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Map Folding
In the mathematics of paper folding, map folding and stamp folding are two problems of counting the number of ways that a piece of paper can be folded. In the stamp folding problem, the paper is a strip of stamps with creases between them, and the folds must lie on the creases. In the map folding problem, the paper is a map, divided by creases into rectangles, and the folds must again lie only along these creases. credits the invention of the stamp folding problem to Émile Lemoine. provides several other early references. Labeled stamps In the stamp folding problem, the paper to be folded is a strip of square or rectangular stamps, separated by creases, and the stamps can only be folded along those creases. In one commonly considered version of the problem, each stamp is considered to be distinguishable from each other stamp, so two foldings of a strip of stamps are considered equivalent only when they have the same vertical sequence of stamps. For example, there are six ways t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics Of Paper Folding
The discipline of origami or paper folding has received a considerable amount of mathematical study. Fields of interest include a given paper model's flat-foldability (whether the model can be flattened without damaging it), and the use of paper folds to solve mathematical equations up to the third order. Computational origami is a recent branch of computer science that is concerned with studying algorithms that solve paper-folding problems. The field of computational origami has also grown significantly since its inception in the 1990s with Robert Lang's TreeMaker algorithm to assist in the precise folding of bases. Computational origami results either address origami design or origami foldability."Lecture: Recent Results in Computational Origami". ''Origami USA: We are the American national society devoted to origami, the art of paperfolding''. Retrieved 2022-05-08. In origami design problems, the goal is to design an object that can be folded out of paper given a specific targ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carpenter's Rule Problem
The carpenter's rule problem is a discrete geometry problem, which can be stated in the following manner: ''Can a simple planar polygon be moved continuously to a position where all its vertices are in convex position, so that the edge lengths and simplicity are preserved along the way?'' A closely related problem is to show that any non-self-crossing polygonal chain can be straightened, again by a continuous transformation that preserves edge distances and avoids crossings. Both problems were successfully solved by . The problem is named after the multiple-jointed wooden rulers popular among carpenters in the 19th and early 20th centuries before improvements to metal tape measures made them obsolete. Combinatorial proof Subsequently, to their work, Ileana Streinu provided a simplified combinatorial proof formulated in the terminology of robot arm motion planning. Both the original proof and Streinu's proof work by finding non-expansive motions of the input, continuous transform ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Paperfolding Sequence
In mathematics the regular paperfolding sequence, also known as the dragon curve sequence, is an infinite sequence of 0s and 1s. It is obtained from the repeating partial sequence by filling in the question marks by another copy of the whole sequence. The first few terms of the resulting sequence are: If a strip of paper is folded repeatedly in half in the same direction, i times, it will get 2^i-1 folds, whose direction (left or right) is given by the pattern of 0's and 1's in the first 2^i-1 terms of the regular paperfolding sequence. Opening out each fold to create a right-angled corner (or, equivalently, making a sequence of left and right turns through a regular grid, following the pattern of the paperfolding sequence) produces a sequence of polygonal chains that approaches the dragon curve fractal: Properties The value of any given term t_n in the regular paperfolding sequence, starting with n=1, can be found recursively as follows. Divide n by two, as many times as possi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theoretical Computer Science (journal)
''Theoretical Computer Science'' (''TCS'') is a computer science journal published by Elsevier, started in 1975 and covering theoretical computer science. The journal publishes 52 issues a year. It is abstracted and indexed by Scopus and the Science Citation Index. According to the Journal Citation Reports, its 2020 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a type of journal ranking. Journals with higher impact factor values are considered more prestigious or important within their field. The Impact Factor of a journa ... is 0.827. References Computer science journals Elsevier academic journals Academic journals established in 1975 {{comp-sci-theory-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
NP-hard
In computational complexity theory, a computational problem ''H'' is called NP-hard if, for every problem ''L'' which can be solved in non-deterministic polynomial-time, there is a polynomial-time reduction from ''L'' to ''H''. That is, assuming a solution for ''H'' takes 1 unit time, ''H''s solution can be used to solve ''L'' in polynomial time. As a consequence, finding a polynomial time algorithm to solve a single NP-hard problem would give polynomial time algorithms for all the problems in the complexity class NP. As it is suspected, but unproven, that P≠NP, it is unlikely that any polynomial-time algorithms for NP-hard problems exist. A simple example of an NP-hard problem is the subset sum problem. Informally, if ''H'' is NP-hard, then it is at least as difficult to solve as the problems in NP. However, the opposite direction is not true: some problems are undecidable, and therefore even more difficult to solve than all problems in NP, but they are probably not NP- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theory And Applications
A theory is a systematic and rational form of abstract thinking about a phenomenon, or the conclusions derived from such thinking. It involves contemplative and logical reasoning, often supported by processes such as observation, experimentation, and research. Theories can be scientific, falling within the realm of empirical and testable knowledge, or they may belong to non-scientific disciplines, such as philosophy, art, or sociology. In some cases, theories may exist independently of any formal discipline. In modern science, the term "theory" refers to scientific theories, a well-confirmed type of explanation of nature, made in a way consistent with the scientific method, and fulfilling the criteria required by modern science. Such theories are described in such a way that scientific tests should be able to provide empirical support for it, or empirical contradiction ("falsify") of it. Scientific theories are the most reliable, rigorous, and comprehensive form of scientific know ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
NP-complete
In computational complexity theory, NP-complete problems are the hardest of the problems to which ''solutions'' can be verified ''quickly''. Somewhat more precisely, a problem is NP-complete when: # It is a decision problem, meaning that for any input to the problem, the output is either "yes" or "no". # When the answer is "yes", this can be demonstrated through the existence of a short (polynomial length) ''solution''. # The correctness of each solution can be verified quickly (namely, in polynomial time) and a brute-force search algorithm can find a solution by trying all possible solutions. # The problem can be used to simulate every other problem for which we can verify quickly that a solution is correct. Hence, if we could find solutions of some NP-complete problem quickly, we could quickly find the solutions of every other problem to which a given solution can be easily verified. The name "NP-complete" is short for "nondeterministic polynomial-time complete". In this name, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial Time
In theoretical computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by the algorithm, supposing that each elementary operation takes a fixed amount of time to perform. Thus, the amount of time taken and the number of elementary operations performed by the algorithm are taken to be related by a constant factor. Since an algorithm's running time may vary among different inputs of the same size, one commonly considers the worst-case time complexity, which is the maximum amount of time required for inputs of a given size. Less common, and usually specified explicitly, is the average-case complexity, which is the average of the time taken on inputs of a given size (this makes sense because there are only a finite number of possible inputs of a given size). In both cases, the time complexity is gener ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Valley Fold
A valley is an elongated low area often running between hills or mountains and typically containing a river or stream running from one end to the other. Most valleys are formed by erosion of the land surface by rivers or streams over a very long period. Some valleys are formed through erosion by glacial ice. These glaciers may remain present in valleys in high mountains or polar areas. At lower latitudes and altitudes, these glacially formed valleys may have been created or enlarged during ice ages but now are ice-free and occupied by streams or rivers. In desert areas, valleys may be entirely dry or carry a watercourse only rarely. In areas of limestone bedrock, dry valleys may also result from drainage now taking place underground rather than at the surface. Rift valleys arise principally from earth movements, rather than erosion. Many different types of valleys are described by geographers, using terms that may be global in use or else applied only locally. Format ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mountain Fold
A mountain is an elevated portion of the Earth's crust, generally with steep sides that show significant exposed bedrock. Although definitions vary, a mountain may differ from a plateau in having a limited summit area, and is usually higher than a hill, typically rising at least above the surrounding land. A few mountains are isolated summits, but most occur in mountain ranges. Mountains are formed through tectonic forces, erosion, or volcanism, which act on time scales of up to tens of millions of years. Once mountain building ceases, mountains are slowly leveled through the action of weathering, through slumping and other forms of mass wasting, as well as through erosion by rivers and glaciers. High elevations on mountains produce colder climates than at sea level at similar latitude. These colder climates strongly affect the ecosystems of mountains: different elevations have different plants and animals. Because of the less hospitable terrain and climate, mountains te ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics Of Origami
The discipline of origami or paper folding has received a considerable amount of mathematical study. Fields of interest include a given paper model's flat-foldability (whether the model can be flattened without damaging it), and the use of paper folds to solve mathematical equations up to the third order. Computational origami is a recent branch of computer science that is concerned with studying algorithms that solve paper-folding problems. The field of computational origami has also grown significantly since its inception in the 1990s with Robert Lang's TreeMaker algorithm to assist in the precise folding of bases. Computational origami results either address origami design or origami foldability."Lecture: Recent Results in Computational Origami". ''Origami USA: We are the American national society devoted to origami, the art of paperfolding''. Retrieved 2022-05-08. In origami design problems, the goal is to design an object that can be folded out of paper given a specific targe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
