Albrecht Durer (1471—1528) was a great german painter. He studied theo­ret­ical ques­tions of art, in partic­ular, prob­lems of perspec­tive. A part of his book «The art of measure­ment with a ruler and compasses, plane and spatial bodies», 1525, he devoted to the study of prop­er­ties of geomet­rical objects, in partic­ular, poly­hedra and their nets.

An net consists of a number of poly­gons placed without inter­sec­tions in a single plane and a number of gluing rules of the polygon edges. If some cut of a poly­he­dron along its edges allows to use a single polygon without breaking the rule of no inter­sec­tions, such a net is called connected.

On the pages of his book Durer provides different connected nets of some­times quite compli­cated poly­hedra. It's doubtable that he was thinking of whether a single polygon is always enough, but the following assump­tion is often called after him. The Durer conjec­ture tells us that every convex poly­he­dron has at least one connected net.

Why do we stay inter­ested in poly­he­dron nets for centuries? The matter is that a net contains the internal geom­etry of the poly­he­dron, namely, the infor­ma­tion a point-like crea­ture living strictly on the poly­he­dron may get. This crea­ture may only measure distances between points. Having some math­e­mat­ical abil­i­ties, the crea­ture can also deter­mine angles between direc­tions and compute areas, …

For some purposes using a net is «more comfort­able» that using the poly­he­dron. For example, if you want to send a model of a poly­he­dron to another town, you need to send a package. But to send a net of a poly­he­dron it's suffi­cient to send a construct the convex poly­he­dron himself. If you think that trans­porta­tion of poly­hedra is a rare oper­a­tion, you are wrong! The result we see in everyday life when buying a box of milk or juice…

The Durer conjec­ture is about convex poly­hedra. It has neither been proved nor disproved yet. But if the orig­inal problem is hard to solve, one should change some condi­tions and try to solve the resulting problem. In our case it's natural to study the analogue of the conjec­ture including non-convex poly­hedra.

It's easy to build a non-convex poly­he­dron possibly having non-convex faces that has no connected nets. Take a inter­sect the star. Thus, the base should be sepa­rated from the side faces and the net will not be connected.

It's not so easy to construct a poly­he­dron with convex faces not having a single connected net. The first example was given only in 1999.

Move from the vertices of a tetra­he­dron along all the edges to a small common self-inter­secting.

After consid­ering a non-convex coun­terex­ample return to its orig­inal setting: convex poly­hedra.

The simplest convex poly­he­dron is a trian­gular pyramid: it has 4 vertices and 4 faces. Even in this simplest case there are connected nets. There is still no convex poly­he­dron constructed having only self-inter­secting connected nets.

Recently N. P. Dolbilin has stated a problem about the «anti-Durer» conjec­ture. It says that for every $k$ there is a convex poly­he­dron such that every its net placed without inter­sec­tions on a plane should have at least $k$ pieces.

We should mention that if the Durer conjec­ture is false, there are two completely different cases.

The bounded case: every convex poly­he­dron has a net consisting of no more that $K$ pieces. More­over, the bounding number $K$ can be chosen the same for all the convex poly­hedra, i.e. not depending on the partic­ular example.

The other case is more inter­esting: that the number of pieces is not bounded from above for convex poly­hedra.

The «anti-Durer» conjec­ture says that we are in the unbounded case.

No so far its analogue for non-convex poly­hedra (in the unbounded case) has been proved by russian math­e­mati­cians.

You may try to construct a convex poly­he­dron that has only self-inter­secting connected nets. Or to prove that such a poly­he­dron doesn't exist. And if you succeed, you will add a new vivid page to geom­etry.

Other etudes in “Polyhedra’s inner geometry”