Jump to content

Amram

Members
 View Profile  See their activity

  • Posts

    135

  • Joined

  • Last visited

 Content Type 

Posts posted by Amram

    You are abducted by aliens. You may ask 3 questions. What do you ask?

    in Science & Spaceflight

    Posted

    1) Having studied our species, what technologies do you possess that we do not, including all prerequisites, provided in english, on a storage medium compatible with human computers, or a suitable device that i may keep. One question, provided there are no omissions or falsehoods, and I will possess the complete difference in our civilisation's respective knowledge bases. Everything from Materials sciences, to medical, to physics to, well, whatever it is they know that we don't.

    2) Why me?

    3) What will happen now with respect to myself and my planet(with what I know, it is my planet, its current inhabitants just don't realise that yet) As per OP, i know i'm safe, at least for now, i'd like to know when i go home, if I go home, and if that home will be subjugated, eliminated, or to co-exist in peace or be forgotten. In short, how free am i to pursue my newfound ambitions.

    Sensor Jamming in Space

    in Science & Spaceflight

    Posted · Edited by Amram

    A modern military example or a passive IR sensor:Vampir MB. it passively detects hostile threats to the ship, but its range is dwarfed by the radars. In space it might outperform a radar since it would be freed from atmospheric absorption of IR energy. Then again, the radar would also be freed of atmospheric absorption.

    Passive systems do allow for triangulation, but for one unit that is alone, its hard to achieve enough angular separation between the sensors for good triangulation, and if you can't, then target motion analysis can provide a good estimate, but that takes time, and every time your target maneuvers, or you maneuver, the TMA solution goes out the window. which in active combat may make it useless. its hard to say how like submarine warfare space warfare will be. i think it'll be more akin to combat between surface ships than subs personally, other than the three dimensional maneuvering obviously.

    If all you have are passive systems, you will be hard pressed to put up a real fight one to one I think. Your opponent will be able to locate you with the same precision from greater range in less time, even though this may give his presence away. Unable to accurately and very rapidly determine range, its almost impossible to judge the target speed and how much lead your weapons need.

    To be fair, of course you could have your ships stay in groups and triangulate together. Your accuracy will still be less than with an active sensor given the techniques that can be utilised to improve resolution beyond what wavelength alone can provide, but it may be sufficient for combat.

    So, consider that you know your target relies on IR to see you much as you do to see him, how important will reducing IR signature be in the design of new ships? A specially designed canopy or plating at the front of the ship, with an active cooling system to manage the temperature and as a result the difference to your surroundings could result in making the craft invisible to an IR only sensor. Given that IR is line of sight, they could have enormous radiators hidden away behind a thermally controlled canopy. From the side or behind, you couldn't miss their approach, from the front they are impossible to see as line of sight only allows you to look at the thermally controlled surface of the canopy, which near perfectly matches its surroundings.

    Then there's tricks like approaching with a star at your back to hide in the emissivity of the star. you may need to actually increase your own emissivity to match so that you don't become a dark spot to an IR sensor.

    Or killing a ship that's known to be planned to go to you, and jamming their communication attempts to alert you. you detect a craft approaching at the right time, and by the time you can see that its not who you thought it was, it may be too late to adequately respond to the attack. kinetic weapons launched at a facility from long range are such a situation.

    Just communication alone may be reason enough for jamming to be alive and well, and still very useful since it is a send/receive type of system

    I think relying solely on passive systems would be asking to get blindsided. Instead i think its more likely that active systems will still have their place to work in harmony with the passive systems.

    Defeating EM emissions is challenging, but doable. Defeating IR is challenging, but doable. Defeating both is exponentially more difficult than merely defeating one or the other because the constraints of designing for one will interfere with those of designing for the other.

    Given the technical difficulties involved, and which systems will see the most use, I suspect that ships will find themselves more focused on IR stealth, with dedicated EM warfare vessels still in use to deal with those instances where active systems must be dealt with, and some degree of EM warfare built into every craft, much as it is with ships today.

    All that said, i do think IR, and other passive systems will take over for early warning. they won't be perfect, since they'll have exploitable weaknesses, but if your target is 8 light-days away, then an active system needs at minimum 16 days to alert you to their location if your turn it own now, or 8 days if it was on at this time 8 days ago so emitted radiation is arriving there now, while a passive system may alert you this very moment if they have been within your detection range for more than 8 days.

    Sensor Jamming in Space

    in Science & Spaceflight

    Posted · Edited by Amram

    flares aren't a very successful answer against modern seekers. The problem lies in what are sometimes called two color seekers, or rather, those that can use both long and short wavelength IR.

    The long wavelength, 3-5µm is the more common, and has been in use since the beginning, and this is what flares fool with relative ease. in fact, early sidewinders were known for deciding they liked the sun as a target better than whatever they were shot at. These are limited to tail shots though, as they require a very hot source to home in on.

    More modern seekers can use the short wavelength IR frequencies, from 8 to 13µm, and this allows them some seriously upgraded capabilities. For one they can detect the difference in temperature between the surrounding air and the targets skin. These are what allowed for all aspect missile shots, since they don't need to see the hot engines to lock on, they can see all they need easily enough from any angle.

    Flares can be made to trick the 8-13µm wavelength seekers, but those are near always paired with the 3-5µm wavelegnth capability as well, and fooling both in one flare is nearly impossible.

    Worse still, truly modern seekers use what is known as IIR, or imaging infrared, they don't just see a hot spot, they see shapes, they know what a plane looks like, and flares aren't it. These seekers are especially dangerous given that they can selectively hit whatever part of your aircraft they want to. Want to put the missile down the intake, you can do that, if your missile is agile enough and the angles allow for such a shot. Want to always aim for the pilot, you can do that too. Prefer center of mass, that's also doable, and even though its offset from the significant head sources, its usually pretty easy to determine where that is.

    For example, this is from an AIM-9X seeker during testing. if they'll share this much, i'd wager it can do considerably better now

    AIM-9x, Iris-T, Python-5, AAM-5, or R-73, either of these decides it likes you for a target and you have some seriously huge problems, because flares are iffy at best, useless at worst. There's a reason why you throw lots and lots of flares in the hopes even one provides a good decoy. if you think you've thrown enough flares, throw more.

    regarding dishes vs arrays, keep in mind that sometimes an array is straight up overkill and definitely far more expensive. This may not be sufficient deterrent, but i'll provide an example. a firecontrol illuminator. They generally only need to light up one target at a time, and they need to light them up real good so the weapons being launched can see it.

    To be fair though, things only get cheaper with time, and beyond a certain point, the cheaper dish isn't really that much cheaper to warrant giving up the strengths of an active array, so I can see the dish finally dying out. I wouldn't see a near future spacecraft with only arrays in use as unrealistic. maybe a tad more expensive than it needs to be, but, unlike with ships, I don't think the radars will be a major part of the cost. Currently the electronics suite of a combat ship can actually exceed half the total cost of the ship. The cost of building a ship in space might change that balance significantly.

    Agreed on burning out sensors. its not easy. Pretty much the realm of lasers, or giant emitters with high gain focused on a small little receiver that simply cannot deal with the power its being given, think SPY-1 or bigger putting everything it has into something as big around as your head at ranges of less than 10km. Issue there is that you need the average power to get out of hand.

    Radars and ESM's are designed to handle pulses, but if you can get the sustained energy level above what you it can cope with, you can damage it. I don't know near enough to hazard a semi accurate guess as to the ranges and power levels that would take, but I suspect if any modern system can do it, it'd be SPY-1. Brute force and a tightly focused beam, its the magnifying glass of radars. 6000 times more power than your microwave, and it can focus that into a spot the size of your microwave.........I think the only question is at what range can it fry electronics as opposed to whether or not it can.

    Off topic but neat fact. They actually had a problem when they first started using the SPY-1 system. it was too good. Seriously. It was tracking non-targets, such as insect swarms. Aegis automatically detects and marks tracks, so the operators pretty much had to tolerate this until an 'upgrade' was rolled out to stop that.

    Sensor Jamming in Space

    in Science & Spaceflight

    Posted · Edited by Amram
    finsihed a sentence I derped and forgot to write the rest of.

    I'll preface with, all of this is public knowledge, so if I go and say something you never knew, don't go assuming im sharing secrets, cause its all easily found with a little google-fu and a little reasoning.

    This is a bit long, so if your not inclined, i don't care, skip this at will, its just info.

    I'll focus on radar as it is in the modern world, and the assumption that much of the techniques used will apply to most active electromagnetic sensors in the future for some time. Radar, if used in space will certainly be vulnerable to these very techniques just as it is now. I will also skip passive systems.

    So to begin:

    Active sensors work by sending out a structured signal specifically designed in such a way as to be indistinguishable from noise so that you know with certainty when you hear an echo(found something). A very good example pretty much everyone is aware of is Sonar. Ping! You make noise, and you hear it echo off the target. Radar does the same thing, but instead of sound, its radio waves, and its quite a bit faster.

    This signal needs to be indistinguishable. You don't want random noise generating a spike that looks exactly like a target. Known as the false alarm rate, this is the ration at which the system will falsely identify noise as a target. So you use a structured signal that is just complex enough that the odds of nature every producing it when your listening for it are so low as to be effectively zero.

    Thats all well and good, but what about when you have two units near each other with the same system, they'll hear each others echoes and transmissions, and wont know whose echo is whose, and whether the transmissions are targets(since the other guy can hear your transmission, and vice versa) or not. So you need to make the different enough to distinguish.

    So you use different frequencies. A radar might be designed with a frequency range from 9.8ghz to 10.1ghz for example. So if you encounter significant interference on 9.85, you switch to something else, like maybe 10.05. Its somewhat more complex than that, but that's sufficient an example. this allows you and many friends to use the same radar near each other without blinding each other with interference.

    So that covered, how would a jammer disrupt it:

    There are quite a few ways actually, and not all of them are easily tracked.

    first and foremost, lets begin with intel. We need to know about our enemy and their capabilities.

    If we do not know anything at all, we have very limited choices for jamming.

    If for example, you don't know where your enemy is, but you know they are near and they must be jammed, then you cannot aim an antenna at them, as their location is not known. you must instead use an omnidirectional transmission. This hurts your effectiveness since power at the target point is greater reduced, but it allows you to jam a target whose location is not known.

    If we do not know their frequencies in use, then we are further hampered. We must now jam a very wide range of frequencies in an attempt to also cover the frequencies they are using. This further hurts our average power at the target, as power is being spend on frequencies other than those used by the target.

    Suppose we do know where they are, but we don't know about their systems.

    Now you can use a directional antenna to great effect. the antennae might give you, say 35dBsm of gain, well, that's 3162.28 times the signal strength, pretty significant. This allows either that much more noise at the target point, or the same noise at a target point that is roughly 7.5x further away. pretty big deal. This also applies to their sensor. The antenna magnifies their signal, it also magnifies received noise just the same. They also aren't perfect, they still hear some noise from directions other than where they are pointed, but good antenna hear VERY little of such noise. Still, sufficient power will influence the system even when its not aimed at you, this can result in completely blinding the system no matter where it points.

    Barrage jamming is pretty inefficient as far as the other types go, so lets up the intel.

    Suppose we also know their approximate frequency bands in use.

    Now you can focus your power into a tighter range of frequencies. This has advantages other than in jamming as well. if you must barrage jam from 1ghz to 20ghz just to ensure you have an effect on a potential target, then you might be detected by a passive sensor that is sensitive to 4ghz to 6ghz. If your target is only using 9.8 to 10.1 and you know that, so you jam from 9.75 to 10.15, that same passive sensor will not detect your jamming now, as your not using a frequency it is sensitive to.

    This tightening of the frequencies increases the noise the target experiences because you aren't spreading it around anymore on frequencies you don't need to jam. Im fairly sure the math is incorrect, but the principle at least is correct: if you had 1Mw of jamming spread across 1ghz your putting 0.1w per hz, if you tighten that to 9.75 to 10.15, your focused into a 400mhz band, 2.5w per hz for a noise increase of 25x.

    But what if we know even more about our target, suppose we also know the rotation speed of their antenna?

    Older systems must track back and forth predictably. The signal goes where the antenna points, so you listen to their radar, and you can tell its rpm by the detections per minute, since it will only hit you with strong signals when its aimed at you. Phased arrays can electronically steer the beam, so they can aim the signal at points other than where the array is pointed, which can make this impossible to achieve since the hostile radars pattern of painting you might be very random.

    Now you can upgrade to spot jamming. Same basic idea so far, but you don't have to spend all of your time transmitting, now you can transmit only when they look at you. This allows you to take advantage of something they might have done in the design of their own ship. They don't really want their passive sensors that detect radar to also hear their target echoes from their own radar, so they might not be sensitive to those frequencies. A simple solution is to be sensitive, but ignore returns on the same bearing the radar is aimed at, so now your jamming might be ignored since your jamming will only be detected when the radar is aimed at you.

    A more advanced system will still detect your jamming, as they used a very specific signal structure for the radar, and you're transmitting noise, they are easy to tell apart.

    if you know when the radar is aimed at you, its frequency bands, and its transmission length, you can go even better still.

    now you can perform sweep jamming in conjunction with spot jamming. This is where you do not continuously jam a range of frequencies, instead, you 'scan' through them with your noise. You do not stay on any one frequency at a time, so you do not completely blind a system, but you also become harder to detect and localise as your noise is not all in one frequency making a nice big spike, nor is it a block of frequencies. You may even cross into frequencies they can't listen to causing them to lose your noise intermittently as well. You could randomly bounce(frequency hopping) from frequency to frequency while spitting out noise, causing interference and potentially making the error correction systems on the target go nuts. This can make the system decide its false alarm rate is too high so its become defective, and throw error messages which can mislead crew into not trusting the output of a good sensor.

    Yet we can still do even better. What if we know the target's signal structure as well as everything else?

    So now that we know exactly what the target systems transmission will look like, we can offset our jamming by 180° in phase. A quick explanation is in order. Picture waves in the ocean, they are very similar to electromagnetic waves in radar for example. All frequencies do this, all signals do this, you can influence it a bit, square it more than leaving it a smoothly curving peak, but it still must have a peak. A well designed jammer can throw noise that is perfectly inverted, where there were peaks it throws troughs and where troughs it throws peaks. The result is noise. They will still detect something, plenty of power will go back to the radar(you did just double it), but it looks nothing like what they sent out, its noise. They know something is that way, but not how far, or fast. This is the same principle as that behind noise cancelling headphones. And since the noise only happens when the radar is pointed at you, the definitely know a jammer is somewhere in your direction.

    Before I proceed to an even better result, i'll diverge with a quick bit of ECCM.

    They know you will try these tricks, and so they design radars with some neat features. For example frequency hopping and large bandwidth. The very simply version of bandwidth, is that is the range of frequencies in use simultaneously by a system. For example if the above system were to use 9950mhz as the center frequency, with a bandwith of 40mhz, it would transmit and listen across everything form 9930mhz to 9970mhz. if you can only jam a band that is 20mhz wide, then the best you can do is to blot out half their signal with noise. At best that's only a 41% decrease in their effective range against you if your just throwing noise. If they can frequency hop, they can simply change to 9970 to 10010 for the next pulse and not experience any noise until you adjust to match, then change again when you do. it could be purely random, 9970-10010 one pulse, 9704-9744 the next, and 9872-9912. if they can do that every pulse, you'll never match them without also know how the number generator is picking its pseudo-random numbers, so your stuck with covering a range to hamper them.

    Back to jammers.

    Suppose you know where they are, what frequencies they can use, that they cannot frequency randomly hop from pulse to pulse, can cover their full frequency range and bandwidth, and know precisely what their signal looks like.

    Now you have some neat tricks you can pull. In effect you can talk directly to their radar, and lie to it, because you can now form a signal that looks exactly like that receive wants to see as an echo.

    So a little more detail on how they work. In order to detect a target behind another target, use what is called a range-gate. The easiest way to explain it is to think of the radar scanning with a grid. Everything from 100,000meters to 100,005m for example, is maybe considered to be one target, so if it receives two returns that are 4 meters apart, it might not see them as separate targets, it might only see one target that is twice as big, since its detecting two echoes as one echo. the range gate is related to the wavelength of the radar, and its signal structure as there are tricks to improve range resolution. So knowing that, here's the next trick.

    Deception jamming.

    In deception jamming you have to be very quick, and very capable, and know very much of your target system's specifics. if you do, now you can start lying to it. In effect, you can read their signals to learn what you need to know, for example which direction they are in, maybe how far away they are(by looking at how strong the signal you received is-modern LPI radars can modify that on the fly per target, so its useless there), and where they think you are. This is not worth it if they haven't found you already for certain. there is no point to giving them a good track when they have none and are unaware you are near. The best indication they have found you will be a switch in the radar signal to a firecontrol frequency/pattern, or a sudden increase in the energy directed your way to provide a stronger signal, or more frequently being painted as the radar spends more time keeping an eye on you specifically.

    Then what you do is on the next pulse once your ready, you spit back an inverted signal turning it to noise but not quite as strong as theirs. the goal isn't to produce noise at the full strength, but to weaken their signal enough that they lose it in the noise that's already there. Then at nearly the same time you send back a signal that is identical to theirs, but stronger and delayed just a tiny bit. Because you weakened and disrupted their signal, you hope that is dropped, and by giving a signal of your own that is stronger, you hope that is accepted by the system as the true signal. By transmitting a bit after initial reception, you also imply that you are further away than you really are. They can tell sorta where you are by your jamming, but not precisely enough to shoot at with guns/lasers, or to guide missiles to you. the missiles own radar can be mislead exactly the same way, and it will believe you are over there when really your right here and it will miss.

    if you possess enough transmission power, you can also mislead them in the other direction, into believing you are closer than you really are because you can make your signal strong enough to be believed as an echo from that close in. You just need to increase your apparent approach rate by gradually transmitting a bit sooner and sooner while turning the real echo to gibberish and trying to keep it from being accepted as a target.

    its important to note the doppler effect at this time. Not only does your timing of the signal matter to where the radar thinks you are, but the doppler effect is very important in determining how fast you are, you need to use this to lie as well. the radar is very likely to reject your signal if you move your self 1000m in 1 second, but the doppler says you only moving at 300m/s. The mismatch will be noted, and it will likely follow that very weak signal that you tried to make it ignore, because that still says 300m/s and is where it expected to find it. Your signal was too different to be believed. you gotta take it slow.

    Speaking of Doppler, you can also lie about your speed. The radar can track you step by step and work out that its wrong, but a missile is likely to work with what it gets per pulse, which means while you won't defeat a tracking radar, a missile might believe you are moving much faster than you really are, or much slower and use an entirely incorrect lead angle. Coupled with an incorrect range-gate and the missile is almost certain to miss. Still, the missile might only care about bearing, and that can be hard to fool.

    there is still one better that can be done that i'm aware of. Its merely the combination of both sweep jamming, and deception. Once you have the radar fully deceived and tracking your ghost return you can start introducing noise in an attempt to make the error correction freak out, and if you can manage it, the system will then paint your track as erroneous and drop it, and because it has stopped tracking your real return and its lost in the noise, you are now effectively gone. its easier to keep a weak track than to find one which is why this works, make it stop actually tracking you by lying to it, misleading it with its own signals, while making yourself too weak a return to be redetected, then break up the ghost with noise and mislead the operator into thinking the system has malfunctioned and make him less likely to trust it until it has been checked and found to be operating fine.

    1 to 1, you can screw anyone up right good if you can jam them effectively. they'll be unable to determine your exact position even if the jamming itself can give them an approximate position. if they must rely on a guided weapon at this point to complete the kill since those can be sent to approximate locations and then search on their own, if you can also mislead that, it too becomes useless against you, and in the end, whether they know you are jamming them or not, it matters not because they can't locate you well enough to hit you, they might as well be blind for all the good it does them.

    if your capable of achieving enough width, rapidly enough, on enough frequencies with sufficient power simultaneously, say multiple independant jamming systems, or one spectacularly capable system you can simultaneously do this to multiple sensors at once, and blind multiple units, and then even triangulation is ineffective at more than 'somewhere near there', which isn't enough to shoot at. They then need either enough systems to overwhelm your ability to jam them all, a system that operates outside your specialties so they can remain unaffected, or a sensor that can home in on your jamming.

    its worth noting your jammer can pull all the same tricks the radar can so that a seeker that's homing on your jamming may need to listen over a very wide range to effectively home in on your jammer since you might be chasing target radars across many frequencies seemingly at random. This means the bandwidth on the seeker may need to be large to accomodate your potential frequencies and keep you detected, this means a lot of noise, which reduces the strength of its detection. if its tightly focused it might not hear much of your jamming and will have difficulty keeping a lock.

    of course, you could always stop jamming for a moment to see if the missile switches to an active sensor. If it does, reach into your bag of tricks and lie to it to, it'll keep using the radar not realising its being lied to, and will miss. if it doesn't, resume jamming until you must stop to blind the weapon, then maneuver. they'll get a good lock, but your distant, they'll need time to hit you, you need only a few moments to evade, then resume jamming efforts.

    To put the ball back in the radar's court, AESA(active electronically scanned array) systems make much of this useless. They can almost as a rule frequency hop at random, they can steer the beam allowing for randomising the time between transmissions aimed at a given target, they can use potions of the array for different tasks simultaneously allowing for directing energy at more than one target, or at the same target on very different frequencies or patterns, or to be a jammer and radar in effective operation simultaneously, and they can vary the power output so that the signal is never stronger than it needs to be. These can make detecting the system incredibly difficult, and predicting it near impossible. if you can't predict it, you can't deceive it, nor cancel it, nor effectively counter it with noise on the same frequencies without covering a much wider band.

    They aren't immune to the above techniques, just very capable of working despite them.

    I'll approach the ending with a word on passive systems.

    These are troublesome. they never give you anything, and use what you give them against you. it doesn't much matter what you give them, because its always something. There are few ways I am aware of to deal with such systems.

    Barrage jamming it with so much noise it cannot find targets. But this leaves a great big spike in the right direction, and if you can't throw energy down the bearings of multiple sensors, you can't blind them all, and they only need them to cross to know sorta where, and two to cross that aren't jammed to know exactly where. Your problem is know where they are, because they are passive, they give you nothing to help you find them unlike with radar that must transmit first.

    ideally you will know what range to expect them to be able to detect you, and shortly before that you blind them. they are going to know something is over there in a moment anyways, why let them know who, what, or how many?

    Depending on the system, and how it functions, it may be possible to flood it with false positives much like with radar, but being a passive system, they don't really care what they see, just so long as its something much of the time so good luck with that.

    I think this is the concept behind IR jammers, they paint the seeker head of a missile with the laser and watch the scintillation of the return then try to keep the laser painted on the same spot on the seeker head to mislead the missile as to where it needs to go. Since the point is off center, it thinks it needs to turn, and with tracking the seeker head, it continues to turn and then loses the original target entirely.

    For example, turning left and climbing while painting the lower right of the seeker head so it dives and goes right. Timed right it might be unable to correct and hit you even if it never did lose sight of you.

    Lastly, your left with burning it out. if you can throw enough energy at it, you can potentially damage the electronics which are very sensitive and may not be able to handle it, which is a semi permanent solution to the problem. An array will fail gracefully, which means as pieces fail capability diminishes rather than an abrupt termination of function. Think of that like the pixels of an LCD dying one by one instead of the whole screen crapping out in one instant. Diminished sufficiently the result is the same as effective jamming, but considerably more effort is required.

    I'll end with:

    this makes it seem easier than it really is, but if you have a significant technological advantage, or excellent intel, you can make their radar operators go crazy. Old jammers have almost no hope of even interfering with a modern system let alone successfully jamming it in any way thats useful, and old radars have an equally dim chance of coping with modern jammers. When your on equal footing, the possibilities are endless.

    Given that radar is nothing more than an active electromagnetic energy detection system with limits on its spectrum, i would wager that these principles will apply to any system in the near future, possibly also distant future that also uses electromagnetic energy as its medium, and relies on transmit/receive as its method of operation.

    purely passive systems are your problem there

    And finally, im 100% certain the militaries of the world have quite a few tricks no one has ever heard of.

    Disable automatic part symmetry

    in KSP1 Suggestions & Development Discussion

    Posted

    while I agree it would be great to prevent the autosymmetry at times, you will definitely find this to be mostly a minor annoyance once your more in the habit of using the keys than clicking the button.

    You want 3x symmetry.

    Pick up part, move to rocket.

    Moused over 6x symmetrical part while getting near final postion, game adjusts to 6x symmetry automatically.

    position part, press shift-x to reduce symmetry back down to 3x

    place part.

    The four fours.

    in Forum Games!

    Posted · Edited by Amram
    link fix, i've been on a roll lately

    yep, tested a couple values at wolframalpha to confirm.

    1*2*3 = Γ(4) = 6

    1*2*3*4 = Γ(5) = 24

    1*2*3*4*5 = Γ(6) = 120 (which is why you'll find me doing Γ(Γ(4)) quite a bit, since its equivalent to Γ(6), which is 120. Useful at these numbers.

    I think you understand it well enough, and put like that makes way more sense to me than how wolframalpha explains it, lol. I look at their definition page for gamma and my head explodes.

    151 = Γ(Γ(4)) + 4! + Γ(4) + !(√4)

×
×
  • Create New...