Record sealing is the process of making public records inaccessible to the public. In many cases, a person with a sealed record gains the legal right to deny or not acknowledge anything to do with the arrest and the legal proceedings from the case itself. Records are commonly sealed in a number of situations: Sealed birth records (typically after adoption or determination of paternity) Juvenile criminal records may be sealed Other types of cases involving juveniles may be sealed, anonymized, or pseudonymized ("impounded"); e.g., child sex offense or custody cases Cases using witness protection information may be partly sealed Cases involving trade secrets Cases involving state secrets == Filing under seal in US court == Normally, records should not be filed under seal without a court permission. However, FRCP 5.2 requires that sensitive text – like Social Security number, Taxpayer Identification Number, birthday, bank accounts, and children’s names – should be redacted off the filings made with the court and accompanying exhibits. A person making a redacted filing can file an unredacted copy under seal, or the Court can choose to order later that an additional filing be made under seal without redaction. Alternately, the filing party may ask the court’s permission to file some exhibits completely under seal. When the document is filed "under seal", it should have a clear indication for the court clerk to file it separately – most often by stamping words "Filed Under Seal" on the bottom of each page. Person making filing should also provide instructions to the court clerk that the document needs to be filed "under seal". Courts often have specific requirements to these filings in their Local Rules. == Difference from expungement == Expungement, which is a physical destruction, namely a complete erasure of one's criminal records, and therefore usually carries a higher standard, differs from record sealing, which is only to restrict the public's access to records, so that only certain law enforcement agencies or courts, under special circumstances, will have access to them. A record seal will greatly improve the chance of employment, as employers will not have access to damning records. There are occasions, like expungement, where one can truthfully state under oath that they have never been convicted before. Most of the time, a record seal has more relaxed requirements than an expungement. If an expungement is not allowed with a case, then sealing a record may be the best bet. Different states have different terms for what constitutes sealing of a record. == Cybersecurity incidents involving sealed records == Several cybersecurity incidents have demonstrated that sealed court documents are not always secure in practice, with vulnerabilities and data breaches exposing sensitive information. In January 2021, following the SolarWinds cyber attack, the U.S. Bankruptcy Court United States District Court for the District of Nevada announced that its Case Management/Electronic Case Files CM/ECF system had been potentially compromised. The judiciary stated that additional safeguards were being implemented to protect filings, and that the review of the incident and its impact was ongoing. Reports noted that the breach raised concerns about exposure of highly sensitive and sealed documents submitted through the CM/ECF system. In 2023, security researcher Jason Parker, following a tip from an activist, identified flaws in online court systems that exposed sealed records including confidential testimony and medical records through publicly accessible portals. In 2024, a cyber intrusion targeting attorneys in a civil case involving Representative Matt Gaetz led to the unauthorized access and leak of sealed depositions and related records. The breach exposed confidential testimony and financial records, some of which were later reported by news outlets, raising concerns about the security of electronically stored legal materials and the handling of sealed filings. In 2025, multiple reports confirmed that the federal judiciary's CM/ECF and PACER (law) filing system was compromised, exposing sealed indictments, confidential informant information, and other sensitive filings. Some courts temporarily reverted to paper-based filing to mitigate the risks of further disclosure. The FBI later confirmed that the breach had exposed sealed records, and investigators suspected foreign state actors were involved. == GAO publications referencing sealed records == Closed Criminal Plea and Sentencing Proceedings (1983) – Reviewed Department of Justice policies on closing plea and sentencing hearings. GAO noted that sealed transcripts should be unsealed once the reasons for closure no longer applied. Information on Plea Agreements and Settlements in Defense Procurement Fraud Cases (1992) – Examined outcomes of procurement fraud prosecutions. GAO observed that in some instances the results were sealed from public access. Military Recruiting: More Needs to Be Done to Better Screen Applicants and Detect Fraud (1999) – Investigated fraudulent enlistments in the armed forces. The report highlighted that sealed juvenile records often prevented recruiters from discovering prior offenses. Social Security Numbers: Governments Could Do More to Reduce Display in Public Records (2004) – Analyzed risks associated with SSN availability in state and local records. GAO pointed out that some categories of records, such as adoption proceedings, were sealed and less likely to expose identifiers. Social Security Numbers: Stronger Safeguards Needed to Protect Privacy (2005 testimony) – Testimony before Congress reiterating concerns over SSN exposure in public records, while noting that sealed categories (e.g., adoption) were exceptions. U.S. Supreme Court: Policies and Perspectives on Video and Audio Coverage of Appellate Court Proceedings (2016) – Surveyed appellate court policies on courtroom media coverage. The report acknowledged distinctions between public filings, confidential submissions, and sealed materials. Evictions: National Data Are Limited and Challenging to Collect (2024) – Examined nationwide eviction data. GAO reported that in some states eviction records may be sealed or expunged, limiting researchers' ability to compile datasets. DOD Fraud Risk Management: Enhanced Data and Collaboration Could Improve Efforts (2024) – Reviewed Department of Defense fraud-risk management. GAO noted that some adjudicative records in its dataset were sealed, restricting completeness of oversight data.
Clubhouse (app)
Clubhouse is an American social audio app for iOS and Android developed by Alpha Exploration Co. that enables users to participate in real-time, audio-only communication within virtual "rooms". Launched in March 2020 by Paul Davison and Rohan Seth, the platform is characterized by its "drop-in" nature, where users can join live discussions on a wide range of topics as either listeners or speakers. The application gained attention in early 2021, operating on an invite-only model and featuring appearances from public figures such as Elon Musk, Oprah Winfrey, and Mark Zuckerberg. During this period, Clubhouse reached a reported valuation of approximately $4 billion and contributed to the expansion of similar social audio features like Twitter Spaces and Spotify Greenroom. The app later expanded to Android in May 2021 and removed its waitlist in July 2021, opening access to the general public. == History == Clubhouse began as an invite only social media startup by Paul Davison and Rohan Seth in Fall 2019. Originally designed for podcasts with the name Talkshow, the app was rebranded as "Clubhouse" and officially released for the iOS operating system in March 2020 and as of May 2021 the Android systems as well. Clubhouse was valued at $100 million after receiving funding from notable angel investors. These investors included Ryan Hoover (Founder, Product Hunt), Balaji Srinivasan (Former CTO, Coinbase), James Beshara (Co-Founder, Tilt.com), and several venture capitalists, including a $12 million Series A investment from the venture capital firm, Andreessen Horowitz, in May 2020. The app gained popularity in the early months of the COVID-19 pandemic. It had 600,000 registered users by December 2020. In January 2021, CEO Paul Davison announced that the active weekly user base on the app consisted of approximately 2 million individuals. The company announced that it would start working on an Android version of the app. In that month, the app became widely used in Germany when German podcast hosts Philipp Klöckner and Philipp Gloeckler began an invite-chain over a Telegram group. It brought German influencers, journalists, and politicians to the platform. Clubhouse raised their Series B at a $1 billion valuation. On February 1, 2021, Clubhouse had an estimated 3.5 million downloads on a global level which grew rapidly to 8.1 million downloads by February 15. This significant growth in popularity was because celebrities such as Elon Musk and Mark Zuckerberg made appearances on the app. In the same month, Clubhouse hired an Android Software Developer. A year after the app's release, the number of weekly active users was greater than 10 million, but the user base declined 21% during three weeks from late February to early March. This decline was reportedly caused by a decrease in the number of Clubhouse users after its initial release. During its initial roll out, the app was accessible only by invitation, and invitation codes on eBay were selling at up to $400. On April 5, 2021, Clubhouse partnered with Stripe to launch its first monetizing feature called Clubhouse Payments. Although testing began with only 1,000 users, after a week, the company rolled out the functionality to another 60,000 or more users in the US. In the same month, Twitter entered in discussions to purchase Clubhouse for $4 billion. The talks ended with no acquisition. Later, the company raised their Series C round of funding at a $4 billion valuation. The app also received interest in a partnership, with the National Football League announcing a content deal that month; Twitter Spaces later poached Clubhouse's exclusive NFL deal with 20 official NFL Spaces scheduled for the 2021-22 season. Finally, On May 9, 2021, Clubhouse launched a beta version of the Android app for users in the US, and on May 21, 2021, Clubhouse became available worldwide for Android users. In July 2021, Clubhouse announced a partnership with TED to offer exclusive talks. and on July 21, 2021, the company discarded its invitation system and made the application available to all, though a wait list for registration was still applied in order to manage new traffic. As of the time of the announcement, the company stated it had 10 million users on the wait list. On September 23, 2021, the company announced a new feature named "Wave". In October 2021, Clubhouse rolled out new features called "Replays and Clips". In April 2023, the company announced it was reducing its staff by half amid a "resetting" due to post-pandemic market shifts. == Features == === Rooms === The primary feature of Clubhouse is real-time virtual "rooms" in which users can communicate with each other via audio. Rooms are divided into different categories based on levels of privacy. Moderator roles are denoted by a green star that appears next to the user's name. When a user joins a room, they are initially assigned to the role of a "listener" and cannot unmute themselves. Listeners can notify the moderators of their intent to join the stage and speak by clicking on the "raise hand" icon. Users who are invited to the stage become "speakers" and can unmute themselves. Users can exit a room by tapping the "leave quietly" button or with the help of peace sign emoji. === Houses === In August 2022, Clubhouse announced a feature called Houses, an invite-based version of the rooms. === Events === A lot of conversations in Clubhouse are of spontaneous nature. However, users can schedule conversations by creating events. While scheduling an event, users can first name the event and then set the date and time at which the conversation will begin. Users can also add co-hosts to help moderate the event. Once the event has been created, it is added to the Clubhouse "bulletin". The bulletin shows upcoming scheduled events and allows users to set notifications for events by clicking the bell icon corresponding to the event. Users can access the bulletin by clicking on the calendar icon at the top of the home page. === Clubs === At the Clubhouse, clubs are user communities that regularly discuss a common interest. Many clubs are present in Clubhouse which represents a wide array of topics. Users can find clubs by name under the search tab. A club consists of three categories of users: "Admin", "Leader", and "Member". Members can create private rooms and invite more users into the club. Leaders have all the privileges of a member. Apart from that, they are authorized to create/schedule club-branded open rooms. An admin can modify club settings, add/delete users, change user privileges and create/schedule any type of room. There are three types of clubs: "Open", "By Approval", and "Closed" for membership. Any user can join an open club by pressing the "Join The Club" button on the club profile. In case of approval, users need to apply and wait for membership by clicking the "Apply To Join" button on the club profile. The admins of the respective club are privileged to accept or reject the user's request. In a closed club, membership is limited to users selected by the club admin. All users of a club will be notified when a public room within the club is created. The club creation is restricted to active users and whoever creates the club will become the club admin. Eligible users can create a club by going to their profile, press the "+" sign present in the "Member of" section. Clubs in which a user is a member are shown on their profile page. The first club to half a million members was the Human Behavior Club founded by The Digital Doctor (Dr. Sohaib Imtiaz). === Backchannel === Backchannel is the messaging function which allows users to interact individually or within a group via text. The Backchannel feature was initially leaked on June 18, 2021, in response to the launch of Spotify Greenroom. This is notable step because, until this point, Clubhouse was voice only with no way to hyperlink or message. It was entirely dependent on Instagram and Twitter for text messaging. The feature was initially leaked in the App Store, which the company says was an accident on Twitter. A month later, after multiple failed attempts, the Clubhouse Backchannel finally launched on July 14, 2021. === Explore === The homepage of Clubhouse provides access to ongoing chat rooms, which are recommended based on the people and clubs that are followed by the user. As the users tap on the magnifying glass icon, they will be redirected to the explore page. On that page, users can search for people and clubs to follow and also find conversations categorized by topics. === Clubhouse Payments === This is the direct payment service provided by the app, which allows users to send money to content creators. It includes those users who had enabled this functionality in their profile. Money can be sent from users to the creator by clicking on their profile. Press "Send Money" then enter the amount you want to send. When a user does this for the first time, they'll be prompted to reg
Immediate mode (computer graphics)
Immediate mode is an API design pattern in computer graphics libraries, in which the client calls directly cause rendering of graphics objects to the display, or in which the data to describe rendering primitives is inserted frame by frame directly from the client into a command list (in the case of immediate mode primitive rendering), without the use of extensive indirection – thus immediate – to retained resources. It does not preclude the use of double-buffering. Retained mode is an alternative approach. Historically, retained mode has been the dominant style in GUI libraries; however, both can coexist in the same library and are not necessarily exclusive in practice. == Overview == In immediate mode, the scene (complete object model of the rendering primitives) is retained in the memory space of the client, instead of the graphics library. This implies that in an immediate mode application, the lists of graphical objects to be rendered are kept by the client and are not saved by the graphics library API. The application must re-issue all drawing commands required to describe the entire scene each time a new frame is required, regardless of actual changes. This method provides on the one hand a maximum of control and flexibility to the application program, but on the other hand it also generates continuous work load on the CPU. Examples of immediate mode rendering systems include Direct2D, OpenGL and Quartz. There are some immediate mode GUIs that are particularly suitable when used in conjunction with immediate mode rendering systems. == Immediate mode primitive rendering == Primitive vertex attribute data may be inserted frame by frame into a command buffer by a rendering API. This involves significant bandwidth and processor time (especially if the graphics processing unit is on a separate bus), but may be advantageous for data generated dynamically by the CPU. It is less common since the advent of increasingly versatile shaders, with which a graphics processing unit may generate increasingly complex effects without the need for CPU intervention. == Immediate mode rendering with vertex buffers == Although drawing commands have to be re-issued for each new frame, modern systems using this method are generally able to avoid the unnecessary duplication of more memory-intensive display data by referring to that unchanging data (via indirection) (e.g. textures and vertex buffers) in the drawing commands. == Immediate mode GUI == Graphical user interfaces traditionally use retained mode-style API design, but immediate mode GUIs instead use an immediate mode-style API design, in which user code directly specifies the GUI elements to draw in the user input loop. For example, rather than having a CreateButton() function that a user would call once to instantiate a button, an immediate-mode GUI API may have a DoButton() function which should be called whenever the button should be on screen. The technique was developed by Casey Muratori in 2002. Prominent implementations include Omar Cornut's Dear ImGui in C++, Nic Barker's Clay in C and Micha Mettke's Nuklear in C.
Glow (app)
Glow is a fertility awareness and period-tracking app. It is part of a suite of mobile apps focused on women's reproductive health and childcare, which includes Eve by Glow (a dedicated period tracker), Glow Nurture (a pregnancy tracker), and Glow Baby (a baby development tracker). The Glow company also operates an online shop that sells several fertility-related products, including ovulation test strips, pregnancy tests, and wearable breast pumps. In 2024, Glow was reported to have approximately 25 million users across its various apps and community message boards. == History == Glow debuted in August 2013 as an iOS app. It was founded by Michael Huang and Max Levchin and launched with $6 million in Series A funding from venture capital firms Founders Fund and Andreesen Horowitz. In 2014, Glow raised an additional $17 million in Series B funding, with Formation 8 joining existing investors. In 2015, Glow launched Ruby, an app dedicated to sexual health. That year, Wired reported that the company had added features to their apps allowing men to monitor their fertility. Glow subsequently released an additional set of apps focused on pregnancy tracking and infant development. In 2016, Glow reported that it had a total of approximately 3 million users; by 2018, this had grown to 15 million. Vox described it as one of the “big two” period and fertility tracking apps and the one that had started the “boom” in the femtech space. == Application and features == Glow was initially described as a fertility application that applied data-driven methods to menstrual and ovulation tracking. Core features include cycle logging, ovulation prediction, and symptom tracking. The app also provides educational content related to reproductive health and childcare, as well as a set of online message boards that allow individuals to share experiences and seek peer support. == Privacy and legal issues == Glow has received significant media attention for its privacy and security practices. In 2016, Consumer Reports identified potential exploits in the Glow app that they claimed could have exposed private user data to hackers. Glow subsequently reported that it had fixed the vulnerabilities and told The Washington Post they had no evidence that user data had been compromised. In September 2020, the California Attorney General announced a settlement with Glow related to Consumer Reports’ findings, which included a $250,000 civil penalty. Following the US Supreme Court's 2022 Dobbs v. Jackson ruling, which legalized state-level bans on abortion, Glow (and other fertility trackers, such as Clue and Flo) came under additional scrutiny over concerns that user data on abortions could be reported to law enforcement. After this surge of media interest, a research team affiliated with the University of New South Wales conducted an investigation into the privacy practices of several popular fertility apps, including Glow. Their review of Glow was mixed, noting that they provided several privacy settings and de-identified sensitive data, but that user information could still be disclosed in the future if the app was sold. Glow rejected that claim, telling the Australian Associated Press that it "did not share" personal data. The company also cited several internal security measures it had implemented and its apps' offline data protection setting, which allows users to permanently delete their health-related data. == Reception == In 2014, Fast Company reported that 20,000 women had used Glow to conceive. Later that year, The Guardian included Glow Nurture on its list of the best iPhone apps of 2014. Media coverage often praised Glow's array of menstrual tracking options, although some reviews also noted that fertility apps are not birth control tools and cautioned against relying on them for that purpose. In 2019, Cosmopolitan singled Glow's community of users as one of its standout features.
ParkMobile
ParkMobile is a mobile and web app providing parking payments in North America. Headquartered in Atlanta, Georgia, users can pay for on-street and off-street parking via app on their smartphone, web browser, or through calling a phone number. ParkMobile also offers parking reservations at stadiums or venues for concerts and sporting events, and in metro area garages. == History == ParkMobile was founded in the United States in 2008 by Albert Bogaard after originally starting in the Netherlands. The initial product served only zone (on-demand) parkers and payment for the parking spot was made via a phone call through an IVR system. In 2009, the ParkMobile app was released and the product launched in its first city, Grand Rapids, Michigan. Parking payments have since been accepted through a user's account by connecting a credit card. ParkMobile deployed in Washington, D.C., in 2011. As of 2023, ParkMobile now has over 50 million users. Parking reservations were introduced in 2017, allowing users to reserve parking in advance. In 2018, the company recapitalized with BMW as the shareholder. ParkMobile was then acquired by a joint venture with BMW and Daimler. Under this joint venture, ParkMobile parking payment functionality was available and integrated with BMW's navigation system in many of its 2018 models. EasyPark Group, the Swedish-based parking solutions company, acquired ParkMobile in 2021 and is the current owner rebranded as Arrive. In 2022, ParkMobile launched in the City of Boston with a city-wide parking app, ParkBoston, powered by ParkMobile. == Operations == === Products === ParkMobile's product offerings include zone (on-demand) parking payments, parking reservations, and a self-service reporting engine. Zone parking is the company's most widely used service. Users can use the app on their smartphone to pay parking fees. In 2017, ParkMobile began offering parking reservations. The service is provided in addition to on-demand parking options at stadiums and venues, as well as metro area parking garages. After launching the reservations feature, ParkMobile became the first mobile parking app provider in North America to have a consolidated app with both on-demand and reservations parking in one. ParkMobile 360, the company's self-service management and reporting platform for operators, launched in 2018. It is a web-based application for parking operators to manage parking inventory, adjust rates, create special parking events, and track analytics. In 2020, ParkMobile began offering an option to pay for parking with Google through integrating the ParkMobile experience with Google Maps In 2021, ParkMobile launched its web application, allowing users to complete their parking transactions directly from the mobile website without having to download the app or have an account. ParkMobile integrates with parking gate equipment so customers can use their app to pay for parking and scan to enter and exit the garage. === Locations === ParkMobile has over 50 million users across the United States, Canada, and Puerto Rico. The app is available in over 550 cities in the U.S. and over 150 colleges and universities. == Controversies == === Predatory towing and excessive ticketing === Since all paid parking sessions from a single supplier are able to be viewed together, the ease of viewing and enforcing parking violations has caused controversy. Parking Enforcement Services in Birmingham, Alabama, has been the subject complaints by users of the ParkMobile app who had paid for a parking session and still had their vehicle towed. Customers often use old or expired license plates and forget to update to the correct number, or mistype when entering their information into the ParkMobile app. The complaints are that the towing companies offer no lenience for these mistakes. They return to their car as the session expires, and find their car has been towed. Additionally, other municipality across the country have received complaints about excessive parking ticket issuing when inputting their information incorrectly in the ParkMobile app. In Stone Harbor, New Jersey, parking ticket violations increased by over 1,600% from the previous year since launching with the ParkMobile app. Police officers refute complaints of being "too strict" on writing tickets by admitting the ParkMobile system allows officers to "more seamlessly enforce" the city's parking laws. === Data security breach === In March 2021, ParkMobile suffered a cybersecurity incident "linked to a vulnerability in a third-party software," potentially exposing users' email addresses, phone numbers, and license plate numbers. ParkMobile responded by launching an investigation and notifying law enforcement authorities and affected municipalities. The investigation concluded "no sensitive data or Payment Card Information was affected" but ParkMobile confirmed that basic account information, such as license plate numbers and possibly email addresses or phone numbers, was accessed.
GitHub Codespaces
GitHub Codespaces is a cloud-based online integrated development environment developed by GitHub. It allows users to create and manage development environments directly within the browser or through Visual Studio Code desktop. Codespaces is tightly integrated with GitHub repositories and enables on-demand coding, debugging, and testing in a full-featured development container hosted in the cloud. == Features == Instant development environments integrated with GitHub Browser-based and desktop access via Visual Studio Code Configurable Dockerfile or devcontainer.json environments Built-in support for GitHub Copilot, extensions, snippets, and SSH. == Licensing == GitHub Codespaces is proprietary software and available to GitHub users under various subscription plans. Codespaces includes a monthly usage quota for free tier users of 120 hours, and expanded access for GitHub education, Pro, Team, and GitHub Enterprise plans. == GitHub Classroom == GitHub Classroom is an educational tool developed by GitHub to streamline the process of managing programming assignments and coursework. Integrated with GitHub repositories, it allows instructors to distribute starter code, automate grading workflows, and track student progress. GitHub Classroom is widely used in computer science education and supports integration with GitHub Codespaces for cloud-based development environments. == Programming languages supported == == Extensions == Some of the popular extensions include:
Space partitioning
In geometry, space partitioning is the process of dividing an entire space (usually a Euclidean space) into two or more disjoint subsets (see also partition of a set). In other words, space partitioning divides a space into non-overlapping regions. Any point in the space can then be identified to lie in exactly one of the regions. == Overview == Space-partitioning systems are often hierarchical, meaning that a space (or a region of space) is divided into several regions, and then the same space-partitioning system is recursively applied to each of the regions thus created. The regions can be organized into a tree, called a space-partitioning tree. Most space-partitioning systems use planes (or, in higher dimensions, hyperplanes) to divide space: points on one side of the plane form one region, and points on the other side form another. Points exactly on the plane are usually arbitrarily assigned to one or the other side. Recursively partitioning space using planes in this way produces a BSP tree, one of the most common forms of space partitioning. == Uses == === In computer graphics === Space partitioning is particularly important in computer graphics, especially heavily used in ray tracing, where it is frequently used to organize the objects in a virtual scene. A typical scene may contain millions of polygons. Performing a ray/polygon intersection test with each would be a very computationally expensive task. Storing objects in a space-partitioning data structure (k-d tree or BSP tree for example) makes it easy and fast to perform certain kinds of geometry queries—for example in determining whether a ray intersects an object, space partitioning can reduce the number of intersection test to just a few per primary ray, yielding a logarithmic time complexity with respect to the number of polygons. Space partitioning is also often used in scanline algorithms to eliminate the polygons out of the camera's viewing frustum, limiting the number of polygons processed by the pipeline. There is also a usage in collision detection: determining whether two objects are close to each other can be much faster using space partitioning. === In integrated circuit design === In integrated circuit design, an important step is design rule check. This step ensures that the completed design is manufacturable. The check involves rules that specify widths and spacings and other geometry patterns. A modern design can have billions of polygons that represent wires and transistors. Efficient checking relies heavily on geometry query. For example, a rule may specify that any polygon must be at least n nanometers from any other polygon. This is converted into a geometry query by enlarging a polygon by n/2 at all sides and query to find all intersecting polygons. === In probability and statistical learning theory === The number of components in a space partition plays a central role in some results in probability theory. See Growth function for more details. === In geography and GIS === There are many studies and applications where Geographical Spatial Reality is partitioned by hydrological criteria, administrative criteria, mathematical criteria or many others. In the context of cartography and GIS - Geographic Information System, is common to identify cells of the partition by standard codes. For example the for HUC code identifying hydrographical basins and sub-basins, ISO 3166-2 codes identifying countries and its subdivisions, or arbitrary DGGs - discrete global grids identifying quadrants or locations. == Data structures == Common space-partitioning systems include: BSP trees Quadtrees Octrees k-d trees Bins == Number of components == Suppose the n-dimensional Euclidean space is partitioned by r {\displaystyle r} hyperplanes that are ( n − 1 ) {\displaystyle (n-1)} -dimensional. What is the number of components in the partition? The largest number of components is attained when the hyperplanes are in general position, i.e, no two are parallel and no three have the same intersection. Denote this maximum number of components by C o m p ( n , r ) {\displaystyle Comp(n,r)} . Then, the following recurrence relation holds: C o m p ( n , r ) = C o m p ( n , r − 1 ) + C o m p ( n − 1 , r − 1 ) {\displaystyle Comp(n,r)=Comp(n,r-1)+Comp(n-1,r-1)} C o m p ( 0 , r ) = 1 {\displaystyle Comp(0,r)=1} - when there are no dimensions, there is a single point. C o m p ( n , 0 ) = 1 {\displaystyle Comp(n,0)=1} - when there are no hyperplanes, all the space is a single component. And its solution is: C o m p ( n , r ) = ∑ k = 0 n ( r k ) {\displaystyle Comp(n,r)=\sum _{k=0}^{n}{r \choose k}} if r ≥ n {\displaystyle r\geq n} C o m p ( n , r ) = 2 r {\displaystyle Comp(n,r)=2^{r}} if r ≤ n {\displaystyle r\leq n} (consider e.g. r {\displaystyle r} perpendicular hyperplanes; each additional hyperplane divides each existing component to 2). which is upper-bounded as: C o m p ( n , r ) ≤ r n + 1 {\displaystyle Comp(n,r)\leq r^{n}+1}