Marek14 wrote:What if you use something like ((III)(II))(II), though -- there's no unique "first marker" to choose in the first part.
That's a very good question, Marek. I haven't explored lathing the tigroid prisms much, though I suspect they won't be too far off from what we see with cyltorinder ((II)I)(II). By that, I refer to how a cyltorinder can also be described as a duocylinder torus. So, let's take a looksy, here:
The simplest example is (tiger,circle)-prism : ((II)(II))(II)
Tiger is circle->duoring, and extruding into 5D makes ((II)(II))I, (tiger,line)-prism. This shape can be decomposed into cylinder->duoring, the extruded tiger. This prism has two flat endcaps of tigers, laced by an extruded version of circle->duoring. So, in between tiger || tiger , we get a [line-torus]-->duoring, the linear connection surface joining two parallel tigers. Spherating such a shape leads to (((II)(II))I) , torus->duoring, which conforms with what we already know, in how (II)I is cylinder and ((II)I) makes torus.
But, before spherating, let's lathe (tiger,line)-prism ((II)(II))I around a bisecting hyperplane, to make (tiger,circle)-prism. In the orientation of ((xy)(zw))v , a bisecting rotation around hyperplane xyzw will send the tiger endcaps tumbling end over end, along a circular path into 6D. This effectively creates a tiger->circle (((II)I)(II)) as one of the surtopes.
For the other surtope on ((II)(II))I , we have the [line-torus]->duoring. This linear attaching shape has a bisecting rotation, and becomes torus->duoring, the (((II)(II))I) surtope. This transformation follows the same rules as a bisecting rotation of a cylinder into duocylinder. The cylinder has two flat circle endcaps laced by a line-torus, the hollow tube. In the orientation (xy)z , rotating around bisecting hyperplane xy will turn the circles into a torus, and the line torus into another torus, ((II)I)+((II)I) the two ortho bound toruses.
So, resting on the surface of ((II)(II))(II) , the (tiger,circle)-prism, we have (((II)I)(II))+(((II)(II))I), a tigritorus ortho bound to a toratiger. Since our names are cris-crossing definitions, I'll just use (((II)I)(II))+(((II)(II))I). As for how to derive these surtopes in the new Surtope Algorithm, let's explore:
Starting off, we have ((II)(II))(II) again.
1) rewrite to form ((xI)(II))(II)
2) take the circle parameter and move in place of X, making (((II)I)(II)), the first surtope as a tigritorus
3) now use the form ((II)(II))(xI)
4) take the entire tiger parameter, and move in place of X, making (((II)(II))I), the second surtope as a toratiger
5) and, Voi-la ! We have both surtopes, correctly derived! Yay! (((II)I)(II))+(((II)(II))I)
---- Another important point to make is that lathing cylinder-->duoring will also make duocylinder-->duoring, by definition. This is pretty much what ((II)(II))(II) is, starting with (!!)(II) and inflate a duoring with a duocylinder. Minds blown, yet?
So, it would seem that ((II)(II))(II) has only two distinct solid shapes as parameters. Following the " replace first marker with X, then move other parameter in place of X " seems to conform fairly well, no matter how many parameters the tiger has. Though, I suppose BOTH circle parameters in the tiger can have the X wherever, and still produce the same result, given the commutative property of cartesian products.
Now, let's explore that ((III)(II))(II) , (cylspherintigroid,circle)-prism:
Cylspherintigroid decomposes into circle->(sphere x circle). A prism of this will produce ((III)(II))I, cylspherintigroid || cylspherintigroid, laced by a [line-torus]-->(sphere x circle). As stated before, the ((III)(II))I can also be decomposed into cylinder-->(sphere x circle).
In the orientation ((xyz)(wv))u , rotating around bisecting hyperplane XYZWV will, again, tumble the cylspherintigroid endcaps along a circular path, making a cylspherintigroid-->circle, (((II)II)(II)).
As for the linear connecting surface [line-torus]-->(sphere x circle), it predictably becomes torus-->(sphere x circle) , the (((III)(II))I) surtope. We all know how this works, it's the sock-rolling of the hollow tube, but a [hollow tube]-->(sphere x circle) prism. And, again, ((III)(II))(II) will lathe the inflating cylinder part into a duocylinder, making duocylinder-->(sphere x circle).
So far, it looks like ((III)(II))(II) will have a (((II)II)(II))+(((III)(II))I) as the two orthogonal bound surtopes.
Now, this brings me to my next point: A cylspherintigroid has TWO distinct types of parameters, and thus two types of toruses, (((II)II)(II)) type-1, and (((II)I)(III)) type-2. According to the orientation of ((xyz)(wv))u , it would seem that we have TWO distinct circular paths we can take during this bisecting rotation. One makes either cylspherintigroid torus type-1 or type-2. It depends on which parameter, sphere or circle, we tumble the cylspherintigroid endcaps along the rotating circular path. So, I guess ((III)(II))(II) has two states, depending on which lathing direction we use. This is new to me in regards to bisecting rotations. I just recently discovered this relationship with the topratope notation, the two types of cylspherinder toruses.
So, in the Cylspherintigroid Torus Type-1 case, let's apply it to the surtope algorithm:
1) Rewrite to form ((xII)(II))(II)
2) Move the circle parameter in place of X, making (((II)II)(II)) as the first surtope
3) Rewrite to form ((III)(II))(xI)
4) Move the cylspherintigroid parameter in place of X, making (((III)(II))I) as the second surtope
5) And, we end up with (((II)II)(II))+(((III)(II))I) a {cylspherintigroid-->circle} + {torus-->(sphere x circle)}, same as above derivations
In the case of Cylspherintigroid Torus Type-2:
1) Rewrite to form ((III)(xI))(II) , a new modification of the algorithm to address the recent findings
2) Move the circle parameter in place of X, making ((III)((II)I)) as the first surtope
3) Rewrite to form ((III)(II))(xI)
4) Move the cylspherintigroid parameter in place of X, making (((III)(II))I) as the second surtope
5) And, we end up with (((II)I)(III))+(((III)(II))I) a {cylspherintigroid-->circle} ortho bound to a {torus-->(sphere x circle)}, the OTHER surtopes of a ((III)(II))(II). This isn't all that startling after all . It follows the same principle of the type-1 and type-2 cylspherinder toruses, only they become the surtopes. Notice how both ((III)(II))(II) maintain the same (((III)(II))I) surtope, only differ in the (((II)I)(III)) and (((II)II)(II)) surtopes. This coming about from a [circle-->(sphere x circle)-->circle] being either a (((II)I)(III)) or a (((II)II)(II)).
In conclusion, an open toratope with different parameters in a tigroid factor will have as many distinct surtope pairs as the tigroid parameter has. Now, for a REAL challenge, we could try this method on something like a ((III)(II))(((II)I)(II)) with FOUR distinct factor parameters!!! In this case, we would get a combinatorial collection of different ortho bound surtopes. Not to mention an open toratope with three factors, as in a ((II)I)(III)(II), a (cylspherinder,circle)-prism --> circle. This would have only three surtopes, (((II)I)I)+(((III)I)I)+((II)II), all ortho bound to each other. Only in the case of an open toratope with a tigroid factor, and only in the case of that tigroid factor having different parameters, will we see alternate surtopes of the same shape come about.
And, yet again, we run into even more ambiguity when playing with high-D shapes. Only serving to further develop our cause of classification and computation, of course !!
-- Philip
PS: I can only hope that one of these posts will be notable enough for me to become a notable green person one day....