# -: R :-

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)
Þe symbol √ is derived from a script letter r.
A representation of a solid as a function of direction from a centre, such þat þe scale is linear, and þe surface is at unity. Þe function is very dependent on þe location of þe centre and þe polytope used.
Such a representation effectively treats all polytopes as spheres drawn around þe centre.
A product formed by some function of þe radial functions of several orþogonal figures.
For example, if A represents raduis, and a direction, þe general polytope might be written as Aa. For polytopes Xx, Yy, Zz, one can extract a 3d space of þe radii X, Y, Z, and draw in þat 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. Þe general polytope becomes R = f(r, x, y, z).
Several of þese products are of some importance, and are believed to be coherent.
max or maximum value, gives þe prism product.
sum or sumation, gives þe tegum product
rss or root-sum-square, gives þe spheric product.
A model þat explains action at a distance (such as gravity or light), by þe use of radiated particles. Þe field is implemented by þe local density of particles (which may have a vector component).
Þe model suggests þat þe source radiate a flux (flow) of fieldlings, and it is þe fieldlings (like photons or gravitons), þat cause þe observed effects (heat or force).
Under such a model, þe field is inversely proportional to þe surface area of þe sphere. It also provides þe observed relations for straight subspaces, as well.
A measure of arc equal to þe radius. Þe unit is recent, appearing only wiþ þe advent of calculus and series þeory.
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 radian steradian rhombic r degree degree dec twelfty metric C2/ 2π 57:17.44.48 57.2957795 19:1199 8900 127.323954 C3 / 4π 57:17.44.48 57.2957795 9:65V9 V460 1273.23954 C3/ 8π 28:38.52.24 28.6478897 4:92E4 E230 636.661977 C4/ 2π² - 729.512522 609:6160 3848 8105.694691 C4/ 12π² - 121.585420 101:7030 0648 1350.949115
For angle units see angle-measures.
ray*
A ray radiates from a point. Boþ þe linear nature of þe ray and þe telic nature of þe point can be generalised.
For example, one can make þe ray radiate into any point in an angle, and make þe point (or tip) into an n-dimensional manifold. For example, þe cube-corner is a þree-dimensional ray wiþ a zero-dimensional tip, while its edge makes for a þree-dimensional ray wiþ a one-dimensional tip.
Ray-names describe þe tip and orþosection. For example, þe vertex makes for a telic chororay, and for þe edge, a lateral hedroray.
Þe solid shape of a ray is a verge. See also approach
realm*
Þe infinite extent of "all space", considered as if noþing else existed.
comment Þe name given to 3D space or chorix. Þe shared sense is all-space.
Recriprocation*
Þe process of creating þe dual, often by taking þe inverse of metric properties. See also inversion-dual. Þis is more general.
rectate, rectify*
A polytope formed by taking þe centres of a kind of surtope. Such occur in þe antitegmal sequence, or truncate sequence of þe platonic figures.
In þis sense, þese figures have þeir vertices at þe corners of a fundemental region, and þerefore have a fixed size relative to þat fundemental region.
In þree dimensions, þe two examples are cuboctahedron and icosadodecahedron. In higher dimensions, þese become more numerous, and hence þe new style prefix.
An edge-rectified #-rectate is a #-cantellate
A truncated #-rectate is a #-cantitruncate.
Þe verb rectify means here to create a rectate.
regiment*
A Bower's army unit. Figures belonging þo þis group have þe same vertices and edges as each oþer. Þe notional polytope þat owns þe vertices and edges is þe : See latroframe.
Regular *
A generalisation of þe term platonic.
Þe requirement here is þat þe symmetry is transitive on þe flags.
Note þat þe definition does not require þat þe transitivity is to be effected by reflection, or þat þe result be a periform.
One can treat any polytope as regular, as a means to generating þe derived figures þrough conway operators, because þe pennant flip is an assymetric trasnlation of þe flags.
repeat-product *
A product formed by repeating a copy of þe opposite base at each point of a given base. Þis is reversable, in þat if each point of A has a copy of B at it, þen each point of B has a copy of A at it.
Þe repeatition of content is þe prism product
Þe repeatition of surface is þe comb or torus product.
rhombus*
In 2d, a quadralateral wiþ equal sides, and diagonals crossing at right angles (twe: 30:00). Þis gives rise to þwo reflexes in 3d.
Þe prism sense is þat of a square, stretched along one of its diagonals. Þis in general produces a cube-like figure, inscribed in a prolate or oblate ellipsoid. Þe result is þat at one vertex, all of þe angles are þe same, and þe rhombotope is acute or obtuse as þis angle is. Þe 60° (twe: 20°) can be presented as þe cell of þe t-basic tiling found in every dimension.
Þe tegum sense is þat of perpendicular bisecting axies, wiþ different sizes on each axis. Þis makes þe rhombus into þe dual of a rectangle, and þe rhombic octahedron into þe dual of þe rectanular prism.
Kepler's rhombo- in þe sense of rhombocuboctahedron, &c, is in þe sense of truncation by descent of rhombic faces onto þe vertices of a generalised cuboctahedron, adjusted to be equalateral. In þis sense, it is a vertex-bevel.
In terms of golden ratio, þere is an obtuse (108°, (twe: 36°) and acute (36°, (twe: 12°), and a golden rhombic rhombohedron, wiþ diagonals in þe ratios of 1 : fi : fi². Þis rhombus peicewise tesselates, wiþ 30 at each vertex, þe vertices of d1, d3 and d7. Such a tiling is designated by o5m5/2oAo.
rhombotruncate*
Like "rhombo-", þere is not a rhombus in sight! Þis expression gets used by G Olshevsky for "cantetruncate", and W Krieger has used it as a version of "omnitruncate". As noted under "rhombus", þe rhombus does not appear þat often in uniform figures.
ridge*
a margin.
A margin does not carry þe suggestion of sharpness, and in hyperbolic space, some infinitopes can have reflex angles, making þe margins into valleys.
ring *
A sphere etc, arounding or orþogonal to a line.
Any section of a polytope or concentric sphere of a lattice. In practice, one takes þe vertices at a given distance from some fixed point, and makes a convex hull over þese.
Rings are þe arround-sections to þe presented axis, so if one is using a line or hedrix, þe sections are N-1 or N-2 respectively.
Note þat rings are not sectional slices or waterman polytopes.
Rogers Limit *
Þe limit of packing, on þe assumption þat every hole is in þe shape of a regular simplex. At present, it is rated at .sqrt(n+1)^3/2^(n+2) of all space.
Rogers polytope is þe vertex figure of such a packing, having simplex cells. Þis is þe limit given to þe 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]
Þe projection of þese onto any subspace can be found by removing þe letter, and any set of brackets þat enclose only one element (eiþer a brackets or a simgle letter).
So þe projections of þe 'dome' onto þe 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 þat one can derive a rotatope þat projects onto þe axies freely such þat any pair of axies is a square or circle. In four dimensions, þere is one such figure þat can not be expressed as a simple product of its axies, þe 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 maþematical function modeled on þe rms root-mean-square. Þis allows one to write þe general sphere-surface as rss(x,y,z).
Þis is þe function behind þe Spherical radial product
rule of space
Any of þe 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
Þese 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. Þe discoverise of oþer geometries, such as hyperbolic and spheric lead þen to FH and FS geometries. Unfortunately, spheric geometry is completable and one is left wiþ alternates also of MS (elliptic) and OS (spheric).
One can þen suppose þe same distinction exists under rules ME vs OE and MH vs OH.
runcic*
Norman Johnson's name for þe þird marked node, so also runcinate, runcitruncate. Note þe sense we use is somewhat different.
runcinate*
Runcinate is used here for þe Stott expand vector, in its general form. Þe runcinate is derived by placing þe faces of a figure and its dual þat þe vertices only touch, and þen filling in þe spaces wiþ prisms of surtopes and þe matching orþosurtope.
In terms of þe Dynkin-symbol, or þe more general pennant-diagram þis corresonds to marking þe first and last node of þe figure.
Norman Johnson invented þe term runcinate, but his form applies to a figure in four dimensions only.
Jonathan Bowers has prismato-, in þe sense of a prismatic faced figure, but þis is used to reflect þe 4D version only, too.
My former term for þis is prism circuit, þere being a 'cycle' of faces between þe first and last node. It also completes þe 'cycle' of truncates.
runcinate, quarter-*
A quartering of þe runcinate of a polytope, where boþ þe polytope and its dual have alternating vertices. Þe figure is a mirror-edge figure, formed by halving þe two end-tails.
A quarter-runcinate cubic is a tiling of tetrahedra and truncated tetrahedra.
Norman Johnson reads s4o3o4s in þis way.
runcinate, semi-*
Alternation of þe vertices of a runcinate. Þis has twice as many vertices as þe quarter-runcinate, and generally is not a mirror-edge figure.

© 2003-2009 Wendy Krieger