id: 05957915
dt: j
an: 2012a.00555
au: Langer, Joel C.; Singer, David A.
ti: Reflections on the lemniscate of Bernoulli: the forty-eight faces of a
mathematical gem.
so: Milan J. Math. 78, No. 2, 643-682 (2010).
py: 2010
pu: Springer (Birkhäuser), Basel
la: EN
cc: H75 G75 G45 G55
ut: lemniscate; octahedral symmetry; geodesic triangulation; disdyakis
dodecahedron
ci:
li: doi:10.1007/s00032-010-0124-5
ab: This article provides an extensive study of Bernoulli’s lemniscate and
its symmetries as curve in complex projective space. It can be viewed
as an immersed copy of the Riemann sphere on which the rotational
octahedral group $\cal O$ and the full octahedral group $\hat\cal O$
act. The octahedral vertices correspond to the three complex double
points, the edges are the fix-point curves of those antiholomorphic
reflections that also fix four of the vertices. The $\hat\cal O$-orbit
of the real lemniscate consists of six curves which triangulate the
lemniscate as {\it tetrakis hexahedron.} Adding the octahedral edges
produces a triangulation as {\it disdyakis dodecahedron.} The
lemniscate can also be viewed as a Riemann surface with $\cal
O$-invariant metric. Here, the disdyakis dodecahedron triangulation is
produced by simple closed geodesics. The vertex orbits with respect to
$\hat\cal O$ have lengths six, eight, and twelve and correspond to
critical points of the Gaussian curvature~$K$. Short and concise proofs
of these results are not the author’s aim. They rather take the
reader on a leisurely tour through many different fields such as
complex mappings (inversion, stereographic projection, Joukowski maps,
Schwarzian reflection), linkages, groups, Lagrangian subspaces, and
quadrics. Mathematical truth and proofs evolve at a slow pace but leave
a deep impression. The text is accompanied by attractive and
informative pictures that nicely illustrate the lemniscate’s
symmetries in different models. Source code for {\tt Mathematica}
animations is given in an appendix.
rv: Hans-Peter Schröcker (Innsbruck)