# -: K :-

k symmetry *
The laminate symmetry formed by the product of the apeirogon and the planar t group. It corresponds to the zeroth trigonal group.
K2
K2 is the representation of Klein distance, or the distance from the focus a point appears in the klein projection.
kaleidoscope*
In common use, a kaleidoscope is a tube made of several mirrors. One looks through the end of the tube, to reveal whatever is at the other end. Normally, there are coloured crystals there.
The idea of a zone reflected accurately describes reflective groups, and one might speak of reflective groups as kaleidoscopes.
kaleidoscope letter*
The letter-designation of kaleidoscope groups. The designation is a separate letter for each different class of mirror, a class being a set of mirrors that become marked when any one of its members is marked.
The current designations are s (simplex), h (half-cube), hr (cubic), hh (3,4,3), f (pentagonal) and g (gosset). The horocyclic groups are t, q, qr, qrr, qq, v, y.
ki symmetry *
A symmetry of atoms etc, formed by layers of t, with inversion (and prehaps mirrors). Such have packings that occupy two stations of the 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 *
The measure K2 = 4 E2 (4+E2)/(2+E2)² = 4 - 16/(2+E2)²
A measure E2 from the focus of a klein projection gives on a disk of diameter2 of 4, gives K2. One can rewrite it to show that the horizon disk is very small, even for moderate distances.
Klein projection*
The projection also hight Beltrami-Klein.
One effects this projection by standing at the centre of a sphere, mapping the intercepts of a ray through the sphere and a bollosphere. Where the sphere is infinite radius, it appears as a horosphere, and the projection falls on a euclidean space.
By way of the construction (which is identical to that of the spherical azymthal projection), one notes that straight lines and circles become straight lines and circles, but angle and distances are not preserved. The projection is point-centric, the point being where the ray is perpendicular to the bollosphere and the space it is projected on.
Because regardless of where one is, one sees things in some kind of projection-on-a-sphere, a bollosphere at distance appears as a klein projection.
The projection is not as asthetic as the Poincare projection, which preserves local angles whilst distorting space in a general escheresque way. Features tend to get compressed quite heavily.