Multipole analysis in cosmic topology

Low multipole amplitudes in the Cosmic Microwave Background CMB radiation can be explained by selection rules from the underlying multiply-connected homotopy. We apply a multipole analysis to the harmonic bases and introduce point symmetry.We give explicit results for two cubic 3-spherical manifolds and lowest polynomial degrees, and derive three new spherical 3-manifolds.Comment: 15 pages, with figure

Topology of Platonic Spherical Manifolds: From Homotopy to Harmonic Analysis

We carry out the harmonic analysis on four Platonic spherical three-manifolds with different topologies. Starting out from the homotopies (Everitt 2004), we convert them into deck operations, acting on the simply connected three-sphere as the cover, and obtain the corresponding variety of deck groups. For each topology, the three-sphere is tiled into copies of a fundamental domain under the corresponding deck group. We employ the point symmetry of each Platonic manifold to construct its fundamental domain as a spherical orbifold. While the three-sphere supports an~orthonormal complete basis for harmonic analysis formed by Wigner polynomials, a given spherical orbifold leads to a selection of a specific subbasis. The resulting selection rules find applications in cosmic topology, probed by the cosmic microwave background.Comment: 29 pages, 4 figure

Commentary: From 'sense of number' to 'sense of magnitude' - The role of continuous magnitudes in numerical cognition

Unlike abstract ones in mathematics, concrete sets of elements in the real world have continuous physical properties, such as overall area and density. The dominant view has it that humans can estimate the discrete numerosities of such sets independently of the co-varying continuous magnitudes; i.e., that humans have a “sense of number”. It has indeed been claimed that various animals, ranging from monkeys to tiny fish, have this sense too. A recent paper by Leibovich et al. (2016) questions all of this (see also Gebuis et al., 2016; Morgan et al., 2014) and argues convincingly that numerosity estimation is not independent from continuous magnitudes but relies on them; that we have not a “sense of number” but a “sense of magnitude”. Yet the authors fail to cite a classic article that made the very same argument 25 years ago, and—unlike Leibovich et al.—supported it with a quantitative model (Allik and Tuulmets, 1991). Although neither density, nor overall area, nor any other single continuous magnitude can provide reliable information about numerosity, Leibovich et al. imply that all of them together can; they suggest that “statistical learning” will take care of extracting this information and turn numerosity estimates out of it. How statistical learning achieves this feat and whether the resulting numerosity estimates will fit observed ones remains, unfortunately, unclear. Allik and Tuulmets’s alternative “occupancy” model has its limits (e.g., Bertamini et al., 2016; Kramer et al., 2011) but it is specific, it is quantitative, and it predicts observed numerosity estimation surprisingly well with just a single free parameter