Digital sculpting, also known as sculpt modeling or 3D sculpting, is the use of software that offers tools to push, pull, smooth, grab, pinch or otherwise manipulate a digital object as if it were made of a real-life substance such as clay. == Sculpting technology == The geometry used in digital sculpting programs to represent the model can vary; each offers different benefits and limitations. The majority of digital sculpting tools on the market use mesh-based geometry, in which an object is represented by an interconnected surface mesh of polygons that can be pushed and pulled around. This is somewhat similar to the physical process of beating copper plates to sculpt a scene in relief. Other digital sculpting tools use voxel-based geometry, in which the volume of the object is the basic element. Material can be added and removed, much like sculpting in clay. Still other tools make use of more than one basic geometry representation. A benefit of mesh-based programs is that they support sculpting at multiple resolutions on a single model. Areas of the model that are finely detailed can have very small polygons while other areas can have larger polygons. In many mesh-based programs, the mesh can be edited at different levels of detail, and the changes at one level will propagate to higher and lower levels of model detail. A limitation of mesh-based sculpting is the fixed topology of the mesh; the specific arrangement of the polygons can limit the ways in which detail can be added or manipulated. A benefit of voxel-based sculpting is that voxels allow complete freedom over form. The topology of a model can be altered continually during the sculpting process as material is added and subtracted, which frees the sculptor from considering the layout of polygons on the model's surface. After sculpting, it may be necessary to retopologize the model to obtain a clean mesh for use in animation or real-time rendering. Voxels, however, are more limited in handling multiple levels of detail. Unlike mesh-based modeling, broad changes made to voxels at a low level of detail may completely destroy finer details. == Uses == Sculpting can often introduce details to meshes that would otherwise have been difficult or impossible to create using traditional 3D modeling techniques. This makes it preferable for achieving photorealistic and hyperrealistic results, though, many stylized results are achieved as well. Sculpting is primarily used in high poly organic modeling (the creation of 3D models which consist mainly of curves or irregular surfaces, as opposed to hard surface modeling). It is also used by auto manufacturers in their design of new cars. It can create the source meshes for low poly game models used in video games. In conjunction with other 3D modeling and texturing techniques and Displacement and Normal mapping, it can greatly enhance the appearance of game meshes often to the point of photorealism. Some sculpting programs like 3D-Coat, Zbrush, and Mudbox offer ways to integrate their workflows with traditional 3D modeling and rendering programs. Conversely, 3D modeling applications like 3ds Max, Maya and MODO are now incorporating sculpting capability as well, though these are usually less advanced than tools found in sculpting-specific applications. High poly sculpts are also extensively used in CG artwork for movies, industrial design, art, photorealistic illustrations, and for prototyping in 3D printing. == 3D print == Sculptors and digital artists use digital sculpting to create a model (or Digital Twin) to be materialized through CNC technologies including 3D printing. The final sculptures are often called Digital Sculpture or 3D printed art. While digital technologies have emerged in many art disciplines (painting, photography), this is less the case for digital sculpture due to the higher complexity and technology limitations to produce the final sculpture. == Sculpting Process == The best way to learn sculpture is by understanding primary, secondary and tertiary forms. First, break down the object you want to make down its basic shapes, such as a sphere or cube. Focus on making the large, overall shape of the object. After that, work on the bigger shapes on top of or inside the object. These can be protrusions or cut outs. Then, do a final detail pass, such as pores or lines to break up the shape. == Sculpting programs == There are a number of digital sculpting tools available. Some popular tools for creating are: Traditional 3D modeling suites are also beginning to include sculpting capability. 3D modeling programs which currently feature some form of sculpting include the following:
MobileNet
MobileNet is a family of convolutional neural network (CNN) architectures designed for image classification, object detection, and other computer vision tasks. They are designed for small size, low latency, and low power consumption, making them suitable for on-device inference and edge computing on resource-constrained devices like mobile phones and embedded systems. They were originally designed to be run efficiently on mobile devices with TensorFlow Lite. The need for efficient deep learning models on mobile devices led researchers at Google to develop MobileNet. As of June 2025, the family has five versions, each improving upon the previous one in terms of performance and efficiency. == Features == === V1 === MobileNetV1 was published in April 2017. Its main architectural innovation was incorporation of depthwise separable convolutions. It was first developed by Laurent Sifre during an internship at Google Brain in 2013 as an architectural variation on AlexNet to improve convergence speed and model size. The depthwise separable convolution decomposes a single standard convolution into two convolutions: a depthwise convolution that filters each input channel independently and a pointwise convolution ( 1 × 1 {\displaystyle 1\times 1} convolution) that combines the outputs of the depthwise convolution. This factorization significantly reduces computational cost. The MobileNetV1 has two hyperparameters: a width multiplier α {\displaystyle \alpha } that controls the number of channels in each layer. Smaller values of α {\displaystyle \alpha } lead to smaller and faster models, but at the cost of reduced accuracy, and a resolution multiplier ρ {\displaystyle \rho } , which controls the input resolution of the images. Lower resolutions result in faster processing but potentially lower accuracy. === V2 === MobileNetV2 was published in March 2019. It uses inverted residual layers and linear bottlenecks. Inverted residuals modify the traditional residual block structure. Instead of compressing the input channels before the depthwise convolution, they expand them. This expansion is followed by a 1 × 1 {\displaystyle 1\times 1} depthwise convolution and then a 1 × 1 {\displaystyle 1\times 1} projection layer that reduces the number of channels back down. This inverted structure helps to maintain representational capacity by allowing the depthwise convolution to operate on a higher-dimensional feature space, thus preserving more information flow during the convolutional process. Linear bottlenecks removes the typical ReLU activation function in the projection layers. This was rationalized by arguing that that nonlinear activation loses information in lower-dimensional spaces, which is problematic when the number of channels is already small. === V3 === MobileNetV3 was published in 2019. The publication included MobileNetV3-Small, MobileNetV3-Large, and MobileNetEdgeTPU (optimized for Pixel 4). They were found by a form of neural architecture search (NAS) that takes mobile latency into account, to achieve good trade-off between accuracy and latency. It used piecewise-linear approximations of swish and sigmoid activation functions (which they called "h-swish" and "h-sigmoid"), squeeze-and-excitation modules, and the inverted bottlenecks of MobileNetV2. === V4 === MobileNetV4 was published in September 2024. The publication included a large number of architectures found by NAS. Inspired by Vision Transformers, the V4 series included multi-query attention. It also unified both inverted residual and inverted bottleneck from the V3 series with the "universal inverted bottleneck", which includes these two as special cases. === V5 === MobileNetV5's architecture was published shortly after the release of Gemma 3n in June 2025. While the announcement stated a technical report on MobileNetV5 would be available soon, this has not yet materialised. The network is 10 times larger than the largest V4 variant.
Mata v. Avianca, Inc.
Mata v. Avianca, Inc. was a U.S. District Court for the Southern District of New York case in which the Court dismissed a personal injury case against the airline Avianca and issued a $5,000 fine to the plaintiffs' lawyers who had submitted fake precedents generated by ChatGPT in their legal briefs. == Background == In February 2022, Roberto Mata filed a personal injury lawsuit in the U.S. District Court for the Southern District of New York against Avianca, alleging that he was injured when a metal serving cart struck his knee during an international flight. The plaintiff's lawyers used ChatGPT to generate a legal motion, which contained numerous fake legal cases involving fictitious airlines with fabricated quotations and internal citations. Avianca's lawyers notified the Court that they had been "unable to locate" a few legal cases cited in the legal motion. The Court could not locate the cases either and ordered the plaintiff's lawyers to provide copies of the cited legal cases. Mata's lawyers provided copies of documents purportedly containing all but one of the legal cases, after ChatGPT assured that the cases "indeed exist" and "can be found in reputable legal databases such as LexisNexis and Westlaw." == Opinion == In May 2023, Judge P. Kevin Castel dismissed the personal injury case against Avianca and ordered the plaintiff's attorneys to pay a $5,000 fine. Judge Castel noted numerous inconsistencies in the opinion summaries, describing one of the legal analyses as "gibberish." Judge Castel held that Mata's lawyers had acted with "subjective bad faith" sufficient for sanctions under Federal Rule of Civil Procedure Rule 11. == Impact == In July 2024, the American Bar Association issued its first formal ethics opinion on the responsibilities of lawyers using generative AI (GAI). The 15-page opinion outlines how the Rules of Professional Conduct apply to the use of GAI in the practice of law. Experts caution that lawyers cannot reasonably rely on the accuracy, completeness, or validity of content generated by GAI tools. Due to the continued usage of GAI in the practice of law, Mata has been described as a landmark case by legal professionals, as it is frequently cited by courts in cases where usage of GAI during the course of proceedings leads to the creation and citation of nonexistent caselaw.
Trevor Paglen
Trevor Paglen (born 1974) is an American artist, geographer, and author whose work covers mass surveillance and data collection. In 2016, Paglen won the Deutsche Börse Photography Foundation Prize and he has also won The Cultural Award from the German Society for Photography. In 2017, he was a recipient of a MacArthur Fellowship. On March 17, 2026, Paglen was awarded the 2026 LG Guggenheim Award (a collaboration between LG and Guggenheim New York). == Early life and education == Paglen earned a B.A. degree in religious studies in 1998 from the University of California at Berkeley, a M.F.A. degree in 2002 from the School of the Art Institute of Chicago, and a Ph.D. in Geography in 2008 from the University of California at Berkeley. While at UC Berkeley, Paglen lived in the Berkeley Student Cooperative, residing in Chateau, Fenwick, and Rochdale co-ops. == Work == Sean O'Hagan, writing in The Guardian in 2015, said that Paglen, whose "ongoing grand project [is] the murky world of global state surveillance and the ethics of drone warfare", "is one of the most conceptually adventurous political artists working today, and has collaborated with scientists and human rights activists on his always ambitious multimedia projects." His visual work such as his "Limit Telephotography" and "The Other Night Sky" series have received widespread attention for both his technical innovations and for his conceptual project that involves simultaneously making and negating documentary-style truth-claims. Paglen’s work relies on contemporary technology in two meaningful ways. Firstly, the views he photographs would be impossible to shoot without media tech, that includes the cameras, the microscopes, and even helicopters. But interestingly enough, the shots would not be possible if not for the existence of the subject. The contrasts between secrecy and revelation, evidence and abstraction distinguish Paglen's work. With that the artist presents not so much "evidence" as admonitions to awareness. He was an Eyebeam Commissioned Artist in 2007. In 2008 the Berkeley Art Museum devoted a comprehensive solo exhibition to his work. In the next year, Paglen took part in the Istanbul Biennial, and in 2010 he exhibited at the Vienna Secession. Autonomy Cube was a project by Paglen and Jacob Appelbaum that placed relays for the anonymous communication network Tor in traditional art museums. He contributed to the Oscar-winning documentary film Citizenfour (2014), directed by Laura Poitras. Paglen features in the nerd-culture documentary Traceroute (2016). Orbital Reflector was a reflective, mylar sculpture by Paglen intended to be the first "purely artistic" object in space. The temporary satellite, containing an inflatable mylar balloon with reflective surface, launched into space 3 December 2018. A mid-career survey in 2018–2019, Trevor Paglen: Sites Unseen, was a traveling exhibition shown at the Smithsonian American Art Museum in Washington DC and the Museum of Contemporary Art San Diego. In September 2020, Pace Gallery in London held an exhibition of Paglen's work, exploring "the weird, partial ways computers look back at us". His work is included in the permanent collections of the San Francisco Museum of Modern Art, the Columbus Museum of Art, and the Metropolitan Museum. === Experimental Geography === Paglen is credited with coining the term "Experimental Geography" to describe practices coupling experimental cultural production and art-making with ideas from critical human geography about the production of space, materialism, and praxis. The 2009 book Experimental Geography: Radical Approaches to Landscape, Cartography, and Urbanism is largely inspired by Paglen's work. == Publications == Paglen has published a number of books. Torture Taxi (2006) (co-authored with investigative journalist A. C. Thompson) was the first book to comprehensively describe the CIA's extraordinary rendition program. I Could Tell You But Then You Would Have to be Destroyed by Me (2007), is a look at the world of black projects through unit patches and memorabilia created for top-secret programs. Blank Spots on the Map: The Dark Geography of the Pentagon's Secret World (2009) is a broader look at secrecy in the United States. The Last Pictures (2012) is a collection of 100 images to be placed on permanent media and launched into space on EchoStar XVI, as a repository available for future civilizations (alien or human) to find. === Publications by Paglen === I Could Tell You But Then You Would Have to be Destroyed by Me. Brooklyn, NY: Melville House, 2007. ISBN 1-933633-32-8. Blank Spots on the Map: The Dark Geography of the Pentagon's Secret World. New York: Dutton, 2009. ISBN 9781101011492. Invisible: Covert Operations and Classified Landscapes, Photographs by Trevor Paglen. New York: Aperture, 2010. ISBN 9781597111300. With an essay by Rebecca Solnit. The Last Pictures. Oakland, CA: University of California, 2012. ISBN 9780520275003. Trevor Paglen. London: Phaidon, 2018. ISBN 0714873446. With essays by Laren Cornell, Julia Bryan-Wilson, Omar Kholeif. === Publications co-authored === Torture Taxi. Co-authored with A. C. Thompson. Brooklyn, NY: Melville House Publishing, 2006. ISBN 1-933633-09-3. Icon, 2007. ISBN 9781840468304. === Publications with contributions by Paglen === Experimental Geography: Radical Approaches to Landscape, Cartography, and Urbanism. Brooklyn, NY: Melville House, 2009. ISBN 978-0091636586. Edited by Nato Thompson. With essays by Paglen, Thompson, and Jeffrey Kastner. Trevor Paglen and Jacob Appelbaum – Autonomy Cube. Revolver, 2016. ISBN 978-3957633026. Essays by Luke Skrebowski and Keller Easterling on Autonomy Cube, a piece of sculpture by Paglen and Jacob Appelbaum. In English and German. == Exhibitions == Bellwether Gallery, New York, November–December 2006 The Other Night Sky, Berkeley Art Museum, 2008 A Compendium of Secrets, Cologne Still Revolution: Suspended in Time, Museum of Contemporary Canadian Art, Toronto, May–June 2009. Group exhibition with Paglen, Barbara Astman, Walead Beshty, Mat Collishaw, Stan Douglas, Idris Khan, Martha Rosler, and Mikhael Subotzky A Hidden Landscape, Aksioma, Ljubljana, Slowenia Geographies of Seeing, Lighthouse, Brighton, England, October–November 2012 The Last Pictures, New York, 2012–13 Trevor Paglen, Altman Siegel gallery, San Francisco, CA, March–May 2015 The Octopus, Frankfurter Kunstverein, Frankfurt am Main, 2015 Autonomy Cube, Edith-Russ-Haus, Oldenburg, Germany, October 2015 – January 2016. Sculpture by Paglen and Jacob Appelbaum. Deutsche Börse Photography Foundation Prize 2016, The Photographers' Gallery, London, April–July 2016. Deutsche Börse Photography Prize shortlist with Paglen, Erik Kessels, Laura El-Tantawy, and Tobias Zielony. Radical Landscapes, di Rosa, Napa, February–April 2016 L’Image volée, Americas II, Bahamas Internet Cable System (BICS-1) and Globenet, Fondazione Prada, Milan (group exhibition), 2016 A Study of Invisible Images, Metro Pictures, New York, September–October 2017 == Awards == 2014: Pioneer Award from the Electronic Frontier Foundation. 2015: The Cultural Award from the German Society for Photography (DGPh) 2015: Academy Award as cameraman and director for the documentary film Citzenfour. 2016: Deutsche Börse Photography Foundation Prize 2017: MacArthur Fellowship, John D. and Catherine T. MacArthur Foundation, Chicago, IL 2018: Nam June Paik Art Center Prize == Films about Paglen == Unseen Skies (2021) == Works ==
The Quantum Thief
The Quantum Thief is the debut science fiction novel by Finnish writer Hannu Rajaniemi and the first novel in a trilogy featuring the character of Jean le Flambeur; the sequels are The Fractal Prince (2012) and The Causal Angel (2014). The novel was published in Britain by Gollancz in 2010, and by Tor in 2011 in the US. It is a heist story, set in a futuristic Solar System, that features a protagonist modeled on Arsène Lupin, the gentleman thief of Maurice Leblanc. The novel was nominated for the 2011 Locus Award for Best First Novel, and was second runner-up for the 2011 Campbell Memorial Award. == Setting == Several centuries after the technological singularity largely destroyed Earth, various posthuman factions compete for dominance in the Solar System. Though sentient superintelligent AGI has never been successfully developed, civilization has been greatly transformed by the proliferation of Hansonian brain emulations (termed "gogols" in reference to Nikolai Gogol, and in particular his novel Dead Souls). An alliance of powerful gogol copies rule the inner system from computronium megastructures housing trillions of virtual minds, laboring to resurrect the dead in religious devotion to the philosophy of Nikolai Fedorov. This alliance, the Sobornost, has been in conflict with a community of quantum entangled minds who adhere to the "no-cloning" principle of quantum information theory, and so do not see the Sobornost's ultimate goal as resurrection, but death. Most of this community, the Zoku, was devastated when Jupiter was destroyed with a weaponized gravitational singularity. Among the last remnants of near-baseline humanity exist on the mobile cities of Mars, where advanced cryptography and an obsessive privacy culture ensure that the Sobornost cannot upload their citizens' minds. The most notable of these cities is the Oubliette, where time is used as a currency. When a citizen's balance reaches zero their mind is transferred to a robotic body to serve the needs of the city for a set period, before being returned to their original body with a restored balance of time. == Plot summary == Countless gogols of the legendary gentleman thief Jean Le Flambeur are trapped in a virtual Sobornost prison in orbit around Neptune, playing an iterated prisoner's dilemma until his mind learns to cooperate. A warrior from the Oort Cloud, which has been settled by Finnish colonists, successfully retrieves one of the Le Flambeur gogols and uploads it into a real-space body. Acting on behalf of a competing Sobornost authority, this Oortian, Mieli, ferries the thief to the Martian city known as The Oubliette, where he has stored his memories for later recovery. The two intend to recover his memories so that he may return to an operating capacity sufficient to serve his Sobornost benefactor in a theft and repay his liberation. On the Oubliette, the young detective Isidore Beautrelet helps vigilantes catch Sobornost agents illicitly uploading human minds. These vigilantes are revealed to be in the service of a local colony of Zoku. Beautrelet is employed to investigate the arrival of Le Flambeur, and in the process becomes aware that the Oubliette's cryptographic security was always compromised. The memories of its citizens are fabrications, and the "King of Mars" long believed ousted in a revolution, still reigns behind the scenes. This King, who is another copy of Jean Le Flambeur, is defeated in the ensuing conflict. Le Flambeur fails to recover all of his memories, which he had locked with a quantum entangled revolver that required him to kill several of his old friends to open his stored memory. He and Mieli escape a liberated Mars having recovered only a mysterious "Schrödinger’s Box" from the Memory Palace. == Themes == Themes central to The Quantum Thief are the unreliability and malleability of memory and the effects of extreme longevity on an individual's perspective and personality. Prisons, surveillance and control in society are also major themes. In the book, the people living in the Oubliette society on Mars have two types of memory; in addition to a traditional, personal memory, there is the exomemory, which can be accessed by other people, from anywhere in the city. Memories about personal experiences can be stored in the exomemory and partitioned, with different levels of access granted to different people. These memories can be used, among other things, as an expedient form of communication. The Oubliette society has an economy where time is used as currency. When an individual's time is expended, their consciousness is uploaded into a "Quiet". The Quiet are mute machine servants who maintain and protect the city. Although the quiet seem to have little interest in the world outside their occupations, they do seem to retain some traces of their former personalities and memories. The conspiracy central to the plot involves the hidden rulers, called the "cryptarchs", manipulating and abusing the exomemory and through the citizens' transformations to quiet and back, the traditional memory as well. In the book, the Oubliette society is compared to a panopticon; a prison, where every action of the dwellers can be scrutinized. == History and influences == The first chapter of The Quantum Thief was presented by Rajaniemi's literary agent, John Jarrold, to Gollancz as the basis for the three-book deal that was eventually secured. Rajaniemi has stated that he had "come up with an outline that had every single idea I could cram into it, because I wanted to be worthy of what had happened." The outline eventually expanded into three parts, and the first part became The Quantum Thief. The novel's plot was inspired by one of Rajaniemi's favorite characters in fiction, Maurice Leblanc's gentleman thief Arsène Lupin, who operates on both sides of the law. What intrigued Rajaniemi were the cycles of redemption and relapse Lupin goes through as he tries to go straight, always falling short. Besides LeBlanc, Rajaniemi mentioned Roger Zelazny as a strong influence. Ian McDonald was the other science fiction author he mentioned as influential, plus Frances A.Yates's book The Art of Memory, for memory palaces. In an interview, Rajaniemi said he wasn't trying to write the novel as hard science fiction: "For me, the more important consequence of having a scientific background is a degree of speculative rigour: trying hard to work out the consequences of the assumptions one begins with." == Reception == The novel has received generally positive reviews. Gary K. Wolfe writes in his Locus review that Rajaniemi has "spectacularly delivered on the promise that this is likely the most important debut SF novel we'll see this year". James Lovegrove, reviewing the book in his Financial Times column, notes that "many an anglophone author would kill to turn out prose half as good as this, especially on their maiden effort." Eric Brown, reviewing for The Guardian, finds the novel to be "a brilliant debut", while alluding to the "apocryphal" (and incorrect) myth that "this novel sold on the strength of its first line." Sam Bandah, at SciFiNow, praises the novel for "its engaging narrative and characters backed by often almost intimidatingly good sci-fi concepts." Criticism for the novel has generally centred on Rajaniemi's sparse "show, don't tell" writing style. Brown notes that "the author makes no concessions to the lazy reader with info-dumps or convenient explanations." Niall Alexander, of the Speculative Scotsman, states that "had there been some sort of index, [he] would have gladly (and repeatedly) referred to it during the mind-boggling first third of The Quantum Thief", while proclaiming the novel to be "the sci-fi debut of 2010." == Awards == Nominee for the 2011 Locus Award for Best First Novel. Third place for the 2011 John W. Campbell Memorial Award for Best Science Fiction Novel
CinePlayer
CinePlayer is a software based media player used to review Digital Cinema Packages (DCP) without the need for a digital cinema server by Doremi Labs. CinePlayer can play back any DCP, not just those created by Doremi Mastering products. In addition to playing DCPs, CinePlayer can also playback JPEG2000 image sequences and many popular multimedia file types. There are two versions of CinePlayer available, standard and Pro. The standard version supports playback of non-encrypted, 2D DCP's up to 2K resolution. The Pro version supports playback of encrypted, 2D or 3D DCP's with subtitles up to 4K resolution. == Supported formats == === Containers === AVI MOV MXF MPG TS WMV M2TS MTS MP4 MKV === Video codecs === JPEG2000 ProRes 422 DNxHD YUV Uncompressed 8-10 bits DIVX XVID MPEG4 AVC / H-264 VC-1 MPEG2 === Supported image sequences === BMP TIFF TGA DPX JPG J2C === Supported audio files === WAV MP3 WMA MP2
Type-2 fuzzy sets and systems
Type-2 fuzzy sets and systems generalize standard type-1 fuzzy sets and systems so that more uncertainty can be handled. From the beginning of fuzzy sets, criticism was made about the fact that the membership function of a type-1 fuzzy set has no uncertainty associated with it, something that seems to contradict the word fuzzy, since that word has the connotation of much uncertainty. So, what does one do when there is uncertainty about the value of the membership function? The answer to this question was provided in 1975 by the inventor of fuzzy sets, Lotfi A. Zadeh, when he proposed more sophisticated kinds of fuzzy sets, the first of which he called a "type-2 fuzzy set". A type-2 fuzzy set lets us incorporate uncertainty about the membership function into fuzzy set theory, and is a way to address the above criticism of type-1 fuzzy sets head-on. And, if there is no uncertainty, then a type-2 fuzzy set reduces to a type-1 fuzzy set, which is analogous to probability reducing to determinism when unpredictability vanishes. Type1 fuzzy systems are working with a fixed membership function, while in type-2 fuzzy systems the membership function is fluctuating. A fuzzy set determines how input values are converted into fuzzy variables. == Overview == In order to symbolically distinguish between a type-1 fuzzy set and a type-2 fuzzy set, a tilde symbol is put over the symbol for the fuzzy set; so, A denotes a type-1 fuzzy set, whereas à denotes the comparable type-2 fuzzy set. When the latter is done, the resulting type-2 fuzzy set is called a "general type-2 fuzzy set" (to distinguish it from the special interval type-2 fuzzy set). Zadeh didn't stop with type-2 fuzzy sets, because in that 1976 paper he also generalized all of this to type-n fuzzy sets. The present article focuses only on type-2 fuzzy sets because they are the next step in the logical progression from type-1 to type-n fuzzy sets, where n = 1, 2, ... . Although some researchers are beginning to explore higher than type-2 fuzzy sets, as of early 2009, this work is in its infancy. The membership function of a general type-2 fuzzy set, Ã, is three-dimensional (Fig. 1), where the third dimension is the value of the membership function at each point on its two-dimensional domain that is called its "footprint of uncertainty"(FOU). For an interval type-2 fuzzy set that third-dimension value is the same (e.g., 1) everywhere, which means that no new information is contained in the third dimension of an interval type-2 fuzzy set. So, for such a set, the third dimension is ignored, and only the FOU is used to describe it. It is for this reason that an interval type-2 fuzzy set is sometimes called a first-order uncertainty fuzzy set model, whereas a general type-2 fuzzy set (with its useful third-dimension) is sometimes referred to as a second-order uncertainty fuzzy set model. The FOU represents the blurring of a type-1 membership function, and is completely described by its two bounding functions (Fig. 2), a lower membership function (LMF) and an upper membership function (UMF), both of which are type-1 fuzzy sets! Consequently, it is possible to use type-1 fuzzy set mathematics to characterize and work with interval type-2 fuzzy sets. This means that engineers and scientists who already know type-1 fuzzy sets will not have to invest a lot of time learning about general type-2 fuzzy set mathematics in order to understand and use interval type-2 fuzzy sets. Work on type-2 fuzzy sets languished during the 1980s and early-to-mid 1990s, although a small number of articles were published about them. People were still trying to figure out what to do with type-1 fuzzy sets, so even though Zadeh proposed type-2 fuzzy sets in 1976, the time was not right for researchers to drop what they were doing with type-1 fuzzy sets to focus on type-2 fuzzy sets. This changed in the latter part of the 1990s as a result of Jerry Mendel and his student's works on type-2 fuzzy sets and systems. Since then, more researchers around the world are writing articles about type-2 fuzzy sets and systems. == Interval type-2 fuzzy sets == Interval type-2 fuzzy sets have received the most attention because the mathematics that is needed for such sets—primarily Interval arithmetic—is much simpler than the mathematics that is needed for general type-2 fuzzy sets. The literature about interval type-2 fuzzy sets is large, whereas the literature about general type-2 fuzzy sets is much smaller. Both kinds of fuzzy sets are being actively researched by an ever-growing number of researchers around the world and have resulted in successful employment in a variety of domains such as robot control. Formally, the following have already been worked out for interval type-2 fuzzy sets: Fuzzy set operations: union, intersection and complement Centroid (a very widely used operation by practitioners of such sets, and also an important uncertainty measure for them) Other uncertainty measures [fuzziness, cardinality, variance and skewness and uncertainty bounds Similarity Subsethood Embedded fuzzy sets Fuzzy set ranking Fuzzy rule ranking and selection Type-reduction methods Firing intervals for an interval type-2 fuzzy logic system Fuzzy weighted average Linguistic weighted average Synthesizing an FOU from data that are collected from a group of subject == Interval type-2 fuzzy logic systems == Type-2 fuzzy sets are finding very wide applicability in rule-based fuzzy logic systems (FLSs) because they let uncertainties be modeled by them whereas such uncertainties cannot be modeled by type-1 fuzzy sets. A block diagram of a type-2 FLS is depicted in Fig. 3. This kind of FLS is used in fuzzy logic control, fuzzy logic signal processing, rule-based classification, etc., and is sometimes referred to as a function approximation application of fuzzy sets, because the FLS is designed to minimize an error function. The following discussions, about the four components in Fig. 3 rule-based FLS, are given for an interval type-2 FLS, because to-date they are the most popular kind of type-2 FLS; however, most of the discussions are also applicable for a general type-2 FLS. Rules, that are either provided by subject experts or are extracted from numerical data, are expressed as a collection of IF-THEN statements, e.g., IF temperature is moderate and pressure is high, then rotate the valve a bit to the right. Fuzzy sets are associated with the terms that appear in the antecedents (IF-part) or consequents (THEN-part) of rules, and with the inputs to and the outputs of the FLS. Membership functions are used to describe these fuzzy sets, and in a type-1 FLS they are all type-1 fuzzy sets, whereas in an interval type-2 FLS at least one membership function is an interval type-2 fuzzy set. An interval type-2 FLS lets any one or all of the following kinds of uncertainties be quantified: Words that are used in antecedents and consequents of rules—because words can mean different things to different people. Uncertain consequents—because when rules are obtained from a group of experts, consequents will often be different for the same rule, i.e. the experts will not necessarily be in agreement. Membership function parameters—because when those parameters are optimized using uncertain (noisy) training data, the parameters become uncertain. Noisy measurements—because very often it is such measurements that activate the FLS. In Fig. 3, measured (crisp) inputs are first transformed into fuzzy sets in the Fuzzifier block because it is fuzzy sets and not numbers that activate the rules which are described in terms of fuzzy sets and not numbers. Three kinds of fuzzifiers are possible in an interval type-2 FLS. When measurements are: Perfect, they are modeled as a crisp set; Noisy, but the noise is stationary, they are modeled as a type-1 fuzzy set; and, Noisy, but the noise is non-stationary, they are modeled as an interval type-2 fuzzy set (this latter kind of fuzzification cannot be done in a type-1 FLS). In Fig. 3, after measurements are fuzzified, the resulting input fuzzy sets are mapped into fuzzy output sets by the Inference block. This is accomplished by first quantifying each rule using fuzzy set theory, and by then using the mathematics of fuzzy sets to establish the output of each rule, with the help of an inference mechanism. If there are M rules then the fuzzy input sets to the Inference block will activate only a subset of those rules, where the subset contains at least one rule and usually way fewer than M rules. The inference is done one rule at a time. So, at the output of the Inference block, there will be one or more fired-rule fuzzy output sets. In most engineering applications of an FLS, a number (and not a fuzzy set) is needed as its final output, e.g., the consequent of the rule given above is "Rotate the valve a bit to the right." No automatic valve will know what this means because "a bit to the right" is a linguistic expression, and a valv