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Tammes Problem
In geometry, Tammes problem
Tammes problem
is a problem in packing a given number of circles on the surface of a sphere such that the minimum distance between circles is maximized. It is named after a Dutch botanist who posed the problem in 1930 while studying the distribution of pores on pollen grains. It can be viewed as a specialization of the generalized Thomson problem. See also[edit]Spherical code Kissing number problemBibliography[edit]Journal articlesTammes PML (1930). "On the origin of number and arrangement of the places of exit on pollen grains". Diss. Groningen.  Tarnai T; Gáspár Zs (1987). "Multi-symmetric close packings of equal spheres on the spherical surface". Acta Crystallographica. A43: 612–616. doi:10.1107/S0108767387098842.  Erber T, Hockney GM (1991). "Equilibrium configurations of N equal charges on a sphere" (PDF). Journal of Physics A: Mathematical and General. 24: Ll369–Ll377
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Geometry
Geometry
Geometry
(from the Ancient Greek: γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer. Geometry
Geometry
arose independently in a number of early cultures as a practical way for dealing with lengths, areas, and volumes
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ArXiv
arXiv (pronounced "archive")[2] is a repository of electronic preprints (known as e-prints) approved for publication after moderation, that consists of scientific papers in the fields of mathematics, physics, astronomy, computer science, quantitative biology, statistics, and quantitative finance, which can be accessed online. In many fields of mathematics and physics, almost all scientific papers are self-archived on the arXiv repository. Begun on August 14, 1991, arXiv.org passed the half-million article milestone on October 3, 2008,[3][4] and hit a million by the end of 2014.[5][6] By October 2016 the submission rate had grown to more than 10,000 per month.[6][7]Contents1 History 2 Peer review 3 Submission formats 4 Access 5 Copyright status of files 6 Controversy 7 See also 8 Notes 9 References 10 External linksHistory[edit]A screenshot of the arXiv taken in 1994,[8] using the browser NCSA Mosaic
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Torus
In geometry, a torus (plural tori) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis coplanar with the circle. If the axis of revolution does not touch the circle, the surface has a ring shape and is called a torus of revolution. Real-world examples of toroidal objects include inner tubes. A torus should not be confused with a solid torus, which is formed by rotating a disc, rather than a circle, around an axis. A solid torus is a torus plus the volume inside the torus. Real-world approximations include doughnuts, many lifebuoys, and O-rings. In topology, a ring torus is homeomorphic to the Cartesian product
Cartesian product
of two circles: S1 × S1, and the latter is taken to be the definition in that context. It is a compact 2-manifold of genus 1
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Microsoft PowerPoint
Microsoft
Microsoft
PowerPoint is a presentation program,[4] created by Robert Gaskins and Dennis Austin[4] at a software company named Forethought, Inc.[4] It was released on April 20, 1987,[5] initially for Macintosh computers only.[4] Microsoft
Microsoft
acquired PowerPoint for $14 million three months after it appeared.[6] This was Microsoft's first significant acquisition,[7] and Microsoft
Microsoft
set up a new business unit for PowerPoint in Silicon Valley
Silicon Valley
where Forethought had been located.[7] PowerPoint became a component of the Microsoft
Microsoft
Office suite, first offered in 1989 for Macintosh[8] and in 1990 for Windows,[9] which bundled several Microsoft
Microsoft
apps
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Portable Document Format
The Portable Document
Document
Format (PDF) is a file format developed in the 1990s to present documents, including text formatting and images, in a manner independent of application software, hardware, and operating systems.[3][4] Based on the PostScript
PostScript
language, each PDF file encapsulates a complete description of a fixed-layout flat document, including the text, fonts, vector graphics, raster images and other information needed to display it
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International Standard Book Number
"ISBN" redirects here. For other uses, see ISBN (other).International Standard Book
Book
NumberA 13-digit ISBN, 978-3-16-148410-0, as represented by an EAN-13 bar codeAcronym ISBNIntroduced 1970; 48 years ago (1970)Managing organisation International ISBN AgencyNo. of digits 13 (formerly 10)Check digit Weighted sumExample 978-3-16-148410-0Website www.isbn-international.orgThe International Standard Book
Book
Number (ISBN) is a unique[a][b] numeric commercial book identifier. Publishers purchase ISBNs from an affiliate of the International ISBN Agency.[1] An ISBN is assigned to each edition and variation (except reprintings) of a book. For example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, and 10 digits long if assigned before 2007
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Special
Special
Special
or specials may refer to:Contents1 Music 2 Film and television 3 Other uses 4 See alsoMusic[edit] Special
Special
(album), a 1992
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Bibcode
The bibcode (also known as the refcode) is a compact identifier used by several astronomical data systems to uniquely specify literature references.Contents1 Adoption 2 Format 3 Examples 4 See also 5 ReferencesAdoption[edit] The Bibliographic Reference Code (refcode) was originally developed to be used in SIMBAD
SIMBAD
and the NASA/IPAC Extragalactic Database
NASA/IPAC Extragalactic Database
(NED), but it became a de facto standard and is now used more widely, for example, by the NASA Astrophysics Data System
Astrophysics Data System
who coined and prefer the term "bibcode".[1][2] Format[edit] The code has a fixed length of 19 characters and has the form YYYYJJJJJVVVVMPPPPA where YYYY is the four-digit year of the reference and JJJJJ is a code indicating where the reference was published
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Proceedings Of The Royal Society A
Proceedings of the Royal Society
Royal Society
is the parent title of two scientific journals published by the Royal Society. Originally a single journal, it was split into two separate journals in 1905:Series A: which publishes research related to mathematical, physical, and engineering sciences Series B: which publishes research related to biologyThe two journals are the Royal Society's main research journals. Many celebrated names in science have published their research in the Proceedings of the Royal Society, including Paul Dirac,[1] Werner Heisenberg,[2] Ernest Rutherford,[3] and Erwin Schrödinger.[4] All articles are available free[citation needed] at the journals' websites after one year for Proceedings B and two years for Proceedings A
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Digital Object Identifier
In computing, a Digital Object Identifier or DOI is a persistent identifier or handle used to uniquely identify objects, standardized by the International Organization for Standardization
International Organization for Standardization
(ISO).[1] An implementation of the Handle System,[2][3] DOIs are in wide use mainly to identify academic, professional, and government information, such as journal articles, research reports and data sets, and official publications though they also have been used to identify other types of information resources, such as commercial videos. A DOI aims to be "resolvable", usually to some form of access to the information object to which the DOI refers. This is achieved by binding the DOI to metadata about the object, such as a URL, indicating where the object can be found. Thus, by being actionable and interoperable, a DOI differs from identifiers such as ISBNs and ISRCs which aim only to uniquely identify their referents
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Pollen
Pollen
Pollen
is a fine to coarse powdery substance comprising pollen grains which are male microgametophytes of seed plants, which produce male gametes (sperm cells). Pollen
Pollen
grains have a hard coat made of sporopollenin that protects the gametophytes during the process of their movement from the stamens to the pistil of flowering plants, or from the male cone to the female cone of coniferous plants. If pollen lands on a compatible pistil or female cone, it germinates, producing a pollen tube that transfers the sperm to the ovule containing the female gametophyte. Individual pollen grains are small enough to require magnification to see detail
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Apollonian Sphere Packing
Apollonian sphere packing
Apollonian sphere packing
is the three-dimensional equivalent of the Apollonian gasket. The principle of construction is very similar: with any four spheres that are cotangent to each other, it is then possible to construct two more spheres that are cotangent to four of them. The fractal dimension is approximately 2.473946 (±1 in the last digit).[1] Software for generating and visualization of the apollonian sphere packing: ApolFrac.[2] References[edit]^ Borkovec, M.; De Paris, W.; Peikert, R
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Conway Puzzle
Conway's puzzle, or Blocks-in-a-Box, is a packing problem using rectangular blocks, named after its inventor, mathematician John Conway. It calls for packing thirteen 1 × 2 × 4 blocks, one 2 × 2 × 2 block, one 1 × 2 × 2 block, and three 1 × 1 × 3 blocks into a 5 × 5 × 5 box.[1]Contents1 Solution 2 See also 3 References 4 External linksSolution[edit]A possible placement for the three 1×1×3 blocks. The vertical block has corners touching corners of the two horizontal blocks.The solution of the Conway puzzle
Conway puzzle
is straightforward once one realizes, based on parity considerations, that the three 1 × 1 × 3 blocks need to be placed so that precisely one of them appears in each 5 × 5 × 1 slice of the cube.[2] This is analogous to similar insight that facilitates the solution of the simpler Slothouber–Graatsma puzzle. See also[edit]Soma cubeReferences[edit]^ Weisstein, Eric W. "Conway Puzzle". MathWorld.  ^ Elwyn R
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Slothouber–Graatsma Puzzle
The Slothouber–Graatsma puzzle is a packing problem that calls for packing six 1 × 2 × 2 blocks and three 1 × 1 × 1 blocks into a 3 × 3 × 3 box. The solution to this puzzle is unique (up to mirror reflections and rotations). It was named after its inventors Jan Slothouber and William Graatsma. The puzzle is essentially the same if the three 1 × 1 × 1 blocks are left out, so that the task is to pack six 1 × 2 × 2 blocks into a cubic box with volume 27
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Set Packing
Set packing is a classical NP-complete
NP-complete
problem in computational complexity theory and combinatorics, and was one of Karp's 21 NP-complete
NP-complete
problems. Suppose one has a finite set S and a list of subsets of S. Then, the set packing problem asks if some k subsets in the list are pairwise disjoint (in other words, no two of them share an element). More formally, given a universe U displaystyle mathcal U and a family S displaystyle mathcal S of subsets of U displaystyle mathcal U , a packing is a subfamily C ⊆ S displaystyle mathcal C subseteq mathcal S of sets such that all sets in C displaystyle mathcal C are pairwise disjoint. The size of the packing is C displaystyle mathcal C
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