number of legs a 4D chair would need

[X-soldier]

I was thinking about this and i imagine a 4D chair would be allot different from our 3d chairs. The main difference would be the number of legs a 4D chair would need to stand up. Our chairs really only need 3 legs but i imagine a 4D chair would need more then that since it has an extra dimension to support.

Lets take a 2D chair which only needs 2 legs to stand up since it can only fall forward or back but doesn't have the risk of falling side ways since "side ways" doesn't exist for the 2 dimensional man . But if you take that stable 2D chair with 2 legs and bring it into the 3rd dimension then it would undoubtedly fall over. All you would need to do is add an extra leg to support the direction it would want to fall in and it would stand up just fine (also assuming the chair becomes 3 dimensional itself after passing into our realm. if the theory brought up in the 'The Danger Fred Could Pose to Bob, and Bob to Emily ' thread is true it would be quite painful to sit in a 2D chair ). anyway, so you only need 3 legs for a 3d chair, but if you brought that chair into the 4th dimension it would have a new direction to fall in and would probably need a minimum of 4 legs to stand. (I say 4 because you only needed one extra leg to make a 2d chair stand in 3d. stuff like that tends to stick to patterns so a 4D would also only need one extra leg.)

I also imagine the shape of the chair itself would be quite different since i think gravity would probably work different in the 4th dimension as well.

any thoughts? (and sorry if this has been brought up before )

[Keiji]

Yeah, for anything to stand up on legs, it needs a minimum of the same number of legs as the number of dimensions it's in. Adding extra legs stabilizes it more, for example most 3D chairs have four legs, and those swivelly office chairs have 5.

I also imagine the shape of the chair itself would be quite different since i think gravity would probably work different in the 4th dimension as well.

Not really. Gravity always acts one-dimensionally. The seat and back (if it had one) would have to be swock-like (i.e. 3 significant dimensions plus one thin dimension), otherwise a 4D person would fall off of it like we could fall off of a log.

[wendy]

The number of legs depends on stability type issues. In essence, it corresponds to the number of vertices the base has, or alternately, edges etc.

For latrid legs (line-shaped), four [tetrahedron] is minimal (cf stools with three legs in 3d). Six (triangular prism) or eight (cube) might be more appropriate.

You could have something like 12 [icosahedron] (cf office chairs, which sit on a pentagonal array), or even, prehaps [cuboctahedron] or 14 [rhombic dodecahedron] might also pass.

For hedrous legs (ie in the shape of a 2-flat), you could get away with three or four. For adventure, one can place the legs at different angles (eg two in wx dir, two in wy direction). Hedrous legs are known in 3d too.

For chorous legs, one usually has one (wall-mounted) leg, cf wall-mounted and rail-mounted chairs in 3d.

[Bobbie6Tee]

Sitting on a 4D chair would support my 4th dimension, whatever that looks like, which begs the question; do we have a 4th dimension ? or is that only for 4D thinggy's. Perhaps we do have more than 3 D's but are unable to experince them with 3D senses ! ! Can we have or learn a 4D sensory mechanism ?

[papernuke]

No, we do not "have" a 4D o_0
However, if you beleive, there is a 4D in which we are not in. We are in three dimensions.
We cannot use all of the legs on a 4D chair because we would only be using however many it takes to support our three dimensional weight.
The legs which are used to support 4D weight are not being used if we sit on it.

My own question is- if a 2D being sat on a 3D chair, wouldn't it simply split it in half? because the 2D being has no breadth (or whatever) so it'll just cut the chair lengthwise. Would we do that for a 4D chair?

[Keiji]

No, we'd simply pass straight through it, thanks to discreteness.

[Tamfang]

Adding extra legs stabilizes it more, for example most 3D chairs have four legs, and those swivelly office chairs have 5.

Well, sorta. I reckon the reason for five legs on office chairs is this: when one wheel falls off, the chair's stem (which is normally nearly under the center of mass) is within the convex hull of the remaining wheels. More legs wouldn't improve that by much.

How many would you need in hyperspace? An octahedral arrangement of six feet fails, for the same reason as four feet on a 2-plane, but I guess a 3-prism will work. How counter-intuitive, that lower symmetry is the way to go.

Gravity always acts one-dimensionally.

That's why it'll never be a star.

[anderscolingustafson]

Adding extra legs stabilizes it more, for example most 3D chairs have four legs, and those swivelly office chairs have 5.

Well, sorta. I reckon the reason for five legs on office chairs is this: when one wheel falls off, the chair's stem (which is normally nearly under the center of mass) is within the convex hull of the remaining wheels. More legs wouldn't improve that by much.

That means a 4d equivalent of a swivelly office chair would probably have 12 legs since the 3d equivalent of the pentagon is the dodecahedron,which has 12 sides, since all of its sides are pentagons.

A 4d swivelly office chair might also have 20 sides since the icosahedron, which has 20 sides, is sort of another 3d equivalent of the pentagon but might be much easier to spin than the dodecahedron base in 4d.

[Tamfang]

If I'm right about why office chairs have five feet, then in 4D twelve (let alone twenty) is far more than necessary.

Say Wendy, if the feet are at vertices of a triangular prism, what should be the aspect ratio of the rectangular side? Should the edges be equal, or should the faces have equal area, or some other rule?

[quickfur]

If I'm right about why office chairs have five feet, then in 4D twelve (let alone twenty) is far more than necessary.

Say Wendy, if the feet are at vertices of a triangular prism, what should be the aspect ratio of the rectangular side? Should the edges be equal, or should the faces have equal area, or some other rule?

Why do they need to be at the vertices of a triangular prism? They could just as well be at the vertices of a triangular antiprism (i.e., octahedron). A more symmetrical shape leads to a better distribution of load on each leg, which I presume would be a good thing.

[Tamfang]

With six feet at the vertices of a 3-prism, the chair can lose two (non-adjacent) feet and still be stable because the stem is within the convex hull of the remaining feet. In our world, five-wheeled office chairs have this property.

With feet arranged as the vertices of a cross-polytope, losing one foot puts the stem on the boundary of said convex hull, thus unstable.

[quickfur]

With six feet at the vertices of a 3-prism, the chair can lose two (non-adjacent) feet and still be stable because the stem is within the convex hull of the remaining feet. In our world, five-wheeled office chairs have this property.

With feet arranged as the vertices of a cross-polytope, losing one foot puts the stem on the boundary of said convex hull, thus unstable.

Interesting point, never really thought about that. But wouldn't that depend on which two feet are lost, though? I imagine losing two feet from the same triangular face would be worse than losing two feet from opposite triangular faces (one from each).

[Tamfang]

That's why I said "non-adjacent". Losing two non-adjacent feet from the same triangle would be a problem, but of a more ontological kind.

[quickfur]

... wouldn't that depend on which two feet are lost, though? I imagine losing two feet from the same triangular face would be worse than losing two feet from opposite triangular faces (one from each).

That's why I said "non-adjacent". Losing two non-adjacent feet from the same triangle would be a problem, but of a more ontological kind.

Oops, brainfart there. I know that all vertices in a triangle are adjacent. (Or do I??? )