(real projective space $\,\mathbb{R}P^1$)
quaternionic projective space$\,\mathbb{H}P^1$
In ancient times, Musica Universalis or Music of the Spheres referred to a philosophical concept that regards proportions of orbits in the solar system as exhibiting a cosmic harmony (Plato 380BC).
Johannes Kepler proposed that this is governed by the shape of the Platonic solids (Kepler 1596). (See also at multiverse.)
More recently, it was suggested that the term refers to the stable homotopy groups of spheres $\pi_\bullet(\mathbb{S})$ and the absolute E-β geometry over Spec(S) as seen by the MU-Adams spectral sequence.
My initial inclination was to call this book The Music of the Spheres, but I was dissuaded from doing so by my diligent publisher, who is ever mindful of the sensibilities of librarians. (Ravenel 86, preface)
Modern chromatic homotopy theory makes contact between this kind of music and physics:
the A-hat genus (partition function of the spinning particle) appears in the first chromatic level of the spheres;
the Witten genus (partition function of the spinning string) appears in the second chromatic level of the spheres.
Modern equivariant stable homotopy theory makes contact between this kind of music and Keplerβs proposal:
For whatever itβs worth, all these items come together in the Equivariant cohomology of M2/M5-branes
Wikipedia, Musica universalis
Doug Ravenel, Complex cobordism and stable homotopy groups of spheres, 1986/2003
Last revised on September 10, 2020 at 10:17:33. See the history of this page for a list of all contributions to it.