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**Implied S***-
A measure of efficiency.
The unit multipiles the solid fraction taken by the spheres by 2**(n/2)/√(n+1).
This accomidates the vast emptyness of space of packed spheres in the higher
dimensions.

A unit of x would place x tegmal units at each vertex of a simplex of edge 2. In the maximum case, where all of the holes are simplex ones, the implied S would actually give S.

For example, in eight dimensions, the sphere of diameter 2 has a volume of π**4/24. A tiling of efficiency 1 would then occupy a packing of 24/&pi**4;, of nearly 528/2143( . The effect of using implied S units will be to multiply this by 16/3, giving dec 2816/2143*twe:*448/17V3)( .*twe:*2356/17V3)

This implies that the simplex has a solid angle of greater than 46/35 t7r.

In any case, the efficiencies of the 8d and 24d simplexes to greater than 4/3 and 8/5 respectively. Since in tegmal radians, the solid angle of a simplex S(n) is greater than S(n-1), it means that even in 120 dimensions, the solid angle of a simplex is known to a factor of 2. **in***-
Being part of a solid region in some space, such that it is neither part of
a dividing surface, nor part of a region that contains a point at infinity.

Note that in practice, for convex regions, the interior is taken to be the area where the shorter or sole lines connecting the surface points fall. **incidence ***-
In a polytope, a surtope is incident on a second surtope, if one is
completely part of the other. Incidence is denoted by the notion of
corner, and angluotope.

A**direct incidence**is where the surtope and the angulotope are of adjacent dimensions.

**Incidence down**means that there exists a chain of direct incidences from the nulloid to the surtope, that passes through the angulotope. This means that the angulotope is part of the surtope.

**Incidence up**means that the node representing the angulotope is in a chain of direct incidences from the surtope to the content-node. This implies that the surtope is part of the angulotope.

An**incidence matrix**hight angluotope matrix.

An**incidence antitegum**is the shape formed where surtopes represent vertices, and direct incidences represent edges. All surtopes that are incident on each other are at opposite axies of some surtope of the antitegum. **infinito- ***-
The prefix [infinito] is taken to represent infinity as a number. So an
infinitotope has infinty faces, with no requirements or statement on closure
or curvature.

A polygon {π} would wind around its centre infinitly, and so be an infinitogon.

Infinitotopes are normally designated by the shortchord-square, eg {w2.25} would be the regular polygon, where the shortchord is 1.5 times the edge. The polygon {w4}, formed by segments of a horocycle, is a horogon. **infinity, ∞ ***-
A number large enough, that using adjacent values suffice. Note this
does not imply something larger than any largest known number. The
effect is akin to
*far away*.

The term is best avoided, because it collects a multitude of sins.- apero mean unbounded, in the sense of without perimeter. A glomos apeirotope is finite and unbounded.
- peicewise means infinitely dense, but for things incident on surtopes, etc, it is finite.
- horo conveys the sense of horizon-centered. The distance to the centre of a {3,6} is a horo-ray (ie a line connecting a real point to a point on the horizon)

**inno- ***-
Chordal, in the sense that the surface of an innotope is part of the
surface of the polytope. Note that the sharing can also be partial.

This prefix is used mainly in the construction task of completing a polytope given surtopes to a given level.

The idiom is rather like tacking on faces to complete a given frame of vertices and edges of the kind that Leonardo di Vinci would draw. **innoanalysis**-
The finding of polygons, etc where the veritices, edges, etc belong to
a disclosed set.

For example, if one is given the vertices and the edges of an icosahedron, one might close this as either an icosahedron, or a great dodecahedron. This is because the icosahedral edge-frame encloses 'chordal' polygons, of both 20 triangles and 12 pentagons. Other irregular sets of polygons might also provide complete closure, and hence a polyhedron. **innohedron ***-
A polygon for which the edges are part of the polytope. Surhedra are
those innohedra chosen to form the surface.

An innohedron can also be outside the polyhedron, eg the triangles of the icosahedron are outside the great dodecahedron. **inversion***-
A azythmal projection whereby (R, θ) is replaced by (1/R, θ).
This inverts space, so that the point at infinity is replaced by the point
at zero.

The process preserves isocurves and angles. Interestingly, the geometry is that of Euclid's, and the set of circles passing through a point can be used as a Euclidean geometry in Non-Euclidean space. Correspondingly, anything that can be made in Euclidean geometry can be done in any other iso-curve geometry.

If a given polytope has an isocurve tangential to all surtopes of the same order (eg all vertices or all edges), then one might associate to it points that represent each surtope, so that on inversion, the points associated with the matching orthosurtope become coplanar. Because this is a very common feature of the studied polytopes, one might suppose it is always true. It is not. **inversive geometry ***-
A geometry derived from the horizon of the hyperbolic space, also from
considering inversion as an isometry.

One can consider for a given sphere, every circle to be straight. Any three points define a unique straight line.

When one embeds such a sphere in an ordinary space, one then realises straight as the intersection of a plane E2 and the inversive sphere I2.

Parallelism then arises from the various ways that straight lines E2 cross a common E1.- E1 does not cross sphere: The E2 intersect the sphere as concentric circles [lattitude], with poles at the tangency.
- E1 is tangent to sphere: The E2 cross the I2 to give cotangent circles.
- E2 cross the sphere at two points: a set of lines crossing at each of two points.

*straight*for some geometry, if E2 contains a fixed point U, which is either inside the sphere (spheric), on the surface (euclidean) or outside (hyperbolic). As the point is involved with the parallelisms above, one gets all of the parallelisms and other allied elements.

Projective geometry can then be derived by taking a plane, not containing U, and letting planes containing U fall on P. The effect of this projection is to merge antipodal points into the same point (since R, R' and U fall in the same line, and intersect the plane P at a point R").

The interesting observation of this model is that the exteriors of the poincare and klien disks are different. Poincare restricts itself to the inversion-sphere surface, and so the inversion of the interior gives the antipodes.

The exterior of the klein disk corresponds to the orthoinversion. That is, for a flat of m dimensions, the dual is N-m-1, the point nearest the centre of the sphere being taken as the common point of inversion. It is this way we find the 'centre' of a straight line. **iso***-
Iso means equal. It devolves into a number of different senses, but in
the PG,

**equi-**means of equal measure. This implies that the surtope is the same shape as the second surtope.

**homo-**means alike. The implied meaning is that also incident surtopes (the ones that contain it) are also transitive.

**iso-**means equal. This means that the whole polytope is transitive on the surtope. **isocircle ***- A line of uniform curvature, such as a circle or line. The progression of equally spaced points on an isocurve form an isosequence.
**isocurve***-
A curve that is point transitive: that is, every point in the surface can
be replaced by any other point.

**glome**is the general class of sphere.

**horoglome**is the isocurves of zero curvature, corresponding to a centre on the geometric horizon. In Euclidean space, these appear as straight, but they are as crooked as a metric ruler in hyperbolic space.

**bolloglome**is the isocurves of negative curvature. One should not confuse bolloglomes with the horn-shaped negative curve (which is not an isocurve). One sees projections of bolloglomes in the poincare and beltrami-klein projections.

**plane**is an isocurve having the same curvature as all-space. **isogon***-
A polytope having equal iso vertices.

Do not confuse this with a polygon. A cube is isogonal.

gon is related to*knee*, which is not the general rule in higher dimensions. **isopower***-
The N't term in an isosequence beginning with 2, k, written as k^^n

In an isosequence on k, any sequence derived by taking members n steps apart form an isosequence with a constant k^^n. **isosequence***- The general sequence defined by T(n+1) + T(n-1) = k.T(n). The series is important, since the progression of chords of any polygon, or equally spaced points on an isocurve, form an isosequence.
**isospace***- All-space, when it is an isocurve. This includes the hyperbolic, euclidean and spheric geometries.

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© 2003-2009 Wendy Krieger