# The space of clouds in an Euclidean space

by Jean-Claude Hausmann & Eugenio Rodriguez

THE PAPER:

• The space of clouds in an Euclidean space. pdf-file

• Minfeng Wang, an undergraduate graduate student at Fuzhou University, has redone all our comuptations with other programs. Everything coincides except for some minimal realizations in 8 and 9-gons, where his more powerfull simplex algorithm produced solutions amin in P1 (Equation 4-3 in our paper) with smaller perimeter (i.e. coordinate sum). The new solutions often reflect more the symmetries of the chambers (it may be that these symmetries made our algorithm stop prematurely). These improvements concern 53 cases for 8-gones (2.1%) and 22371 cases for 9-gones (12.7%). Tables for 8 and 9-gons with these better realizations are given below.

One interesting fact is that 14 minimal realizations for P1 obtained by M. Wang for 9-gons have not integral coordinates. For example, the 9-gon with genetic code <94321,9632,964,9721,983> has minimal realization amin=(2,2,3,5,6.5,6.5,8,9,17) with perimeter 59 (our solution was (2,2,3,5,7,7,9,10,18) with perimeter 63). This destroys our conjecture 4.1.(c) (see below for the status of Conjectures 4.1).  Note that all these 14 examples have Z[1/2]-coordinates and integral perimeter. For d=3, where polygon spaces carry a symplectic class ω, this implies that the de Rham cohomology class of ω is integral, see Proposition 6.5 of our paper with A. Knutson The cohomology ring of polygon spaces. Annales de l'Institut Fourier 48 (1998) 281--321, pdf file (By the way, in the published version of this paper, the formula in the proposition has a parasite equal sign after the sign +; such unwanted equal signs unfortunately occur in several places, but not in the above pdf-version).
• In the first line of the table in our Example 6.9, the genetic code of (3,3,3,4,4,5,5) should be <754,763> (instead of <754,762>).
• The geometric description of the 6-gon spaces (Table 6) has been much improved, see Table C in Geometric descriptions of polygon and chain spaces (by J-C. Hausmann), math.GT/0702521. In Topology and Robotics, American Math. Soc. Contemporary Mathematics 438 (2007) 47-57.
• M. Wang has listed the virtual 10-gons. They are 1'214'554'343 in number. Dirk Schütz obtained the same number with another algorithm and found that 52'980'624 (4.3%) are realizable as length vectors (April 2012). M. Wang has confirmed this number and found the number ov virtual 11-gons: 1'706'241'214'185'942 (1.7 x 1015) (May 2012).
• The number of n-gons and of virtual n-gons coincide with number of some classes of Boolean functions of of majority games, see arXiv:1501.07553 [math.GT] .
• Due to counting problems, the second line of the table p. 33 (|Str(R^m_\nearrow)|) contains wrong numbers for m=6,7,8. This was discovered by Lyle Ramshaw who computed the right numbers. These are:

 m \$|Str^m_\nearrow|\$ 3 4 5 6 7 8 9 3 7 21 118 1546 62236 ?

• TABLES :

• 4-gons text file (in the paper)
• 5-gons text file (in the paper)
• 6-gons text file (in the paper)
• 7-gons pdf-file, text file
• 8-gons (obtained by Minfeng Wang) text file
• 9-gons (obtained by Minfeng Wang) text file (12 MB)
• The 131 monogenic virtual genetic codes for 9-gons pdf-file
• appendix: Cuts pdf-file

STATUS OF CONJECTURES 4.1 (p. 38):

As said above, Conjecture 4.1.(c) is destroyed by the examples of Minfeng Wang. Conjecture 4.1.(a) is also false. It is equivalent to the following statement: Any stratum, up to permutation, contains a unique length vector bmin=(b1, ... , bn) (possibly a conventional representative) with integral coordinates and minimal perimeter. A counterexample with n=12 is given by the length vectors (1,3,5,6,8,11,12,23,28,31,31,38) and (1,3,5,7,8,11,12,23,28,31,31,37), which belong to the same chamber and have both minimal perimeter (137). This counterexample was actually found in 1959 by John R. Isbell and appeared in his paper ''On the enumeration of majority games'', Math. Comp. 13 (1959) 21-28 (pdf-file).

Conjecture 4.1.(b) has been solved positively by Matthias Franz. More precisely:

Proposition. A length vector bmin as in Conjecture 4.1.(a) is generic if and only if its perimeter is an odd integer.

Proof. Obviously, an integer vector with odd perimeter is generic. It remains to show that a generic vector v with integral coordinates is not of minimal perimeter if the latter is even. Observe that the perimeter of v is the dot product of v with the vector (1,..., 1). This is congruent Mod 2 to the dot product v⋅h with any vector h = (+/-1, ..., +/-1) orthogonal to an hyperplane delimiting the chamber. Since v is generic with even coordinate sum, v⋅h must be non-zero and even. Decreasing any non-zero coordinate of v by 1 changes each dot product v⋅h only by +/-1. So this gives another vector with integral coordinates in the same chamber, but with smaller perimeter.

PROGRAMS: