0
$\begingroup$

While playing with bar-and-joint linkages, I noticed that the skeleton of a regular 3-dimensional cube can be folded to a single edge (this can be achieved by first flexing the cube to bring it to a star shaped (non-convex) object and then folding the star to a single edge). Other examples of such polytopes are equilateral prisms with convex 2k-gon as base, which can also be folded to an edge (in fact, the cube is a also a special case of such polytopes). I was wondering which other equilateral polytopes have this property? Is it possible to charaterize such polytopes?

Let me note a few observations. In 2 dimension, it is easy to check that an equilateral convex polygon can be folded to an edge if and only if it has even number of vertices. In 3-dimension a necessary condition for foldability to an edge is that the skeleton has no odd cycle, i.e., the skeleton graph is bipartite. I was wondering if this condition is also sufficient, that is, is it true that every equilateral convex polytope with bipartite skeleton can folded to an edge?

Finally, if the answer in 3-dimensions turns out to be easy, I would like to know what happens in higher dimensions, i.e., if such questions can be asked in higher dimension and if they have any interesting answers.

Improve this question
$\endgroup$
5
  • $\begingroup$ Are you asking about continuous folding? The final state for any bipartite skeleton is obvious… $\endgroup$
    – Ilya Bogdanov
    Commented Aug 16, 2023 at 8:26
  • $\begingroup$ @IlyaBogdanov Yes continuous folding (the usual physical folding, i.e. the edges can not cross during folding). Could you please elaborate why it is obvious? $\endgroup$
    – Pritam Majumder
    Commented Aug 16, 2023 at 9:09
  • $\begingroup$ If you colour the vertices in black and white, each edge has vertices of differ4nt colours. So, fixing one edge, all others should be put there in a unique manner. $\endgroup$
    – Ilya Bogdanov
    Commented Aug 16, 2023 at 9:17
  • $\begingroup$ Is it possible to implement a physical argument? Let the skeleton (where vertices are hinges) hand on one fixed vertex $u$. The bipartite structure allows all verticves to fit into a single line (each vertex's distanve to $u$ is just the graph distance), and this clearly priviides the minimum of the pitential energy. The only question is --- if you really hang the skeleton, would it indeed move to that state? (Surely, it iseasy to pass from this state to a one-edge.) $\endgroup$
    – Ilya Bogdanov
    Commented Aug 16, 2023 at 9:21
  • $\begingroup$ @IlyaBogdanov Ah! I see, Thanks. I think that works. So basically, we start with a node $v$, then fold the edges incident to $v$ to a single edge $vv_1$, then fold the rest of the edges incident to $v_1$ to an edge $v_1v_2$, and so on. And this works for any bipartite graph with equal edge lengths (need not be skeleton graph of a polytope) $\endgroup$
    – Pritam Majumder
    Commented Aug 16, 2023 at 15:32

0

Reset to default

You must log in to answer this question.