by Marek14 » Thu Mar 27, 2014 9:58 am
Here's another suggestion for your explorations: two toratopes chained together.
You can plot two toratopes, either as separate graphs, or with the use of fact that if you have two graphs F(x,y,z) = 0 and G(x,y,z) = 0, then F(x,y,z) * G(x,y,z) = 0 is both at once. And you can translate toratopes by replacing the coordinates by (coordinate - constant). For best results for oblique cuts, you should probably translate both of them, putting the origin halfway between their centers.
I think that the possible chains can be explored by cutting them and studying the cuts.
For example, two torispheres won't fit together: if you'd imagine each torisphere in its torus cut, then both toruses start to inflate and they will certainly and inevitably collide.
But a torisphere and a spheritorus would form a nice chain in this case: while the torisphere inflates, the spheritorus grows thinner and can disappear from the hole of torisphere before the torisphere fills the hole.
Another cut of this situation would see torisphere as two concentric spheres and spheritorus as two spheres -- one inside the hollow sphere and one out. Here, the two concentric spheres would merge and disappear before the spheritorus spheres touch.
How about two spheritoruses? Here, starting from the two chained toruses, both would grow thinner and disappear -- and I think they would be actually never truly linked since some other cuts of the same situation (like four spheres/shperoids or two spheres plus one torus) would have the shapes completely unlinked.
How would we go with tigers and ditoruses here?
Let's have a look at tiger first since its 3D cuts are remarkably simple - the 3D vertical stack of toruses.
Here, we could thread another torus through either one or both holes. If we thread it through just one hole, it can still lead to a valid shape if it's a spheritorus that disappears before both toruses of tiger merge. (Torisphere, of course, is right out.)
Another option is to thread the spheritorus through both holes. But the same can be also done with a torisphere -- all we need is for the tiger to disappear before the torisphere cut touches it, which should be possible for a sufficiently large torisphere and sufficiently small tiger.
How about two tigers chained together? We could have a small tiger whose two toruses hang on a single torus of a large tiger projection -- the small tiger would disappear before the large tiger merges. Having doubly-linked tigers also seems as a possibility -- imagine two very thin toruses close to each other for one tiger. Here, however, both could merge and disappear with no regards to the other, which could mean it's not a true link.
Now, the ditorus. It has three different 3D cuts, both made of two toruses.
Torisphere/ditorus: If we put the ditorus in (((xy)z)w) position, then torisphere has three possible orientations: ((xyz)w), ((xyw)z) and ((xzw)y)
Well, with first orientation we could have a ditorus put entirely inside of torisphere, but that's not really what we're looking for. With orientation 2, the w-cut doesn't seem to allow for any simple chaining since all toruses are parallel and z-cut is also suspicious since two concentric spheres can't be really chained with a minor pair of toruses. With orientation 3, however, the chain works:
x-cut is a torus threaded through one of the separated toruses of the ditorus cut. The ditorus disappears before the torisphere torus gets filled.
y-cut is ditorus cut with two separated toruses and one of them is encased in two concentric spheres. The spheres disappear before the two toruses merge.
z-cut is major pair of toruses and perpendicular torus passing through both of them. The ditorus disappears first.
w-cut is minor pair of toruses and perpendicular torus linked through them. The ditorus disappears first.
Spheritorus/ditorus: Once again, we put the ditorus in (((xy)z)w) position. The spheritorus has four possible orientations: ((xy)zw), ((xz)yw), ((xw)yz) and ((zw)xy).
Orientation 1: Doesn't really work, both shapes have parallel main circles.
Orientation 2: While w-cut has a chain, none of the others do -- seems that this is not a true chain.
Orientation 3: Seems similar to orientation 2, some cuts are not chains.
Orientation 4: Doesn't work either -- the problem is that only one cut of ditorus contains separate "outsides" (minor torus pair), while two cuts of spheritorus contain two separate pieces. For this reason, it seems that spheritorus could be manipulated into any position relatively to a ditorus just by moving. I am not 100% sure of this, I can imagine links where spheritorus and ditorus must disappear in certain order (for example torus cut of spheritorus linked to exactly one torus of major pair or of two separated toruses cut of ditorus.
I'll try to have a look at tiger/ditorus and ditorus/ditorus chains later. Do you have any ideas here?