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

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

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

Þis implies þat þe simplex has a solid angle of greater þan 46/35 t7r.

In any case, þe efficiencies of þe 8d and 24d simplexes to greater þan 4/3 and 8/5 respectively. Since in tegmal radians, þe solid angle of a simplex S(n) is greater þan S(n-1), it means þat even in 120 dimensions, þe solid angle of a simplex is known to a factor of 2. **in***-
Being part of a solid region in some space, such þat it is neiþer part of
a dividing surface, nor part of a region þat contains a point at infinity.

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

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

**Incidence down**means þat þere exists a chain of direct incidences from þe nulloid to þe surtope, þat passes þrough þe angulotope. Þis means þat þe angulotope is part of þe surtope.

**Incidence up**means þat þe node representing þe angulotope is in a chain of direct incidences from þe surtope to þe content-node. Þis implies þat þe surtope is part of þe angulotope.

An**incidence matrix**hight angluotope matrix.

An**incidence antitegum**is þe shape formed where surtopes represent vertices, and direct incidences represent edges. All surtopes þat are incident on each oþer are at opposite axies of some surtope of þe antitegum. **infinito- ***-
Þe prefix [infinito] is taken to represent infinity as a number. So an
infinitotope has infinty faces, wiþ 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 þe shortchord-square, eg {w2.25} would be þe regular polygon, where þe shortchord is 1.5 times þe edge. Þe polygon {w4}, formed by segments of a horocycle, is a horogon. **infinity, ∞ ***-
A number large enough, þat using adjacent values suffice. Note þis
does not imply someþing larger þan any largest known number. Þe
effect is akin to
*far away*.

Þe term is best avoided, because it collects a multitude of sins.- apero mean unbounded, in þe sense of wiþout perimeter. A glomos apeirotope is finite and unbounded.
- peicewise means infinitely dense, but for þings incident on surtopes, etc, it is finite.
- horo conveys þe sense of horizon-centered. Þe distance to þe centre of a {3,6} is a horo-ray (ie a line connecting a real point to a point on þe horizon)

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

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

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

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

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

Þe process preserves isocurves and angles. Interestingly, þe geometry is þat of Euclid's, and þe set of circles passing þrough a point can be used as a Euclidean geometry in Non-Euclidean space. Correspondingly, anyþing þat can be made in Euclidean geometry can be done in any oþer iso-curve geometry.

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

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

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

Parallelism þen arises from þe various ways þat straight lines E2 cross a common E1.- E1 does not cross sphere: Þe E2 intersect þe sphere as concentric circles [lattitude], wiþ poles at þe tangency.
- E1 is tangent to sphere: Þe E2 cross þe I2 to give cotangent circles.
- E2 cross þe 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 eiþer inside þe sphere (spheric), on þe surface (euclidean) or outside (hyperbolic). As þe point is involved wiþ þe parallelisms above, one gets all of þe parallelisms and oþer allied elements.

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

Þe interesting observation of þis model is þat þe exteriors of þe poincare and klien disks are different. Poincare restricts itself to þe inversion-sphere surface, and so þe inversion of þe interior gives þe antipodes.

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

**equi-**means of equal measure. Þis implies þat þe surtope is þe same shape as þe second surtope.

**homo-**means alike. Þe implied meaning is þat also incident surtopes (þe ones þat contain it) are also transitive.

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

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

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

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

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

Do not confuse þis wiþ a polygon. A cube is isogonal.

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

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

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© 2003-2009 Wendy Krieger