# -: T :-

t-*
stations A prefix used to define a class of honeycombs having the "t" symmetry. This is the one formed on treating the edges of a simplex as vectors. This has n+1 stations, arranged in a polygon.
tangential *
Two isocurves are tangential, if the line between their surfaces at the same point: that is, they share a common normal at a common point.
A surtope tangential is the isocurve that is tangential to all of the named surtopes of a polytope.
t-basic*
A lattice formed by allowing translations of the edges of a simplex. The vertex-figure is a runcinated simplex. The efficiency in q-units is 1/sqrt(n+1) The cells of this is the simplex and its rectates. This is cut by planes parallel to the faces of a simplex, and spaced at the height of a simplex. Because of this, it gives rise to a series of laminate non-Wythoffian figures, being the prismatoreflects.
t-catseye*
A tiling of rhombotopes, n+1 to a vertex. Such rhombotopes are the simplex antitegums formed by the faces of a simplex, taken from the centre.
The shape also can be made from a projection of n+1 dimensional cubics projected down a simplex-axis. Because the reflection in n orthogonal planes return light to its source direction, such a mirror is a cat's eye mirror. Cats-eye mirrors are used in safety devices, designed to return light to the direction of its sender.
Projected onto a plane, a catseye surface gives a tiling hight t-catseye. It is the dual of the truncate-form, or one with two consecutive mirror-nodes, ie m3m3o3o..3z
t-diamond*
A position of points corresponding to two consecutive stations of a t-group, ie the vertices of the compound xo3ox3oo3...oo3z.
One can effect this from the t-basic, by taking vertices + centres of simplexes pointing in a fixed direction.
The shame resembles the placement of atoms in the substance diamond.
t-rhombic*
This is the dual of the t-basic. The cell is the strombiated simplex, formed by replacing each face of a simplex with an obtuse rhombotope or parallelotope.
This is the voronii cells of the t-group.
t-stations*
Any of the vertices of the fundamental region of the simplex. In terms of the 60° rhombotope, the t-stations divide the long diagonal symmetrically into N+1 points.
t-truncate*
A Wythoff construction based on a pair of consecutive marked nodes. This is a tiling of the simplex and its polytruncates.
teelix
A zero-dimensional manifold. See hedrix
What one might use teelices for is unknown, but the word forms regularly from teelon.
teelon *
A mounted zero-dimensional polytope. One might for example, describe a 1D polytope with three vertices as a triteelon.
See hedron for examples.
In practice, the word [tele] means far, and [telos] a proverbial end. We use the telos meaning here, but add an extra 'e'.
Guy Inchbald derived telon from greek [telos] end (figurative). The change of stem is to avoid clashes with [talon] bird-claw , and [telix] device for sending messages.
A unit of angle equal to the tegmic power of a radius, either in surface or volume.
The vertex-angle of a simplex lies between 1 and sqrt(n+3)/2 tegmic radians.
tegum *
The regular product by draught of surface.
Tegum products include the nulloid but not the bulk terms of the surtope polynomial.
• [Tegum] is derived from a latin word to cover. The sense is that the bases of the product form an internal framework, and the surface is a new skin covering this arrangement. The original new name for it was (tent), but somehow [tabernacle] is already loaded in meaning.
tegum-types
anti-, exoto-, mod-, poly-, sur-, #-
tegum product *
A radial product formed by the sum function. When applied to line segments of length sqrt(2), it yields the cross polytopes of unit length in every dimension.
A surtope product formed by the product of the surtopes, ommitting the volume. This is identical to the above function.
The product defines coherent units: the tegum-volume of a tegum-product is the product of the tegum-volumes of the bases.
t1 = diagonal inch = unit-length line
t2 = rhombic inch = square of unit diagonal
t3 = octahedral inch = octahedron of unit diameter
t4 = tetrategmic inch = 16-choron of unit diameter
terix *
A four-dimensional manifold. It is used in the same way as hedrix.
teron *
A mounted 4d polytope, or a 4d 'hedron'
tesseract*
The usual name for the tetraprism {4,3,3}
comment: Sometimes restricted to right-angled {4,3,3} only. Also called octachoron.
tetra-*
A prefix meaning four or fourfold, in the senses of #.
tiling*
A tiling is a series of mounted polytopes that cover the full extent of some space, either all-space or some lesser part of it. A tiling is solid when it fulls all-space.
Tilings also hight aperitopes
time*
The role of time in geometry is to focus on a sequence of diverse events as if they were snapshots of something in motion. One must be aware that what is presented as the same thing is in fact different but similar things, and the desired equality must be proven.
tope*
A figure that is solid in # dimensions, and bounded by lesser surtopes. This is the general #-dimensional polytope.
torus*
#- In three dimensions, a figure formed by connecting the top and bottom of a tall cylinder. The surface is topologically a plane, with opposite sides identified.
1: The torus-product of two circles.
2: The surface of such a figure.
3: In four dimensions, a name for a kind of glomic coordinate, being the comb product of two circles.
4: The torus-product of two spheres, or generally any number of polytopes.
torus product *
A of figures, reducing the dimension for each application of the product: the torus-product is polygons (2d solid) is a polyhedron (a 3d solid).
While the surtope consist of a torus-product is order-independent, the topological shapes are order-dependent.
One can add a new figure either by the sock or hose method. These add to the two ends of the product: that is, if one end is a hose-end, the other is the sock-end.
The Sock method adds the new figure in the manner of rolling down a sock. The surface of the original becomes in its orthospace, the centre of the added figure, the skin encloses the named-base.
The Hose method adds the new figure in the manner of connecting a hose: One covers the added figure, rather like covering the water in the hose.
transverse*
[Transverse] means across.
A transverse projection is a mapping that presents some straight line across as a straight line. That is, distortion is in the lines across the constant line. A famous example of this sort of projection is the Mercator projection.
A transverse symmetry is the symmetry across the altitude of a simplex or lace product. For example, the transverse symmetry of a pentagonal antiprism is that of the pentagonal base.
Tri-
A prefix meaning three or thrice, eg in the meaning of #.
Trigional groups
The Euclidean groups that have mirrors set at only 90° and 60° only. These can be built up from layers of the t-basic, by advancing one (t), two (q) and three (y) stations. While other numbers are possible, eg 0, 4, these do not produce any new groups. The version on 0 simply is a comb product of the aperigon and the t-basic, and the 4 and higher dimensions are alternate presentations. The primary lattice, built on a base of n, and advancing x stations has x²+(2-x)n stations, the efficiency being 1/sqrt(s) in q-units.
true length*
The length measured along a straight line. In practice, the horolength is used, since this follows Euclidean concepts (eg scaling without change).
true space*
A special kind of all-space which is generally regarded as encompassing all-space. Eg, for the Earth, all-space is taken as the 2D spheric surface, and true-space is the 3D Euclidean space. When considering the aperitopes, the true-space is taken to be co-incident with the smooth surface, and the all-space is taken to have some kind of interior.
truncate*
The operator that removes vertices through a vertex-bevel. The truncate is usually the portion until the edges are exhausted by the beveling. In regular figures, the #truncate means the planes have removed all the #-1 edges, but not the #-edge.

© 2003-2009 Wendy Krieger