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**{w#}***- A polygon having infinite number of sides. Because it can not be represented by its side-count, the style is to give some measure of its shortchord, as in {w4}. The normal reference is shortchord squared.
**wall ***- A surtope that bounds a cell of a tiling.
**walls and balloons ***-
A reference to a multi-dimensional geometry, where one uses solid plane-
sections, and the surface of solid spheres, in the manner of the compass
and straight-edge.

It is interesting to note that edge originally referred to a dividing element of a plane (ie N-2), but custom is that N is taken to be 3, and edge is applied to 1D. **walls and bridges ***-
Walls divide, Bridges unite.

There ought be simple words for relations to solid space, and that for some ends, one thinks of space as being divided by the intersection of planes etc. For example, every equal sign in a definition reduces a dimension.

Bridges unite. There ought be words for relations to fixed dimensions, so there ought be a word for 2D, as well.

The title is from a John Lennon album. **waterman polytope ***-
The convex shell enclosing points of an atom-structure, that are less
than or equal to a certian distance. Such might be regarded as the
convexure of shells up to a certian point. In practice, these tend to
approximate the sphere, because of the accumulation of many shells.

The third ring of the x3o3o5o gives an icosahedron f3o5o. The corresponding section is apiculated by edges from ring two [dodecahedron] to ring four [ID], giving here a triangle, some vertex-first section of the tetrahedron. So the section has faces matching the icosahedral ones.

The waterman section is simply the current convexure of rings 0 to 3. Here ring 2 (dodecahedron, o3o5x), and ring 3 (f3o5o). The result is a rhombic tricontahedron.

Waterman polytopes are the invention of Steve Waterman. See his site at http://www.watermanpolyhedron.com/ **wheel rotations ***-
A form of rotation where all but two axies are held stable. One supposes that
rotation happens in the vertical-forward direction, and that steering happens
by tugging the axle. Rotation in the space containing the axle would mean
that to steer, one would have to continually change the point of tug.

See also clifford and Lissajous rotations. *wrap**-
A process, usually in H2, of increasing the repetition of segments. The
operation is the product that produces the infinite class of polygons,
and in bollohedra, produces all sorts of multiplications. For example,
one may wrap a vertex figure, to make a second tiling with twice as
many faces at each vertex, the vertex figure being a double-wrap of the
original vertex figure.

One can wrap any sort of polygon, including snub cells, glides, etc. *wythof**- The mirror-edge construction, and the reduction of these by removal of alternating vertices (snub). These produces all but one of the uniform polytopes. The process can produce a guaranteed uniform figure for every combination of mirrors if the symmetry group is a simplex.
**Wythoff Construction ***- For any Wythoff group, one might place a vertex in any reflective region, and drop perpendiculars to each mirror. When reflected through all the symmetries, this generates a isogonal mirror-edge polytope.
**Wythoff Groups ***- Any group based on reflections in the walls of a simplex. Such groups lead to many useful constructions, including Dynkin Symbols and Stott vectors.
*Wythoff Mirror-Edge*-
The process or result of Wythoff's construction.

Note there is no need for the edges to be equal: any rectangle is a Wythoff mirror-edge polygon, ie x2x

See also mirror-edge *Wythoff Mirror-Margin*-
A polytope generated by preserving walls of the reflective simplex as margins.
Such polytopes are the duals of the mirror-edge figures.

The mirror-margin figures make suitable dice, since they are isofacial.

Mirror-margin figures are the dual of mirror-edge figures. **Wythoff Notation**-
A notation by Coxeter, Longeut-Higgens and Miller in their 1954 monograph on
uniform polyhedra. The unmarked symbol (eg 3 2 4) refers to a
Schwarz-triangle.

The placement of the vertex is to divide the symbol into which mirrors the vertex is off | on. So the 2|34 has a vertex off the mirror opposite the 2-angle, and on the 3-angle and 4-angle. The face consist can be derived by replacing pairs of dots by a marker. The number of markers before the bar is multiplied by the remaining number, to reveal the kind of face, so *|*3 is a triangle, and **|3 is a hexagon. The position of the number has no effect on the formed polygon.tetrahedron 3 | 2 3 trunc tetrahedron 3 2 | 3 octahedron 4 | 2 3 trunc octahedron 4 2 | 3 cube 3 | 2 4 trunc cube 3 2 | 4 icosahedron 5 | 2 3 trunc icosahedron 5 2 | 3 dodecahedron 3 | 2 5 trunc dodecahedron 3 2 | 5 cuboctahedron 2 | 3 4 icosadodecahedron 2 | 3 5 rhombocubocta 3 4 | 2 rhomboicosahedron 3 5 | 2 rhombotrunc CO 3 4 2 | rhombotrunc icosad 3 5 2 | snub cube | 2 3 4 snub dodecahedron | 2 3 5

Note: Wythoff has nothing to do with this notation. He discovered the use of mirror-edge constructions of polytopes, and applied these to the 15 derived from the {3,3,5}. The name was applied by Coxeter et al, in his honour. **Wythoff Polytope ***-
Any polytope that might be constructed by any of the following:
- Mirror-edge figures
- Mirror-margin figures
- Snubs

**Wythoff Snub ***-
A figure formed by removing alternate vertices of the omnitruncate of a
Wythoff group. This produces snub of the faces, and simplex faces.

The figure has as many edge kinds as a simplex with N vertices, and N degrees of freedom. An equalateral solution can be found always for three dimensions but not for any higher.s3s3s = icosahedron s3s3sAs snub 24choron s3s4s = snub cube s3s3sAsBs snub {3,4,3,3} s3s5s = snub dodecahedron sEs3s3sAsBs snub {3,4,3,3,3}

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