ndl wrote: I think I understand what your calling non-bisecting rotate, I imagine it as a double bend of the square: into a cylinder and then the top edges bending towards each other inside (or outside really). But in 4D that seems to only create the torinder and the spheritorus.
ndl wrote:The concept of spherating described in the wiki doesn't seem like a valid operation. All the other operations add one dimension to the existing toratope, but spherating adds an arbitrary amount and doesn't always represent a basic shape, for example spherating a line gets you a cylinder with two half spheres on the end.
ndl wrote:I also was wondering about tapering, starting with a square isn't there a possibility of tapering of only one of the 2 dimensions to end up with the triangular prism? A square pyramid is more like a double tapering (the same way a torus is a double bending). If explained like this could tapering become commutable with the other operations?
ICN5D wrote:Spheration is tricky to explain. Think of it as more of a symbolic manipulation, that has a few different geometric and algebraic definitions, depending on what's being spherated. I trust that you've seen the open toratope notation symbols, like : (II)I , (II)(II) , (II)(II)(II) , III , etc. To spherate these shapes, simply add another pair of parentheses to the outside : ((II)I) , ((II)(II)) , ((II)(II)(II)) , (III) , etc. These are now the closed toratope forms.
ICN5D wrote:Tapering is meant to be a dimension adding operation, that's like a modified extrusion. So, instead of simply dragging a shape across n+1 dimensions, tapering means to shrink it to a point while dragging. Or, another way is to connect all points of a shape to a single point, into +1 higher dimension (convex hull with a point). But, if you assume unit edge shapes, a tesseract pyramid cannot extend into 5D, because of the whole degenerate pyramid height problem (derived through the pythagoras and long diagonal equations), and so remains in 4D, even though you've technically tapered it.
As far as I know, tapering is only commutable with the bisecting rotate. You can think of the two together as one operation that adds +2 dimensions. A cone is the only 3D example. You could either spin a line into a circle, then taper to cone, or taper the line into a triangle, then spin into a cone. Either way works. The extrude also commutes with bisecting rotate, so shapes like IO>IIO (6D cone x cylinder), have 4 forms : IO>IIO , I>OIIO , IO>IOI , I>OIOI , that are equal. But, extrude does not commute with taper.
wendy wrote:rss and spheration are two entirely different things.
Hi.gher. Space Wiki wrote:Each "I" represents a variable and parentheses represent the root-sum-square operation; the toratope is then the surface defined by the resulting polynomial. For example, (II) is the circle √(x2 + y2) − r = 0.
Hi.gher. Space Wiki wrote:Each vertical line (written as the capital letter I) represents a digon and parentheses represent spheration.
ndl wrote:Ok, after reading more carefully I think I understand the spheration described in the wiki as the "rss" operation, it's not something that adds dimension at all just rounds out what you already have by extrusion. It's interesting how the cylinder rounds out into a torus.
ndl wrote: Instead of just the idea of tapering to a point, what about tapering to another polytope of less dimension than the starting one. The simplest is a square to a line which makes a triangular prism. Also a cube to a line (square pyramid prism) or cube to square (3,4 duoprism). These should all be valid operations.
ICN5D wrote:R. Klitzing adopted the symbol || (the double pipe) ...
Klitzing wrote:There is a lot being found right in this very forum, spread out on various threads of discussion.
I myself found it always inspiring to follow, even so I contributed not so much in the field of round shapes.
P. Wright, M. Četrnáct, and W. Krieger mainly invented the first type of constructive symbols,
while P. Pugeau and Quickfur invented a somewhat different type of constructive symbols.
At some time I tried to line out those constructions to myself and started to aggregate the easiest shapes for examplifying survey,
cf. https://bendwavy.org/klitzing/explain/round.htm. Perhaps this might serve to ease your entry too.
--- rk
wendy wrote: The other thing to note is there may be some confusion between Keiji's brick polytopes (which are by general products of lines), and In5cd's torus-polytopes.
ndl wrote:Has anyone ever described the shape circle||torus? It's like a cone version of a torinder. What would be a name for it?
Klitzing wrote:If I got your symbols right, then that cone torus, aka circle || torus, ought be
|o>(o) = |>o(o)
ain't it?
--- rk
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