![]() ![]() ![]() ![]() You can buy ps subscription and play games like god of war 4, days gone some other great old titles but the monthly games sucks I haven't played I single monthly game may dirt 5 when my setup will be installed but my point here is that you end up spending 2k to 4k every month which means around 24k to 48k in a year.īut one thing is for sure you will never be disappointed after playing games on your ps5.īcz you lose track of your controller battery level and you don't want to pause your game in the middle of a boss fight and recharge your controller. When we got the tech demo up and running the whole studio came around and everyone wanted a go. Dirt 5 also utilises ps5's dual sense Controller features, including haptic feedback and. Dirt 5 Update Has Improved Visuals In 120 FPS Mode On Xbox Series Bringing It Equal To PS5 Ali Haider Dirt 5 has been updated on the Xbox Series consoles for the 120 FPS performance mode in order to address a bug that leads to a visual downgrade. Summary: On PS5, experience technical advances brought to life in an amplified off-road racer, such as faster loading times, greater graphical fidelity and a 120FPS option. It should also have HDR bcz that makes colour s to pop up. Dirt 5 will also have a 120FPS mode on PS5 LantWar Forums Discussion Gaming Forum 1 2 3 6 Next LantWar Classic Anus Game The Fallen 19,491 1. Mixed or average reviews based on 161 Ratings. The power of the PS5 also means that we will have an option for 120fps, which is very exciting for the racers who know that every frame counts, teases Rob Karp, Dirt 5’s development director. You have brought your ps5 but you should have a good display to show it's potential so I highly recommend this console to the people who have 4k tv or planning to buy a 4k tv DIRT 5 PlayStation 5 Codemasters, Deep Silver Release Date: Also On: PC, PlayStation 4, Stadia, Xbox One, Xbox Series X Summary Critic Reviews User Reviews Details & Credits Trailers & Videos Metascore 80 Generally favorable reviews based on 11 Critic Reviews What's this User Score 5. when I try to change the video options in dirt 5 to high frame rate mode for 120 FPS the cars look very dark with a metallic look and you can't see them in the game. I have the dirt 5 Amplified digital edition. DIRT 5 I own a Samsung q90t with a certified HDMI 2.1 cable. The smooth 60 fps and 120fps in some games like fortnite makes gaming so much PS5 Dirt 5 120fps High Frame Rate Mode Samsung Q90T. Instead you can buy a 4k oled tv and a ps5įor the same price and enjoy games at stunning visuals. You will end up spending more than 2 lakhs. Recorded at 1080p 120fps HDR.Click here to buy it. It must have RTX series GPU with minimum 8gb vram DIRT 5 (PS5) High Frame Rate Mode Gameplay 120FPS HDR - YouTube This is DIRT 5 on the PS5 running on the high frame rate mode. Were so demanding my pc maxed at 40 fps on medium set of graphics. It's been 2 months since I bought this thing.īefore I got my PS5 I was using my laptop for playing games on my tv but I was not able to play games with hassle free experience.I always have to down some settings to get stable frame rates. I am currently shifting so all my products are packed but the game Assassin creed vallhalla you are looking at is running on my ps5. ![]()
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![]() ![]() These techniques are used to create special-purpose diodes that perform many different functions. Semiconductor diodes in the forward direction also need to surpass a threshold voltage before being able to conduct electricity.Ī semiconductor diode's current–voltage characteristic can be tailored by selecting the semiconductor materials and the doping impurities introduced into the materials during manufacture. And its high resistance to current flowing in the reverse direction suddenly drops to a low resistance when the reverse voltage across the diode reaches a value called the breakdown voltage. For instance, a diode's forward-direction voltage drop varies only a little with the current, and is more so a function of temperature this effect can be used as a temperature sensor or as a voltage reference. However, diodes can have more complicated behavior than this simple on–off action, because of their nonlinear current–voltage characteristics. As rectifiers, diodes can be used for such tasks as extracting modulation from radio signals in radio receivers. This unidirectional behavior is called rectification and is used to convert alternating current (AC) to direct current (DC). As such, the diode can be viewed as an electronic version of a check valve. The most common function of a diode is to allow an electric current to pass in one direction (called the diode's forward direction), while blocking it in the opposite direction (the reverse direction). A common variant of a diode is a light-emitting diode, which is used as electric lighting and status indicators on electronic devices. The obsolete thermionic diode is a vacuum tube with two electrodes, a heated cathode and a plate, in which electrons can flow in only one direction, from cathode to plate.Īmong many uses, diodes are found in rectifiers to convert alternating current (AC) power to direct current (DC), demodulation in radio receivers, and can even be used for logic or as temperature sensors. Today, most diodes are made of silicon, but other semiconducting materials such as gallium arsenide and germanium are also used. The discovery of asymmetric electrical conduction across the contact between a crystalline mineral and a metal was made by German physicist Ferdinand Braun in 1874. Semiconductor diodes were the first semiconductor electronic devices. It has an exponential current–voltage characteristic. It has low (ideally zero) resistance in one direction, and high (ideally infinite) resistance in the other.Ī semiconductor diode, the most commonly used type today, is a crystalline piece of semiconductor material with a p–n junction connected to two electrical terminals. The filament itself may be the cathode, or more commonly (as shown here) used to heat a separate metal tube which serves as the cathode.Ī diode is a two- terminal electronic component that conducts current primarily in one direction (asymmetric conductance). Electron flow is the reverse of conventional current flow. In most diodes, a white or black painted band identifies the cathode into which electrons will flow when the diode is conducting. The square silicon crystal can be seen between the two leads. The anode is on the right side the cathode is on the left side (where it is marked with a black band). Short of foil i have experimented with conductive paint (and used the paint as a binder for foil ) with varying results but foils are better, as to which ( tin or copper) is better that's for others to debate.Close-up view of a silicon diode. With bridge grounds it becomes a crap shoot of conditions that stem from the fact most rely on pressure between the bridge and the wood body of the guitar which can shrink or the finish softens or becomes corroded due to moisture.Īnd most bridges are nickle plated and you can't solder to them so insuring a good mechanical connection becomes a bit of a trick. I think bad bridge grounds and lack of cavity shielding covers the majority of noise troubles with single coil guitars.ĭoes cavity shielding help? you bet your a** it does! even backside of the pick guard foil helps so long as it see's ground properly. After pouring a glass and thinking to myself "what would constitute the basics?" ![]() ![]() ![]() Luck hasn’t indicated a timetable for his return but doesn’t seem likely to be back on the field until October for the Colts, who were also missing center Ryan Kelly (out, foot) and star cornerback Vontae Davis (out, groin) last week. at 10:40 pm ET 2 min read Through 3 Quarters Their last head-to-head September of 2017 was close, and so far it looks like that's how the Indianapolis Colts and the Arizona. Jacoby Brissett, who drove the team to its lone score against the Rams, split time with Tolzien at practice during the week and Pagano may turn to their recent acquisition from the New England Patriots sooner rather than later. The Cardinals may also be minus speedy wide receiver John Brown, who missed practice during the week with a quad injury and is listed as questionable.ĪBOUT THE COLTS (0-1): Tolzien threw two pick-sixes last week and his stint under center may be short-lived. Cat’s record in print stands at 652-562 (53.8 percent) against the spread. David Johnson’s loss cannot be understated as he led the league with 2,118 yards from scrimmage last season and scored 20 touchdowns. 30 in NFL power ranking (ahead of Houston Texans, Indianapolis Colts) Jay Sanin wrote: 'The Arizona Cardinals are tied for the worst odds to win the Super Bowl alongside. The Wildcat returns for his 25th season in the Bettor’s Guide. the Lions) will start at running back on Sunday. pass from Carson Wentz (Michael Badgley kick), 7 3:03, ARI - Chase Edmonds 2 yd. ABOUT THE CARDINALS (0-1): Arizona brought back running back Chris Johnson earlier in the week to help fill the void created by David Johnson’s injury, but Kerwynn Williams (5 carries, 10 yards, TD vs. 22 CARDINALS 16 FINAL DecemCardinals QB Murray passes for 235 yds, 1 TD but can't get the W State Farm Stadium Glendale, AZ NFL > Scores > Indianapolis Colts vs Arizona. Scoring Summary 1st Quarter 6:11, IND - T.Y. ![]() ![]() ![]() Harris’s lack of achievements was not the result of her not having the capacity for the job, according to the book. In another Oval Office meeting, this time with Joe Manchin shortly after Harris did a television interview with a West Virginia station that Manchin interpreted as a power play, she seemingly snubbed Manchin, not offering a handshake or acknowledgment as she left the room so the West Virginia senator could meet privately with Biden. In one meeting with Republican senators intended to entice them to support bipartisan infrastructure legislation, she ended up chiding Roger Wicker of Mississippi about the state of the economy under Donald Trump, leading to Cabinet secretaries wanting to alternately pump their fists and cover their eyes. The result, Foer reports, is that Harris “lets the criticism guide her” and thus fails to rack up meaningful achievements.įoer, an author and staff writer at The Atlantic, describes an administration where Harris is a comparatively minor player and seemingly manages to always put the wrong foot forward despite her best efforts. ![]() The new book depicts Harris as having “rabbit ears” - acutely sensitive to “any hint of criticism” and “instantly aware” of the slightest bit of discontent. The book, which has been eagerly anticipated by officials in Washington, comes out on Tuesday. It is the deepest dive so far into a White House that has been far more guarded with the press and averse to leaks than past administrations. ![]() This revelation is one of many in Franklin Foer’s new book, The Last Politician: Inside Joe Biden’s White House and the Struggle for America’s Future, which is the result of two years of reporting inside the Biden administration. Eventually, a staffer caught wind of the gathering and she had to leave home and go back to work. The only person missing was Vice-President Kamala Harris, who had already left the White House for the day. In November 2021, top Biden-administration officials gathered in the residence of the White House as the House was about to finally vote on the Build Back Better legislation that had been their objective since taking office. ![]() ![]() ![]() Players may also set challenges for each other on the game's forums, such as visiting all five moons of Jool (the in-game analogue of Jupiter), or use mods to test each other's aircraft in air combat tournaments. ![]() Missions (either player-set or assigned "contracts") involve goals such as reaching a certain altitude, escaping the atmosphere, reaching a stable orbit, landing on a certain planetary body, rescuing stranded astronauts, capturing asteroids, and creating space stations and surface bases. Spacecraft maneuver 'nodes' can be plotted, timed, and executed, and objects can be selected and 'targeted' to facilitate flybys, rendezvous, and docking. To visualize vehicle trajectories, the player is provided a 'map' mode that displays the current vehicle's trajectory as well as that of celestial bodies and other spacecraft, as well as their orbital parameters. Provided it maintains sufficient thrust and fuel, a spacecraft can enter orbit or even travel to other celestial bodies. Players control flight with little assistance other than a Stability Assist System (SAS) to keep their rocket oriented. Constructed craft can be launched from the space center's launch pad or runway to accomplish various tasks while avoiding partial or catastrophic failure (such as lack of fuel or structural failure). From the space center players can build various vehicles such as rockets, aircraft, spaceplanes, and rovers from a provided set of components. The player administers a space program operated by Kerbals, a species of small green humanoids, who have constructed a spaceport on their home planet of Kerbin. In-game Kerbals, Valentina Kerman and Jebediah Kerman, standing on the launchpad ![]() Gameplay A rocket (preloaded ship "Kerbal X") sitting on the launchpad with the Vehicle Assembly Building and Mun, Kerbin's nearest moon, in the background. A sequel, Kerbal Space Program 2, was released in Early Access on February 24, 2023. Two expansions for the game have been released as downloadable content: Making History in March 2018 and Breaking Ground in May 2019. An Enhanced Edition was released for Xbox One and PlayStation 4 in January 2018, and for PlayStation 5 and Xbox Series X/S in September 2021 by Private Division, a publishing subsidiary of Take-Two Interactive. In May 2017, Squad announced that the game had been purchased by video game company Take-Two Interactive, who will help support Squad in keeping the console versions up-to-date alongside the personal computer versions. The game has garnered commendation from spaceflight industry figures such as NASA, ESA, science communicator Scott Manley, ULA CEO Tory Bruno, SpaceX CEO Elon Musk, and Rocket Lab CEO Peter Beck. Popular mods have received support and inclusion in the game by Squad. Kerbal Space Program has support for user-created mods that add new features. The game was released out of beta on 27 April 2015. The first public version was released digitally on Squad's Kerbal Space Program storefront on 24 June 2011, and joined Steam's early access program on 20 March 2013. The game features a realistic orbital physics engine, allowing for various real-life orbital maneuvers such as Hohmann transfer orbits and orbital rendezvous. In the game, players direct a nascent space program, staffed and crewed by green humanoid aliens known as "Kerbals". Kerbal Space Program ( KSP) is a space flight simulation video game developed by Mexican studio Squad for Linux, OS X, Windows, PlayStation 4, PlayStation 5, Xbox Series X/S and Xbox One. ![]() ![]() But, yes, most of the at-home tools will work to some degree, but only slightly-and not without risks.īasically, as long as you have pores on your face, you'll have to deal with them filling back up and forming blackheads (unless you change your skincare routine more on that below). Our top picks for the best blackhead removal tools of 2023:Īs to whether or not blackhead removal tools really work, it depends on your desired outcome and what tool you're referring to, because "blackhead removal tools are quite variable in their efficacy," says Dr. ![]() ![]() We updated this article in February 2023 to add new blackhead removal tools the internet won't stop talking about, deleted outdated oldies, and checked in with experts for the latest news and products, just for you. ✔️ P.S., This is new, up-to-date info, bb. But before you drop $$$ on some fad pore vacuum or ultrasonic skin scrubber that may or may not damage your skin, take a beat and keep reading, because I chatted with board-certified dermatologists Mona Gohara, MD, and Rachel Westbay, MD, for all the deets on the best blackhead removal tools and how to use them below. That said, there are products that can help you treat your blackheads safely (see: salicylic acid), and even a few blackhead removal tools endorsed by dermatologists. Because the reality is that nobody can have pore-less or perfectly clear skin all skin has texture and bumps, and all humans have blackheads (even if some have more than others). And I will also (hesitantly) admit that in order to get rid of clogged pores, I've indulged in some really bad habits, like poking and prodding at my face with blackhead removal tools until I'm left with inflamed skin-and the same blackheads I started with. ![]() I'll be the first to admit that I often dream of having clear, smooth skin without a single blackhead or pimple. ![]() ![]() ![]() Let's look at some of the best Black Butler characters that contribute to the show's unique appeal. ![]() ![]() Characters such as the flamboyant Grim Reaper Grell Sutcliff, the enigmatic Undertaker, and the loyal Phantomhive staff members bring interest and diversity. The series is known for its captivating blend of dark themes, humor, and supernatural elements. RELATED: Black Butler New Season Announced For 2024 Sebastian becomes Ciel's butler and bodyguard, always on hand and available. The story follows Ciel Phantomhive, a young aristocrat who makes a pact with a demon, Sebastian Michaelis, to avenge his parents' death. These characters add depth and appeal to the story with their complex backstories, unique abilities, loyal relationships, humorous moments, and intriguing dynamics, making Black Butler a must-watch for fans of the genre.īlack Butler is a popular anime and manga series set in Victorian-era England. Spears, Grell Sutcliff, Undertaker, Sebastian Michaelis, and Ciel Phantomhive. The series stands out for its dark themes, humor, and supernatural elements, brought to life by diverse and interesting characters like Alois Trancy, Mey-Rin, Bardroy, Finny, Elizabeth Midford, William T. This is a very clean easy to see theme and could become a favorite as it develops.Black Butler is a captivating anime and manga series set in Victorian-era England, featuring a young aristocrat named Ciel Phantomhive who seeks revenge with the help of his demon butler, Sebastian Michaelis. but where is the output routing button on the master track cheers henrik. I guess Im still missing a switch for input fx. You could have the record arm and monitor buttons show in the narrow states instead.īut you are on the right track so to speak. This will be my new default theme, thanks for the great work. In my opinion that item is real estate you could use for something else when the volume fader is visible. Third, I especially like how narrow you can make the tracks in the TCP, although I would like to see the volume fader change to a volume knob instead just showing the volume readout in db. The second is I'd like some more layouts for the MCP specifically the ability to indent folders and or add colour borders around them so they stand out better. ![]() This is a deal breaker for me - I use it all the time to slow multitrack orchestral midi down to find the one dissident note I hear. One issue for me is that the playback rate slider is missing form the transport bar. By the way, you installed reaper for this, remember?Īdded Playrate control and Time signature Im planning on making a 4k-version when the 1080-version is done. In general, I hope this theme will find its users and create comfortable conditions for creating new music. Its still a work-in-progress, and it only supports 1920x1080 yet. However, I was very attentive to details and was able to solve most of the problems so that everything worked smoothly and did not look clumsily. I am not a designer or a programmer, so I did not manage to implement everything that I wanted. It was also important for me to make the interface convenient and understandable so that it would not take long to get used to it. The colors in this theme are chosen so that it is not too colorful, not too contrasty, not too bright and not too dark. But I liked what it turned out, I wanted to share it with everyone and now, after many months, you see the result of my work here. It was hard, long and I gave it up a few times. However, I was very interested in customizing, and then I decided that I would create my own theme for reaper. When I first looked into the theme tweaker, I didn't know I would get this far. ![]() ![]() ![]() It’s a great resource for beginning and amateur photographers until your editing needs outgrow the limitations. You can use the limited version as long as you like for free! All the basic editing features you need are included in the free version. The full mobile app is included in the basic photography plan as well. For the premium features of the mobile version, you’ll have to buy a subscription. It comes with most of the features offered in Lightroom, but not all. The mobile version of Lightroom is free for anyone to use. Luckily, this next free way to use Lightroom doesn’t come with a limited trial run. Not very practical for long-term use, right? Okay, so free access to all of Lightroom’s features is cool and all…but it only lasts for 7 days. Once the trial is over, you can start a subscription to continue using all the features of the program. This is a risk-free way to find out if you like Lightroom or not. You can test-drive all the features of Lightroom, including Lightroom presets and other features that are included with the software. This 7-day trial gives you complete access to Lightroom. Once your free trial ends you may qualify for the 60% discount that Adobe offers on their All Apps subscription.įill out the form that pops up next with your details and you’re ready to download and use the trial version. If you are a student or teacher, switch to that tab. On the next screen, choose which version of Adobe’s subscriptions you would like to try out. You can take the quiz on this page to find out which is the best fit for you.īut for the free version, you’ll want to click on Free trial. If there are other Adobe apps that interest you, you might want one of the other bundles. There are various bundles that you can choose from that include different combinations of Adobe’s apps.įor example, the basic Photography plan includes Photoshop and both the desktop and mobile versions of Lightroom. You’ll arrive on the landing page that describes the key features of Lightroom.Īdobe offers Lightroom as part of its Creative Cloud subscription service. Go to the Adobe website and enter the Photographer’s section under the Creativity tab. The first method is to take advantage of the free 7-day trial that Adobe offers. It will be cheaper in the long run, I promise. ![]() Instead, I recommend you stick with the two legal ways to download Lightroom. You could end up with a virus that destroys your computer (or costs you a pretty penny to fix). If you scour the Internet, no doubt you’ll find various pirated versions of Lightroom that you can download. Two Ways to Legally Get Lightroom for Free Two Ways to Legally Get Lightroom for Free. ![]() ![]() ![]() You can upload files of up to 50 MB and export videos in 720p. The free account also gives you two hours of automatic translations, 20 GB storage, and unlimited projects. Most of all, you can get the majority of these features with the free account as long as you’re okay with the small VEED icon in the corner. What’s more, each feature is organized in an easy to study manner on the site with a vast knowledge bank and video tutorials that walk you through how to edit videos for YouTube with their tool. More on the plus size: the customer service team is a customer-applauded one. This helps you not only establish a solid brand identity but also save editing time. This way, all your video (and other projects) will feature the same branded elements. To boot, with the brand kit, you can add your custom fonts, colors, logo, watermarks, and more. ![]() VEED also offers collaboration on video creation so you can add teammates and edit YouTube videos with them for best results. You can also create GIFs from your YouTube videos – helpful for building your personal brand and marketing videos on social media. Other notable features include video frame speed converter, video cutter, video splitter, video joiners, and video effects and filters. Auto subtitling with subtitle editor and translator.Removing noise from video’s background audio.Adding pictures, emojis, and stickers to videos.This web software is the easiest video editor you can find – giving you a plateful of both basic and advanced video editing features. I LOVE IT!” - Julia B., CEO (via Capterra) It is the only software I found that can transcribe capitals automatically. All my content has subtitles and it is a huge effort to enter it manually, especially because our language has capital nouns. “It saves me SO much time and is extremely easy to use! I create loads of study material for YouTube, Instagram and TikTok to teach German as a foreign language. “No learning curve, doesn’t hog your laptop's processor like most video editing software, really quick to put high-quality videos together” – Alan W., Content Marketer (via Capterra) ⭐️ Import YouTube videos using just a URL People who want to make videos like a pro without a learning curve VEED Best video editing software for YouTubers Also, if you’ve an entry-level device, then the installation and rendering can be slow as per reviewers.Ģ. However, if you’re a beginner and looking for basic editing functionality then it’s not worth the investment. Overall, the price point is comparatively reasonable than other software such as Adobe. ![]() Plus, all your edited videos are watermarked. The free plan, PowerDirector Essential, has limited features. ![]() Speaking of which, keep in mind that all the perks and features discussed so far are those that the paid version offers. Additionally, you can’t upgrade from the free trial to the paid plan unless you download and reinstall the software. However, with too many program icons in the start menu, you might get overwhelmed. You can also easily add edited graphics, photographs, and animated images directly into your video projects. Using the Picture in Picture (PiP) editing option that breaks down the video into different grids, you can edit each grid separately. Work from different workstation options such as auto mode, storyboard mode, and slideshow creator. In fact, these are all updated monthly (mostly) as are the features in the paid versions. You also get over 8,000,000 stock photos, video and music clips powered by iStock and Shutterstock. I use it for trimming, adding audio, transitions,its and animations, and text overlays.” Deb P, Karmic Coach ( via G2.com )ĬyberLink PowerDirector 365 is an easy-to-use YouTube video editing software that offers most of the key features you’d need. It can do so much more than the basic editing that I use it for. “It has very good tools and powerful enough to get a professional result, if you are new to editing and want to progress while learning, this software is quite useful to get into the world of editing.” – Luis M., Project Designer ( via Capterra ) Lifetime license PowerDirector 20 Ultimate: $139.99.Lifetime license PowerDirector 20: $99.99.Subscription plan for marketers: $299.99.□The variety of options in the interface can be overwhelming. □ Videos are watermarked in the free plan. ⭐️ 360-degree editing, Picture in Picture (PiP) editing, LUT color presets, and motion tracking. It’s an easy to learn video editing software for creators including: Video editing software for Mac and Windows users. ![]() ![]() ![]() Antennas broadcast radio waves in different directions depending on design choices. Moving on from toy scenarios-the hairy ball theorem actually imposes tangible limitations on radio engineers. Does this violate the hairy ball theorem? No, because drilling a hole transformed the ball into a doughnut! Even doughnuts with unusually long, narrow holes flout the rules of the theorem-contradiction averted. What if we bored a tiny hole through the ball exactly along that axis to remove the stationary points? It seems then that every point would be moving. A spinning ball rotates around an invisible axis, and the points on either end of that axis don’t move. Upon further reflection, this might seem obvious. Spinning is a continuous motion, so the hairy ball theorem applies and assures a point with no speed at all. Again, we associate a tangent vector with each point based on the direction and speed at that point on the ball. There will always be a point on the surface that has zero velocity. To observe another weird ramification of the theorem, spin a basketball any which way you want. This neat online tool depicts up-to-date wind currents on Earth, and you can clearly spot the swirly cowlicks. ![]() ![]() A cowlick could occur in the eye of a cyclone or eddy, or it could happen because the wind blows directly up toward the sky. (Vector magnitudes don’t need to represent physical lengths, such as those of hairs.) This meets the premises of the theorem, which implies that the gusts must die somewhere (creating a cowlick). The wind flows in a continuous circulation around the planet, and its direction and magnitude at every location on the surface can be modeled by vectors tangent to the globe. Here’s a curious consequence of the hairy ball theorem: there will always be at least one point on Earth where the wind isn’t blowing across the surface. This doughnut shape is covered in small lines resembling hairs that are all combed in the same direction, with no tufts resulting. ![]() Doughnuts are also distinct from spheres, so a hairy doughnut-an unappetizing image, no doubt-can be combed smoothly. So sadly, math can’t excuse your bedhead. A scalp on its own can be flattened into a surface and combed in one direction like the fibers on a shag carpet. Something that is not equivalent to a sphere is your scalp. (You could mold them all from a ball of Play-Doh without violating the rubbery rules.) This means that the hairy ball theorem automatically applies to hairy cubes, hairy stuffed animals and hairy baseball bats, which are all topologically equivalent to spheres. If one shape can be smoothly deformed into another without doing these things, then those shapes are equivalent, as far as topologists are concerned. Although that rubber is capable of molding into other forms, it is incapable of tearing, fusing or passing through itself. In the field of topology, mathematicians study shapes, as they would in geometry, but they imagine these shapes are made from an ever elastic rubber. This claim extends to all sorts of furry figures. Tufts on either side demonstrate the hairy ball theorem. This sphere is covered in small lines resembling hairs that are all combed in the same direction. In full jargon: a continuous nonvanishing tangent vector field on a sphere can’t exist. If we stitch these criteria together, the theorem says that any way you try to assign vectors to each point on a sphere, something ugly is bound to happen: there will be a discontinuity (a part), a vector with zero length (a bald spot) or a vector that fails to be tangent to the sphere (Alfalfa). In other words, the arrangement of vectors on the sphere must be continuous, meaning that nearby hairs should change direction only gradually, not sharply. Also, we want a smooth comb, so we don’t allow the hair to be parted anywhere. Combing the hair flat against the sides of the coconut would form the equivalent of tangent vectors-those that touch the sphere at exactly one point along their length. A vector, often depicted as an arrow, is just something with a magnitude (or length) and a direction. In more technical language, think of the coconut as a sphere and the hairs as vectors. Of course, mathematicians don’t refer to coconuts or cowlicks in their framing of the problem. Here, “cowlick” can mean either a bald spot or a tuft of hair sticking straight up, like the one the character Alfalfa sports in The Little Rascals. Juvenile humor aside, the theorem has far-reaching consequences in meteorology, radio transmission and nuclear power. Perhaps even more surprising, this silly claim with an even sillier name, “the hairy ball theorem,” is a proud discovery from a branch of math called topology. You might be surprised to learn that you can’t comb the hairs flat on a coconut without creating a cowlick. ![]() |
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