Probably the best way to learn the cut algorithm is to forget about the open toratope cuts for now, and nail down the closed type first. I posted the open cuts here
viewtopic.php?f=24&t=801&start=210#p20089, only to display the function of the algorithm, to show that it also works. But, you have to change the rules for it to work. So, trying to learn both simultaneously will cause endless confusion. It may be in your powers to do both, but the learning curve will be steeper.
Odd number pairings cannot be made by the cuts of any toratope, as far as I know. They are always an even number or a single one, tracing back to the primordial cut of a circle as two points. Cutting a hollow circle is analogous to cutting any toratope through a radius, exposing the hole in the middle of
two cut-shapes. Marek's system uses cartesian products along with spheration. It also happens to be in the opposite order of diameters to mine, except for tigroids. I'm using linear construction along with cartesian products with hollow shapes. Both can be seen in each other, but are applied differently. Cartesian products are easy to see, but spheration is more abstract and complex in its physical transformation. From what I gather, spheration is more of a sock-rolling method to any (n-sphere,line) or (toratope,line)-prism, like the cylinder, spherinder, glominder, pentaspherinder, torinder, etc. All must be rolled up like a sock to achieve the correct shape. If using a hose connection method, these types turn into the toratopic dual of a sock-rolling. For the {n-sphere,m-cube}-prisms, where m≥2, spheration is even more abstract. All flat ends are snipped, leaving behind the toratope surcell. It's always the m-cube(n-sphere) toratope (in my notation), but its transformation always adds a dimension and smooths out the minor shape into an m+1-sphere.
Using my notation with toratopes,
{n-sphere,m-cube}-prism :
• Has 2 x M number of {n-sphere,(m-1)-cube}-prisms
• Has an m-cube(n-sphere) toratope surcell
• Spheration makes the (m+1)-sphere(n-sphere) toratope, (m+1)-sphere is the meat, (n-sphere) is the bones
Take the cylinder for instance: (II)I , we have two flat circle endcaps connected by a hollow tube, the line torus. This is the prism of the edge of a disk. The line torus is the linear extension between the edge of both disks. It has a radius and a linear length. The sock rolling method will snip out the two circle endcaps, leaving a hollow tube behind. If we then take one end and curl it back, rolling it down, we can attach the two cut ends together, and make a torus. This turns the linear extension into a new, minor radius, the circle torus. The original major radius stays the same. The hose link also produces the same shape with a cylinder.
(II)I - cylinder
((II)I) - torus, sock-roll
((II)I) - torus, hose-link
Sock-Roll
• Cylinder: [ |O-2 , |(O) ], two circles |O-2, and a line torus |(O)
• Snip out two circles leaves line torus |(O)
• Roll line torus back to make circle out of line, circle torus |O(O)
Hose-Link
• Cylinder: [ |O-2 , |(O) ], two circles |O-2, and a line torus |(O)
• Snip out two circles leaves line torus |(O)
• Bend linear extension around and join ends making circle major radius out of line, switching the original major into the new minor radius, torus |O(O)
This also works with the torisphere, being made by spherating a spherinder (III)I --> ((III)I). The spherinder has two flat sphere endcaps, connected by a single linear extension of the surface of both spheres. The surface of a sphere can be described as having an infinite number of points, and so becomes a point torisphere. When we extrude this we get a line torisphere, the prism of the edge of a sphere. Now, we cut the spheres out, leaving just the line torisphere: the 4D analog of a cardboard tube. One could wrap spherical paper towels around this kind of hollow tube. One could also grab one open end and curl it back, rolling the linear extension into a circular extension, the new minor radius. This makes a circle along surface of sphere, the torisphere. The original major radius is spherical, the minor part a line. By rolling it back and joining the snipped ends, the line is turned into a circle.
(III)I - spherinder
((III)I) - torisphere, sock-roll
((II)II) - spheritorus, hose-link
Sock-Roll
• Spherinder |OO| = [ |OO-2 . |(OO) ] , two spheres |OO-2, and a line torisphere |(OO)
• Snip out the sphere ends, leaving hollow tube of line torisphere |(OO)
• Roll linear extension back into a circle, torisphere |O(OO)
Hose-Link
• Spherinder |OO| = [ |OO-2 . |(OO) ] , two spheres |OO-2, and a line torisphere |(OO)
• Snip out the sphere ends, leaving hollow tube of line torisphere |(OO)
• Bend linear extension around and join ends making circle major radius out of line, switching the original major into the new minor radius, spheritorus |OO(O)
However, this method changes the outcome with certain open toratopes. If we hose-link a spherinder, we get the spheritorus. If we sock-roll a spherinder, a torisphere results. By sock-rolling a cubinder, we are snipping out the flat cylinder ends, leaving only a hollow square torus tube. By rolling this square torus back onto itself, we add a spin operator to the square cross cut, and get a cylinder torus, or torinder. If we hose-link a cubinder, we are snipping out two cylinder ends, leaving two cylinders and a square torus behind, with two openings separated by a linear extension. Then, we bend this linear part around into a circle to make a torinder.
(II)II - cubiner, |O||
((II)II) - spheritorus, |OO(O)
((II)I)I - torinder
Sock-Rolling
• Cubinder |O|| = [ |O|-2 , |O|-2 , ||(O) ] , four cylinders |O| and a square torus ||(O)
• Snip cylinders out, leaving hollow square torus tube ||(O)
• Roll square part back, adding circular extension, making cylinder torus ||O(O), or a torinder |O(O)|
Hose-Linking, not sure about this one, but I'll try
• Cubinder |O|| = [ |O|-2 , |O|-2 , ||(O) ] , four cylinders ||O and a square torus ||(O)
• Snip one cylinder pair out, leaving two openings, 2 cylinders and square torus |O|-2 , ||(O)
• Bend open ends around and join in a circle, makes torinder/cylinder torus ||O(O) = [ |O(O)-2 , |(O)(O) ]
Spheration
• Cubinder |O|| = [ |O|-2 , |O|-2 , ||(O) ] , four cylinders |O| and a square torus ||(O)
• Snip cylinders out, leaving hollow square torus tube ||(O)
• Add 1 dimension to square crosscut making cube III, then spherate cube to sphere, as part of spheritorus |OO(O)
In writing this post, I noticed a new algorithm can be made that defines sock-rolls and hose-links for all rotopes! It's actually quite simple, but requires that one identify the type of curved face on them, especially for the sock-rolling. This is where my line torus, square torus, square torisphere, etc notations come into play. Identifying the crosscut will allow us to create the next minor radius shape, by adding a spin operator. Sock rolling requires that one cut out all flat ends, leaving only the equivalent hollow tube(s) behind. Hose linking joins linear extensions into circular ones, smoothing them out into one circular major radius. I have elaborated on it above.
As for how all this fits in with the cut algorithm, I think I got lost along the way, and went off on a tangent. So, I'll have to detail that after I get some sleep. But, maybe it has some insights worthy of dissecting.
-- Philip
