Peano surface

The Peano Surface, a counterexample to some criteria of local maxima of the 19th century

In mathematics, the Peano surface is a real algebraic surface, the graph of the two-variable function

$f(x,y)=(2x^{2}-y)(y-x^{2}).$

It was proposed by Giuseppe Peano as a counterexample to several conjectured criteria for the existence of maxima and minima of surfaces.

Background

In 1886 Joseph Alfred Serret published a paper with a proposed criteria for the extremal points of a surface given by $z=f(x_{0}+h,y_{0}+k)$

"the maximum or the minimum takes place when for the values of $h$ and $k$ for which $d^{2}f$ and $d^{3}f$ (third and fourth terms) vanish, $d^{4}f$ (fifth term) has constantly the sign — , or the sign +."

It is assumed that the linear terms vanish and the Taylor series of $f$ has the form $z=f(x_{0},y_{0})+Q(h,k)+C(h,k)+F(h,k)+...$ where $Q(h,k)$ is a quadratic form like $ah^{2}+bhk+ck^{2}$ and $C(h,k)$ is a cubic form with cubic terms in $h,k$ and $F(h,k)$ is a quartic form with a homogeneous quartic polynomial in $h,k$. Serret proposes that if $F(h,k)$ has constant sign for all points where $Q(h,k)=C(h,k)=0$ then there is a local maximum or minimum of the surface at $(x_{0},y_{0})$.

Counterexample to Serret's conditions

Peano's example full example was $f(x,y)=(y^{2}-2px)(y^{2}-2qx)$ with $p>q>0$. At the point $x_{0}=0,y_{0}=0$ the local expansion of $f(0+h,0+k)$ is $z=4pqh^{2}-2(p+q)hk^{2}+k^{4}$. Here the quadratic form is $Q(h,k)=4pqk^{2}$, the cubic form is $C(h,k)=-2(p+q)hk^{2}$ and the quartic form is $F(h,k)=k^{4}$. The quadratic and cubic forms both vanish along $h=0$ (note in Emch its the set k=0 which is considered) and quartic form is positive along this set. So this surface meets Serret conditions. But the surface clearly does not reach a maximum at the origin by considering the curves $y^{2}=2lx$. The expansion becomes $z=((2l-2p)x)((2l-2q)x)$ which is positive when $l>p$ or $l<q$ but negative when $p>l>q$.

Intersection of Peano's surface with a plane $y=ax$ (a plane containing the z-axis) showing a maxima at the origin.

One feature of Peano's surface is that every plane through the z-axis, cuts the surface in a quartic curve with a maximum at the point (0, 0, 0). If you just considered the intersection with such planes then it would be natural to conclude that the point (0, 0, 0) was a local maximum.

Discoverer

Giuseppe Peano

According to https://fabpedigree.com/james/mathmen.htm#Peano , Peano's calculus textbook (Calcolo differenziale e integrale) was: "best calculus textbook of his time". and also that Peano: "He was the champion of counter-examples, and found flaws in published proofs of several important theorems."

see also: http://dolecki.perso.math.cnrs.fr/Peano=vulgarization_1307.pdf

Models

Models of Peano's surface have appeared in a number of collections of physical models.

Goettingen http://modellsammlung.uni-goettingen.de/index.php?r=7&sr=32&m=443&lang=en

Dresden http://www.math.tu-dresden.de/modellsammlung/karte.php?ID=314

http://www.math.tu-dresden.de/modellsammlung/karte.php?ID=341

Mathematical Models: Photograph Volume and Commentary (Advanced Lectures in Mathematics Series) Hardcover – December 1, 1986 by Gerd Fischer ISBN-13: 978-3528089917

Not sure if it features in Schilling catalog yet. Schilling, Martin (undefined). Catalog mathematischer Modelle für den höheren mathematischen Unterricht veröffentlicht durch die Verlagshandlung. Halle a. S – via Internet Archive. {{cite book}}: Check date values in: |date= (help)

External links

• Weisstein, Eric W. "Peano Surface". MathWorld.
• https://archive.org/search.php?query=subject%3A%22Giuseppe+Peano%22

Category:Calculus Category:Surfaces Category:Paradoxes Category:Mathematical optimization