So I was reading about the example of hanging a 2d safe on a 3d nail, and a 3d safe on a 4d nail and it got me thinking: If you were to hammer a 3d nail through a 2d object, it would leave a circular hole in the middle of the 2d object, so if something were to hammer a 4d nail through a 3d object, it should leave a spherical hole in the middle of the 3d object correct?  Any thoughts/improvements? 

A nail hammered through an object with one less dimension than it self would only be required to leave a hole if the object was not hollow.  If the object was hollow then the nail with one more dimension than the object would not need to travel through the edges of the object to get to the hollow center.  If the object was solid then the nail would leave a hole assuming it didn't cause the object to shatter first and if the object was hollow then the nail could leave a hole if it went through the solid part of the object but it would not be required to leave a hole in the object when going through the object. 

As safes are usually hollow a nail with one more dimensions than the safe would not need to leave a hole in a safe if it was hammered through a safe as it could go through the center of the safe without leaving a hole in the safe. 

If a nail does leave a hole in a object with one less dimension than it self then the hole will indeed be in the shape that is the equivalent of a circle in the number of dimensions the object has so a 2d object will have a circular hole, a 3d object will have a spherical hole, and a 4d object will have a glome hole. 

If you want to hang 3d picture on the wall using 4d nail, what shape of the rope should you use?  Linear rope with spherical cross-section will not fit: it'll slide from the nail, and two points of attachment may be not enough to hold a picture. 

4D ropes are of very limited use, because they can't be tied into a knot (they just come undone when you pull them no matter how they are tied).  You need 2D sheets if you want to make any knots. 

Having said that, though, it is possible to hold things up with ropes if you pierce holes through the object being hung so that the ropes can't slide off.  In 3D, both the string and the loop through which it is inserted are 1D objects (1D string + 1D ring).  In 4D, however, the loop cannot be merely a loop; it must be at least a 2D sphere through the center of which the string passes. ( You can obtain the topological equivalent by piercing through the center of a cube in the ana/kata direction.) 

As for how many points of attachment you need: minimum 3, but assuming a cubical (or cuboidal) picture frame, probably 4. 

How about if the nail was a torinder?  That should allow you to hang 1D ropes from it. 

I cannot quite picture it at the moment, but would the action of mounting a picture frame suspended from a 1D rope on a torindric nail be a sliding motion akin to sliding a 1D rope in a 3D universe between two pegs? 

Keiji wrote:How about if the nail was a torinder?  That should allow you to hang 1D ropes from it. 

I cannot quite picture it at the moment, but would the action of mounting a picture frame suspended from a 1D rope on a torindric nail be a sliding motion akin to sliding a 1D rope in a 3D universe between two pegs? 

The torinder is just the extrusion of the torus, right?  Are we talking about a solid torus or a hollow one?  A torinder made from a solid torus is topologically equivalent to a ring, and so won't knot with a 1D rope: it'd just slide off.  If the torinder is made from a hollow torus, however... I'm not sure I can quite imagine what that looks like, but it would have two kinds of holes... and one of them may be able to hold a 1D rope without slipping off. 

Basically, in order for the rope to hang tight, you need the equivalent of the extrusion of a hollow cup in 3D, such that the hollow inside the cup is where the rope will be inserted.  If the top is also closed off (making it equivalent to a sphere), then the rope will be trapped inside and won't be detachable; if the top is open, then the rope can be removed by lifting it off (akin to lifting a picture off the wall in 3D). 

Keiji wrote:My thinking was that in 3D, a rope threaded through a torus cannot be removed (without passing one end through the hole). 

Therefore, if you extrude the torus into a torinder, you should be able to "slide" the rope "on" to the torinder, so that the cross-section would be the above. 

Having a torindric nail, i.e. with a head that prevents sliding on one end, and inserted into a wall on the other, should prevent the rope from "sliding off". 

However, in this case, wouldn't this stop one from being able to hang/remove the rope like one does in 3D, using gravity to keep the rope resting against the nailhead? 

See, the thing is, in 4D a 1D rope behaves like a point.  So the actual analogy is that you're trying to prevent a point (a small ball, if you will) from rolling off a nail.  The only way you can do that is to "catch" the point by having a cup-shaped surface. 

I like the balloon idea!  So basically the nail is a spherinder with a stub at the end to prevent the "balloon net" from slipping off it.  The interconnections within the net itself prevent individual strings from slipping off the nail (because if you only had, say, one loop of string to hold the basket under the balloon, it would just slip off the round surface of the balloon, but if you have a net, then the strings hold each other in place).  So such a net will indeed be able to hang on a nail without falling off. 

This is a very interesting idea... because a net is essentially a skeletal version of a 2D surface, and we know that in 4D, a surface is needed to make a knot.  Of course, unlike a 3D balloon net, the 4D net will need to have strings glued together at their crossing points, since otherwise the extra dimension allows the strings to slip past each other and the net will come undone. 

And for 1D rope you can use a half of solid torinder (cut it by horizontal 3D-plane passing through rotational plane).  In 3D section it will look like a rope hanging on half of torus.  But it's not very stable construction: for example, if you have half-torinder tube between two walls, you can hang rope loop on it without disconnecting it.  But if the picture is heavy enough, it shouldn't be a problem...

The most stable "hook" I can think of is actually also very simple: take a 3D cube, punch a cylindrical hole in it halfway deep, then extrude the result.  Sharpen one end to nail into the wall, and block off the other end with a stub.  Now a 1D spherindrical rope will hang on it without slipping off. 

Actually, now that I think of it, this is probably identical to your half of solid torinder. 

P.S. Looking over the previous posts, it seems that we're really looking at the same thing from two perspectives: basically there's a 1D object expanded into the form of a spherinder, and another object which is basically a spherinder with a groove cut in it that the spherinder.  The inside of the groove essentially provides a 2D barrier to hold the 1D object in place.  IOW, it's a 1D object being held in place by a 2D barrier, and 1+2=3=4-1.  In 3D, two criss-crossing 1D objects will be held in place (1+1=2=3-1).  In 4D, one of the objects needs to have a 2D boundary in order to hold things in place. 

So I'm guessing that in 5D we will need either two 2D objects or a 1D object and a 3D barrier in order to hold things in place.  The pattern seems to generalize to 2D too: in 2D, a single point is enough to hold a 1D rope in place (0+1=1=2-1). 

For the hook we can use large sector (like 270 deg) of torispherinder (is it how you call it?  Product of large 2-sphere and small circle).  It will work in the same way as our hooks with shape of 3/4 of torus: rope goes around the small circle, and rests in the bottom part of the thick 2-sphere.  Another question is how to attach rope to the object?  With developed technology they can use some kind of clutch (fix rope between two blocks), but what could they do before such method is invented?  Looks like they should invent the glue even before the wheel