linkedin post 2016-12-18 07:22:21

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VIOLATED LOGIC. "By the “logic” of space we mean those spatial relations among physical objects which are necessary, and which when violated result in visual paradoxes, sometimes called optical illusions. Escher understood that the geometry of space determines its logic, and likewise the logic of space often determines its geometry." https://lnkd.in/d_HZ4K8 View in LinkedIn
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linkedin post 2016-12-18 07:15:47

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THE PRINT GALLERY. "Note how the scale of the grid grows continuously in a clockwise direction. And note especially what this trick entails: A hole in the middle. A mathematician would call this a singularity, a place where the fabric of the space no longer holds together. There is just no way to knit this bizarre space into a seamless whole, and Escher, rather than try to obscure it in some way, has put his trademark initials smack in the center of it." https://lnkd.in/d_HZ4K8 View in LinkedIn
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linkedin post 2016-12-18 07:10:31

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DISTORTED TOPOGRAPHY. "In addition to Euclidean and non-Euclidean geometries, Escher was very interested in visual aspects of Topology, a branch of mathematics just coming into full flower during his lifetime. Topology concerns itself with those properties of a space which are unchanged by distortions which may stretch or bend it—but which do not tear or puncture it—and topologists were busy showing the world many strange objects." https://lnkd.in/d_HZ4K8 View in LinkedIn
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linkedin post 2016-12-18 07:04:51

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FIGURATIVE SPACE. "Among the most important of Escher’s works from a mathematical point of view are those dealing with the nature of space itself. His woodcut Three Intersecting Planes is a good place to begin a review of these works, for it exemplifies the artist’s concern with the dimensionality of space, and with the mind’s ability to discern three-dimensionality in a two-dimensional representation." https://lnkd.in/d_HZ4K8 View in LinkedIn
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linkedin post 2016-12-19 07:39:18

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TAPE OF LIFE. "Gould argued that contingency renders evolution fundamentally quirky and unpredictable, and he famously suggested that replaying the “tape of life” from some point in the distant past would yield a living world far different from the one we see today." http://www.pnas.org/content/105/23/7899.full View in LinkedIn
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linkedin post 2016-12-18 07:00:11

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STELLATIONS. "There are many interesting solids that may be obtained from the Platonic solids by intersecting them or stellating them. To stellate a solid means to replace each of its faces with a pyramid, that is, with a pointed solid having triangular faces; this transforms the polyhedron into a pointed, three-dimensional star. A beautiful example of a stellated dodecahedron may be found in Escher’s Contrast (Order and Chaos)." https://lnkd.in/d_HZ4K8 View in LinkedIn
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linkedin post 2016-12-21 06:51:19

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WHY SO SLOW? "A citrate-using (Cit+) variant finally evolved in one population by 31,500 generations, causing an increase in population size and diversity. The long-delayed and unique evolution of this function might indicate the involvement of some extremely rare mutation. Alternately, it may involve an ordinary mutation, but one whose physical occurrence or phenotypic expression is contingent on prior mutations in that population." http://www.pnas.org/content/105/23/7899.full View in LinkedIn
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linkedin post 2016-12-21 06:49:08

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SUDDEN APPEARANCE OF CITRATE EVOLUTION. "Here we report that a Cit+ variant finally evolved in one population by 31,500 generations, and its descendants later rose to numerical dominance." For 30,000 generations there were no citrate metabolizes evolving in this defined population. http://www.pnas.org/content/105/23/7899.full View in LinkedIn
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