These days I hardly fold any origami, but when I do, it's usually from either of the books Origami Art or Advanced Origami, both by Michael G. Lafosse & Richard L. Alexander from their Origamido Studio. Their designs have this elegance and beauty that attract me, and leaves me satisfied when I am able to reproduce part of that beauty with my own hands. So when I offered to fold a present for a friend, I asked him to choose from those books. He picked their Humpback Whale. Elephanthide paper, wet-folded
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I improvised this design in class using an envelope that was lying around. I liked being able to use the flap of the envelope as part of the nun's gown - so I was using the envelope "fully" and not simply as a rectangular sheet of paper! Thinking back, I was probably inspired by Vietnamese origami master Giang Dinh's evocative abstract human figures: You must check out his achingly beautiful origami faces, animals and figures. I saw his gorgeous work in a book of photographs from an origami exhibition, titled Masters of Origami: At Hanger-7.
One of my earliest designs, born from a brainwave on creating hexagonal cells from a square grid: One uncut square, designed in 2007.
An early design of mine that arose as I played around with the paper. One uncut square, designed between 2000 to 2007.
Look out for the origami event of the year! Origami Singapore together with NLB is organising an exhibition “Origami : From Traditional to Modern”, which will be a roving exhibition travelling to 3 public libraries. See traditional models as well as the latest complex origami models from around the world including fantasy creatures, animals, insects and geometrical models. More than 200 models will be on display, including the largest collection of models by local designers so far. Come and learn how to fold your own origami models at the public workshops too!
[1.5] Ultrafilters, non-standard analysis, and epsilon managementIn the previous article we saw that fruitful analogies between finitary and infinitary mathematics can allow the techniques of one to shed light on the other. Here we borrow the power of infinitary math—in particular, ultrafilters and non-standard analysis—to simplify proofs of finitary statements.
Prerequisites Readers need some familiarity with sentences like "for every $\varepsilon$ there exists a large $N$ such that..." or its symbolic equivalent, "$\forall\varepsilon: \exists N: \dotsc$". A rough idea of big O notation would also be helpful but is not necessary. A brief introduction: we say that $f(x) = O(g(x))$ (as $x \to \infty$) to mean that the "growth" of $f(x)$ is "bounded above" by $g(x)$; more formally, there exist positive real numbers $M$ and $x_0$ such that $\lvert f(x)\rvert \leq M\lvert g(x)\rvert$ for all $x > x_0$. Note that $\lvert x\rvert$ denotes the absolute value of $x$, not its cardinality as indicated in Part 4. [1.3] Soft analysis, hard analysis, and the finite convergence principle Analysis (something like an advanced calculus) is often differentiated into "hard analysis" ("quantitative", "finitary") and "soft analysis" ("qualitative", "infinitary"). Discrete math, computer science, and analytic number theory normally uses hard analysis while operator algebra, abstract harmonic analysis, and ergodic theory tend to rely on soft analysis. The field of partial differential equations uses techniques from both. Convenient notation (e.g. $O(\:)$) from qualitative analysis can conceal gritty details from quantitative and argue efficiently from the big picture, at the cost of a precise description. Conversely, quantitative analysis can be seen as a more precise and detailed refinement of qualitative analysis. The intuitions, methods and results in hard analysis often have analogues in soft analysis and vice versa, despite their contrasting language. Tao argues this technique transfer can benefit both disciplines. Table 5 features a rough "dictionary" between the notational languages of soft and hard analysis. Kudos to Tao for such an illuminating comparison! Table 5: "Translating" soft analysis to hard analysis [1.3]
[1.10] The crossing number inequalityA graph $G = (V, E)$ is a set $V$ of vertices ("objects") and a set $E$ of edges ("relationships between two objects"). For example, $V$ could be a set of people and $E$ the set of friendships (Facebook really uses this social graph to analyze user behavior). $V$ could also be the set of airports and $E$ the set of flights from one airport to another. The immense flexibility of this definition allow graphs to model and analyze a tremendous variety of real-life situations, but here we are interested in the abstract representation of a graph, in particular its drawing. The focus is on applying the technique of amplification, which was presented in Part 3 of this series. A drawing of graph $G = (V, E)$ simply draws the vertices in $V$ as dots on the plane, and the edges $E$ as lines (or curves) connecting them. $G$ can have many possible drawings, in which the edges can have different numbers of crossings between pairs of edges (i.e. three concurrent edges have 3 crossings). Fig. 4(a)-(c) features three drawings of the $K_{3, 3}$ graph with different numbers of crossings. |
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