snayperx wrote:Thank you for the answers. In 3D, the truncated tetrahedron P is able to pack in it its small copies p in such a way, that small copies meet in common vertices and vertices of P. For this purpose small copies should be scaled three times with respect to P. My question is how to perform such a procedure for 4D convex polytopes?
Exactly the same as you do in 3D.
In fact, you consider some vertex of your convex uniform polytope. (Thus any would serve.) Now for any incident edge you would elongate that much beyond the other vertex, and consider a perpendicular hyperplane getting from infinity closer and closer to the polytope. (Kind of a sliding caliper.) When it hits the polytope you'll stop. Then you read off from that caliper the respective extend.
As the larger polytope was uniform, the larger edges would be all of the same size. Therefore, your question just comes down to whether all read off measures would be the same for any edge type.
Clearly, whenever the original polytope was quasiregular, i.e. has exactly one node ringed within its Dynkin symbol, then all edges are of the same type. Thus your quest would be fulfilled for free. In any other case your quest comes down to do those measures and to compare those. Most generally those would be different then.
Well, there is no real 4D caliper gauge. Thus you would have to do that by calculation for sure. This can be done as follows. First you consider the right triangle vertex - edge-center - body-center. This shows, if the polytope would be scaled according to unit edges, and further more you would align the polytope such that the edge under consideration would point along the x-axis (while the polytope itself is centered at the origin), that the fixed vertex would have x-coordinate -1/2. Further you would have to calculate the coordinates (at least the x-coord.s) of all other vertices. Then take the maximum thereof. Accordingly the searched for gauge measure for that edge direction would be the maximum value plus 1/2.
Then apply the same for the other edge directions as well, compare those measures, and you are done.
The calculation of the coordinates for all polytopes of consideration, and moreover for any relevant orientation, clearly would be the hardest job of your quest.
BTW., the scaling factor from the large to the small polytope versions would be, if all those direction values coincide, twice that calculated maximum value plus 1.
--- rk