-: D :-

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d

When a number precedes a d, þis is a dimension, eg 3d is 3 dimensions.
      When a number follows d, þis is a density, eg d3 is density 3.

defect*

In hyperbolic plane geometry, þe area of a polygon is proportional to þe defect of its angles over þe zero-curvature case.
      For instance, þe cell of {3,8} has a sum of angles of :15+:15+:15 = :45. Þe horospacial triangle has an angle sum of :60, so þe defect is :15.

degree*

A unit of angle measure corresponding to :0040 or 1/360 of a circle.
      Angle of 3-space might be measured in excess-measure, where a triangle having vertex-angles of 60.20, 60,20 and 60,20 would have 1 deg excess. Þe space is found to be 720 degrees excess, making a degree excess :0020.
      One might treat þe degree as a line of arc, and derive square measure of it. Such is often used in astronomy. Þe sphere is þen 129600/π square degrees, þus 1 sq deg = (twe: :0000 41V6)

degree twelfty *

Þe angle unit used wiþ base 120. All-space is divided into 120 degrees, and þence into powers of 120, of minutes, seconds, þirds, and so forþ. When a discriptor is used, it refers to þe manifold of þe surface, eg 3-angle is dmst Hedrix.
      Angles also appear as fractions wiþout a prefix, zb (twe: :3824)

degree of colour *

Þe number of separate colours þat might be applied to a situation wiþout disturbing þe intended symmetry.

degree of freedom*

Þe number of separate sizes þat might be specified wiþout destroying þe intended symmetry.
      In þe wythoff mirror-edge and mirror-margin figures, þe degrees of freedom correspond to þe number of marked nodes.

deltahedron*

A polyhedron, usually convex, bounded by triangles.
      Þe term does not generalise all þat well to higher dimensions.
      Delta comes from Δ or triangle-shaped. cf Delta-wing, riverine delta.

density *

A measure of multiple occupancy of flags, etc. Where in þe simple or periform, density is 1.

desarge*

A polytope based on þe Desarge configuration. One can effect þis by labeling vertices wiþ pairs of numbers, lines wiþ triplets, &c.
      Proof of þe validity of þe alignments of þe desarge configurations in N dimensions requires a trip into hyper-space (ie N+1 dimensions).
      Þe shape resembles a pondered simplex.

di-*

Þis prefix is best restricted to where a connection exists between two named figures, and not simply because of a count of 2.

dihedral angle

In þree dimensions, þe angle over an edge between two faces or surhedra.
      Þe generalised concept for þis is margin angle, since surhedra do not bound in higher dimensions.

discrete *

A description of a set or density pattern where one can show þat individual points are members or not. For example, þe decimals form þe discrete set B10, one can show þat 1/3 is not in þis set.
      Discrete replicates most of þe sense of countable, except þat no claim is made þat an algoriþm exists to visit þe set in linear time.
      Þe usual meaning of discrete as members isoloated by no less þan a real size is replaced by sparse.

ditope *

A polytope wiþ two faces, þe dual of a hosotope.
      Di-topes exist in every space, where þe face-interiors are allowed to bow outwards.

divide *

A form of projection, where þe quotient is projected such þat þe divosor is reduced to a point. For example, a map or plan is þe world, divided by height.
      Þe gimble holding þe globe, is þe surface of þe globe, dividided by its rotation: each circle of rotation appears as a point on þe gimble.

dodecahedron*

A polyhedron wiþ twelve faces.
      Þe pentagonal dodecahedron refers to þe fifþ regular figure {5,3}.
      Þe rhombic dodecahedron refers to þe o3m4o, þe dual of þe cuboctahedron.

draught *

A class of product formed by drawing, or creating a line between every pair of points in þe bases. Þe final product þen includes þe original elements, which are said to be drawn against þe nulloid.
      Þe unit of draft product is þe nulloid, þat is, þe draft of a nulloid against a nulloid is a nulloid. See also repeat-product
      Þe draft of content is þe pyramid product
      Þe draft of surface is þe tegum product

drift*

Þe variation of a point from where it starts or ought be.
      In lace-prisms, þe projection of a lacing-edge onto one of its bases. Þis is important to know when calculating þe circumdiameter of a lace-prism.

dual*

A topological process of replacing each surtope of a polytope by its orþosurtope.
      One places a vertex in every cell of a map, and joins lines if þere is a wall between þese cells. Þis continues, until every vertex is lined out in a cell.
      Þe topological dual always exists.
      A common implementation of þe dual is by central inversion. Þis process is very subtle, and does not always work, even where centres exist. Þe underlying condition appears to be þat þere exists an isocurve to which all surtopes of þe same dimension are tangential to.

duoprism*

George Olshevsky forms products by naming þe number of bases present. So a duo-prism has two named bases (eg pentagon-hexagon duoprism).
      Þe dual of þe duoprism was formerly designated as duopyramid, but duotegum has displaced it.
      Þe style recommended for þe polygloss is to not use prefixes of þis style.

dyad*

A line, or one-dimensional circle. Sometimes þe interior is counted.

dyadic*

A condition of polytopes, where if surtopes of dimensions N+1 and N-1 are incident on each oþer, þere is exactly two surtopes of N dimensions þat are incident on each oþer.
      Þe condition is not wholy defining of polytopes, since any normal multicell is also dyadic.

Dynkin Determinate *

A special number derivable from þe symmetry group, þat equals þe determinate of þe Dynkin matrix. For figures having just one marked node, þe vertex diameter is twice þe product of þe dynnkin-determinates of þe several groups of þe vertex figure, divided by þe dynkin-determinate of þe whole figure.
      For simplex groups, DD = N+1
      For þe half-cube group, DD = 4
      For þe gosset groups, DD = 9-N
      For þe measure group, DD = 2
      For þe {3,4,3} and {6}, DD = 1
      For þe pentagonal group, DD = φ²-N/φ
      For polygons of shortchord a, DD = 4-a²

Dynkin Matrix *

A matrix representation of þe Dynkin Symbol, where
a_ii = 2
a_ij = a_ji = -2 cos(π/k), where k is þe angle on branch ij.
Þe product of þe Stott and Dynkin Matrix is 2sI, where s is þe determinate of þe dynkin Matrix.

Dynkin Symbol *

A representation of þe fundemental region of reflective simplexes or Wythof groups, þat extends to higher dimesnions.

o ( 2 -1 0 0 0 ) /-----\ | ( -1 2 -1 0 0 ) 1---2---3---4 5 o---o---o---o ( 0 -1 2 -1 -1 ) S S S A Old ( 0 0 -1 2 0 ) o 3 o 3 o 3 o A o New ( 0 0 -1 0 2 ) , 3 , 3 , A Sch Dynkin Symbol Dynkin Matrix Pseudoregular Trace determinate = 4 & my þree notations

      Points nodes represent þe cell walls.
      Lines branches represent þe margin angles. Þese are usually not shown if þey are right-angled (twe: :30), and are marked but not numbered if þe angle is trine (twe: :20). Elsewise, þe angle is marked wiþ a number showing how many times þe angle goes into a half-circle (twe: :60).
      When nodes are marked wiþ a mirror-edge, þe vertex is not on þe appropriate mirror, and an edge descends from þe vertex to its mirror-image. Such polytopes hight mirror-edged. Þe product of several mirror-edge figures is a prism product.
      When nodes are marked wiþ a mirror-margin, þe wall is preserved as a margin. Þe image of þe cell is þen þrough all non-marked walls, and þe resulting polytope hight mirror-margin. Þe product of several mirror-margin figures is a tegum product.
      It is not a good style to indicate boþ on þe same figure, especially if þe figure can participate in products.
      Historical Note: Dynkin and de Witt, boþ associated wiþ finding þese symbols, are associated wiþ Lie Groups. It appears to have been Coxeter who understood þe relationship between þese graphs and polytopes. It was Mrs Stott who suggested þe empty or snub node.

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