so the crossings at the pentagon are not vertices.

equal sign *

A subspace might be set by several equal signs. A plane is set by something like one equal sign, eg x=0. A boundary on a plane needs two: x=0, y=0. The idea here is that 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.
      The circle and sphere are treated as polytopes, {O} and {O,O}, using the letter O. According to the Schlafli symbol, it implies the first node is marked.
      To represent ellipsoids, one makes use of the old style / and \ marks, which suggest rising and falling edges. These are translated without any alteration. One is not allowed to mix the 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)
      The sphere then participates in assorted products, which are represented in the 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, this is read that all of the 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. The snub 24choron {;3;4,3} is equihedral.

equidistant *

Two isocurves are equidistant, if any perpendicular ray through one is perpendicular to the other.

equilateral *

Having equal edges.
      The concept corresponding to the features of the dual is equimarginal, is not in general circulation.

equimarginal *

Having equal margins. The implied sense is that the same size insphere can be inscribed in every cell, in such a way that it touches every wall of the cell.
      In the case of polytopes, this usually implies equal margin angles.
      This is the dual of equilateral.

Euclidean*

A name in common use to refer to space of zero curvature.
      For this space the fifth postulate and its equivalents are true, eg there exists a triangle with a corner sum of :60, or the isocurve equidistant from a straight line is also straight.
      Zero-curvature is designated by the prefix horo-, related to the infinite horizon.

Eutactic *

A eutactic asterix is a set of vectors radiating from a point, from which one forms zonehedra. Coxeter introduced the term under the name eutactic star in his book Regular Polytopes. It is used in conjunction with projections of the measure polytope from higher dimensions.
      The span of vectors in a eutactic asterix make a eutactic lattice.
      For mirror groups, the eutactic asterix is taken as the orthogonals to the mirror-planes. The corresponding eutactic lattice is either sparce or a peicewise finite tiling.
      Every mirror-edge polytope of integral edges can be reproduced in the Eutactic lattice for its symmetry. Thus if one can reproduce the vectors in the asterix, one can reproduce any derived mirror-edge figure.

excess*

The area of a spherical polygon is proportional to the excess of its angles over the euclidean or zero-curvature value. The whole of the sphere is two circles, and the dimension is linear: so one might talk of 12° 22' of Excess

exon *

A mounted 6d polytope, or a 6d 'hedron'
      This spelling is depreciated in favour of ecton. In part this was due to the word exix for ectix.

exoskeleton *

A proposed name for the outer or visiable parts of a polytope where the surface crosses itself. See periform

Exotic *

Exotic means foreign or out-landish.
      An exotic polytope has a surtope with ambiguous margins.
      The 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 the gaps.
      For example, if the edges of a cube are radially moved out, the vertices become triangles, and the square faces become octagons. Expanding the faces makes the cube into a rhombocuboctahedron, the vertices become triangles, and the edges become new squares.
      The process of contraction undoes an expand.
      A Conway operator corresponding to Stott's face expand. The effect of this operator is to make a runcinate. The dual operator is ortho.

extrude *

To extend a subset of faces of a polytope to create a new polytope.
      This corresponds to the process of inscribing a figure: for example, as a cube is inscribed in a dodecahedron, so is an octahedron extruded from the icosahedron.

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