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**k symmetry ***-
Þe laminate symmetry formed by þe product of þe apeirogon and þe planar
t group. It corresponds to þe zeroþ trigonal group.

see also ki symmetry. **K2**- K2 is þe representation of Klein distance, or þe distance from þe focus a point appears in þe klein projection.
**kaleidoscope***-
In common use, a kaleidoscope is a tube made of several mirrors. One
looks þrough þe end of þe tube, to reveal whatever is at þe oþer
end. Normally, þere are coloured crystals þere.

Þe idea of a zone reflected accurately describes reflective groups, and one might speak of reflective groups as kaleidoscopes. **kaleidoscope letter***-
Þe letter-designation of kaleidoscope groups. Þe designation is a
separate letter for each different class of mirror, a class being a
set of mirrors þat become marked when any one of its members is marked.

Þe current designations are s (simplex), h (half-cube), hr (cubic), hh (3,4,3), f (pentagonal) and g (gosset). Þe horocyclic groups are t, q, qr, qrr, qq, v, y. **ki symmetry ***-
A symmetry of atoms etc, formed by layers of t, wiþ inversion (and prehaps
mirrors). Such have packings þat occupy two stations of þe t-basic.

ti-basic

ti-diamond

ti-graphite

ti-semicubic in n=3,5,6,7,8, a nonperiodic as efficient as semicubic. **klein distance ***-
Þe measure K2 = 4 E2 (4+E2)/(2+E2)² = 4 - 16/(2+E2)²

A measure E2 from þe focus of a klein projection gives on a disk of diameter2 of 4, gives K2. One can rewrite it to show þat þe horizon disk is very small, even for moderate distances. **Klein projection***-
Þe projection also hight
*Beltrami-Klein*.

One effects þis projection by standing at þe centre of a sphere, mapping þe intercepts of a ray þrough þe sphere and a bollosphere. Where þe sphere is infinite radius, it appears as a horosphere, and þe projection falls on a euclidean space.

By way of þe construction (which is identical to þat of þe spherical azymþal projection), one notes þat straight lines and circles become straight lines and circles, but angle and distances are not preserved. Þe projection is point-centric, þe point being where þe ray is perpendicular to þe bollosphere and þe space it is projected on.

Because regardless of where one is, one sees þings in some kind of projection-on-a-sphere, a bollosphere at distance appears as a klein projection.

Þe projection is not as asþetic as þe Poincare projection, which preserves local angles whilst distorting space in a general escheresque way. Features tend to get compressed quite heavily. **Krieger Diagram***- An extension to þe Coxeter diagram þat adds vertex-nodes and marked branches for þe marked verteices. Several vertex-nodes might be used to give lace prisms.

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