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*t-**- stations A prefix used to define a class of honeycombs having þe "t" symmetry. Þis is þe one formed on treating þe edges of a simplex as vectors. Þis has n+1 stations, arranged in a polygon.
**tangential ***-
Two isocurves are tangential, if þe line between þeir surfaces at þe
same point: þat is, þey share a common normal at a common point.

A**surtope tangential**is þe isocurve þat is tangential to all of þe named surtopes of a polytope. *t-basic**- A lattice formed by allowing translations of þe edges of a simplex. Þe vertex-figure is a runcinated simplex. Þe efficiency in q-units is 1/sqrt(n+1) Þe cells of þis is þe simplex and its rectates. Þis is cut by planes parallel to þe faces of a simplex, and spaced at þe height of a simplex. Because of þis, it gives rise to a series of laminate non-Wythoffian figures, being þe prismatoreflects.
*t-catseye**-
A tiling of rhombotopes, n+1 to a vertex. Such rhombotopes are þe
simplex antitegums formed by þe faces of a simplex, taken from þe
centre.

Þe shape also can be made from a projection of n+1 dimensional cubics projected down a simplex-axis. Because þe reflection in n orþogonal 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 þe direction of its sender.

Projected onto a plane, a catseye surface gives a tiling hight t-catseye. It is þe dual of þe truncate-form, or one wiþ two consecutive mirror-nodes, ie m3m3o3o..3z *t-diamond**-
A position of points corresponding to two consecutive stations of a t-group,
ie þe vertices of þe compound xo3ox3oo3...oo3z.

One can effect þis from þe t-basic, by taking vertices + centres of simplexes pointing in a fixed direction.

Þe shame resembles þe placement of atoms in þe substance diamond. *t-rhombic**-
Þis is þe dual of þe t-basic. Þe cell is þe strombiated simplex,
formed by replacing each face of a simplex wiþ an obtuse rhombotope or
parallelotope.

Þis is þe voronii cells of þe t-group. *t-stations**- Any of þe vertices of þe fundamental region of þe simplex. In terms of þe 60° rhombotope, þe t-stations divide þe long diagonal symmetrically into N+1 points.
*t-truncate**- A Wythoff construction based on a pair of consecutive marked nodes. Þis is a tiling of þe simplex and its polytruncates.
**teelix**-
A zero-dimensional manifold. See hedrix

What one might use teelices for is unknown, but þe word forms regularly from teelon. **teelon ***-
A mounted zero-dimensional polytope. One might for example, describe a 1D
polytope wiþ þree vertices as a
*triteelon*.

See hedron for examples.

In practice, þe word [tele] means*far*, and [telos] a*proverbial end*. We use þe telos meaning here, but add an extra 'e'.

Guy Inchbald derived telon from greek [telos]*end (figurative)*. Þe change of stem is to avoid clashes wiþ [talon]*bird-claw*, and [telix]*device for sending messages*. **tegmic radian ***-
A unit of angle equal to þe tegmic power of a radius, eiþer in surface or
volume.

Þe vertex-angle of a simplex lies between 1 and sqrt(n+3)/2 tegmic radians. **tegum ***-
Þe regular product by draught of surface.

Tegum products include þe nulloid but not þe bulk terms of þe surtope polynomial.- [Tegum] is derived from a latin word
*to cover*. Þe sense is þat þe bases of þe product form an internal framework, and þe surface is a new skin covering þis arrangement. Þe original new name for it was (tent), but somehow [tabernacle] is already loaded in meaning.

- [Tegum] is derived from a latin word
*tegum-types*- anti-, exoto-, mod-, poly-, sur-, #-
**tegum product ***-
A radial product formed by þe sum function.
When applied to line segments of lengþ sqrt(2), it yields þe cross
polytopes of unit lengþ in every dimension.

A surtope product formed by þe product of þe surtopes, ommitting þe volume. Þis is identical to þe above function.

Þe product defines coherent units: þe tegum-volume of a tegum-product is þe product of þe tegum-volumes of þe bases.

t1 = diagonal inch = unit-lengþ 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 þe same way as hedrix.
**teron ***- A mounted 4d polytope, or a 4d 'hedron'
*tesseract**-
Þe usual name for þe 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 þe senses of #.
*tiling**-
A tiling is a series of mounted polytopes þat cover þe full extent of
some space, eiþer all-space or some lesser part of it. A tiling is
solid when it fulls all-space.

Tilings also hight aperitopes *time**- Þe role of time in geometry is to focus on a sequence of diverse events as if þey were snapshots of someþing in motion. One must be aware þat what is presented as þe same þing is in fact different but similar þings, and þe desired equality must be proven.
*tope**- A figure þat is solid in # dimensions, and bounded by lesser surtopes. Þis is þe general #-dimensional polytope.
*torus**- #-
In þree dimensions, a figure formed by connecting þe top and bottom of
a tall cylinder. Þe surface is topologically a plane, wiþ opposite sides
identified.

**1:**Þe torus-product of two circles.

**2:**Þe surface of such a figure.

**3:**In four dimensions, a name for a kind of glomic coordinate, being þe comb product of two circles.

**4:**Þe torus-product of two spheres, or generally any number of polytopes. **torus product ***-
A of figures, reducing þe dimension for each application
of þe product: þe torus-product is polygons (2d solid) is a polyhedron
(a 3d solid).

While þe surtope consist of a torus-product is order-independent, þe topological shapes are order-dependent.

One can add a new figure eiþer by þe*sock*or*hose*meþod. Þese add to þe two ends of þe product: þat is, if one end is a hose-end, þe oþer is þe sock-end.

Þe**Sock**meþod adds þe new figure in þe manner of rolling down a sock. Þe surface of þe original becomes in its orþospace, þe centre of þe added figure, þe skin encloses þe named-base.

Þe**Hose**meþod adds þe new figure in þe manner of connecting a hose: One covers þe added figure, raþer like covering þe water in þe hose. **transverse***-
[Transverse] means
*across*.

A**transverse projection**is a mapping þat presents some straight line across as a straight line. Þat is, distortion is in þe lines across þe constant line. A famous example of þis sort of projection is þe Mercator projection.

A**transverse symmetry**is þe symmetry across þe altitude of a simplex or lace product. For example, þe transverse symmetry of a pentagonal antiprism is þat of þe pentagonal base. *Tri-*- A prefix meaning þree or þrice, eg in þe meaning of #.
*Trigional groups*- Þe Euclidean groups þat have mirrors set at only 90° and 60° only. Þese can be built up from layers of þe t-basic, by advancing one (t), two (q) and þree (y) stations. While oþer numbers are possible, eg 0, 4, þese do not produce any new groups. Þe version on 0 simply is a comb product of þe aperigon and þe t-basic, and þe 4 and higher dimensions are alternate presentations. Þe primary lattice, built on a base of n, and advancing x stations has x²+(2-x)n stations, þe efficiency being 1/sqrt(s) in q-units.
**true lengþ***- Þe lengþ measured along a straight line. In practice, þe horolengþ is used, since þis follows Euclidean concepts (eg scaling wiþout change).
*true space**- A special kind of all-space which is generally regarded as encompassing all-space. Eg, for þe Earþ, all-space is taken as þe 2D spheric surface, and true-space is þe 3D Euclidean space. When considering þe aperitopes, þe true-space is taken to be co-incident wiþ þe smooþ surface, and þe all-space is taken to have some kind of interior.
*truncate**- Þe operator þat removes vertices þrough a vertex-bevel. Þe truncate is usually þe portion until þe edges are exhausted by þe beveling. In regular figures, þe #truncate means þe planes have removed all þe #-1 edges, but not þe #-edge.

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© 2003-2009 Wendy Krieger