Fixing the length of edges during transformation?


I hope this question is clear enough. I am trying to create a 3D model of a tesselated origami model. What I want to do is translate a vertex and have the surrounding vertices translate such that edges connecting them to the translated vertex keep their original length.

Here is an image of the origami model:

[Origami Model]

Here is an image of the 3D model of the flat tesselated pattern:

[3D Model]

I hope you can see here that I would like to translate the centre points of the concave hexagons (the vertices currently selected) and have the connected edges keep their length as if it was made of a rigid material.

Thank You

I’d go parenting the selected vertices to an empty or an armature, then applying a cloth sim.
Hopefully, somebody has another idea.