How to model a couchtable in the shape of an irregular polyhedron

In order to not have twisty surfaces for the sides you have to do it a certain way.

As is the case with nearly every challenge, there are multiple ways to approach it. This is what I would do.

(Edit to add this much easier way to do it, but the other way (below) is fun too and gives good practice with surfaces.)

Easy method:

Start with a sketch on the top plane that is the shape you wish it to be.

Extrude it height you want the table to be.

Now, simply use the draft feature, one at a time on each side of the table, using the top as the 'neutral plane' and angle each side inward or outward depending on your design preference.

That should give the blank size of each face of the table.

Shelling it will make a single solid, and the miter challenge is pretty much the same as the original solution below.

I think it might be true to say that the number of sides of the top and bottom polygons need to be equal, but I'm not sure of that. If I was trying to make the table in my basement they would need to be equal, for sure.

Alternate method:

Start with a sketch of just the top shape. Putting that sketch on the 'top' plane makes the most sense.

Create planes (one at a time) one through each segment of the top surface and at angles relative to the top plane. They will angle inward or outward, depending on design preference. These represent the outer surfaces of the sides of the table.

Construct oversized planar surface features on each of the planes, using the appropriate top segment in the first sketch for the top edge of each planar surface feature, and 'intersection curves' in the sketches to minimize surface overlaps.

Create surface trim features to establish the overall height of the table and remove the excess surface geometry that was sketched oversized in the previous step. This should fully define the blank area for each side piece.

Create top and bottom planar surfaces to define the blank sizes for those pieces. If the bottom is open, only the top surface is needed.

At this point creating a 'thickened surface' feature from each of the surfaces will give you the solid geometry of each blank piece, pre-mitering.

I'd need to stop here and figure out how I'd want to approach the mitering, but it's probably just brute force construction of planes and cuts from here for me since the overall edge to edge interface count is somewhat low, but there's probably a more elegant way to do the miters than that.

Good luck! Looks like a fun little part to model!

Do you mean something like this? I made a very rough model to get the idea.

To complete Robert's procedure

The "TABLE POLYEDRE ok.SLDPRT" file is simpler then jpegs

You likely already have a design in mind, but if you made it like this most famous polyhedron, it might be a good way to boost sales:
http://www.albrechtdurerblog.com/wp-content/uploads/2013/10/melencolia_i.jpg

For furniture construction, I'd avoid the use of mitered corners. They tend to shrink and open over the years.
I don't have a better solution in mind at the moment, but I'd look closely at the construction methods of other furniture in the same price range and borrow some ideas from those.