And this is a net?!

There is a whole variety of nets for the most usual poly­hedra. But is it really possible to fold a regular tetra­he­dron of this sheet of card­board?

Unfold the tetra­he­dron into a tradi­tional net.

Draw a segment connecting a vertex of the the big triangle and the center of the oppo­site side (which is a vertex of the initial tetra­he­dron) and cut the card­board along this segment. Turn a part of the net around the point that repre­sents the vertex of the tetra­he­dron. Doing that we'll glue two edges, but in the initial tetra­he­dron they were glued in the same way, so we didn't break the gluing condi­tions. Now we have an addi­tional part of the border that we'll mark as red.

Let's repeat this oper­a­tion.

Once again, draw a segment from the corner to the center of the oppo­site side and Turn and glue. We get the same sheet of card­board we saw in the begin­ning of the movie!

Let's make sure that the resulting sheet of card­board is a net of the initial poly­he­dron. In the left upper part of the triangle there are pieces that remain unmoved from the begin­ning. One of the small trian­gles corre­sponds to a part of the initial tetra­he­dron's base. Let's match them.

And now we'll wind the figure round the tetra­he­dron. As we see, every­thing matches!

All the segments of the red «false» edges connect the trian­gles that lie in the same plane, that means that after gluing they will disap­pear. Those segments that were painted in yellow lie on the edges of the tetra­he­dron and are the real edges.

The ques­tion whether one can fold a convex poly­he­dron out of the given sheet of card­board is answered by a theorem of a great Russian math­e­mati­cian Alexander Danilivich Alexan­drov. It is possible to figure out where the future vertices are. But we still don't know how will the real edges go from vertex to vertex. But it is another story for another movie…

Other etudes in “Polyhedra’s inner geometry”