# Work on “Churning Polygons”

Background of this question is that I recently stumbled over the problem of deforming polygons in area-preserving way, i.e. modifying the angles between adjacent edges while preserving edge-lengths, topological equivalence to a circle and size of enclosed area.

Questions:

• Has that problem been encountered and worked on before?
I'am looking for existing work, because I encountered some interesting questions when investigating the problem, e.g. what the minimum number of edges of polygons allowing such deformations is (I conjecture that it must be six).

• What to do with the further questions I encountered?

Any pointers to articles or blogs related to the problem would be of help to me.

Remark:
I chose the preliminary term "churning" as an analogy to the approximately surface-area and volume preserving deformations of the stomach; I am however no native English speaker, so a more appropriate verb may exist.

• I have no references, although Unsolved problems in geometry (edited by Croft and Guy I think?) may have something. I would think the term "adjusting" preferable to "churning". Finally, a nonconvex regular pentagon should be adjustable, as well as some convex ones: fix an edge and consider a linkage in which the opposite vertex corresponds to an area, and note that this gives a continuous function with lengthy level curves for all but the extreme values. Gerhard "Proving Propositions Through Picturing Pantographs" Paseman, 2017.06.25. – Gerhard Paseman Jun 25 '17 at 14:58
• @GerhardPaseman thanks for the reply; designing such polygons isn't the problem; I have found some interesting examples, which I can share. What fascinates me about those polygons is that they yield a primitive model for the motion of crustaceans filled with an incompressible liquid. – Manfred Weis Jun 25 '17 at 15:08
• That is an interesting application, and probably has been explored by companies producing digital animation. You might consider looking for technical reports from such companies that are available through a web search. Also, the pentagon linkage has more degrees of freedom and thus there are up to four different possible values for area with some locations for the vertex opposite the fixed edge. However a quadrilateral linkage has essentially less than two degrees of freedom, so five is the minimal number of edges. Gerhard "State Spaces Can Be Weird" Paseman, 2017.06.25. – Gerhard Paseman Jun 25 '17 at 15:21

If I understand your question correctly, you're asking about (signed) area-preserving deformations of planar polygonal linkages. One place to start reading about polygonal linkages is this chapter by Connelly and Demaine. You might be interested about questions about the configuration space of such linkages, for that see "On the moduli space of polygons in the Euclidean plane" by Kapovich and Millson and the papers which cite this.

Let me turn to your comment / question:

[What is] the minimum number of edges of polygons allowing such deformations [...] (I conjecture that it must be six).

The dimension of the moduli space of a planar $n$-gonal linkage (with plane isometries quotiented out) is $n-3$. Preserving the area function gives one more constraint on this space. Therefore, at a generic value of the area $A$, the dimension of the space of polygonal linkages with area $A$ (if nonempty) should be $n-4$. Thus (as I see Gerhard Paseman has already pointed out in the comments) nontrivial area-preserving deformations exist for $n\geq 5$.

The Morse-theoretic properties of the area function have been the subject of study. At certain special values of $A$, the area function is critical (i.e. the space of fixed-area polygonal linkages will have some singularity). It turns out that the "typical" critical configurations of the polygon are cyclic (i.e. vertices lie on a circle)! This is a result of Panina and Khimshiashvili. There's quite a bit of followup work on this result, to give one random recent example, see this recent preprint of Panina and Siersma.

• A concrete example for $n=5$ would be very desirable to me. – Manfred Weis Jun 25 '17 at 17:35
• The provided pointers to the resources answer my question regarding previous/related work sufficiently; than you very much! – Manfred Weis Jun 25 '17 at 17:39
• For a concrete example when $n=5$, consider a polygon with all side lengths equal to 1 but not in the regular configuration (as that maximizes the area and hence cannot be "churned"). Suppose vertex 1 is at the origin and vertex 2 is at $(1,0)$. Then we have 6 variables $(x_i,y_i)$ for $i=3,\dots,5$ which must satisfy 4 more equations preserving unit lengths of sides 2 through 5. The area of the polygon can be written in terms of the vertex coordinates using e.g. en.wikipedia.org/wiki/Shoelace_formula and this gives one more equation. – j.c. Jun 25 '17 at 17:55
• This is 6 variables satisfying 5 equations so you can set one more condition freely, e.g. tweak the angle at vertex 2 or change the distance between vertices 1 and 3. Hopefully that is sufficiently concrete that you can test this out with your favorite polynomial system solver. – j.c. Jun 25 '17 at 17:58