# -: M :-

#-manifold *
A surface piecewise topologically equilivant to n-dimensional space.
maps*
This is a series of maps, that can project one geometry onto some other surface. Because non-euclidean geometry has a definite size, these techniques are also important in the 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
orthogonal *       orthogonal         (nature)     area

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

```
Marek Snub*
A H2 snub-operation that 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 that bounds a face.
• Margin acquires some of the sense of edge, especially in the senses of divide. For example, the knife-edge and the leading edge are both margins that divide, not lines that connect.
• The margin angle is the angle over a margin, formed by rays falling in the faces, and perpendicular to the margin. It generalises the concept of dihedral angle.
• When a prefixing number is applied, eg 2-margin, it means a margin bounding a margin.
margin angle
The angle measured between two faces, measured across their common angle. In 3d this hight dihedral angle
margin-uniform *
matrix dot*
A matrix-operation applied to two vectors: the first vector is multiplied by the matrix, and the result dotted with the second vector: ie the matrix dot of V and W is sum(i,j) (S_ij V_i W_j). Matrix dots are used when the defining vectors are not at right-angles to each other.
matrix norm*
The matrix-dot of a vector and itself.
max() *
The maximum value of several supplied values, eg max(3,5,2) = 5. When applied as a radial function, the product produces the prism product.
measure*
The notion of measuring things suggest that an item can be moved about the space, and placed beside separate cases. What holds for time applies here as well. Units of measure of content are the polyprism, polytegmal, polyglomal units.
measure polytope*
A polyprism. See #prism. This figure is usually rated the first of the uniform polytopes, and has a special name: square, cube, tesseract, &c.
Mercator Projection*
A transverse map of spheric space, that preserves the equator in nature, lines perpendicular to the equator as straight lines, and angles. Because of this, it preserves the exact compass rose at every point, it is much favoured in navigation, and is the usual presentation of the earth.
It does have great elongation at high lattitudes, though.
A straight line on a mercator map represents a lexidrome, or 'map-runner'.
One might also map the hyperbolic geometry in this format as well. Such a map would be of finite height, and infinite equator. The lexidromes would on such a map be a finite length, but would represent an infinite length, lieing between thwo lines of longitude.
meta*
One of the Conway Operators that replaces every face by its flags. It is the dual of bevel.
Mete-star
The compound of 120 pentachora that have the same face-planes of a 600chora, and the vertices of a 120-chora. The name comes from the fact that 4-angle is measured in the symmetries of the pentachora and the 120-chora. This compound, and its two daughter compounds, do not belong to the standard constructions of compounds in four dimensions. See also s-unit, f-unit.
The mete-star is considered to be one of the few euclidean examples of compounds more common in the hyperbolic world eg: {7,3}[24{14,7}]{3,7}, or the euclidean tiling {4,4}[36{4,4}]{4,4}.
mirror*
A surface that does inversion.
mirror-edge*
A polytope whose every edge is perpendicularly bisected by a mirror. Many, but not all, of the mirror-edge figures can be constructed by Wythoff's construction. The exceptions are those in non-simplex mirror-groups.
There is no requirement that the edges of a mirror-edge figure be equal. Any rectangle is a mirror-edge figure.
mirror-margin*
A polytope whose every margin lies in a mirror-plane of the figure. This is the dual of the mirror-edge figure. A mirror-margin figure can be constructed by converting some mirrors into margins. The face then becomes a reflection of the mirror-margin in different faces.

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

mobius*
An adaption of the Dynkin Symbol, designed to reflect the Mobius polygon. It can be distinguished by a leading loop-node. Branches represent angles pi/n. Unconnected nodes represent sides that do not intersect. This is also called a Hatch Loop and the leading node a Hatch Node.
Mobius Group
A reflective group, where the fundamental region is a polygon with more than three sides. In higher dimensions, any group with a non-simplex reflective group.
Mobius Mirror (edge/mirror)
A figure based on a mirror-edge construction on a mobius group. In H2 tilings, the class is super-infinite.
Mobius Omnitruncate
A bolloohedron resulting from placing a vertex in the 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 the density function is reduced modulo 2.
Mod-2 polytopes are different to the easier-to-render XOR'ed polytopes, where each endocell alternates where planes cross.
motion*
The notion of motion is resolved into identifying similarities in a sequence of different geometric situations. The snapshots of motion are really different geometric situations, the equality of which needs to be shown. (Moving a square on a Mecatour projection will reduce the area the more it is moved away from the equator).
mount*
The act of making a polytope a surtope, or part of a larger polytope.
The idea of "mounted" is the same as one might mount something on a wall, &c. A polytope is mounted, that, if it shares any of the interior of some surtope with another polytope, than the whole of that surtope is a surtope of both polytopes.
mulli- *
This stem was formerly used to denote the polytope made by the outline of another one. For example, the mullitope of a pentagram is the decagonal zigzag that bounds it.
The sense is now transferred to peri-, in the senses of perimeter and periphery. Likewise, we convert apeiro- to aperi, without perimeter in the space where the cells are solid.
The word mulli- is derived from the heraldic term mullet, the outline of a pentagram.
multi- *
Many, without the sense of closure. For example, one might have many polygons mounted together, without the sense of making a polyhedron or higher. See also poly.
multicell *
Many mounted solid polytopes, without 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, without the sense of closure. For example, a net of a polytope is itself not a polytope, so would be a multitope.