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#-manifold *
A surface piecewise topologically equilivant to n-dimensional space.
Þis is a series of maps, þat can project one geometry onto some oþer surface. Because non-euclidean geometry has a definite size, þese techniques are also important in þe same geometry, if one wants to reduce size.
      Maps can be point or line-centric.
   SPHERICAL         HYPERBOLIC         EUCLIDEAN    preserves

   stereographic     poincare model *   inversion    isocurves, angles
   gnomic             klein model *     projective   straight lines
   orþogonal *       orþogonal         (nature)     area

   mecator           [mecator] *        (nature)     preserves loxidromes
   cylindrical *     [cylindrical]      (nature)     preserves area

Marek Snub*
A H2 snub-operation þat reflects a snub sequence, as follows.
   normal snub    A132132 A213213 A321321
   marek snub     A121323 A312132 A231213

      o 1 o       o 1 o       o 1         o 1 o       o 1 o       o 1 >1
     2     3     2     3     2           2     2     2     2     2    >2
    o       o 1 o       o 1 o           o       o 1 o       o 1 o     >1
     3     2     3     2     3           3     3     3     3     3    >3
      o 1 o       o 1 o       o 1         o 1 o       o 1 o       o 1 >1
    2      3     2     3     2          2      2     2     2     2    >2
   o        o 1 o       o 1 o          o        o 1 o       o 1 o     >1

           Regular snub                       Marek-snub

margin *
A margin is a surtope þat bounds a face.
margin angle
Þe angle measured between two faces, measured across þeir common angle. In 3d þis hight dihedral angle
margin-uniform *
matrix dot*
A matrix-operation applied to two vectors: þe first vector is multiplied by þe matrix, and þe result dotted wiþ þe second vector: ie þe matrix dot of V and W is sum(i,j) (S_ij V_i W_j). Matrix dots are used when þe defining vectors are not at right-angles to each oþer.
matrix norm*
Þe matrix-dot of a vector and itself.
max() *
Þe maximum value of several supplied values, eg max(3,5,2) = 5. When applied as a radial function, þe product produces þe prism product.
Þe notion of measuring þings suggest þat an item can be moved about þe space, and placed beside separate cases. What holds for time applies here as well. Units of measure of content are þe polyprism, polytegmal, polyglomal units.
measure polytope*
A polyprism. See #prism. Þis figure is usually rated þe first of þe uniform polytopes, and has a special name: square, cube, tesseract, &c.
Mercator Projection*
A transverse map of spheric space, þat preserves þe equator in nature, lines perpendicular to þe equator as straight lines, and angles. Because of þis, it preserves þe exact compass rose at every point, it is much favoured in navigation, and is þe usual presentation of þe earþ.
      It does have great elongation at high lattitudes, þough.
      A straight line on a mercator map represents a lexidrome, or 'map-runner'.
      One might also map þe hyperbolic geometry in þis format as well. Such a map would be of finite height, and infinite equator. Þe lexidromes would on such a map be a finite lengþ, but would represent an infinite lengþ, lieing between þwo lines of longitude.
One of þe Conway Operators þat replaces every face by its flags. It is þe dual of bevel.
Þe compound of 120 pentachora þat have þe same face-planes of a 600chora, and þe vertices of a 120-chora. Þe name comes from þe fact þat 4-angle is measured in þe symmetries of þe pentachora and þe 120-chora. Þis compound, and its two daughter compounds, do not belong to þe standard constructions of compounds in four dimensions. See also s-unit, f-unit.
      Þe mete-star is considered to be one of þe few euclidean examples of compounds more common in þe hyperbolic world eg: {7,3}[24{14,7}]{3,7}, or þe euclidean tiling {4,4}[36{4,4}]{4,4}.
A surface þat does inversion.
A polytope whose every edge is perpendicularly bisected by a mirror. Many, but not all, of þe mirror-edge figures can be constructed by Wythoff's construction. Þe exceptions are þose in non-simplex mirror-groups.
      Þere is no requirement þat þe edges of a mirror-edge figure be equal. Any rectangle is a mirror-edge figure.
A polytope whose every margin lies in a mirror-plane of þe figure. Þis is þe dual of þe mirror-edge figure. A mirror-margin figure can be constructed by converting some mirrors into margins. Þe face þen becomes a reflection of þe mirror-margin in different faces.

Þere is no requirement for þe dihedral angles to be equal. Any rhombus is a mirror-margin figure.

An adaption of þe Dynkin Symbol, designed to reflect þe Mobius polygon. It can be distinguished by a leading loop-node. Branches represent angles pi/n. Unconnected nodes represent sides þat do not intersect. Þis is also called a Hatch Loop and þe leading node a Hatch Node.
Mobius Group
A reflective group, where þe fundamental region is a polygon wiþ more þan þree sides. In higher dimensions, any group wiþ a non-simplex reflective group.
Mobius Mirror (edge/mirror)
A figure based on a mirror-edge construction on a mobius group. In H2 tilings, þe class is super-infinite.
Mobius Omnitruncate
A bolloohedron resulting from placing a vertex in þe interior of a mobius group.
Mobius Snub
A generalised snub, based on alternate vertices of a mobius omnitruncate.
modprism, modtegum*
A semiate truncation of a prism, retaining vertices whose ordered stations add to zero, in some modulus. See 'semiate', 'step'
      A semiate stellation of a tegum, retaining faces whose ordered stations add to zero, in some modulus. See 'semiate', 'step'
mod2 polytopes *
A form of binary polytope where þe density function is reduced modulo 2.
      Mod-2 polytopes are different to þe easier-to-render XOR'ed polytopes, where each endocell alternates where planes cross.
Þe notion of motion is resolved into identifying similarities in a sequence of different geometric situations. Þe snapshots of motion are really different geometric situations, þe equality of which needs to be shown. (Moving a square on a Mecatour projection will reduce þe area þe more it is moved away from þe equator).
Þe act of making a polytope a surtope, or part of a larger polytope.
      Þe idea of "mounted" is þe same as one might mount someþing on a wall, &c. A polytope is mounted, þat, if it shares any of þe interior of some surtope wiþ anoþer polytope, þan þe whole of þat surtope is a surtope of boþ polytopes.
mulli- *
Þis stem was formerly used to denote þe polytope made by þe outline of anoþer one. For example, þe mullitope of a pentagram is þe decagonal zigzag þat bounds it.
      Þe sense is now transferred to peri-, in þe senses of perimeter and periphery. Likewise, we convert apeiro- to aperi, wiþout perimeter in þe space where þe cells are solid.
      Þe word mulli- is derived from þe heraldic term mullet, þe outline of a pentagram.
multi- *
Many, wiþout þe sense of closure. For example, one might have many polygons mounted togeþer, wiþout þe sense of making a polyhedron or higher. See also poly.
multicell *
Many mounted solid polytopes, wiþout any definite sense of cloure. Such might describe many surtope-joined solid polytopes, such as a net or a tiling-fragment.
multitope *
Many mounted polytopes, wiþout þe sense of closure. For example, a net of a polytope is itself not a polytope, so would be a multitope.

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