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r √ *

When prefixed in front of a number, r is taken to be square-root, √.
      r2 = √2 (twe: 2:4984 8104,3529 0779,4628 3031,0365 V712,4341 E393)
      r2 = √2 (dec: 1.414213562373095048801688724209698078569)
      r3 = √3 (twe: 1:87V1 6395,54V7 8969,65V1 0284,4966 3953,V372 91V9)
      r3 = √3 (dec: 1.732050807568877293527446341505872366942)
The symbol √ is derived from a script letter r.

radial function *

A representation of a solid as a function of direction from a centre, such that the scale is linear, and the surface is at unity. The function is very dependent on the location of the centre and the polytope used.
      Such a representation effectively treats all polytopes as spheres drawn around the centre.

radial product *

A product formed by some function of the radial functions of several orthogonal figures.
      For example, if A represents raduis, and a direction, the general polytope might be written as Aa. For polytopes Xx, Yy, Zz, one can extract a 3d space of the radii X, Y, Z, and draw in that space (it is an octant or for positive X, Y, Z), assorted figures, or derive a new radial function, where X,Y,Z make a polytope Rr. The general polytope becomes R = f(r, x, y, z).
      Several of these products are of some importance, and are believed to be coherent.
      max or maximum value, gives the prism product.
      sum or sumation, gives the tegum product
      rss or root-sum-square, gives the spheric product.

radiant field *

A model that explains action at a distance (such as gravity or light), by the use of radiated particles. The field is implemented by the local density of particles (which may have a vector component).
      The model suggests that the source radiate a flux (flow) of fieldlings, and it is the fieldlings (like photons or gravitons), that cause the observed effects (heat or force).
      Under such a model, the field is inversely proportional to the surface area of the sphere. It also provides the observed relations for straight subspaces, as well.

radian*

A measure of arc equal to the radius. The unit is recent, appearing only with the advent of calculus and series theory.
      1 radian = 57° 17:44:48:22:29:22:22 = 57.295779513082320 = (twe: 19° 1199 88E9 E337 2403)
      Solid and higher angles might be measured in terms of prismatic or tegmatic radians.

unit	per sph	degree	degree dec	twelfty	metric
radian	C2/ 2π	57:17.44.48	57.2957795	19:1199 8900	127.323954
steradian	C3 / 4π	57:17.44.48	57.2957795	9:65V9 V460	1273.23954
rhombic r	C3/ 8π	28:38.52.24	28.6478897	4:92E4 E230	636.661977
cubic r	C4/ 2π²	-	729.512522	609:6160 3848	8105.694691
octa r	C4/ 12π²	-	121.585420	101:7030 0648	1350.949115

For angle units see angle-measures.

ray*

A ray radiates from a point. Both the linear nature of the ray and the telic nature of the point can be generalised.
      For example, one can make the ray radiate into any point in an angle, and make the point (or tip) into an n-dimensional manifold. For example, the cube-corner is a three-dimensional ray with a zero-dimensional tip, while its edge makes for a three-dimensional ray with a one-dimensional tip.
      Ray-names describe the tip and orthosection. For example, the vertex makes for a telic chororay, and for the edge, a lateral hedroray.
      The solid shape of a ray is a verge. See also approach

realm*

The infinite extent of "all space", considered as if nothing else existed.
      comment The name given to 3D space or chorix. The shared sense is all-space.

Recriprocation*

The process of creating the dual, often by taking the inverse of metric properties. See also inversion-dual. This is more general.

rectate, rectify*

A polytope formed by taking the centres of a kind of surtope. Such occur in the antitegmal sequence, or truncate sequence of the platonic figures.
      In this sense, these figures have their vertices at the corners of a fundemental region, and therefore have a fixed size relative to that fundemental region.
      In three dimensions, the two examples are cuboctahedron and icosadodecahedron. In higher dimensions, these become more numerous, and hence the new style prefix.
      An edge-rectified #-rectate is a #-cantellate
      A truncated #-rectate is a #-cantitruncate.
      The verb rectify means here to create a rectate.

regiment*

A Bower's army unit. Figures belonging tho this group have the same vertices and edges as each other. The notional polytope that owns the vertices and edges is the : See latroframe.

Regular *

A generalisation of the term platonic.
      The requirement here is that the symmetry is transitive on the flags.
      Note that the definition does not require that the transitivity is to be effected by reflection, or that the result be a periform.
      One can treat any polytope as regular, as a means to generating the derived figures through conway operators, because the pennant flip is an assymetric trasnlation of the flags.

repeat-product *

A product formed by repeating a copy of the opposite base at each point of a given base. This is reversable, in that if each point of A has a copy of B at it, then each point of B has a copy of A at it.
      See also draught-products
      The repeatition of content is the prism product
      The repeatition of surface is the comb or torus product.

rhombus*

In 2d, a quadralateral with equal sides, and diagonals crossing at right angles (twe: 30:00). This gives rise to thwo reflexes in 3d.
      The prism sense is that of a square, stretched along one of its diagonals. This in general produces a cube-like figure, inscribed in a prolate or oblate ellipsoid. The result is that at one vertex, all of the angles are the same, and the rhombotope is acute or obtuse as this angle is. The 60° (twe: 20°) can be presented as the cell of the t-basic tiling found in every dimension.
      The tegum sense is that of perpendicular bisecting axies, with different sizes on each axis. This makes the rhombus into the dual of a rectangle, and the rhombic octahedron into the dual of the rectanular prism.
      Kepler's rhombo- in the sense of rhombocuboctahedron, &c, is in the sense of truncation by descent of rhombic faces onto the vertices of a generalised cuboctahedron, adjusted to be equalateral. In this sense, it is a vertex-bevel.
      In terms of golden ratio, there is an obtuse (108°, (twe: 36°) and acute (36°, (twe: 12°), and a golden rhombic rhombohedron, with diagonals in the ratios of 1 : fi : fi². This rhombus peicewise tesselates, with 30 at each vertex, the vertices of d1, d3 and d7. Such a tiling is designated by o5m5/2oAo.

rhombotruncate*

Like "rhombo-", there is not a rhombus in sight! This expression gets used by G Olshevsky for "cantetruncate", and W Krieger has used it as a version of "omnitruncate". As noted under "rhombus", the rhombus does not appear that often in uniform figures.

ridge*

a margin.
      A margin does not carry the suggestion of sharpness, and in hyperbolic space, some infinitopes can have reflex angles, making the margins into valleys.

ring *

A sphere etc, arounding or orthogonal to a line.
      Any section of a polytope or concentric sphere of a lattice. In practice, one takes the vertices at a given distance from some fixed point, and makes a convex hull over these.
      Rings are the arround-sections to the presented axis, so if one is using a line or hedrix, the sections are N-1 or N-2 respectively.
      Note that rings are not sectional slices or waterman polytopes.

Rogers Limit *

The limit of packing, on the assumption that every hole is in the shape of a regular simplex. At present, it is rated at .sqrt(n+1)^3/2^(n+2) of all space.
      Rogers polytope is the vertex figure of such a packing, having simplex cells. This is the limit given to the kissing spheres problem.

rotatope *

A class of figures typically formed by nested prism and spheric products of unit edges. One finds here figures like: () is spheric product, [] is prism-product
      sphere = ( x, y, z)
      cylinder = [ (x, y), z]
      'dome' = ( [x, y), z)
      cube = [ x, y, z]
      The projection of these onto any subspace can be found by removing the letter, and any set of brackets that enclose only one element (either a brackets or a simgle letter).
      So the projections of the 'dome' onto the xy, xz and yz axies are
      xy: [(x,y)] = (x,y) = circle
      xz: [(x),z] = [x,z] = square
      yz: [(y),z] = [y,z] = square
      Marek Ctrnact showed that one can derive a rotatope that projects onto the axies freely such that any pair of axies is a square or circle. In four dimensions, there is one such figure that can not be expressed as a simple product of its axies, the simplest expression was found by Richard Klitzing as [(x,y),(y,z),(z,x)], where [] is max[] and () is rss() functions.

rss() Root Sum Square *

A mathematical function modeled on the rms root-mean-square. This allows one to write the general sphere-surface as rss(x,y,z).
      This is the function behind the Spherical radial product

rule of space

Any of the sets of alternate valid rules, which result from collapses of statements.
      Rule F : fragment, space is orientable and lines cross once - non-complete
      Rule M : monocross, space is complete and lines cross once - non-orientable
      Rule O : orientable, space is complete and orientable - lines cross twice
These rules apply to curvature.
      Rule H : bollous, allspace is negative curvature
      Rule E : horrous, allspace is zero curvature
      Rule S : glomous, allspace is positive curvature.
Euclid's elements describe a fragment of euclidean space, ie rules FE. The discoverise of other geometries, such as hyperbolic and spheric lead then to FH and FS geometries. Unfortunately, spheric geometry is completable and one is left with alternates also of MS (elliptic) and OS (spheric).
      One can then suppose the same distinction exists under rules ME vs OE and MH vs OH.

runcic*

Norman Johnson's name for the third marked node, so also runcinate, runcitruncate. Note the sense we use is somewhat different.

runcinate*

Runcinate is used here for the Stott expand vector, in its general form. The runcinate is derived by placing the faces of a figure and its dual that the vertices only touch, and then filling in the spaces with prisms of surtopes and the matching orthosurtope.
      In terms of the Dynkin-symbol, or the more general pennant-diagram this corresonds to marking the first and last node of the figure.
      Norman Johnson invented the term runcinate, but his form applies to a figure in four dimensions only.
      Jonathan Bowers has prismato-, in the sense of a prismatic faced figure, but this is used to reflect the 4D version only, too.
      My former term for this is prism circuit, there being a 'cycle' of faces between the first and last node. It also completes the 'cycle' of truncates.

runcinate, quarter-*

A quartering of the runcinate of a polytope, where both the polytope and its dual have alternating vertices. The figure is a mirror-edge figure, formed by halving the two end-tails.
      A quarter-runcinate cubic is a tiling of tetrahedra and truncated tetrahedra.
      Norman Johnson reads s4o3o4s in this way.

runcinate, semi-*

Alternation of the vertices of a runcinate. This has twice as many vertices as the quarter-runcinate, and generally is not a mirror-edge figure.

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