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Discrete differential geometry is an active mathematical terrain where differential geometry and discrete geometry meet and interact. This article discusses the beautiful tale of how discrete differential geometry is linked to modern approaches to computational design for architecture, as well as fabrication and rationalization of freeform designs. B02 discrete multidimensional integrable systems and b07 lagrangian multiform structure and multisymplectic discrete systems. Rather unexpectedly, the very basic structures of discrete differential geometry turn out to be related to the theory of integrable systems. Subdivision exterior calculus for geometry processing. A new field of discrete differential geometry is presently emerging on the border between differential and discrete geometry. This is one of the first books on a newly emerging field of discrete differential geometry and an excellent way to access this exciting area. Our approach adapts the numerical framework of discrete exterior calculus dec from the polygonal to the subdivision setting by exploiting the refinability of subdivision basis functions. One of the main goals of this book is to reveal this integrable structure of discrete differential geometry. It is now an outgrowth of two projects from the first funding period of crc. Books on differential and riemannian geometry whsmith. Discrete differential geometry graduate studies in. This paper introduces a new computational method to solve differential equations on subdivision surfaces.