# -: D :-

d
When a number precedes a d, this is a dimension, eg 3d is 3 dimensions.
When a number follows d, this is a density, eg d3 is density 3.
defect*
In hyperbolic plane geometry, the area of a polygon is proportional to the defect of its angles over the zero-curvature case.
For instance, the cell of {3,8} has a sum of angles of :15+:15+:15 = :45. The horospacial triangle has an angle sum of :60, so the 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. The space is found to be 720 degrees excess, making a degree excess :0020.
One might treat the degree as a line of arc, and derive square measure of it. Such is often used in astronomy. The sphere is then 129600/π square degrees, thus 1 sq deg = (twe: :0000 41V6)
degree twelfty *
The angle unit used with base 120. All-space is divided into 120 degrees, and thence into powers of 120, of minutes, seconds, thirds, and so forth. When a discriptor is used, it refers to the manifold of the surface, eg 3-angle is dmst Hedrix.
Angles also appear as fractions without a prefix, zb (twe: :3824)
degree of colour *
The number of separate colours that might be applied to a situation without disturbing the intended symmetry.
degree of freedom*
The number of separate sizes that might be specified without destroying the intended symmetry.
In the wythoff mirror-edge and mirror-margin figures, the degrees of freedom correspond to the number of marked nodes.
deltahedron*
A polyhedron, usually convex, bounded by triangles.
The term does not generalise all that 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 the simple or periform, density is 1.
desarge*
A polytope based on the Desarge configuration. One can effect this by labeling vertices with pairs of numbers, lines with triplets, &c.
Proof of the validity of the alignments of the desarge configurations in N dimensions requires a trip into hyper-space (ie N+1 dimensions).
The shape resembles a pondered simplex.
di-*
This 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 three dimensions, the angle over an edge between two faces or surhedra.
The generalised concept for this is margin angle, since surhedra do not bound in higher dimensions.
discrete *
A description of a set or density pattern where one can show that individual points are members or not. For example, the decimals form the discrete set B10, one can show that 1/3 is not in this set.
Discrete replicates most of the sense of countable, except that no claim is made that an algorithm exists to visit the set in linear time.
The usual meaning of discrete as members isoloated by no less than a real size is replaced by sparse.
ditope *
A polytope with two faces, the dual of a hosotope.
Di-topes exist in every space, where the face-interiors are allowed to bow outwards.
dodecahedron*
A polyhedron with twelve faces.
The pentagonal dodecahedron refers to the fifth regular figure {5,3}.
The rhombic dodecahedron refers to the o3m4o, the dual of the cuboctahedron.
draught *
A class of product formed by drawing, or creating a line between every pair of points in the bases. The final product then includes the original elements, which are said to be drawn against the nulloid.
The unit of draft product is the nulloid, that is, the draft of a nulloid against a nulloid is a nulloid. See also repeat-product
The draft of content is the pyramid product
The draft of surface is the tegum product
drift*
The variation of a point from where it starts or ought be.
In lace-prisms, the projection of a lacing-edge onto one of its bases. This is important to know when calculating the circumdiameter of a lace-prism.
dual*
A topological process of replacing each surtope of a polytope by its orthosurtope.
One places a vertex in every cell of a map, and joins lines if there is a wall between these cells. This continues, until every vertex is lined out in a cell.
The topological dual always exists.
A common implementation of the dual is by central inversion. This process is very subtle, and does not always work, even where centres exist. The underlying condition appears to be that there exists an isocurve to which all surtopes of the same dimension are tangential to.
duoprism*
George Olshevsky forms products by naming the number of bases present. So a duo-prism has two named bases (eg pentagon-hexagon duoprism).
The dual of the duoprism was formerly designated as duopyramid, but duotegum has displaced it.
The style recommended for the polygloss is to not use prefixes of this style.
A line, or one-dimensional circle. Sometimes the interior is counted.
A condition of polytopes, where if surtopes of dimensions N+1 and N-1 are incident on each other, there is exactly two surtopes of N dimensions that are incident on each other.
The condition is not wholy defining of polytopes, since any normal multicell is also dyadic.
Dynkin Determinate *
A special number derivable from the symmetry group, that equals the determinate of the Dynkin matrix. For figures having just one marked node, the vertex diameter is twice the product of the dynnkin-determinates of the several groups of the vertex figure, divided by the dynkin-determinate of the whole figure.
For simplex groups, DD = N+1
For the half-cube group, DD = 4
For the gosset groups, DD = 9-N
For the measure group, DD = 2
For the {3,4,3} and {6}, DD = 1
For the pentagonal group, DD = φ²-N/φ
For polygons of shortchord a, DD = 4-a²
Dynkin Matrix *
A matrix representation of the Dynkin Symbol, where
a_ii = 2
a_ij = a_ji = -2 cos(π/k), where k is the angle on branch ij.
The product of the Stott and Dynkin Matrix is 2sI, where s is the determinate of the dynkin Matrix.
Dynkin Symbol *
A representation of the fundemental region of reflective simplexes or Wythof groups, that 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 three notations ```

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