Tipping a cube over

Here is a bit of sample code. Place it on a block and use the arrow keys:

 #pragma strict
 
 private var parent : Transform;
 private var dirs = [Vector3.up, Vector3.down, Vector3.left, Vector3.right, Vector3.forward, Vector3.back];
 var speed = 90.0; 
 
 function Start() {
     parent = new GameObject().transform;
 }
 
 function Update() {
     if (Input.GetKeyDown(KeyCode.LeftArrow)) {
         FlopTheBlock(Vector3.left);
     } 
     else if (Input.GetKeyDown(KeyCode.RightArrow)) {
         FlopTheBlock(Vector3.right);
     }
     else if (Input.GetKeyDown(KeyCode.UpArrow)) {
         FlopTheBlock(Vector3.forward);    
     }
     else if (Input.GetKeyDown(KeyCode.DownArrow)) {
         FlopTheBlock(Vector3.back);
     }
 }
 
 function FlopTheBlock(direction : Vector3) {
     var down = WhatsDown();
     var dir = transform.InverseTransformDirection(direction);
     
     var pos = dir * 0.5 + down * 0.5;
     pos = transform.TransformPoint(pos);
     
     var axis = Vector3.Cross(Vector3.down, direction);
     
     transform.parent = null;
     parent.rotation = Quaternion.identity;
     parent.position = pos;
     transform.parent = parent;
     
     var qTo = Quaternion.AngleAxis(-90.0, axis);
     
     while (parent.rotation != qTo) {
         parent.rotation = Quaternion.RotateTowards(parent.rotation, qTo, speed * Time.deltaTime);
         yield;
     }
 }
 
 function WhatsDown() : Vector3 {
     for (var i = 0; i < dirs.Length; i++) {
         if (Vector3.Dot(transform.TransformDirection(dirs[i]), Vector3.down) > 0.9)
             return dirs[i];
     }
 }

Note to solve this problem, there are a couple of sub-problems that need to be solved. First the rotation around an axis not through the pivot of the object. This code places an empty game object on an edge and make the visible object a child. So rotating the parent rotates the child around an edge.

To find the right edge, the code first finds which side of the cube is facing down. Then it translates the direction the cube is to be pushed into local coordinates. Combining the two gives a local coordinate of the edge point. Converting to world coordinates handles the scaling.