Hi, my name is Becca and I am a senior in highschool with a math puzzel of sorts. I dreamed this up one day and I have consulted many people on this and every one of them has given me a different answer. My question is this: If it takes two nails to secure a plank from rotating on a wall in the third spatial dimension, how many nails would it take if you were attempting to secure the board from rotating in the fourth spatial dimension? Friction between plank and nail doesn't come into play in this scenario. Even if this question has no answer, I would like the input of someone who knows about this sort of thing. My email address is becca_bober@hotmail.com. I would really appreciate a reply. Thank you for your time. Becca Hi Becca,If you are securing a board to a wall (with nails) then you really are REDUCING the problem joining two 2dimensional surfaces  the wall surface and the wall side of the board. When ONE nail is in, you are left with only one degree of freedom  the rotation about the nail. The nail itself is, effectively, giving a 3D axis for a single motion of the plank (relative to the wall). It is a 'thick' (rotationally symmetric) piece of a line  ie. a cylinder. For 4space, there would be several answers  depending on whether you are NOW using '4D' nails or '3D' nails. A 4D nail would leave just one degree of freedom between the plank and the wall. It would be like the 4D axis for a single rotation  i.e. a thick (rotationally symmetric) piece of a plane  form of cylinder in 4space. With this in place, you just need one more attachment to hold the plank still  could be another 4D nail, OR a 3D nail, or even a 2D nail (usually called a ball joint in 3space). All of these are redudant (do more than necessary). IF you are using 2D nails (ball joints) in 3space. Then it is possible that you actually need THREE such nails to hold two 3D objects together. (Sure, with a flat wall and a flat board you can TRY to use the friction and pressure of flatness to hold them from the other rotations after you install a ball joint. However, if you put TWO ball joints between the wall and the board, along one edge, it would still have a rotation about the line through the two ball joints.) Now consider using 3D nails in 4space. One such nail would leave several degrees of rotation. In a cross section (a 3D space perpendicular to the line of the nail) it would look and feel like a universal ball joint in 3space. An additional 3D nail inside THIS space, would still hold the plank to the board. [The two 3D nails in 3space did MORE than was necessary  and they still are enough in 4space.] There ARE general counting principles for this type of problem, in any dimension. The KEY is to break down the effect of something complicated, like a nail, into a set of individual constraints or linear equations. I will give a brief outline for the way a structural engineer might count things  using statics and local counts. In the plane, each 'body' has 3 degrees of local freedom
To hold two full 2D bodies together, we need to remove all three of these possible motions between them. Therefore we need THREE constraints. However a 2D nail is 2 constraints  so 2 x 2 (2D nails) = 4 which is more than enough. In 3D, a full 'body' has 6 degrees of freedom
To hold two 3D objects together you need to remove all 6 motions between them. We need SIX constraints. However, a 3D nail is FIVE constraints (leaves only one motion if used correctly). 5 x 2 (two nails) = 10 > 6 so two nails will do. IF you use 2D nails (ball joints) each of these is THREE constraints. It would appear that 3 x 2 = 6 is enough BUT the two nails SHARE a constraint so two 2D nails actually end up like a 3D nail  removing 5 dgrees of freedom. [If you know matrices and linear algebra, I am saying that the two sets of three equations share a common equation, and a row reduction of the matrices would give a zero row etc.] In 4D, a body has 10 degrees of freedom
A 4D nail would remove 9 degrees of freedom. A 3D nail could remove 5 degrees of freedom (maybe up to 7 depending on how it was made). A 2D nail (a point of attachment) would probably remove only 4. If I am getting a bit less precise here, it is because there IS ambiguity about exactly WHAT a nail is when manufactured in 4D! That ambiguity probably explains why you get different answers. There ARE assumptions being built in at each stage of the discussion. One way out is to concentrate on the 'single' constraints: what an engineer would call a 'bar' with 'universal joints' on the two ends. Then the count is clear, and we only need to make sure the 'bars' are spread out well. IN the plane, you join two bodies by three bars  making sure the three bars do NOT pass through a point. In 3D you join two bodies by six bars, making sure the six lines are independent (not easy to explain that unless you know projective geometry  line complexes. These days, most engineers don't know the precise way to say that  though everyone knows ONE way to fail have all six lines of the bars intersect a single line.) The SAME problem comes up in reverse in ROBOTICS. How many pistons do you need to move an aircraft simulator around to simulate full 3D motions? 6 (in something called the Stewart Platform). ALSO how many hinge joints to you need in a robot arm to ensure that the fingers of the hand can take all positions and orientations in a region. Six. (They often use seven to give easier planning.) In 4D the count says 10 bars properly spread out. In 5D the count says 15 bars. For ND the count is (N+1)(N)/2 bars. Check that this works for the previous cases. It was a GREAT QUESTION  and leads lots of different directions! Ambiguity is, in fact, normal in these kinds of investigations  meaning you still have choices left to make. Keeps life interesting and IS part of applied mathematics at all levels.
Walter Whiteley
