# -: O :-

oa *
The middle chord of an octagon, also the unit of the Z4 or octagonal numbers, equal to √2+1.
(twe: 2:4984 8104,3529 0779,4628 3031,0365 V712,4341 E393)
(dec: 2.414213562373095048801688724209698078569)
octagon*
octagon-cubic*
A group of four non-Wythoffian hyperbolic aperitopes that share a lot of symmetry with {3,4,3,8} and its sections.
• The laminatruncated lt{;4;3,8}. This normal truncate has faces {3,8} that are smooth. Such faces can be used as mirrors, to have space completely filled with t{;4;3}, the vertex-figure is the octagonal tegum formed by rotating an octahedron 45° [15:00] around an axis. The edge corresponds to a {3,8}. The symmetry group contains the {8,3,4}, formed by one of the octahedra.
• The extendotruncated xt{;4;3,8,2} is a tiling of bt{3;4;3}. The vertex figure of this is an bi-octagon tegum, any octagon by a side of the other side is the vertex figure of a {3;4;3,8}. The edge corresponds to a {3,8}.
• The dual of the xt{;4;3,8,2} is xt{;2;8,3,4}, a tiling of bioctagon prisms, 288  to a vertex. The vertex figure is o3m4m3o, the convex hull of dual 24-chora. The edge is that of {4,8} = {8,6}
• The o8o4xp3xr. Without the p and r, the vertex figure of this is a distorted octagon prism xq8oo&#q, the top octagon being of edge 1, and the base and sloping edges being of edge r2=1.414. The trapezium sides are the vertex figure of the rhombocuboctahedron {;4,3;}, the top and bottom being {3,8} and {4,8} respectively.
The effect of the p operator is to replace the top with a pyramid of sloping edge r2, representing triangular prisms. The r operator reflects this in the base, giving a figure bounded by 16 triangles and 16 trapezia.
The vertex figure consists of the two poles, and 24 vertices being the cuboctahedral vertex figure of {8,4,A} and the same rotated 45 degrees [15:00] around one of its square axies. The equator and the lines of longitude are all octagons edge r2, the two lines of latitude is an octagon of edge 1.
Octagonal Ball
The vertex figure of the o8o4xp3xr is an exotic tower oxqxo8ooooo&q, this has 32 faces. It looks like a globe, with lines of longitude and latitude at every 45 degrees.
One can construct this figure, by taking a cuboctahedron, and its image rotated 45 degrees over a square axis, and two vertices on the axis of rotation.
Octagonal Barrel
A figure made from the intersection of a rhombocuboctahedron, and the same rotated 45 degrees (0:15 circle) around a square face. Such is the dual of the octagonal ball, and tiles H3 as o8o4mpAmr.
octagonny*
The special name for the bitruncated 24-chora, bt{3;4;3}. This has 48 truncated cubes as faces. This figure discretely tiles 4-space.
The faces of the octagonny represents one of the modular sets of faces under the Quarterion multiplication (along with {p}{p},{4,3,3}, {3,4,3} and {5,3,3}.
Octagrammy
The quasitruncate 24-choron bt{3;4/3;3}, having a density of 73 and having 48 quasitruncated cubes as faces. This figure tiles 4-space discretely.
off*
A point not in some all-space. Points off a space are neither in or out of elements of the surface. For example, a point may be off a line, plane.
old-style notation*
The first of Wendy Krieger's three presentations, characterised by the use of letters for the more common branches.
omnitruncate*
A figure with a vertex in the interior of every flag of the source.
On
A point is on a surface, if it is a part of a surface which is a proper subset of all-space.
ortho-*
An orthogonal space intersects a given space at a point.
The orthospan of an orthospace is all points that lie in planes defined by three distinct points of the two crossing spaces.
For spaces to be mutually orthogonal, each space is othogonal to the orthospan of the others. This prevents, for example, three lines crossing at a point in a plane from being othogonal.
The sense preserves the meaning of othogonal in the affinite geometry, although the sense of right-angled is usually taken. If one wants to specify a particular meaning, the adjectives right, oblique and general can be used.
orthocontract *
To take an orthogonal section. Usually this is done in some figure's ortho-space.
orthoexpeand *
To extend a figure into one or more higher dimensions, so that every equidistant section is identical to the figure. For example, a triangle in the xy plane, when orthoextended, gives rise to a triangular column.
orthogonal projection*
This is a point-centric projection, where the circumference of circles centred on a fixed point is preserved. In both the Euclidean and Spheric geometries, this is the result of projecting the thing from afar onto a plane. Hyperbolic circles tend to grow very fast in terms of the radius, the area of a circle divided by the circumference can not exceed a fixed amount.
orthospace *
Space that is essentially perpendicular to a given space at a point. In terms of a vector space, the only vector common to orthospaces is the zero-vector.
• The z axis is the orthospace to the x-y plane, but one may replace the vector z by any ax+by+z. While this space is no longer orthogonal to it, it is still an orthospace.
orthosurtope *
The corresponding surtope of the dual to a surtope. This is the dual of the surtope figure.
orthotope*
A name for the measure polytope.
van Oss polytope*
The girthing polytope. One can prove the non-existence of a polytope by showing its van Oss polytope fails to close.
out*
A point is outside a figure, if it lies in the same space that the figure is solid in, but is not part of the figure.
out-vector*
A notional vector that points from interior to exterior of the polytope, and has a magnitude equal to the density of the surtope it passes through.
In a normal periform, the surface is singular, and so it behaves like a unit pressure vector acting on the inside of the figure to the outside.
When multiple crossings are allowed, the size of the vector changes. In the case of the stellated dodecahedron, the out-vector pointing from the core d3 dodecahedron into the d1 points has a magnitude of d2, the same as that of the density of the core pentagon of a pentagram.
Transverse outvectors operate in the surface of a polytope. The idea being that across the margins, there should be no net transverse outvector: a d2 vector one way should match a d2 vector from the other side.
In polytopes with skew marginals, the direction of the outvector changes. Often the skew figures are implemented as binary or XOR'ed figures.