Adding mirrors to the {3,5,4} honeycomb
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- Опубликовано: 12 ноя 2024
- The {3,5,4} honeycomb has ultra-ideal vertices. If we pictured this in the Klein model,we would get icosahedral cells whose vertices are outside the Klein ball.
Since the vertices are outside of the hyperbolic space, we see big, gaping holes where the vertices should be.
Let's trucate the icosahedra -- we cut off every vertex with the common plane of symmetry of all the five edges adjacent to this vertex. In effect, we obtain a new honeycomb with soccerball-like cells.
In the first half of the video we are flying through a hyperbolic manifold constructed out of 22 {3,5,4} cells. This is not a closed manifold, since the cells are infinite. In the second half, we put a mirror on each of these planes, and we no longer have holes! We could interpret this as a closed manifold constructed out of 44 truncated {3,5,4} cells.
Cool!
Ultraideal vertices are easy to make visible and make some sense of in the Klein-Beltrami model of hyperbolic space, these are the meeting points of ultraparallel lines. The Klein-Beltrami model is well suited to such schemes as Universal Hyperbolic Geometry (over the reals of course, despite mr Wildberger's fear for them: rational numbers, algebraic numbers and finite field numbers are cool but cannot substitute real, complex or p-adic numbers!), and decomposes into three regions: the ideal circle/sphere at infinity, the hyperbolic, infinitesimally Euclidean, disc/ball inside this circle, and the hyperbolic (i think), infinitesimally Minkowskian (i wrongly called this Lorentzian in another comment), Moebius surface/solid outside this circle/sphere. There are hyperbolic straight lines fully contained in the outside region, connecting certain ultraideal points, these are weird since they are completely invisible in the Poincaré model.
Distances and angles may be defined for any two points or any two lines (except possibly those lying on the circle at infinity or tangential to it).
Edges (pair of points with a line through them both) come in four main classes: both inside the circle at infinity, one inside and one outside, both outside but connected via a line crossing the circle (line inside), and both outside and connected via a line not crossing the circle (line outside).
Special cases are: one point at infinity and one point inside, one at infinity and one outside, both points at infinity.
Vertices (pairs of lines with a point on them both) come likewise in four main classes: both outside the circle at infinity (not crossing it), one outside and one inside the circle, both inside but intersecting in a point outside the circle, both inside and intersecting in a point inside the circle. Special cases are: one line tangent to the circle at infinity and the other outside, one line tangent and the other inside, both lines tangent to infinity.
Distances may be imaginary as well as real, with a special case for distances between -√-1 , 0 and √-1 (dual to the sine of an ordinary angle). Sines of angles may be bigger than 1 or smaller than -1, or imaginary altogether, angles with imaginary sines are dual to ordinary distances. I don't know exactly how the analytical continuation of inverse sine works for real numbers outside the interval from -1 to 1, or for imaginary numbers, but i think some kind of complex angles will work.
All ordinary hyperbolic circles are equidistant curves to a line outside infinity, and all ordinary equidistant curves are circles with center a point outside infinity. Horocycles are circles with center a point lying at infinity, and also equidistant curves to a line tangent to infinity at this very point.
There are also circles/equidistant curves fully outside infinity, and also an outside portion of ordinary equidistant curves - these are circles with center outside infinity, the circles themselves crossing infinity from the outside.
Almost everything is numerically measurable when using similar methods as those employed in Universal Hyperbolic Geometry (computing distance as multivalued signed square root of quadrance, and angle as multivalued complex inverse sine of signed square root of spread).
Wow
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Serisously though what do the {x,y,z} things mean?!
In two dimensions, x is the shape of every face and y is how many meet in every vertex, for example the icosahedron is {3,5} and hex grid is {6,3}.
In thee dimensions, we say {x,y,z} where {x,y} is the shape of every cell and {z} is how many cells are there around every edge (equivalently, {y,z} is the vertex figure).
@@ZenoRogue Cool, thanks!
@@ZenoRogue An extension might be to say that {#} space is any space where {#} fills up 100% of the space.
@@katakana1 I do not understand what you mean? All honeycombs fill 100% of the space, but in {3,5,4} most of this space is filled in a boring way.