Bisecting rotations into 6D happen around a 4 dimensional plane. There are five such planes to rotate around in this manner. The cylconinder has two toratope faces lacing its base to the vertex. This makes two rotations identical out of the five possible 4-planes. Also, due to the pyramid-like nature of the cylconinder, rotating its height axis between base and vertex will carve out a crind-like shape.
5 Rotations of the Cylconinder : |O>|OO , |O>|O
4O
n n = { 1=2 , 3 , 4=5 }
|O>|O4O1- Code: Select all
| O > | O4 O1
1 2 3 4 5 6
[*] -> [*]= [|] ----> [O] = [|O] -----> [>]=[|O>] -----> [|O>] = [|O>|] ---> [O] = [|O>|O] -----> [O1] = [|O>|OO]
----------│---------------│----------------│-------------------│-----------------│--------------------------------------
[*] -> [*]│[*-2]1 -> [(O)]│[*(O)]1-2 -> [*]│[|(O)]1-2 -> [|(O)]│[||(O)]1-2 -> [O]│[||O(O)]1-2 -> [(O)] = [||O(OO)]1-2-6
│[|O-*]3 ---> [|O-*]│[|O|-|]3 ---> [O]│[|O|O-|O]3 ----> [O] = [|O|OO-|OO]3
│[|O>-2]4 -> [(O)]│[|O>(O)]4-5 ---> [O] = [|O>O(O)]4-5
|O>|O
4O
1 = [||O(OO)]1-2-6 , [|O|OO-|OO]3 , [|O>O(O)]4-5
* |O|OO is also the cylspherinder |OO|O
* |O>O(O) is also the sphone-torus |OO>(O), torus with sphone crosscut
* ||O(OO) is a torisphere with cylinder crosscut
Giving us ||O(OO) , |OO|O-|OO , |OO>(O) as the surtope elements.
In this rotation, we held plane 2-3-4-5 still and rotated axis 1 around into 6D. The resulting shape is a cylspherinder |O|OO lacing to a sphere |OO, joined by two new toratope faces. The original cylinder torus ||O(O) had its main circle on the moving axis, thus rotating the 2-spheric radius into a 3-spheric radius with a cylinder crosscut. The original cone torus |O>(O) had its minor crosscut shape on the moving axis, thus rotating the cone into a sphone, leaving the 2-sphere major radius unaffected. The base duocylinder was rotated into a cylspherinder, along with the opposing circle into a sphere. This shape is very identical to the cylsphoninder |OO>|O
5, which is a cylspherinder lacing to a circle, instead of a sphere. Both shapes have the same exact lacing torii ||O(OO) and |OO>(O).
|O>|O4O3- Code: Select all
| O > | O4 O3
1 2 3 4 5 6
[*] -> [*]= [|] ----> [O] = [|O] -----> [>]=[|O>] -----> [|O>] = [|O>|] ---> [O] = [|O>|O] -----> [O3] = [|O>|OO]
----------│---------------│----------------│-------------------│-----------------│----------------------------------------
[*] -> [*]│[*-2]1 -> [(O)]│[*(O)]1-2 -> [*]│[|(O)]1-2 -> [|(O)]│[||(O)]1-2 -> [O]│[||O(O)]1-2 ---> [O] = [||OO(O)]1-2
│[|O-*]3 ---> [|O-*]│[|O|-|]3 ---> [O]│[|O|O-|O]3 --> [(O)] = [|O|O(O)-|O(O)]3-6
│[|O>-2]4 -> [(O)]│[|O>(O)]4-5 ---> [O] = [|O>O(O)]4-5
|O>|O
4O
3 = [||OO(O)]1-2 , [|O|O(O)-|O(O)]3-6 , [|O>O(O)]4-5
* ||OO(O) is also the spherinder torus |OO|(O) , torus with spherinder crosscut
* |O>O(O) is also sphone torus |OO>(O)
* |O|O(O) is the duocylinder torus
Making |OO|(O) , |O|O(O)-|O(O) , |OO>(O) as the surtope elements.
In this rotation the 1-2-4-5 plane was held still, and rotated axis 3 around into 6D. Axis 3 is also the height axis between the duocylinder and circle. This makes two torii out of this pair, where the circle torus is laced into the interior of the duocylinder torus. The resulting lathing figure has all smooth, round sides, composed of complex lunes.
|O>|O4O4- Code: Select all
| O > | O4 O4
1 2 3 4 5 6
[*] -> [*]= [|] ----> [O] = [|O] -----> [>]=[|O>] -----> [|O>] = [|O>|] ---> [O] = [|O>|O] -----> [O4] = [|O>|OO]
----------│---------------│----------------│-------------------│-----------------│-----------------------------------
[*] -> [*]│[*-2]1 -> [(O)]│[*(O)]1-2 -> [*]│[|(O)]1-2 -> [|(O)]│[||(O)]1-2 -> [O]│[||O(O)]1-2 ---> [O] = [||OO(O)]1-2
│[|O-*]3 ---> [|O-*]│[|O|-|]3 ---> [O]│[|O|O-|O]3 ----> [O] = [|O|OO-|OO]3
│[|O>-2]4 -> [(O)]│[|O>(O)]4-5 -> [(O)] = [|O>(OO)]4-5-6
This is the most important and interesting lathing shape. In this construction sequence, plane 1-2-3-5 is held stationary and rotated axis 4 around into 6D. Take notice that this is also the same axis as the last spin. A peculiar effect comes from rotating the same axis over and over again. Especially if you're rotating a shape that has a circle product in it. According to this last sequence:
|O>|O
4O
4 = [||OO(O)]1-2 , [|O|OO-|OO]3 , [|O>(OO)]4-5-6
* ||OO(O) is also the spherinder torus |OO|(O)
* |O|OO is also |OO|O cylspherinder
* |O>(OO) is the cone torisphere, or torisphere with cone crosscut
Which comes out to: |OO|(O) , |O|OO-|OO , |O>(OO) as the surtope elements. The cool thing about these are that they mean this shape is the (sphere,cone)-duoprism! How cool is that? This shape can be done with cartesian products and entirely through linear operations. So, |O>|O
4O
4 = |O>[|OO] = |OO[|O>] . This will add another rule to the commuting spin. Certain ones can go all over the place, in the proper context.
Derivation of the (sphere,cone)-prism done with cartesian product:
- Code: Select all
[ |OO ] x [ |O> ] │ |OO[|O>] = |O>[|OO]
---------------------│------------------
[ *(OO) ] --> |O> │ [ |O>(OO) ]
|OO ----> [ |(O) ] │ [ |OO|(O) ]
|OO ----> [ |O-* ] │ [ |OO|O-|OO ]
can show the conveniently matching surtopes. It's the perfect analogy of a cylconinder, as the spherical version of it. Really, it's all in how the cylspherinder laces to its sphere-vertex. The cylspherinder has only two faces on it, just like the duocylinder. They are a torisphere bound orthogonally to a spheritorus. That is, a sphere innertube along with a spherical torus. So, when these two toratopes, both of which have a spherical attribute, lace to a single sphere, they collapse one of their diameters down to zero. Since a sphere is the endpoint of the lacing, the spherical radius of either toratope cell will be the one that remains unchanged. The other radius, which is 2-spheric for both, is the one that collapses, or
tapers down to nothing.
Cylspherinder : |OO|O = [ |OO(O) , |O(OO) ]
|OO(O) --> |OO = |OO|(O) - spheritorus has spherical minor radius, circular major. Major shrinks to zero while maintaining spherical part, lacing both spheres like a prism into a spherinder torus
|O(OO) --> (OO) = |O>(OO) - torisphere has spherical major radius, circular minor. Minor shrinks to zero while maintaining spherical part, tapering a 2-plane of circles to a 2-plane of points on surface of sphere, making cone-torisphere.
-- Philip