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Q: Rotational mechanics in a gyroscope question I have a homework question about gyroscope rotation, but I will describe my confusion more specifically as follows:
Below is the setup of the problem. I understand the concept that if the rotor is made to spin, then there is an angular momentum generated in the direction given by right-hand rule, in this case to the right of the page. This, combined with the torque into the page given by the object's weight, will result in the gyroscope to start rotating in a circle.
However, what confuses me is that if I am given the period of gyroscope rotation $T$ and the mass and moment of inertia of rotor, how can I then calculate the rate at which the rotor is spinning at? What kind of equation can allow me to equate these two concepts and solve for the rotor spinning speed?

A: First I will state the notation I'll use, to avoid any confusion between the two angular velocities we'll be dealing with. The mass of the gyroscope is $m$, its moment of inertia $I$ and its distance to the pivot point $r$. The angle $\theta$ denotes the angle that the gyroscope traces with respect to the pivot point, and $\Omega = \frac{d \theta}{dt}$ is what I will call the precessional angular velocity. Meanwhile $\omega_s$ denotes the spin angular velocity of the gyroscope itself. As often seen in textbooks $\vec \tau$ denotes torque  and $\vec L$ denotes angular momentum.
The anwer to the question lies on the relation between the spin angular velocity of the gyroscope itself and the precessional angular velocity of the whole system. We first must realize that the spin angular momentum of the gyroscope will be given by:
$$\vec{L} = I \omega_s \hat r$$
where $\hat r$ is a unit vector that points from the pivot where the rod is placed toward the center of the gyroscope at every moment.
Since there is also an precessional angular velocity there will be an orbital angular momentum pointing upward (assuming that the gyroscope stays rotating in the plane perpendicular to the vertical rod), but as we will see the torque will be perpendicular to it and thus it will be constant, so it is uninteresting for the question asked.
We can also note a crucial relation in the problem, which arises in finding the total time derivative of this angular momentum vector (it is important to keep in mind that want to find $\frac{d \vec L}{dt}$ and not $\frac{d |\vec{L}|}{dt}$, since the magnitude of the spin angular momentum will be constant due to the fact that both $I$ and $\omega_s$ are constant; instead we are interested in the change of the vector, to study its rotation, directly related to the precessional angular veolcity $\Omega$) Using:
$$\frac {d \hat r}{dt} = \frac{d \theta}{dt}\hat \theta$$, where $\hat \theta$ is perpendicular to $\hat r$ and in the same plane as it and $\theta$ is what has been established before. We can thus perform the time derivative of the angular momentum vector with result:
$$ \frac {d \vec L}{dt} = I \omega_s \frac{d \theta}{dt} \hat \theta = I \omega_s \Omega \hat \theta$$
Thus the magnitude of this derivative is given by:
$$|\frac{d\vec L}{dt}| = I \omega_s  \Omega $$
But from the definition $\vec \tau =  \vec r \times \vec F = \frac{d \vec L}{dt}$ we also obtain:
$$|\frac {d \vec L}{dt}| = mgr$$
Since here $\vec F = m\vec g$ is the gravitational force. Equating the two:
$$mgr = I \omega_s \Omega$$
Or because $\Omega = \frac{2\pi}{T} $:
$$mgr = \frac{2\pi I \omega_s}{T}$$
And we can solve for  $\omega_s$, the desired quantity:
$$\omega_s = \frac{mgrT}{2 \pi I}$$
In summary the equation than one could have used to relate this two is $\Omega = \frac{\tau}{L_s}$, which relates the torque and the spin angular momentum of the object.

A:
$\dot\psi$ is the rotation of the rotor , $\vartheta=\pi/2$ is your configuration.
you can obtain the solution  of your problem from the  conservation of the energy :
$$E=T+U=~\text{constant}$$
where T is the kinetic energy  and U is the potential energy
for $\vartheta=0$ is $$E_0=m\,g\,h$$ and for $\vartheta=\pi/2$ is the
energy $$E=\frac{1}{2}\,(I_\phi\,\dot{\phi}^2+I_\psi\,\dot{\psi}^2)$$
With
$$E=E_0$$
you can solve this equation and obtain
$$\dot{\psi}=\frac{\sqrt{I_\psi\,(2\,m\,g\,h-I_\phi\,\dot{\phi}^2})}{I_\psi}$$


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Studio Mir Co., Ltd. () — южнокорейская анимационная студия, располагающаяся в Сеуле.

История
Студия основана в 2010 году и названа в честь советской космической станции «Мир», что означает, что компания зародилась из «духа прогресса посредством сотрудничества». На момент основания в студии работало всего 20 сотрудников.

Первой работой компании стал мультипликационный фильм «Легенда о Корре», который вышел в эфир на американском кабельном канале Nickelodeon. Также студия участвовала в производстве четвёртого сезона сериала «Гетто», сериала Netflix «Вольтрон: Легендарный защитник» и «Кипо и Эра Чудесных Зверей», а также таких фильмов: «По ту сторону океана», «Смерть Супермена» и «Mortal Kombat Legends: Scorpion's Revenge».

Фильмография

Анимационные сериалы

Анимационные фильмы

Примечания

Ссылки

Мультипликационные студии

redpajama_set_name: RedPajamaWikipedia


Ellington Week at JOTW! My take on "It Don't Mean a Thing (if it ain't got that swing)"
Looks like I need to get a microphone for this thing.
Don't you worry 'bout a thing (if it ain't got that swing). Get it?
The Rhythm Junkies perform "It Don't Mean A Thing (If It Ain't Got That Swing)" in the Herrang library.
For the Linton Camp Choir to learn.
Lady Gaga Sustained E♭5 in "It Don't Mean a Thing"
The Count Basie Orchestra at Carnegie Hall 1981, featuring Tony Bennet.


Why not take a Sega Master System and Game Gear emulator (MASSAGE), recode it so that each hardware component is in an easily reused OBJ, then emulate a bunch of other systems that mostly use the same hardware and stick them all in the same executable?
Why not take a reasonably compatible emulator, recode it with more generic code so that it's slower, muddle it up a bit so that it's less compatible and add a load of other systems (that nobody wants and have been emulated better elsewhere) resulting in a load of partially emulated systems in an ever increasing size of executable?
"Multi" is effectively Massage V1.2, with a number of extra systems on top.
Most of these are emulated very nicely, except for the NES. The MSX 2 also needs a bit of work.
I've not updated this page for a while... Mainly because I've not done that much on it. SVI-318/328 is partially in, but Sega SF-7000 is completely non-functional. I've fixed up a few graphical issues here and there, but not a great deal. Mostly, I merged as much of the code as possible with X128 so that all the emulators would be established on a portable base, with the option to combine them.
These emulators aren't ready for Windows yet. I'd tried to put a proper audio mixer in the base so that a particular emulation core would have the freedom to pick and choose however many soundchips it wants, instead of the current method of having it hardcoded. The problem was that it was much slower, so it's half finished and there's no point merging this lot in until it is.
Progress ends due to forthcoming work commitments. I tried to make screenshots of NES, GB and MSX 2 games, but I hadn't updated the code to allow it! So, I've had to use the Windows grabbing method and there is some degradation where they've been converted and scaled down.
MSX 2 support (TMS9938, incomplete sprite support and a number of other bugs).
MSX: Can now load loose files by automatically creating a temporary DSK in the background. (Really exciting....).
Improved WD1793 emulation (see x128 WIP page).
SMS/GG: Sprites per line limit and proper sprite collision detection implemented.
I'm considering adding other Z80 based systems to it, like the SVI-318/328 (very similar to the MSX 1). I'm also investigating the PC88 and Sharp X1 systems (although it's just about impossible to get reliable technical information in English). Naturally, the MSX 2 would be a logical move, although it would require changes that would slow down the Z80A core, the code isn't entirely generic.... I could add checks to the peek routine, but not the instruction fetch. That might be a reasonable compromise.
1/3rd of a minute passed so far.
Everything after Burger Time has been dithered.
I prefer the MSX-1 version.
More Castlevania? I can't take it any more!


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FDA Proposes to Revoke Standard of Identity for Frozen Cherry Pie
Constituent Update
The U.S. Food and Drug Administration is proposing to revoke the definition and standards of identity and quality for frozen cherry pie. The standards do not appear necessary to ensure that these products meet consumer expectations, and the FDA has tentatively concluded that they are no longer necessary to promote honesty and fair dealing in the interest of consumers and may limit flexibility for innovation.
The proposal is part of the FDA's Nutrition Innovation Strategy. One of the goals of the NIS is to modernize food standards to maintain the basic nature and nutritional integrity of products while allowing industry flexibility for innovation to produce more healthful foods. The FDA believes it is important to take a fresh look at existing standards of identity in light of marketing trends and the latest nutritional science.
Standards of identity are mandatory requirements related to the content and production of certain food products. Foods for which standards of identity have been established include bread, fruit jams, certain vegetable and fruit juices, and certain types of chocolate. The standard of identity for frozen cherry pie describes the food as unbaked and as being comprised of a filling of mature, pitted, stemmed cherries contained in a pastry shell, which is frozen. It can contain other optional ingredients, but artificial sweeteners are not permitted. The standard of quality includes requirements for the cherries used in these pies, including amount and limitations on blemishes.
The action also responds to a citizen petition from the American Bakers Association.
Comments are due 90 days following publication in the Federal Register. Submit comments electronically to https://www.regulations.gov. Written comments should be addressed to Dockets Management Staff (HFA-305), Food and Drug Administration, 5630 Fishers Lane, Rm. 1061, Rockville, MD 20852.
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text:
Quality and Standards in Digital Imaging Projects
How do you ensure that your digital imaging projects will be done according to the best standards and practices? Whether you are doing the work in-house or outsourcing, there are many details to consider. Amy McCrory will discuss how to work with the resources you have to produce quality digital images, giving examples from projects completed in The Ohio State University Libraries. Freely available sources of information on imaging standards and best practices will be shared. Amy McCrory's PowerPoint is available here.
6:30 – 7:00 p.m. Refreshments, Sign-In & Networking
7:00 – 8:00 p.m. Presentation and Q & A
Kilgour Auditorium on the OCLC Campus in Dublin, Ohio
* See location and map at:
http://www.oclc.org/about/headquarters/maps/default.htm
RSVP with Laura Cheng at yrcheng@kent.edu, Digital Library Content Manager, National Middle School Association, Westerville, OH 43081
Presenter Amy McCrory is the Digital Imaging Specialist in the Preservation Department, Digital Imaging Division of The Ohio State University Libraries. She received her MLS from Clark Atlanta University, and worked at Georgia Southern University and the Savannah College of Art and Design before coming to OSU.
We look forward to seeing you on Wednesday, April 29th, 2009!


text:
Great cities respect their architectural history as they build their futures. Can Portland, Oregon, become a great city?
Rebuilding Benson Polytechnic High School
For 106 years, the Classical Revival west façade of Benson Polytechnic High School has been one of the most noticeable buildings in Northeast Portland. But unless you are more than 70 years old, you never saw the back side of the original building – or ever thought about it.
Now, an extensive remodel underway at Benson has cleared away a clutter of accessory buildings at the rear of the school, returning to view much of the original --and historically tasteful --eastern side. The massive renovation project is expected to cost close to $300 million before it is completed in 2024.
The spendy remodel will include a bunch of whizzbang new features, including a new gymnasium, library and instructional facilities. But from the preservationists' point of view, another benefit is that historical elements of the original building – both front and back – will be restored and preserved.
Eastern facade exposed after many decades
The original building, finished in 1916, had a narrow, rectangular floorplan. However, its designer, architect Floyd Naramore, intended for it to be added onto incrementally as needed. Shop wings were added to the north and south sides at the rear of the building in 1917 and 1918, followed by what is now called the "old gym" in 1925 and an auditorium in 1930. A library addition in 1953 closed off from public view what little could be seen of the original rear façade.
Naramore served as architect for the rapidly expanding school system from 1912 until 1919. During those years, he designed 12 elementary schools (many still in use) as well as Benson and Franklin High School. Portland was growing rapidly at the time.
To meet the needs of growth Naramore devised what were called "unit" plan schools. "Typically constructed of reinforced concrete with brick facing, the schools were designed to be built in units over time," according to a historical assessment prepared for the school system in 2009. "The initial building unit typically featured minimal provisions of classroom spaces. Specialized spaces including auditoriums, gymnasiums, and cafeterias were added in intervals as enrollment grew."
News articles in 1915 indicated that a talented young Portland architect, Folger Johnson, was hired as a consultant on the original Benson building. Regardless, Naramore is always listed as its architect. Naramore left Portland for Seattle in 1919, where he served as architect for the Seattle school system before opening a practice that was heavily engaged in World War II construction work.
Still a long ways to go...western facade
EXCITING UPDATE: After 2 1/2 years of work by volunteers with the Downtown Neighborhood Association, the South Park Blocks have been added to the National Register of Historic Places. The Portland Parks Bureau first suggested a nomination back in 1985, but never finished an application. The current Parks Bureau administration opposed this application in three public hearings. The bureau's motives are best left to speculation.
-----Fred Leeson
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Two Remodels in Historic Districts
2121 NW Glisan St.
Plans for major remodeling projects in two of Portland's historic districts have been approved by the Portland Historic Landmarks Commission.
Both buildings – Silver Dollar Pizza in the Alphabet Historic District and Grace Peck Terrace in the Irvington Historic District – are too recent to qualify as contributing elements of their historic districts. However, the landmarks commission serves as the city's design review agency because the properties lie within district boundaries.
The single-story Silver Dollar building will sprout two stories and add eight small apartments around an open courtyard on the second and third floors. The pizza store will continue its long-term lease as the ground-floor anchor. Entry to the apartments will be on the NW Glisan Street side, while the pizza entry remains on NW 21st Ave.
The building will be encased in light-colored stucco, with gentle Art Deco adornments celebrating the Glisan façade and the apartment entry. Pilasters on the original building will be extended on the new floors. Landmark commission members applauded the project for adding new residences to an existing building rather than going the tear-down route.
The one-story building was built in the 1940s and underwent subsequent renovations that lack a coherent architectural foundation. The renovation includes a rounded cornice and corner at 21st and Glisan, adding to the Art Deco feel.
View from NE 14th Ave. and Schuyler St.
In Irvington, the six-story Grace Peck Manor, a 95-unit subsidized housing complex erected in 1979, suffers from water leaks in its windows and stucco exterior. The project calls for replacing all windows and cladding the building with oko skin, a fiberglass-reinforced concrete product that is expected to survive better in Portland's wet climate. The oko panels will be applied horizontally to reflect the wooden lap siding common to surrounding houses in the neighborhood.
The primary colors will be contrasting gray tones, with three colors of blue standing vertically between the floors.
The plan also calls for remodeling the main entry and creating a new patio on the Hancock Street side close to 14th Avenue that will be recessed a few feet below the level of the adjacent sidewalk. Small balconies on the building that would have disappeared in an earlier version of the plan will be retained. Although small at approximately 20 square feet each, tenants said they valued having them.
"Overall, I think it's going to be a great change for the neighborhood," said Kristen Minor, landmarks chair.
Landmarks commissioners also discussed the possibility of creating a large mural or work of art that would decorate a tall blank building mass above the main entrance at NE 14th and Hancock, but no conclusion has been reached about what it might be. The building is managed by Home Forward, a federally-funded subsidized housing program.
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at March 25, 2022 1 comment:
Albina's 'Not Little' Plan
"Make no little plans. They have no magic to stir men's blood and probably will not themselves be realized." Daniel Burnham, Chicago architect, 1910.
Over the decades, Portland had big plans for what Downtown Portland perceived as the run-down old neighborhood of Albina close-in on the Willamette River's east side. They included Memorial Coliseum, Interstate-5 Freeway and the Emanuel Hospital Urban Renewal Plan.
Hundreds of houses were removed for the freeway, the Coliseum wiped out what once had been a vibrant residential neighborhood, and the Emanuel urban renewal plan fell by the wayside after the heart of Albina's commercial and cultural district had been demolished.
In case you're new to Portland, Albina was the heart of the city's African-American community, where de facto housing and real estate sales policies channeled Black residents after World War II.
Now, after seven years, a non-profit group called the Albina Vision Trust has a highly-interesting plan for 94 acres of Albina. It calls for a neighborhood of small homes at the north end, high-rise housing in the Broadway corridor, parks, access to the waterfront, historical markers, and focal points where activities could focus on the arts, food, wellness and education.
Of course, the 94 acres is a mere fraction of the area of North and Northeast Portland formerly called "Albina" in general terms. Not shown on the pretty map is the city's public works shops and parking lots just to the north that could affect the desirability of a residential neighborhood.
Winta Yohannes, Albina Vision's executive director, said there is "no going back to 1956," when turmoil began that destroyed old Albina. She knows that many Black residents who were forced out by City Hall's big changes are sufficiently bitter that they aren't interested in returning.
Still, the citizen planners hope their ideas will open doors for small and minority contractors, starting with small homes planned for the area now occupied by the massive Portland Public Schools headquarters. Success with the first phase of building that new neighborhood could leverage those small businesses into bigger opportunities as larger projects unfold in later stages.
"People want a sense of a close-knit neighborhood," Yohannes said. The first phase with smaller homes could help the public understand that yes, progress is possible in Albina. The 94-acre plan also includes three new blocks that be built on covers over the I-5 freeway. The covers, if built, would support the weight of three and four story buildings and provide better walking and bicycle connections to neighborhoods now split by the freeway.
The Albina Vision plan is scheduled to be presented to the Portland City Council in late March. Members of the Portland Historic Landmarks Commission and Portland Design Commission spoke highly of the plan at a meeting in February.
"There is so much to like about this," said Kimberly Moreland, a landmarks commission member. "Albina still serves as a cultural and spiritual center. There are so many connections that are still here." She asked the planning team to make careful note of significant buildings that still remain from the historic African-American era.
One thinks that Daniel Burnham, who dreamed big plans and then built them in Chicago, might be impressed. The question now is how Albina's "big plan" can be implemented between now and 2050.
As somebody smarter than your host once said, it takes a special talent to proclaim big plans from the heavens. It takes another kind of talent to tinker them into reality.
David P. Thompson Elk Fountain update:
Last week's article about the potential demise of this iconic fountain drew a record number of readers by an astounding margin. One sophisticated reader thought I was too pessimistic. The matter ultimately will be decided by the City Council. It is not too early to let the city commissioners know your feelings about restoring the fountain.
Mayor Ted Wheeler:
MayorWheeler@portlandoregon.gov
Commissioner Jo Ann Hardesty:
JoAnn@portlandoregon.gov
Commissioner Dan Ryan:
CommissionerRyanOffice@portlandoregon.gov
Commissioner Carmen Rubio:
Comm.Rubio@portlandoregon.gov
Commissioner Mingus Mapps:
MappsOffice@portlandoregon.gov
----Fred Leeson
Join Building on History's mailing list by writing "add me" to fredleeson@hotmail.com
Goodbye, David P. Thompson Fountain
(Regional Arts & Culture Council image)
If you had hoped to see the historic David P. Thompson fountain restored to its rightful location on S.W. Main Street, forget it. Won't happen.
In what seems like a Kafka-esque turn of events, the City of Portland is pursuing a 120-day "demolition delay" for a historic landmark that the city itself hastily demolished in 2020. As a consequence, the Beaux-Arts designed fountain, dating to 1900, will be removed from the city's list of historic landmarks.
This is the case of city bureaucrats prevailing over the city's historic fabric. The Portland Water Bureau didn't like managing the fountain, and the Portland Bureau of Transportation wanted more room on Main Street for a bike lane and bus passage. So say bye-bye to an iconic historic landmark.
The fountain, originally intended to provide water for horses and dogs, provided a pedestal for the iconic elk statue donated by the Portland pioneer entrepreneur and early mayor, David P. Thompson.
The city promises to return the elk – removed after it was damaged slightly during political protests in 2020 – but it will stand on a new pedestal of as-yet undisclosed design. The pedestal will be smaller in size than the original fountain.
As you may recall, the Regional Arts & Culture Council on July 2, 2020 removed the elk statue after minor damage occurred. Within two weeks, the Portland Water Bureau hired a company to demolish portions of the granite fountain and salvage other parts. The speedy demolition appeared to fly in the face of state and city laws intended to protect historic sites.
Gone and not coming back
Meanwhile, the transportation bureau late in 2020 offered drawings showing the fountains site narrowed either by 4 or 8 feet on its northern and southern edges to accommodate its desired traffic plan.
Last fall, a memo from a city preservation planner recommended that the fountain be restored. "Given that the landmark resource is specifically noted as a fountain, staff believes that maintaining this function is important to maintaining the resource's historic character and significance," Hillary Adam wrote. "Staff does not believe that reducing the fountain to a pedestal maintains this character or significance, essentially reducing the landmark by at least half of its cultural and aesthetic value." In an informal meeting, the Portland Historic Landmarks Commission agreed.
But removal of the fountain landmark designation leaves the Landmarks Commission no voice to recommend its preservation. A new pedestal will be reviewed instead by the Portland Design Commission, whose hands will be largely tied by whatever designs are presented to it.
In coming months, the Design Commission will send its recommendation to the Portland City Council, which already will have made up its mind by informal agreements long before the hearing.
Yes, people can testify to the City Council. Then the council will vote and, as it usually does, thank one another for outstanding leadership. Their decision will send a sad message, as noted by Brian Libby, an astute critic who usually writes about modern architecture.
"To keep the statue separated permanently from its fountain would bring a grim reality: that the City of Portland will have done more lasting damage to this landmark than any protester or counter-protester," he wrote. He added, "It tells me there's a problem here that's bigger than Elk: that the City of Portland is not a thoughtful, committed or humble caretaker of its own heritage."
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at March 10, 2022 30 comments:
A Miracle on N.E. Eighth Ave.
It's good news, usually, when politicians and TV cameras show up for a ribbon-cutting ceremony. It was miraculous news for the Allen Temple CME Church that finished a $3 million restoration that took more than six years to finance and finish.
News accounts properly reported on the remarkable fund-raising and volunteer work that ultimately involved more than 70 design, construction, religious, social and financial partners. There was little discussion, however, of the architectural significance and the population changes now well into a "third wave" in the Northeast Portland neighborhood.
Architectural preservation is not specified as a religious mission, but Allen Temple was fortunate in having two architects assist who know the preservation world. Bill Hart and Logan Cravens, affiliated with the Carlteton Hart Architecture firm, preserved the historic envelope of the building and sanctuary while completely updating the building's other "innards." While original repair estimates were put at $400,000, the completed overhaul cost $3 million.
The Allen temple, at 4236 NE Eighth Ave., was one of several wood-framed churches to sprinkle the neighborhood in the early 20th century. Like this one – erected in 1913 as the Second German Congregational Church and enlarged in 1921 and again in 1927 and 1932 -- many were connected to the ethnic and religious backgrounds of the Scandinavian and European immigrants who dominated the neighborhoods composed of smaller, less expensive homes. In physical size, this church was larger than many.
Like the Allen Temple, most of these vernacular buildings were planned and built by carpenters, not architects. They featured the gothic-arch windows, modest spires and crosses that signified a single meaning: "church." These small churches were common to most western towns, where wood was the least expensive building materials and carpenters had the ability to create simple designs. Over time, many fell into disuse or the parishes graduated into more substantial edifices made of brick or stone.
Demographic changes occurred dramatically in North and Northeast Portland after World War II, when thousands of Black workers recruited to the shipyards were funneled into the Albina area by Portland's de facto racial segregation policies adopted by the real estate and lending businesses.
The Allen Temple CME Church was organized by Black parishioners in 1949. As its membership grew, it learned in 1960 of the potential sale of the former German Episcopal building, and has been based there since 1961. In 2015, two electrical fires that started within hours seriously damaged the church interior and its roof. The site could have been cleared and sold for housing.
The Rev. Dr. LeRoy Haynes Jr., pastor since 1997, thought otherwise. As the neighborhood gentrified during the past two decades, he believed it was important to maintain the presence of Black church and its social services in the neighborhood. Most of the other small wooden historic churches in Northeast Portland had long since given way to other uses.
Overall, the restoration was a daringly bold plan that required thoughtful execution. One assumes that its success will provide for decades of religious and social benefits, as well as retaining a neighborhood architectural landmark.
Join Building on History's mailing list by writing "add me" to fredleeson@hotmail.com
(Regional Arts & Culture Council image) If you had hoped to see the historic David P. Thompson fountain restored to its rightful locat...
Going, Going...Soon to be Gone
Portland State University has filed its intention to tear down the Parkway Manor, a 90-year-old former apartment building, now used for st...
buildingonhistory
Fred Leeson has been interested in architectural history and city development for many years. He is a former president of the Architectural Heritage Center.


text:
\section{Unde Venis Et Quo Vadis?}

Formal Concept Analysis (FCA~\cite{ganter2012formal}) is a formalism that establishes a connection between classical binary data (crosstables) and the structure of concepts and rules that can be found in said data. It is very powerful, if underutilised, as it offers well-studied mathematical structures to be exploited by algorithms.

As crosstables are a rather limiting way of representing data, various extensions of the formalism have been proposed to deal with more complex data, such as Pattern Structures~\cite{ganter2001pattern}, Relational Concept Analysis~\cite{rouane2013relational}, fuzzy FCA~\cite{poelmans2014fuzzy} or graph FCA~\cite{ferre2020graph}.
Just like FCA, they are based on lattice theory~\cite{birkhoff1940lattice}.

Triadic Concept Analysis~\cite{lehmann1995triadic} and Polyadic Concept Analysis (PCA)~\cite{voutsadakis2002polyadic} aim to extend FCA to data in the form of $n$-ary relations (\textit{i.e.} multidimensional crosstables) and have the peculiarity of involving $n$-lattices instead of lattices.
Such structures are considerably less known and studied, and results that would be considered basic in lattice theory are missing.
While some would consider this an opportunity to pioneer a whole new field, the heavy notations and lack of clear definitions and impressive applicative results repels researchers
The opportunities are however numerous as multidimensional data is now ubiquitous: RDF datasets, folksonomies, pharmacogenomical knowledge are all inherently at least triadic and transforming them to fit dyadic crosstables only results in lost information. As such data are also understudied in data mining in general, this is a unique occasion to position FCA and its extensions as a leading formalism. Some work has already been done in this direction in the field of multidimensional association rule mining~\cite{missaoui2022computing,bazin2022condensed}.

In this paper, we aim at promoting Polyadic Concept Analysis by discussing the differences with FCA introduced by $n$-lattices, suggesting future research directions, and presenting recent results and open questions on the size of $n$-lattices. Section~\ref{sec:preliminaries} contains definitions, including a proposition for a stable definition of implications. Section~\ref{sec:pcaAndTheFallOfGeneralisation} discusses the impact of the loss of duality between orders in $n$-lattices on the relation between FCA structures, as well as a suggestion for a multidimensional generalisation of Boolean lattices. Section~\ref{sec:countingStuff} presents recent results on the maximal size of concept $n$-lattices found in cubic contexts.

\section{The Basics of Boxes\label{sec:preliminaries}}

We assume that the reader is at least somewhat familiar with FCA. If this is not the case, we refer the reader to the usual book~\cite{ganter2012formal}.
Polyadic concept analysis is a natural extension of FCA in which the underlying relation is $n$-ary. It was first introduced in its triadic form by Lehman and Wille~\cite{lehmann1995triadic}, and then generalised to the $n$-dimensional case by Voutsadakis~\cite{voutsadakis2002polyadic}.
In this setting, an $n$-context is an $(n+1)$-tuple $\contn$ where the $\m S_i$ are sets called dimensions and $\m R\subseteq \prod_{i=1}^{n} \m S_i$ is an $n$-ary relation between them. We shall call \emph{objects} the elements of the first dimension $\m S_1$ while the $(n-1)$-tuples $(s_2,\dots,s_n)$ such that $(s_1,s_2,\dots,s_n)\in \m R$ form the \emph{description} of the object $s_1$. An example of a triadic context is depicted in Fig.~\ref{fig:ncont}.

\begin{figure}[htb]
\centering
\begin{tabular}{c|ccc||ccc||ccc}
     & ~$a$~ & ~$b$~ & ~$c$~ & ~$a$~ & ~$b$~ & ~$c$~ & ~$a$~ & ~$b$~ & ~$c$~ \\
     \hline
    ~$1$~ & \x & & & \x & & \x & & & \x \\
    ~$2$~ & \x & & & \x & & & & & \\
    ~$3$~ & \x & \x & & & \x & & & \x & \x \\
     \hline
     \multicolumn{1}{c|}{} & \multicolumn{3}{c||}{$\alpha$} & \multicolumn{3}{c||}{$\beta$} &
     \multicolumn{3}{c}{$\gamma$}
\end{tabular}
\caption{\label{fig:ncont}A triadic context $(\{\alpha,\beta\},\{1,2,3\},\{a,b,c\},\m R)$.}
\end{figure}

An $n$-concept is then a maximal $n$-dimensional box full of crosses, \textit{i.e.} an $n$-tuple $\concn$ such that $\prod_{i=1}^{n} X_i\subseteq \m R$ and no $X_i$ is such that
$$\prod_{j=1}^{i-1}\m S_j\times X_i\cup \{x\}\times \prod_{k=i+1}^{n}\m S_k\subseteq \m R$$ with $x\not\in X_i$.
For instance, $(\{\alpha,\beta\},\{1,2\},\{a\})$ is an $n$-concept in Fig.~\ref{fig:ncont}. The set $\mathcal C$ of all $n$-concepts in an $n$-context together with the $n$ quasi-orders induced by the inclusion relation on their $n$ components form an $n$-lattice $\mathcal L = (\mathcal C,\lesssim_1,\dots,\lesssim_n)$.
Note that an $n$-lattice is an $n$-ordered set, \textit{i.e.} it respects:
\begin{itemize}
    \item if $\forall i\in \{1,\dots,n\}\setminus \{j\}, A\lesssim_i B$ then $A\gtrsim_j B$ (antiordinal dependency) and
    \item if $\forall i\in\setn$, $A\sim_i B$, then $A = B$ (uniqueness condition).
\end{itemize}

Different definitions of implications in triadic and polyadic contexts have been proposed through the years such as ``Biedermann's implications''~\cite{biedermann1997triadic}, attribute$\times$condition, conditional attribute or attributional condition implications~\cite{ganter2004implications}.
In~\cite{bazin2020implication}, it was proposed to consider all the implications that hold in dyadic contexts resulting from combinations of two transformations of an $n$-context $\m C$ (see Fig.~\ref{fig:transfo}):
\begin{itemize}
    \item $\m C^{(A,B)} = (\prod A,\prod B,\mathcal R^{(A,B)})$, where $A$ and $B$ form a bipartition of the set of all dimensions and $((s_{a_1},\dots,s_{a_k}),(s_{b_1},\dots,s_{b_l}))\in \m R^{(A,B)}$ iff $(s_1,\dots,s_n)\in \m R$ with $\{a_1,\dots,a_k\}\cup \{b_1,\dots,b_l\} = \{1,\dots,n\}$
    \item $\m C_{D} = (\m S_1,\dots,\m S_{d-1},\m S_{d+1},\dots,\m S_n,\m R_D)$, where $D\subseteq \m S_d$ and $$(s_1,\dots,s_{d-1},s_{d+1},\dots,s_n)\in \m R_D \text{ iff }\forall x\in D, (s_1,\dots,s_{d-1},x,s_{d+1},\dots,s_n).$$
\end{itemize}

\begin{figure}[htb]
\centering
\begin{tabular}{c|ccccccccc}
     & ~$(1,a)$~ & ~$(1,b)$~ & ~$(1,c)$~ & ~$(2,a)$~ & ~$(2,b)$~ & ~$(2,c)$~ & ~$(3,a)$~ & ~$(3,b)$~ & ~$(3,c)$~ \\
     \hline
    ~$\alpha$~ & \x & & & \x & & & \x & \x & \\
    ~$\beta$~ & \x & & \x & \x & & & & \x & \\
    ~$\gamma$~ & & & \x & & & & & \x & \x

\end{tabular}

\bigskip
\begin{tabular}{cc}
\parbox{0.4\textwidth}{\hspace{0.18\textwidth}
    \begin{tabular}{c|ccc}
     & ~$1$~ & ~$2$~ & ~$3$~ \\
     \hline
    ~$\alpha$~ & \x & \x & \x \\
    ~$\beta$~ & \x & \x & \\
    ~$\gamma$~ & & & \\
    \end{tabular}
    }
     &
\parbox{0.4\textwidth}{\hspace{0.1\textwidth}
     \begin{tabular}{c|ccc}
     & ~$a$~ & ~$b$~ & ~$c$~ \\
     \hline
    ~$\alpha$~ & \x & & \\
    ~$\beta$~ & & & \\
    ~$\gamma$~ & & & \x \\
     \end{tabular}
     }
\end{tabular}

\caption{\label{fig:transfo}Transformations $\m C^{(\{greek\},\{numbers,latin\})}$ (top), $\m C_{\{a\}}$ (bottom left) and $\m C_{\{1,3\}}$ (bottom right) of Fig.~\ref{fig:ncont}'s triadic context $\m C$.}
\end{figure}

Such implications contain all previously defined types of implications. For instance, in Fig.~\ref{fig:ncont}'s triadic context $\m C$ and if we assume that $numbers = \{1,2,3\}$ is the set of attributes and $latin = \{a,b,c\}$ is the set of conditions,
\begin{itemize}
\item conditional attribute implications $A_2\xrightarrow{C} A_2$ are the dyadic implications in  the $2$-context $\m C_{C}^{(\{greek\},\{numbers\})}$, \textit{e.g.} $\{3\}\xrightarrow{\{a\}} \{1,2\}$,
\item attributional condition implications $C_1\xrightarrow{A} C_2$ are the dyadic implications in the $2$-context  $\m C_{A}^{(\{greek\},\{latin\})}$, \textit{e.g.} $\emptyset\xrightarrow{\{3\}} \{b\}$,
\item and attribute$\times$condition implications $A\rightarrow B$ are simply the dyadic implications in  the $2$-context $\m C^{(\{greek\},\{numbers,latin\})}$, \textit{e.g.} $\{(1,a)\}\rightarrow \{(3,b)\}$.
\end{itemize}

Note that possible implications include those in $\m C^{(\{greek,numbers\},\{latin\})}$ whose support is in a Cartesian product of dimensions.
Such rules (under the more general umbrella of association rules) are already under consideration in the data mining community~\cite{nguyen2011multidimensional,9720169}.
Additionally, if one is only interested in implications that do not contain the first dimension -- the objects -- these implications are all derivable from the implications of $\m C^{(\{\m S_1\},\{S_2,\dots,S_n\})}$ through the application of Armstrong's axioms plus two other axioms, as discussed in~\cite{bazin2020implication}. This means that one only has to reason on a single type of implication.

Just as in the dyadic case, the set of all the implications that do not involve the first (object) dimension and that hold in an $n$-context (or any its implication bases) can be used to reconstruct the set of $n$-concepts restricted to their last $n-1$ components~\cite{bazin2020implication}.

\medskip

We conclude this first section with a small digression about graphical representation.
In two dimensions, both partial orders of the concept lattice $(\m T(\m C), \subseteq_1, \subseteq_2)$ have the decency of being dual, and thus one can be omitted, allowing a concept lattice to be graphically represented by a Hasse diagram.
Starting from three dimensions, concepts are ordered differently.
In $n$ dimensions, there are classes of equivalent concepts with the same $i$th component, and those concepts can then be differentiated in the $n-1$ other quasi-orders.
A graphical representation that clearly shows the equivalence classes and the quasi-order relations is -- at best -- hard to attain.
This is a first open question offered by PCA.

In three dimensions, a tentative graphical representation exists~\cite[Section 3]{lehmann1995triadic}, that combines a geometric representation of equivalence classes together with Hasse-like representation for each quasi-order.
The few graphical representations of $4$-lattices, for example in~\cite[figures 1 and 2]{voutsadakis2002polyadic} can only make one hope for another type of graphical representation, possibly \emph{via} Virtual Reality, as is already proposed for FCA~\cite{sacarea2020improving,breckner2022improving}.

\section{Loss of Duality and the FCA Trinity\label{sec:pcaAndTheFallOfGeneralisation}}

\subsection{Structural Equivalences}

In the dyadic case, the formal context, the concept lattice and the set of implications are equivalent in the sense that they can be computed from one another. This is the reason why the FCA formalism is so useful for data analysis. In the multidimensional case, this equivalence softens, which introduces new challenges, constraints and open questions. Let us consider the two triadic contexts depicted in Fig.~\ref{fig:deuxcont} in which the Greek letters form the first dimension (the objects), and their triadic concepts.

\begin{figure}[htb]
\centering

\begin{tabular}{c | c}

\parbox{0.45\textwidth}{\centering
\begin{tabular}{c|ccc||ccc}
 & ~$a$~ & ~$b$~ & ~$c$~ & ~$a$~ & ~$b$~ & ~$c$~  \\
 \hline
~$1$~ & $\times$ & $\times$ & $\times$ & $\times$ & &  \\
~$2$~ & $\times$ &  & &  & $\times$ &   \\
~$3$~ & $\times$ & &  & &  &  \\
\hline
\multicolumn{1}{c|}{} & \multicolumn{3}{c||}{$\alpha$} & \multicolumn{3}{c}{$\beta$} \\
\end{tabular}
}

&

\parbox{0.45\textwidth}{\centering
\begin{tabular}{c|ccc||ccc||ccc}
 & ~$a$~ & ~$b$~ & ~$c$~ & ~$a$~ & ~$b$~ & ~$c$~ & ~$a$~ & ~$b$~ & ~$c$~ \\
 \hline
~$1$~ & $\times$ & $\times$ & $\times$ & $\times$ & & & $\times$ & & \\
~$2$~ & & & & $\times$ &  & &  & $\times$ & \\
~$3$~ & & & & $\times$ & & & & & \\
\hline
\multicolumn{1}{c|}{} & \multicolumn{3}{c||}{$\alpha$} & \multicolumn{3}{c||}{$\beta$} & \multicolumn{3}{c}{$\gamma$}\\
\end{tabular}
} \\

\multicolumn{2}{c}{}\\
\multicolumn{2}{c}{}\\

\parbox{0.45\textwidth}{\centering
($\emptyset$,$\{1,2,3\}$,$\{a,b,c\}$)

($\{\alpha,\beta\}$,$\emptyset$,$\{a,b,c\}$)

($\{\alpha,\beta\}$,$\{1,2,3\}$,$\emptyset$)

($\{\alpha\}$,$\{1\}$,$\{a,b,c\}$)

($\{\alpha\}$,$\{1,2,3\}$,$\{a\}$)

($\{\alpha,\beta\}$,$\{1\}$,$\{a\}$)

($\{\beta\}$,$\{2\}$,$\{b\}$)
}

&

\parbox{0.45\textwidth}{\centering
($\emptyset$,$\{1,2,3\}$,$\{a,b,c\}$)

($\{\alpha,\beta,\gamma\}$,$\emptyset$,$\{a,b,c\}$)

($\{\alpha,\beta,\gamma\}$,$\{1,2,3\}$,$\emptyset$)

($\{\alpha\}$,$\{1\}$,$\{a,b,c\}$)

($\{\beta\}$,$\{1,2,3\}$,$\{a\}$)

($\{\alpha,\beta,\gamma\}$,$\{1\}$,$\{a\}$)

($\{\gamma\}$,$\{2\}$,$\{b\}$)
}
\end{tabular}

\caption{\label{fig:deuxcont}Two formal contexts $(\{\alpha,\beta,\gamma\},\{1,2,3\},\{a,b,c\},\mathcal R_1)$ (left) and $(\{\alpha,\beta\},\{1,2,3\},$ $\{a,b,c\},\mathcal R_2)$ (right) and their associated $3$-concepts.}
\end{figure}

We observe that the concepts differ only on their first components. Hence, the second and third quasi-orders of both triadic concept lattices are isomorphic while the first quasi-orders are not. This is a most significant change from the bidimensional case: the knowledge of $n-1$ quasi-orders is not enough to know the last one. Let us call the last $n-1$ components of a concept its \emph{feature} and the first component its \emph{extent}, as usual. Then, given a set of concepts known only by their features, there are multiple non-isomorphic ways objects can belong to the extents. This clashes with the usual notions of subsumption.

Let us say that two concept $n$-lattices are equivalent if and only if their $k$th quasi-orders are isomorphic for all $k\in \{2,\dots,n\}$. We denote by $[\mathcal L]$ the equivalence class of the concept $n$-lattice $\mathcal L$. As mentioned in the previous section, implications in $n$-contexts can be used to construct (exactly) the features of the associated $n$-concepts. Hence, all the implication bases of all the $n$-contexts of the lattices in $[\mathcal L]$ allow for the construction of the same features. These implications can, however, differ from $n$-context to $n$-context. For instance, the implication $\{(2,a)\}\rightarrow \{(1,b)\}$ holds in Fig.~\ref{fig:deuxcont}'s first triadic context (as both crosses appear together in the description of the object $\alpha$) but not in the second. From this, we deduce that some implications have no influence on the construction of the features. We thus propose to identify two types of implications:
\begin{itemize}
    \item \emph{Structural implications} that are used to construct the features of all the $n$-concepts of an $n$-context
    \item \emph{Contextual implications} that are not structural but still hold in an $n$-context
\end{itemize}

Structural implications carry information about the features of $n$-concepts while contextual implications carry information about the distribution of the objects in the $n$-concepts. Thus, structural implications are common to all the $n$-contexts in an equivalence class $[\mathcal L]$ while contextual implications are not. In Fig.~\ref{fig:deuxcont}'s contexts, the implications $\{(1,b),(1,c)\}\rightarrow \{(1,a)\}$ and $\{(2,a)\}\rightarrow \{(3,a)\}$ are structural while $\{(2,b)\}\rightarrow \{(1,a)\}$ is contextual in both contexts and $\{(1,b),(2,a)\}\rightarrow \{(3,a),(1,c)\}$ is contextual in the first context only.
In~\cite{bazin2020implication} is explained in an overly formal way, for which the author is very sorry, that constructing the features of all the $n$-concepts of an $n$-context only requires implications between ``boxes'', \emph{i.e.} implications of the form $\prod_{k = 2}^n X_k\rightarrow \prod_{k = 2}^n Y_k$ with $X_k\subseteq Y_k\subseteq \mathcal S_k$. The structural implications are all the implications entailed by these implications between ``boxes''.

In an equivalence class $[\m L]$, there is one $n$-lattice/$n$-context that seems to be of particular interest as it minimises the number of contextual implications with a non-empty support, which could have some use in data mining. It is fairly easy to construct this $n$-context: if two $n$-concept features are such that their intersection is not the feature of an $n$-concept, then the two features must appear in the descriptions of the same objects. This defines equivalence classes of features. Then, each such equivalence class of features is used to describe a different object. Fig.~\ref{fig:structurality} illustrates this. The rectangles $(\{1\},\{a,b\})$ and $(\{1,2\},\{a\})$ are features of triadic concepts while $(\{1\},\{a\})$, their intersection, is not so both rectangles have to appear in the description of the same object. The intersection $(\{3\},\{c\})$ of the rectangles $(\{2,3\},\{c\})$ and $(\{3\},\{b,c\})$ is the feature of the triadic concept $(\{\alpha,\beta\},\{3\},\{c\})$ so the three rectangles are put in the descriptions of different objects.

\begin{figure}[htb]

\begin{tabular}{clc}
\parbox{0.4\textwidth}{\centering
\begin{tabular}{c|ccc||ccc}
 & ~$a$~ & ~$b$~ & ~$c$~ & ~$a$~ & ~$b$~ & ~$c$~  \\
 \hline
~$1$~ & $\times$ & $\times$ &  &  & &  \\
~$2$~ & $\times$ &  & &  &  & $\times$ \\
~$3$~ &  & & $\times$ & & $\times$ & $\times$ \\
\hline
\multicolumn{1}{c|}{} & \multicolumn{3}{c||}{$\alpha$} & \multicolumn{3}{c}{$\beta$} \\
\end{tabular}
}
& $\Longrightarrow$ &
\parbox{0.5\textwidth}{\centering
\begin{tabular}{c|ccc||ccc||ccc||ccc}
 & ~$a$~ & ~$b$~ & ~$c$~ & ~$a$~ & ~$b$~ & ~$c$~ & ~$a$~ & ~$b$~ & ~$c$~ & ~$a$~ & ~$b$~ & ~$c$~ \\
 \hline
~$1$~ & $\times$ & $\times$ &  &  & & &  & & &  & &  \\
~$2$~ & $\times$ &  & &  &  &  &  & & $\times$ &  & & \\
~$3$~ &  & & & & $\times$ & $\times$ &  & & $\times$ &  & & $\times$ \\
\hline
\multicolumn{1}{c|}{} & \multicolumn{3}{c||}{$\alpha$} & \multicolumn{3}{c||}{$\beta$} & \multicolumn{3}{c||}{$\gamma$} & \multicolumn{3}{c}{$\delta$} \\
\end{tabular}
}

\end{tabular}
\caption{\label{fig:structurality}Transformation of a triadic context in a way that minimises the number of contextual implications with a non-empty support without changing the equivalence class of the associated concept lattice.}
\end{figure}

\subsection{A Generalisation of Boolean Concept Lattices\label{sec:boolCL}}

We believe that studying such equivalence classes of concept $n$-lattices, instead of individual $n$-context/$n$-lattice pairs, is the way to go. We suggest to start with classes that seem to be particularly interesting as they generalise Boolean concept lattices. Let $\m B_{j_1,\dots,j_{n-1}}$ be the class of concept $n$-lattices such that every possible feature on $n-1$ dimensions of size $j_1,\dots,j_{n-1}$ appear in an $n$-concept. For instance, Fig.~\ref{fig:extrem} depicts the $3$-context and $3$-concepts of a member of $\m B_{3,3}$. All rectangles in a $3\times3$ table appear as the feature of a $3$-concept (the empty rectangle appears twice as $(\{1,2,3\},\emptyset)$ and $(\emptyset,\{a,b,c\})$.

\begin{figure}[htb]
\centering
\begin{tabular}{c|ccc||ccc||ccc||ccc||ccc||ccc}
     & ~$a$~ & ~$b$~ & ~$c$~ & ~$a$~ & ~$b$~ & ~$c$~ & ~$a$~ & ~$b$~ & ~$c$~ & ~$a$~ & ~$b$~ & ~$c$~ & ~$a$~ & ~$b$~ & ~$c$~ & ~$a$~ & ~$b$~ & ~$c$~  \\
     \hline
    ~$1$~ & \x & \x & & \x & & \x & & \x & \x & \x & \x & \x & \x & \x & \x & & &  \\
    ~$2$~ & \x & \x & & \x & & \x & & \x & \x & \x & \x & \x & & & & \x & \x & \x  \\
    ~$3$~ & \x & \x & & \x & & \x & & \x & \x & & & & \x & \x & \x & \x & \x & \x  \\
    \hline
    \multicolumn{1}{c|}{} & \multicolumn{3}{c||}{$\alpha$} & \multicolumn{3}{c||}{$\beta$} & \multicolumn{3}{c||}{$\gamma$} & \multicolumn{3}{c||}{$\delta$} &
    \multicolumn{3}{c||}{$\epsilon$} &
    \multicolumn{3}{c}{$\zeta$}
\end{tabular}
\adjustbox{max width = \textwidth}{
\begin{tabular}{rrrr}
\multicolumn{2}{c}{$(\{\alpha,\beta,\gamma,\delta,\epsilon,\zeta\},\emptyset,\{a,b,c\})$} &
\multicolumn{2}{c}{$(\{\alpha,\beta,\gamma,\delta,\epsilon,\zeta\},\{1,2,3\},\emptyset)$} \\
$(\{\alpha,\beta,\delta,\epsilon\},\{1\},\{a\})$ &
$(\{\alpha,\gamma,\delta,\epsilon\},\{1\},\{b\})$ &
$(\{\beta,\gamma,\delta,\epsilon\},\{1\},\{c\})$ &
$(\{\alpha,\delta,\epsilon\},\{1\},\{a,b\})$  \\
$(\{\beta,\delta,\epsilon\},\{1\},\{a,c\})$ &
$(\{\gamma,\delta,\epsilon\},\{1\},\{b,c\})$ &
$(\{\delta,\epsilon\},\{1\},\{a,b,c\})$ &
$(\{\alpha,\beta,\delta,\zeta\},\{2\},\{a\})$ \\
$(\{\alpha,\gamma,\delta,\zeta\},\{2\},\{b\})$ &
$(\{\beta,\gamma,\delta,\zeta\},\{2\},\{c\})$ &
$(\{\alpha,\delta,\zeta\},\{2\},\{a,b\})$ &
$(\{\beta,\delta,\zeta\},\{2\},\{a,c\})$ \\
$(\{\gamma,\delta,\zeta\},\{2\},\{b,c\})$ &
$(\{\delta,\zeta\},\{2\},\{a,b,c\})$ &
$(\{\alpha,\beta,\epsilon,\zeta\},\{3\},\{a\})$ &
$(\{\alpha,\gamma,\epsilon,\zeta\},\{3\},\{b\})$ \\
$(\{\beta,\gamma,\epsilon,\zeta\},\{3\},\{c\})$ &
$(\{\alpha,\epsilon,\zeta\},\{3\},\{a,b\})$ &
$(\{\beta,\epsilon,\zeta\},\{3\},\{a,c\})$ &
$(\{\gamma,\epsilon,\zeta\},\{3\},\{b,c\})$ \\
$(\{\epsilon,\zeta\},\{3\},\{a,b,c\})$ &
$(\{\alpha,\beta,\delta\},\{1,2\},\{a\})$ &
$(\{\alpha,\gamma,\delta\},\{1,2\},\{b\})$ &
$(\{\beta,\gamma,\delta\},\{1,2\},\{c\})$ \\
$(\{\alpha,\delta\},\{1,2\},\{a,b\})$ &
$(\{\beta,\delta\},\{1,2\},\{a,c\})$ &
$(\{\gamma,\delta\},\{1,2\},\{b,c\})$ &
$(\{\delta\},\{1,2\},\{a,b,c\})$ \\
$(\{\alpha,\beta,\epsilon\},\{1,3\},\{a\})$ &
$(\{\alpha,\gamma,\epsilon\},\{1,3\},\{b\})$ &
$(\{\beta,\gamma,\epsilon\},\{1,3\},\{c\})$ &
$(\{\alpha,\epsilon\},\{1,3\},\{a,b\})$ \\
$(\{\beta,\epsilon\},\{1,3\},\{a,c\})$ &
$(\{\gamma,\epsilon\},\{1,3\},\{b,c\})$ &
$(\{\epsilon\},\{1,3\},\{a,b,c\})$ &
$(\{\alpha,\beta,\zeta\},\{2,3\},\{a\})$ \\
$(\{\alpha,\gamma,\zeta\},\{2,3\},\{b\})$ &
$(\{\beta,\gamma,\zeta\},\{2,3\},\{c\})$ &
$(\{\alpha,\zeta\},\{2,3\},\{a,b\})$ &
$(\{\beta,\zeta\},\{2,3\},\{a,c\})$ \\
$(\{\gamma,\zeta\},\{2,3\},\{b,c\})$ &
$(\{\zeta\},\{2,3\},\{a,b,c\})$ &
$(\{\alpha,\beta\},\{1,2,3\},\{a\})$ &
$(\{\alpha,\gamma\},\{1,2,3\},\{b\})$ \\
$(\{\beta,\gamma\},\{1,2,3\},\{c\})$ &
$(\{\alpha\},\{1,2,3\},\{a,b\})$ &
$(\{\beta\},\{1,2,3\},\{a,c\})$ &
$(\{\gamma\},\{1,2,3\},\{b,c\})$ \\
\multicolumn{4}{c}{$(\emptyset,\{1,2,3\},\{a,b,c\})$}
\end{tabular}
}
\caption{\label{fig:extrem}$3$-context and $3$-concepts of a concept $3$-lattice in $\m B_{3,3}$.}
\end{figure}

The $n$-lattices in $\m B_{j_1,\dots,j_{n-1}}$ contain $(\prod_{i=1}^{n-1} 2^{j_i}-1)+n-1$ concepts. They generalise the Boolean concept lattices $\m B_{j_1}$ as they are extremal in the sense that all possible features appear in concepts, they do not contain any structural implications (contextual implications may exist) and one of the corresponding $n$-contexts resembles the contranominal scale as, for each object, an element of a dimension is completely missing (see Fig~\ref{fig:extrem}). As such, it is always possible, in a context in which the number of objects is at least equal to the sum of the sizes of all the other dimensions, to produce such an extremal $n$-lattice. It appears that $\m B_{2,2}$ can exist on $3$ objects (Fig~\ref{fig:b22}) while $\m B_{3,3}$ cannot exist on $5$ (see Fig.~\ref{fig:b33} for another such extremal $3$-context on $6$ objects). It is currently unknown whether $\m B_{2,2}$ is an exception or other extremal $n$-lattices can exist on one less object. This problem of finding ``maximally compact'' $n$-contexts producing specific $n$-lattices seems relevant as it is tied to other potentially interesting problems such as finding the maximal number of $n$-concepts in an $n$-context of a given size or maximising the number of contextual implications.

\begin{figure}[htb]
\centering
\begin{tabular}{c|cc||cc||cc}
 & ~$a$~ & ~$b$~ & ~$a$~ & ~$b$~ & ~$a$~ & ~$b$~ \\
 \hline
~$1$~ & \x & \x & & \x & \x & \\
~$2$~ & \x & & \x & \x & & \x \\
\hline
\multicolumn{1}{c|}{} & \multicolumn{2}{c||}{$\alpha$} & \multicolumn{2}{c||}{$\beta$} & \multicolumn{2}{c}{$\gamma$}
\end{tabular}
\caption{\label{fig:b22}A $3$-context producing a member of $\m B_{2,2}$ with only three objects.}
\end{figure}

\begin{figure}[htb]
    \centering
\begin{tabular}{c|ccc||ccc||ccc||ccc||ccc||ccc}
     & ~$a$~ & ~$b$~ & ~$c$~ & ~$a$~ & ~$b$~ & ~$c$~ & ~$a$~ & ~$b$~ & ~$c$~ & ~$a$~ & ~$b$~ & ~$c$~ & ~$a$~ & ~$b$~ & ~$c$~ & ~$a$~ & ~$b$~ & ~$c$~  \\
     \hline
    ~$1$~ & \x & \x &\x &    & \x & \x & \x & \x & \x & \x & \x &    & \x &    & \x & \x & \x & \x  \\
    ~$2$~ & \x & \x &\x & \x & \x & \x & \x &    & \x & \x & \x & \x &    & \x & \x & \x & \x &     \\
    ~$3$~ & \x & \x &   & \x & \x & \x & \x & \x & \x &    & \x & \x & \x & \x & \x & \x &    & \x  \\
    \hline
    \multicolumn{1}{c|}{} & \multicolumn{3}{c||}{$\alpha$} & \multicolumn{3}{c||}{$\beta$} & \multicolumn{3}{c||}{$\gamma$} & \multicolumn{3}{c||}{$\delta$} &
    \multicolumn{3}{c||}{$\epsilon$} &
    \multicolumn{3}{c}{$\zeta$}
\end{tabular}

\caption{\label{fig:b33}Another $3$-context producing a member of $\m B_{3,3}$ with six objects.}
\end{figure}

\FloatBarrier

\section{The Counter-Curse of Dimensionality\label{sec:countingStuff}}

A natural question one can ask themself about such structures as lattices and $n$-lattices is their maximal size. As discussed in the previous subsection, $n$-contexts in which a dimension is as big as the sum of all the others can contain an extremal $n$-lattice, the size of which is known. The question remains open for all other $n$-contexts.

Given an $n$-context $\contn$ with dimensions of equal size $s$, how many $n$-concepts might that context contain, at most?
We call that number $f_n(s)$.
In this section, we discuss this question, first in the bidimensional case where it is already elegantly solved, and then in the general case where it is not.
The state of the art on this question is presented in Fig.~\ref{fig:regionWhereMaxLivesnd}, at the end of the section.

\subsection{All is Clear in 2 Dimensions}

It is well known that $2$-lattices with $2^s$ elements -- powerset lattices, or Boolean lattices -- can be constructed from $2$-contexts of size $s\times s$, ($2$-)contranominal scales.
The $2$-contranominal scale with $s$ objects and $s$ attributes is denoted $\mathbb N_2^c(s)$.
An example of $\mathbb N_2^c(4)$ and its concept lattice is shown in Figure~\ref{fig:2contraScale}.

\begin{figure}[htb]
\input{2contraScale}
\caption{The contranominal scale $\mathbb N_2^c(4)$ and its associated $2$-lattice, a \emph{tesseract} (only the extents are represented).\label{fig:2contraScale}}
\end{figure}

In~\cite{albano2017concept}, Albano and Chornomaz show that not only do $2$-dimensional contranominal scales $\mathbb N_2^c(s)$ give rise to powerset lattices, but that they are the sole culprit in the large size a concept $2$-lattice can have.
A \emph{large} $2$-lattice has as standard context a context in which lurks a \emph{large} contranominal scale.
In their paper, Albano and Chornomaz quantify exactly what \emph{large} means in both its uses.

\subsection{General Case: What About Dimension $n$?}

First, we can identify a naive upper bound on the maximal number of concept in a $n$-context of size $s\times \dots \times s$.
A $n$-concept is always uniquely described by $(n-1)$ of its components, and as such $( 2^{s}-1)^{n-1}+n-1$ is an upper bound for $f_n(s)$.

Before giving a lower bound for $f_n(s)$, it is necessary to recall some information on the construction of contexts.

The direct sum of two contexts $\mathcal C^1=(\mathcal S_1^1,\dots,S_n^1,\mathcal R^1)$ and $\mathcal C^2=(\mathcal S_1^2,\dots,\mathcal S_n^2,\mathcal R^2)$ where the $S_i^1, S_i^2$ are disjoint is the context $\mathcal C =(\mathcal S_1,\dots,\mathcal S_d,\mathcal R)$ constructed in the following way:

\[\mathcal S_i=\mathcal S_i^1\cup\mathcal S_i^2, \forall i\in\{1,\dots,n\}\]

and
$$\m R=\m R^1\cup \m R^2\cup\{(x_1,\dots,x_d)\mid \exists i, j\in\{1,\dots, n\}, i\neq j \mbox{ such that } x_i\in\mathcal S_i^1, x_j\in\mathcal S_j^2\}.$$

The number of concepts in $\m C$ is then the product of the number of concepts in $\m C^1$ and in $\m C^2$.
This construction is illustrated in two and three dimensions in Fig.~\ref{fig:directSum}.

\begin{figure}[htb]
\centering
\input{directSum}
\caption{Illustration of a combination of concepts under a direct sum of two contexts $\m C^1$ and $\m C^2$ in two dimensions (left) and three dimensions (right). Any two concepts of $\m C^1$ and $\m C^2$  -- a maroon and a green one -- can be extended into a new concept of $\m C$ -- through the gray areas --, thus multiplying their number of concepts.\label{fig:directSum}}
\end{figure}

An empty one-cell $n$-context $\m C_{ell} = (1,\dots, 1, \emptyset)$ gives rise to exactly $n$ $n$-concepts.
Adding together $s$ such empty $n$-contexts gives rise to the $n$-contranominal scale $\mathbb N^c_n(s)$, that is an $n$-context of the form $(\m S,\dots,\m S, R)$ with $R = \prod \m S\setminus \{(x,\dots,x)\mid x\in\m S\}$ that has $n^{|\m S|}$ concepts.
This is a lower bound for $f_n(s)$.

Those two bounds are identical in two dimensions, but are increasingly distant with higher dimensions.

\subsection{In Three Dimensions}

There is no clear answer yet for the value of $f_3(s)$.
The two bounds discussed in the previous section amount to looking for $f_3(s)$ between $3^s$ and $4^s$.

The well known triadic contranominal scale and the associated powerset trilattice were studied by Biedermann~\cite{biedermann1998powerset}.
Powerset trilattices are presented as a triadic generalisation of powerset lattices, can be constructed from $3$-contranominal scales $\mathbb N_3^c(s) = (\m S, \m S, \m S, \neq)$ and have $3^s$ elements.
No explicit claim that this class of $3$-lattices is extremal is made, other than the use of the name powerset~\footnote{In~\cite{biedermann1998powerset}, the name powerset trilattice is justified by some algebraic properties of the powerset trilattice.}.

Although the $3$-contexts responsible for powerset trilattices are good generalisations of contranominal scales, the $3$-lattices themselves are not that good of a generalisation of the extremal $2$-dimensional case, since they are not extremal w.r.t. the size.
This is discussed in Section~\ref{sec:boolCL}.

In~\cite{bazin:hal-01847459}, the authors present bounds for the maximum number of $3$-concepts in a $3$-context of size $s\times s\times s$.
Using a small example of a $5\times 5\times 5$ $3$-context that contains more $3$-concepts than a $3$-contranominal scale and the direct sum of contexts, they provide a construction of arbitrarily large contexts with $3.359^s$ concepts.
Using a measure and conquer approach, they also provide an upper bound of $3.384^s$ for $3$-contexts of that size.
The improvement from the naive bounds is represented in Figure~\ref{fig:regionWhereMaxLives3d}.

\begin{figure}[htb]
\centering
\input{regionWhereMaxLives3d}
\caption{While either naive bounds would be a satisfying answer for the maximum number of $3$-concepts in a $3$-context, the real answer seems to be in a small corner near  $3.36^s$.\label{fig:regionWhereMaxLives3d}}
\end{figure}

\subsection{In Four Dimensions}

Now, we extend the work presented in~\cite{bazin:hal-01847459} and provide a lower bound on $f_4(s)$.
The two bounds discussed in the previous section amount, in four dimensions, to looking for $f_4(s)$ between $4^s$ and $8^s$.
Using the same intuition as in~\cite{bazin:hal-01847459}, we build a $3\times3\times3\times3$ $4$-context where the "holes" are a solution to the chess rook problem in $4$ dimensions~\cite{dudeney1970eightrooks}.
This context\footnote{A data file containing the complement of this contexts (its holes) is available at \url{http://giacomo.kahn.science/resources/Crook.dat}} is shown in Table~\ref{tab:4contRook}.

\begin{table}
\centering
\input{4contRook}
\caption{This is a $3\times 3\times 3 \times 3$ $4$-context that we call $\m C_{rook}$. Empty cells are coloured blue for readability. This $4$-context has 112 $4$-concepts. This amounts to around $4.82^3$ concepts. This is, just as in the three dimensional case, a solution to a multidimensional chess rook problem.\label{tab:4contRook}}
\end{table}

Using the context $\m C_{rook}$ depicted in Table~\ref{tab:4contRook}, we can build arbitrarily large $4$-contexts that have $c4.82^s$ concepts, with $c$ a constant.

Let $s$ be a integer greater than $3$.
Then, there exists two integers $k$ and $r$ such that $s= 3k+r$, with $r\in[0,2]$.
To build a $4$-context $\m C$ of size $s \times s \times s\times s$, we add $k$ versions of $\m C_{rook}$ with the aforementioned procedure, and then add a $4$-dimensional contranominal scale of size $r$.
The resulting context has $4^r\times 4.82^{s-r} = (\frac{4}{4.82})^r\times 4.82^s$ concepts.
By setting $c = (\frac{4}{4.82})^2$, we have that $f_4(s)\geq c4.82^s$.

\subsection{Summary}

Figure~\ref{fig:regionWhereMaxLivesnd} summarises the known results related to the size of $n$-lattices, taking into account the contexts from Section~\ref{sec:boolCL} and Section~\ref{sec:countingStuff}.

\begin{figure}[htb]
\centering
\input{regionWhereMaxLivesnd}
\caption[Summary of the known bounds for $f_n(s)$]{\emph{"Old Tom Bombadil is a merry fellow! Bright Blue his jacket is, and his boots are yellow!"}. In the general case, we do not really know where the maximum number of $n$-concepts lives. This uncertainty is represented here by a dancing Tom Bombadil. This figures summarizes the known bounds for $f_n(s)$. In two dimensions the bound is well known and reachable, in three dimension the gap is small.\label{fig:regionWhereMaxLivesnd}}
\end{figure}

\bibliographystyle{unsrt}

redpajama_set_name: RedPajamaArXiv


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