by wendy » Wed Feb 27, 2019 10:09 am
I make use of the 'art of circle-drawing' as a means to avoid advanced hyperbolic and trignometric functions. It allows me to deal with spheric, hyperbolic and eculidean geometries as variations of the same thing. The definition of 'straight' or 'flat' is that a subspace is straight / flat, if it has the same curvature as 'all-space'. This means, that if you draw a circle in these, and a second one on the radius, the ratios of circumferences of the greater to the radial circles must be identical.
A straight line is then a particular instance of an isocurve, which is concentric with all-space, (as great circles are lines and circles centred at the centre of a sphere), and that all iso-curves are defined by three point-like conditions. There is a class of 'parallelisms' that are defined by two point-like conditions. Straight lines contain the point 'U' (at infinity).
So there is at most one line through A, B, and U. A general parallelism is defined by two points. So there are an infinite number of straight lines through C, U, but only one of these will contain a separate point A. There are an infinite set of isocurves through AB, that pass through any point C.
The interesting thing is that one can derive Möbius geometry, if one supposes that space is a sphere, and the point U is the full interior of the sphere. This is what happens when you evaluate the hyperbolic space horizon. Every circle drawn on the sphere passes through U, and has the same curvature as the space it's in: ie every circle is straight. You can then draw a straight line through any three points.
In regards the digon and so forth, the digon arises in reflection symmetry when one considers that the edge of a polyhedron (like a dodecahedron), where the mirror runs along the edge, is not so much o--------o but o======o. Of course, the polygon-rules of area etc still apply, but it is easier to generalise the cycle of polygons, not so much as 3 pentagons, but three-times alternately pentagon, digon. Of course, if one derives the polygon by wythoff's construction, it is easy to replace the edge with a rectangle, vis 8======8, of zero height.
In essence, each node of a CD diagram has both a 'surround' and 'arround' symmetry, where the first contains the vertex-nodes, and forms a visible surtope, while the second contains no vertex-nodes. So you get a zero-height surtope, such as the rectangle, the surround mirrors makes --------- while the around mirrors make the 8 bit. The mirrors that mark out the surtope are 'wall' mirrors, reflect a copy of the surtope onto a different copy.
In low dimensions, it is not necessary to be all that fussy, and digons will do for s=edge, a=edge prisms.