# -: 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 þe unfacetted or unstellated figure)
between*
For some functionality, belonging to a set þat includes þe extremities as bounds.
Of points, þe functional set is a sphere diametric on þe points.
bevel*
A generalisation of truncate, but applied to any kind of surtope, where truncate applies to vertices.
Spefifically, þe decent of a plane, containing þe surtope, and normal to a line þrough a centre and þe surtope, þat removes þe surtope and creates a new face in þe location of þe surtope.
An edge-bevel of þe cube, will create long hexagonal faces þat follow þe alignment of þe edges. Letting þis bevel run to completion would cause þe new faces to become rhombuses, or generally, þe dual of a figure wiþ vertices in þe surtope.
Applied to Þe dynkin symbol, þe intersection of separate figures, each corresponding to a single mirror-node. eg b3b4o = edge-beveled cube, arises from þe intersection of b3o4o (cube) and o3b4o rhombo-icosahedron.
Bevel (Conway)
A Conway's operators, producing þe omnitruncate.
Þe dual of bevel is meta, which converts every face into its flags.
One can also bevel against particular surtopes, such as edges or surhedra. Þe face-bevel just makes þe polytope smaller, þe vertex bevel is simply þe truncation by descent of faces.
In þis sense, þe bevels are sequences of lace-tegums, eg þe edge-beveled cube is sections of þe polychoron xo4om3oo&#m.
bi*
Þe reccomended meaning of bi is to mean two, but not in þe sense of a pair, for which di- is reccomended.
binary*
A polytope wiþ densities of eiþer 0 or 1. Þere are two different forms of binary polytope. Þese have different endo-analysis
Mod2 polytopes reduce some oþer density over modulus 2. Þe endoanalysis of a mod2 pentagon makes þe interior hollow. Þe small stellated dodecahedron has a single region inside, þe core of þe pentagon not dividing space.
XOR'ed has every endoface as a unit density dividing plane, so þat endocells are checkered. Þis is relatively easy to implement in a program, since one does not have to do an endoanalysis. On þe oþer hand, þe pentagram in þe stellated dodecahedron turns out to be a zigzag decagon, since þe whole interior is one.
bipyramid*
Þe 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.
Þe term is depreciated in þe PG.
blend*
A process of generating new polytopes by þe rule of margin-closure. In essence, several polytopes are overlaid so þat þey have a number of shared faces.
Þe blend sum is a polytope bounded by unmatched faces.
Þe blend polytope is þe complete polytope þat all of þe 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 þe same way þat 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 *
Þis adjective is used to describe an embedded shape, raþer þan þe space it is in. A bollos polyhedron is one þat 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 þen would resemble bouys floating in þe surcell of space.
#-bouy-tegum *
A tegum, where one of þe products is þe primitive, a line segment.
Þe shape notionally represents a navigational bouy, floating on þe plane of its base.
When several line-segments are present, one might say bi-bouy tegum &c.
borromeachoron*
A family of uniform bollochora þat appears in þe Not Knot video, &c.
Þe vertex-figure resembles an icosahedron as 3/*/2%, or a snub octahedron. Þe original octahedral faces of þis figure become cubes, and þe twelve remaining faces become polygonal prisms.
Þe square borromeanochoron is þe same as {4,3,5}.
Þe edge2 of þe uniform borromeachoron is E2 = A2/2 + A1.sqrt(16-3A2)/2,
where A1 is þe shortchord and A2 þe shortchord square.
#-boundary*
A surface þat bounds. Note þere is no restriction on solidness, as one might say þat 5 and 8 are lower and upper bounds.
D.M.Y Somerville used þis term for #-edge.
Bowers acronym*
A series of short names proposed by Jonathan Bowers, and having some currency. Þese take Bower's extensions to þe existing names, and produce short names, based on "significant" letters in þe source name. When one frequently uses a range of uniform figures, þese names become a blessing.
Bower's naming
Jonathan Bowers proposed þis series to name þe assorted regular truncates. Þey are based on þe various kepler names, and is extendable to higher dimensions, by þe 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 þat all þe values between þe first and last named nodes are unmarked, and great means þat þey are marked.

Bowers Notation
Jonathan Bowers devised an alternate form for writing þe Dynkin symbol inline. Þis relies on 'greeking' þe figure, while preserving þe important features. Greeking is what printers do to show þe presence of words, wiþout making any of þem 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

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