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A quasiconformal tree T is a (compact) metric tree that is doubling and of bounded turning. We call T trivalent if every branch point of T has exactly three branches. If the set of branch points is uniformly relatively separated and uniformly relatively dense, we say that T is uniformly branching. We prove that a metric space T is quasisymmetrically equivalent to the continuum self-similar tree if and only if it is a trivalent quasiconformal tree that is uniformly branching. In particular, any two trees of this type are quasisymmetrically equivalent. 

Graphical abstract [Formula: see text] 

Abstract We show that if an expanding Thurston map is the quotient of a torus endomorphism, then it has a parabolic orbifold and is a Lattès-type map. 

Abstract Imaging of complex (biological) samples in the near-infrared (NIR) is beneficial due to reduced light scattering, absorption, phototoxicity, and autofluorescence. However, there are few NIR fluorescent materials known and suitable for biomedical applications. Here we exfoliate the layered pigment CaCuSi 4 O 10 (Egyptian Blue, EB) via ball milling and facile tip sonication into NIR fluorescent nanosheets (EB-NS). The size of EB-NS can be tailored to diameters <20 nm and heights down to 1 nm. EB-NS fluoresce at 910 nm and the fluorescence intensity correlates with the number of Cu 2+ ions. Furthermore, EB-NS display no bleaching and high brightness compared with other NIR fluorophores. The versatility of EB-NS is demonstrated by in-vivo single-particle tracking and microrheology measurements in Drosophila melanogaster embryos. EB-NS can be uptaken by plants and remotely detected in a low-cost stand-off detection setup. In summary, EB-NS have the potential for a wide range of bioimaging applications.