by **wendy** » Thu Sep 15, 2022 7:54 am

I'm not particularly sure this is a good idea, since in general, something like the dilateral angle would vary from hedron to hedron in any polytope, and would not particularly describe something unique to the global polytope in question.

For example, x3x4o has hexagons and squares, but the dilateral angle would simply be the margin-angles of the hexagons and squares by themselves, and could be explained such. margin-angle is already a localised down word, so the angle here is simply the margin-angle of the hexagon and square. Note this is a possesive construction, which means that it is a property of these figures.

If you has a locative construction (margin angle at the hexagon), then the general figure has several different margin angles, one is over a hexagon. For example, the octagonny o3x4x3o has two kinds of margin. The first is the octagon, over or at which the margin-angle is 135°, while over the triangles, it is 120°.

Note here that fractions 0:xx are given in successive columns of 120, or base 120. They are expanded to fractions in decimal numbers. The unit in every case is the full sphere, since this is preserved over a section, while radian-measure etc isn't.

Instead, i find the various solid angles useful. Note here, that if these angles are measured at the interior of some surtope, it is the same as the solid angle in the space orthogonal to the surtope. Then it's a matter of simply associating the solid fraction against the full space around a point. So for example, the hedric angle or margin angle of {3,4,3} is 1/3 (0:40). The choric or edge angle of the same is 0:30, or 1/4. The teric or vertex-angle is 0:15, or 1/8. These same fractions apply to the prisms that these form in higher dimensions. We see already that the edge-angle derives from the choric angle by the choric angle being the half-spherical excess, whether this is applied in 3d or 4d. Also, we see that for a convex solid, the angles decrease as the dimensions increase. For {5,3,3}, it's 0:48, 0:42, and 0:38.24 or 2/5, 7/20 and 191/600.