Polyhedra with heptagons do exist if you consider prisms and anti-prisms. Personally I don't find these symmetries very attractive. They are just 2D symmetries transformed once along a new axis. They are not "real" 3D symmetries. It is easy to construct polyhedra with "real" 3D symmetry by attaching 7-order prisms onto a base that has such a symmetry. E.g. like this:
I built a few of these, but such models aren't very exciting, really.
To do something more, you will have to let go of some constraint. A few years ago I found one that was quite exciting (for me) and then I succeeded in finding polyhedra that only consisted of regular heptagons,... with some constraint of course. This constraint was that I allowed the heptagons to be folded over a diagonal.
Some models were on display at Bridges in 2013. E.g.
More can be found here: http://gallery.bridgesmathart.org/exhib ... marcelteun
and I even wrote a short article about it: http://archive.bridgesmathart.org/2013/ ... 3-183.html
The main weakness in the article is that I don't prove that any of these consist of heptagons that are really regular: numerically they only seem to be. In the mean time for some of them I do think I have a (non-published) prove that some of them really consist of regular heptagons.
One other thing that I found a pity at that time was that I didn't have a polyhedron with A5 symmetry; i.e. that only has the rotational symmetry of an icosahedron (i.e. no reflections). I find these models with only rotational symmetry more attractive. Later I did find some polyhedra and one of them wasn't too complex. I started building a model of this in 2014, but I had to quit. There was too much tension in the model that caused it to "collapse" inwards. Probably caused by the thickness of the cardboard I am using (and the shape of the model). I got quite frustrated and I ended up taking a long break from building models (not only because of that).
Later it was Magnus Wenninger's passing away that got me inspired again and some weeks ago I had gathered enough courage to try to build this one again. This time I gave it some more thought: I made some change in the construction and I am confident that I will finish the model soon.
A 3D image can be found here:
http://tunnissen.eu/polyh/local_off.htm ... istance=18
The link above is a link to a 3D online off-viewer (you can load off-files that have the exact same domain name and you can load local off-files, e.g. that you downloaded to your phone). The above URL also contains a link to an off-file that you can download and then load in Stella. You can see that the model doesn't have any intersections.
If anyone is interested in off-files of some other polyhedra that only consist of regular heptagons, I can post them here too.
