# -: B :-

base*
(1) A polytope being used as a argument to a product or function, such as a base of a prism, tegum or lace-prism.
(2) Any polytope representing an unmodified structure (eg the unfacetted or unstellated figure)
between*
For some functionality, belonging to a set that includes the extremities as bounds.
Of points, the functional set is a sphere diametric on the points.
bevel*
A generalisation of truncate, but applied to any kind of surtope, where truncate applies to vertices.
Spefifically, the decent of a plane, containing the surtope, and normal to a line through a centre and the surtope, that removes the surtope and creates a new face in the location of the surtope.
An edge-bevel of the cube, will create long hexagonal faces that follow the alignment of the edges. Letting this bevel run to completion would cause the new faces to become rhombuses, or generally, the dual of a figure with vertices in the surtope.
Applied to The dynkin symbol, the intersection of separate figures, each corresponding to a single mirror-node. eg b3b4o = edge-beveled cube, arises from the intersection of b3o4o (cube) and o3b4o rhombo-icosahedron.
Bevel (Conway)
A Conway's operators, producing the omnitruncate.
The dual of bevel is meta, which converts every face into its flags.
One can also bevel against particular surtopes, such as edges or surhedra. The face-bevel just makes the polytope smaller, the vertex bevel is simply the truncation by descent of faces.
In this sense, the bevels are sequences of lace-tegums, eg the edge-beveled cube is sections of the polychoron xo4om3oo&#m.
bi*
The reccomended meaning of bi is to mean two, but not in the sense of a pair, for which di- is reccomended.
binary*
A polytope with densities of either 0 or 1. There are two different forms of binary polytope. These have different endo-analysis
Mod2 polytopes reduce some other density over modulus 2. The endoanalysis of a mod2 pentagon makes the interior hollow. The small stellated dodecahedron has a single region inside, the core of the pentagon not dividing space.
XOR'ed has every endoface as a unit density dividing plane, so that endocells are checkered. This is relatively easy to implement in a program, since one does not have to do an endoanalysis. On the other hand, the pentagram in the stellated dodecahedron turns out to be a zigzag decagon, since the whole interior is one.
bipyramid*
The name allocated in common usage to a polygonal tegum.
In four dimensions, one might equally call a polygonal pyramid pyramid or a polyhedral tegum a bipyramid.
The term is depreciated in the PG.
blend*
A process of generating new polytopes by the rule of margin-closure. In essence, several polytopes are overlaid so that they have a number of shared faces.
The blend sum is a polytope bounded by unmatched faces.
The blend polytope is the complete polytope that all of the cells form.
bollo- *
Refering to figures inscribed or following negative curvature. Such are commonly called pseudotopes.
Bollo is derived by backformation from hyperbolic, in much the same way that omnibus becomes bus.
• A bollotope is a polytope inscribed in hyperbolic sphere.
• A bollosurtope is a surface bollotope.
bollohedrix *
A hyperbolic manifold or space of two dimensions, such as designated by H2. See hedrix.
A bollohedron is a polytope bounded by a bollohedrix.
bollos *
This adjective is used to describe an embedded shape, rather than the space it is in. A bollos polyhedron is one that would be a tiling in hyperbolic space. See also glomos, horos.
bouy-land *
An implementation of hyperspace where a solid becomes a bipyramid or tegum. Such then would resemble bouys floating in the surcell of space.
#-bouy-tegum *
A tegum, where one of the products is the primitive, a line segment.
The shape notionally represents a navigational bouy, floating on the plane of its base.
When several line-segments are present, one might say bi-bouy tegum &c.
borromeachoron*
A family of uniform bollochora that appears in the Not Knot video, &c.
The vertex-figure resembles an icosahedron as 3/*/2%, or a snub octahedron. The original octahedral faces of this figure become cubes, and the twelve remaining faces become polygonal prisms.
The square borromeanochoron is the same as {4,3,5}.
The edge2 of the uniform borromeachoron is E2 = A2/2 + A1.sqrt(16-3A2)/2,
where A1 is the shortchord and A2 the shortchord square.
#-boundary*
A surface that bounds. Note there is no restriction on solidness, as one might say that 5 and 8 are lower and upper bounds.
See also peri-.
D.M.Y Somerville used this term for #-edge.
Bowers acronym*
A series of short names proposed by Jonathan Bowers, and having some currency. These take Bower's extensions to the existing names, and produce short names, based on "significant" letters in the source name. When one frequently uses a range of uniform figures, these names become a blessing.
Bower's naming
Jonathan Bowers proposed this series to name the assorted regular truncates. They are based on the various kepler names, and is extendable to higher dimensions, by the addition of additional prefixes.
 ``` (number) cello- prismato- rhombi- trunc- ated 5432xtrpc.. ooxoxx tri prismato rhombi ated oxoox bi prismato xxxo rhombi trunc ated ```

Small and great are also used, small means that all the values between the first and last named nodes are unmarked, and great means that they are marked.

Bowers Notation
Jonathan Bowers devised an alternate form for writing the Dynkin symbol inline. This relies on 'greeking' the figure, while preserving the important features. Greeking is what printers do to show the presence of words, without making any of them legible.
```                o  o  @  @
|  |  |  |
o  @  o  @  o  @      2      3     4   4/3   5

xo6o    o  x  8  9  6  \$    space  abbut   '    "    *

o                 @
|                 |                    5
o---o---o---o     @---o---o---o      o---@---o---o

o   o   8   o     x   o   6   o      o   x   o * 5
oo8o               xo6o             oxo*5

```
The notation is not as flexiable as those on the pseudoregular trace, because the primary zoo is much smaller, and the range of nodes is much less
bulk *
The solid interior of a polytope, when reckoned as a surtope.
branch*
A second-series edge, connecting nodes.
One uses the second series names if there is some source of confusion between the two. This might happen if there are two distinct referrant spaces, such as nature and some representation. Things in nature would be in the first series, and the representation would be in the second series (like nodes, branches &c).
The Dynkin symbol uses second-series names.

© 2003-2009 Wendy Krieger