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Source: Computational Materials Science, Volume 267
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The key difference from Euclidean geometry is that a circle of radius \(r\) has circumference \(2\pi\sinh r 2\pi r\), and volumes grow exponentially rather than polynomially: \[\operatorname{Vol}(B(r)) = \operatorname{Vol}(S^{n-1})\int_0^r \sinh^{n-1}(t)\,dt.\]
Source: Computational Materials Science, Volume 267