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…