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In a recent paper, Gurzadyan & Penrose claim to have found directions in the sky around which there are multiple concentric sets of annuli with anomalously low variance in the cosmic microwave background (CMB). These features are presented as evidence for a particular theory of the pre-Big Bang Universe. We are able to reproduce the analysis these authors presented for data from the WMAP satellite and we confirm the existence of these apparently special directions in the newer Planck data. However, we also find that these features are present at the same level of abundance in simulated Gaussian CMB skies, i.e. they are entirely consistent with the predictions of the standard cosmological model., Comment: 10 pages, 5 figures; changed figure 1, revised asymmetry argument in Section 2, added additional information in Sections 2 and 3; v3 fixed typos