What single object can slide exactly through each of the holes shown below without leaving gaps? That is, the object must slip through each hole filling the hole completely.

The solution is the intersection of each of the outlines of the holes extruded along each of the three orthogonal axes in 3 space. This is where CSG based rendering engines come to the rescue, generating this intersection in a facet based modeller would be extremely difficult, it is trivial in a solid modeller.

The following show the three primitives that intersect, a cylinder, a rectangle, and a plus sign. The plus sign is in turn made up of the union of two rectangles.

Below is a transparent version of the above in an attempt to illustrate the intersecting volume.

And finally the actual intersecting volume. It is circular in along one axis (x), square along another axis (z) and a plus sign along the last axis (y).

If you have good woodworking skills and make one of these, probably from wood, please let me know and perhaps send me a copy.

The base into which the object fits might look something like the following.

Again using CSG this is particularly easy to generate using the same forms as described earlier but subtracting them from a rectangular box.

Source

The images above were generated using the RayShade rendering package, the geometric description for the union and intersecting volume is given below.

eyep 1.5 -2 2
lookp 0 0 0
up 0 0 1
fov 60
screen 200 200
light 1 1 1 directional 1 -0.5 1.5
light 0.5 0.5 0.5 directional -1 -0.5 0.7
light 0.2 0.2 0.2 directional 1 0 0
surface surfx
   ambient .2 .05 .05
   diffuse .8 .05 .05
   specular .05 .05 .05
   specpow 20
surface surfy
   ambient .2 .05 .05
   diffuse .05 .8 .05
   specular .05 .05 .05
   specpow 20
surface surfz
   ambient .2 .05 .05
   diffuse .05 .05 .8
   specular .05 .05 .05
   specpow 20

/* Plus sign along the x axis */
name xobject
union
   box surfx -0.16666667 -1 -0.5 0.16666667 1 0.5
   box surfx -0.5 -1 -0.16666667 0.5 1 0.16666667
end

/* Cylinder along the y axis */
name yobject
list
   cylinder surfy .5 -1  0.0  0.0  1  0.0  0.0
   disc surfy .5 -1  0.0  0.0 -1.0  0.0  0.0
   disc surfy .5  1  0.0  0.0  1.0  0.0  0.0
end

/* Box along the z axis */
name zobject
list
   box surfz -0.5 -0.5 -1 0.5 0.5 1
end

/* Create the unio of the three extruded forms */
name solidobject
list
   object xobject
   object yobject
   object zobject
end

/* Create the intersecting volume
name solidobject
intersect
   object xobject
   intersect
      object yobject
      object zobject
   end
end
*/

/* Create the base
name solidobject
difference
   box surfa -2.5 -1 -0.25 2.5 1 0.25
   union
      object xobject rotate 1 0 0 90
      union
         object yobject
         object zobject
      end
   end
end */