Abstract: Generalized Möbius-Listing surfaces and bodies generalize Möbius bands, and this research was motivated originally by solutions of boundary value problems. Analogous to cutting of the original Möbius band, for this class of surfaces and bodies, results have been obtained when cutting such bodies or surfaces. In general, cutting leads to interlinked and intertwined different surfaces or bodies, resulting in very complex systems. However, under certain conditions, the result of cutting can be a single surface or body, which reduces complexity considerably. These conditions are based on congruence and rotational symmetry of the resulting cross sections after cutting, and on the knife cutting the origin

Abstract: Gielis transformations, with their origin in botany, are used to define square waves and trigonometric functions of higher order. They are rewritten in terms of Chebyshev polynomials. The origin of both, a uniform descriptor and the origin of orthogonal polynomials, can be traced back to a letter of Guido Grandi to Leibniz in 1713 on the mathematical description of the shape of flowers. In this way geometrical description and analytical tools are seamlessly combined.

Abstract: Our scientific and technological worldviews are largely dominated by the concepts of entropy and complexity. Originating in 19th-century thermodynamics, the concept of entropy merged with information in the last century, leading to definitions of entropy and complexity by Kolmogorov, Shannon and others. In its simplest form, this worldview is an application of the normal rules of arithmetic. In this worldview, when tossing a coin, a million heads or tails in a row is theoretically possible, but impossible in practice and in real life. On this basis, the impossible (in the binary case, the outermost entries of Pascal's triangle xn and yn for large values of n) can be safely neglected, and one can concentrate fully on what is common and what conforms to the law of large numbers, in fields ranging from physics to sociology and everything in between. However, in recent decades it has been shown that what is most improbable tends to be the rule in nature. Indeed, if one combines the outermost entries xn and yn with the normal rules of arithmetic, either addition or multiplication, one obtains Lamé curves and power laws respectively. In this article, some of these correspondences are highlighted, leading to a double conclusion. First, Gabriel Lamé's geometric footprint in mathematics and the sciences is enormous. Second, conic sections are at the core once more. Whereas mathematics so far has been exclusively the language of patterns in the sciences, the door is opened for mathematics to also become the language of the individual. The probabilistic worldview and Lamé's footprint can be seen as dual methods. In this context, it is to be expected that the notions of information, complexity, simplicity and redundancy benefit from this different viewpoint.

Abstract: abstract not available

Abstract: This chapter explains the materials and techniques employed in the Amsterdam Sunflowers, enabling a comparison with the London version described in chapter 3. Building upon the 2016 article published in the National Gallery Technical Bulletin, it incorporates the latest findings gained by computer-assisted methods used to characterize the canvas support, as well as in-situ campaigns of non-invasive investigation together with further analysis of microscopic paint samples. The chapter sequence follows the steps in Van Gogh's working practice. Starting with the canvas, automated analysis of the weave enables the provenance of the canvas to be traced back to a particular roll of linen ordered by Van Gogh. Combining technical evidence with knowledge of historical manufacturing techniques further allows us to reconstruct the way in which Van Gogh divided his canvas roll into pieces used for Sunflowers and other paintings. We go on to consider how, with the original painting at hand, he used charcoal to transfer the motif of the London Sunflowers onto his blank canvas. Despite careful planning of the composition, an adjustment was required late in the working process, when Van Gogh added a painted wooden strip to extend the background above the flower at the top edge of the canvas. The artist's process of working up the composition in paint is described, paying special attention to his use of colour. The pigments and pigment mixtures used in the Amsterdam Sunflowers have been comprehensively mapped and are compared with the London picture, with discussion of some similarities and differences that account for the distinctive colour scheme of each painting. This understanding of colour application in the Amsterdam Sunflowers lays the foundation for subsequent chapters that will go on to consider the impact of light-induced colour changes that have taken place over time, and the related need to define appropriate lighting guidelines for the future safe preservation of this painting and others made with similar materials (chapters 5 and 7).