[math-fun] Optimal polygonal pasture
For a fixed N, suppose we want to make a polygonal pasture with N sides, with the largest possible area given a fixed perimeter. Is the unique optimum always a regular N-gon? I am looking for a figure of merit that tells how regular a polygon is, and perimeter/area seems promising.
See Graham, R. L. (1975). The largest small hexagon. *Journal of Combinatorial Theory, Series A*, *18*(2), 165-170. The problem of determining the largest area a plane *hexagon* of unit diameter can have, raised some 20 years ago by H. Lenz, is settled. It is shown that such a *hexagon* is unique and has an area exceeding that of a regular *hexagon* of unit diameter by about 4 ... available here: https://ac.els-cdn.com/0097316575900047/1-s2.0-0097316575900047-main.pdf?_ti... Best regards Neil Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com On Fri, Sep 21, 2018 at 4:01 PM, Allan Wechsler <acwacw@gmail.com> wrote:
For a fixed N, suppose we want to make a polygonal pasture with N sides, with the largest possible area given a fixed perimeter. Is the unique optimum always a regular N-gon? I am looking for a figure of merit that tells how regular a polygon is, and perimeter/area seems promising. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
I put three dots after the 4, which you should have noticed Best regards Neil Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com On Fri, Sep 21, 2018 at 4:44 PM, Tomas Rokicki <rokicki@gmail.com> wrote:
an area exceeding that of a regular *hexagon* of unit diameter by about 4
This was shocking to me, absolutely shocking, until I realized somehow Neil omitted an important character.
It should be "by about 4%".
-tom _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
I went looking for this. here's a picture of the two hexagons from Graham's paper https://www.flickr.com/photos/thane/29893161697/in/dateposted-public/ On Fri, Sep 21, 2018 at 1:45 PM Tomas Rokicki <rokicki@gmail.com> wrote:
an area exceeding that of a regular *hexagon* of unit diameter by about 4
This was shocking to me, absolutely shocking, until I realized somehow Neil omitted an important character.
It should be "by about 4%".
-tom _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Thane Plambeck tplambeck@gmail.com http://counterwave.com/
Alan isn’t fixing the diameter, but the perimeter. For that case it’s easy to see why the regular N-gon has the maximum area. -Veit
On Sep 21, 2018, at 4:50 PM, Thane Plambeck <tplambeck@gmail.com> wrote:
I went looking for this. here's a picture of the two hexagons from Graham's paper
https://www.flickr.com/photos/thane/29893161697/in/dateposted-public/
On Fri, Sep 21, 2018 at 1:45 PM Tomas Rokicki <rokicki@gmail.com> wrote:
an area exceeding that of a regular *hexagon* of unit diameter by about 4
This was shocking to me, absolutely shocking, until I realized somehow Neil omitted an important character.
It should be "by about 4%".
-tom _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Thane Plambeck tplambeck@gmail.com http://counterwave.com/ _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (5)
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Allan Wechsler -
Neil Sloane -
Thane Plambeck -
Tomas Rokicki -
Veit Elser