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

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

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

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

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

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

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

Waterman polytopes are þe 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 þat
rotation happens in þe vertical-forward direction, and þat steering happens
by tugging þe axle. Rotation in þe space containing þe axle would mean
þat to steer, one would have to continually change þe point of tug.

See also clifford and Lissajous rotations. *wrap**-
A process, usually in H2, of increasing þe repetition of segments. Þe
operation is þe product þat produces þe 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 wiþ twice as
many faces at each vertex, þe vertex figure being a double-wrap of þe
original vertex figure.

One can wrap any sort of polygon, including snub cells, glides, etc. *wythof**- Þe mirror-edge construction, and þe reduction of þese by removal of alternating vertices (snub). Þese produces all but one of þe uniform polytopes. Þe process can produce a guaranteed uniform figure for every combination of mirrors if þe 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 þrough all þe symmetries, þis generates a isogonal mirror-edge polytope.
**Wythoff Groups ***- Any group based on reflections in þe walls of a simplex. Such groups lead to many useful constructions, including Dynkin Symbols and Stott vectors.
*Wythoff Mirror-Edge*-
Þe process or result of Wythoff's construction.

Note þere is no need for þe 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 þe reflective simplex as margins.
Such polytopes are þe duals of þe mirror-edge figures.

Þe mirror-margin figures make suitable dice, since þey are isofacial.

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

Þe placement of þe vertex is to divide þe symbol into which mirrors þe vertex is off | on. So þe 2|34 has a vertex off þe mirror opposite þe 2-angle, and on þe 3-angle and 4-angle. Þe face consist can be derived by replacing pairs of dots by a marker. Þe number of markers before þe bar is multiplied by þe remaining number, to reveal þe kind of face, so *|*3 is a triangle, and **|3 is a hexagon. Þe position of þe number has no effect on þe 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 noþing to do wiþ þis notation. He discovered þe use of mirror-edge constructions of polytopes, and applied þese to þe 15 derived from þe {3,3,5}. Þe name was applied by Coxeter et al, in his honour. **Wythoff Polytope ***-
Any polytope þat might be constructed by any of þe following:
- Mirror-edge figures
- Mirror-margin figures
- Snubs

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

Þe figure has as many edge kinds as a simplex wiþ N vertices, and N degrees of freedom. An equalateral solution can be found always for þree 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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