Some images in this post have been automatically down-sized, click on them to view the full sized versions:

Some images in this post have been automatically down-sized, click on them to view the full sized versions:

Some images in this post have been automatically down-sized, click on them to view the full sized versions:

EDIT: ARRRRRRRRRRRRRRRRRRRRRRRRRG, you wanted the INSIDE to be the transition. You stinker! Ill do it all again for the inside and post it as a new post. By the way the inside wont be as good looking of a transition as the outside will be. To avoid concave solids you have to make the transition triangle sort of stick out into the space a bit, but it works.

EDIT²: Im not gonna go about reposting all the new info, just sort of follow what I have and look at the vmf and get an idea.

It'd be easiest to go to multiples of 4, since the original "cylinder" has 4 sides. So you could make a transition from 4 to 8 easily and then a transition from 8 to 16 and you should have a relatively smooth transition. It would also be easiest if you could make the box line up on the x and y axes. That was the cylinder's vertices will line up best with the boxes. Steps:

You have your box. Make a cylinder with 8 sides below it. The center of the cylinder should probably line up exactly below the center of the box to make it easier. You could always move it later. You can make the cylinder as big or as small as you want but obviously you are probably going to make it about the same size as the box. Position the cylinder-8 under the box directly so that the top side is flush with the bottom of the box. You should see that 1/2 the cylinders top points are in the middle of the boxes face, while the other half are in line with its corner. Go into vertex manip. mode and select the cylinder. Drag all of the points that are in line with the corner to be in the same spot as the corner of the cylinder (one at a time). Then drag all of the other points so they line up with one of the corners close to them, either will do. As of now we will get an invalid solid structure. To prevent that we need to split all the quadrilater faces into 2 triangles. Its tough to describe this without a picture so here is one. Select the two verts like here and hit ctrl+f. Do this to all the 4 sided faces and you will be golden. Do the same thing with the 8 sided -> 16 sided cylider.

Here is the picture:

And you'll end up with this:

(sorry for the image size, it is needed to get the full solution in prpoer detail)

I even will give you the vmf:

I sure hope this is what you were asking for ...

*adopts whinny valley girl voice*

Paleeeze..

*coughs because whinny valley girl voice is painful*

lets not go there.