16 ms·
This is interesting, but I have to quibble with this: > If you express this value in any other units, the magic immediately disappears. So, this is no coincide
by mistercow 2y ago
This is interesting, but I have to quibble with this:
> If you express this value in any other units, the magic immediately disappears. So, this is no coincidence
Ordinarily, this would be extremely indicative of a coincidence. If you’re looking for a heuristic for non-coincidences, “sticks around when you change units” is the one you want. This is just an unusual case where that heuristic fails.
- karmakurtisaani 2y agoActually no, the whole equation boils down to the definition of meter. Or rather, one of the earlier definitions.
- mistercow 2y agoYeah, I read the post. What I’m saying is “this relationship vanishes when you change units, so it must not be a coincidence” is a bad way to check for non-coincidences in general. For example, the speed of sound is almost exactly 3/4 cubits per millisecond. Why is it such a nice fraction? The magic disappears if you change units… (of course, I just spammed units at wolfram alpha until I found something mildly interesting).
- brians 2y agoBecause the cubit is a measure of what a body can reach
- Bjartr 2y agoHow does that explain the relationship to the speed of sound?
- ValentinA23 2y agoA bit out of topic, however https://www.youtube.com/watch?v=0xOGeZt71sg https://www.youtube.com/watch?v=0xOGeZt71sg Note: I'm more inclined to think this is a coincidence given that it establishes a link between the most commented text and the the most commented building in history. However I don't think these kind of relationships based on "magic thought" should be discarded right away just because they are coincidences, and I'd be very interested in an algorithm that automatically finds them.
- ants_everywhere 2y agoI never thought of the cubit this way. It's an interesting idea, but the cubit is the length of a forearm, whereas you can reach around yourself in a circle the length of your extended arm, from finger tip to shoulder. That would be somewhere between 1.5 to 2 cubits for people whose forearm is about a cubit long. I think the cubit is mainly a measure of one winding of rope around your forearm. That way you can count the number of windings as you're taking rope from the spool. This is the natural way a lot of us wind up electrical cables, and I'm sure it was natural back in the day when builders didn't have access to precise cubit sticks. I don't see the connection with the units and sound that you're making. But it is kind of interesting to know that sound travels about 3/4 of a forearm length per millisecond. That's something that's easy to estimate in a physical space.
- jncfhnb 2y agoX^2 is a lot more interesting than x*0.0000743 or whatever it is
- satvikpendem 2y agoWhy is it more interesting? Is it just more interesting because we use such bases, or can it be interesting inherently? That is the question that is being asked, and why some say it's merely a coincidence.
- jncfhnb 2y agoWell every number is the product of another number and some coefficient. If it’s a nice clean number then that implies it could be the result of some scaling unit conversion. But that should be sort of apparent. And it’s not super interesting if true. If a number is another number squared then that implies some sort of mechanistic relationship. Especially when the number is pi, which suggests there’s a geometric intuition to understanding the definition.
- satvikpendem 2y agoIn other bases, it does not actually imply much, even if it were squared. Maybe it really does make sense if it existed in base 10 but I cannot see much if it were part of other bases.
- mistercow 2y agoOk, then by that thinking, you should find it really interesting that Earth escape velocity is almost exactly ϕ^4 miles per second. In fact, adding exponents here objectively makes it less interesting, because it increases the search space for coincidences. What makes the case in the post most interesting to me is that it looks at first glance like it must be a coincidence, and then it turns out not to be.
- jncfhnb 2y agoYou would need a compelling argument why x^4 is interesting. Especially something like the golden ratio. Not a long of things go to the fourth power in equations we need to use. Pi^2 directly features in periodicity
- Waterluvian 2y agoOr the speed of light being almost a sweet 300 million m/s. Or after-atmosphere insolation being somewhat on average 1kw/m2.
- ta1243 2y agoI always find insolation and insulation to be such an interesting pair of words I guess the equivelent of "change the units" is "change the language". French: insolation et isolation German: Sonneneinstrahlung / Isolierung Spanish: insolación / aislamiento Chinese: 日照 / 绝缘 I guess coincidence
- mananaysiempre 2y agoinsolation < Latin sol, solis m "sun" insulation < Latin insula, -ae f "island" (apparently nobody knows where this one comes from) isolation < French isolation < Italian isolare < isola < Vulgar Latin *isula < Latin insula, -ae f Spanish aislamiento < aislar < isla < Vulgar Latin *isula < Latin insula, -ae f Oh and the English island never had an s sound, but is spelled like that because of confusion with isle, which is an unrelated borrowing from Old French (île in modern French, with the diacritic signifying a lost s which was apparently already questionable at the time it was borrowed), ultimately also from Latin insula.
- CRConrad 2y ago> Oh and the English island never had an s sound, but is spelled like that because of confusion with isle So the German cognate (I assume) Eiland probably hasn't had one either. Makes sense, since the Nordic Eya / Øy / Ö never felt like they should, and they must be just northern variants that lack the -land (literally the same in English) suffix.
- mistercow 2y ago> Or after-atmosphere insolation being somewhat on average 1kw/m2. I’m kind of inclined to say that this one isn’t so much of a coincidence as it is another implicit “unit” in the form of a rule of thumb. Peak insolation is so variable that giving a precise value isn’t really useful; you’re going to be using that in rough calculations anyway, so we might as well have a “unit” which cancels nicely. The only thing that’s missing is a catchy name for the derived unit. I propose “solatrons”.
- euroderf 2y ago32 meters is 35 yards, to within about an eighth of an inch. How's that grab you ?
- GoldenRacer 2y agoMy favorite is 1 mile = phi kilometers with <1% error
- baliex 2y agoThat one’s useful too. If you know a few Fibonacci numbers you can convert miles to kilometres and vice versa with ease. 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 … 21 km is ~13 miles, 13 km is ~8 miles, etc. A 26 mile marathon? Must be ~42km. Same for speed limits too; 34 mph is ~55 kmh
- eesmith 2y agoI use that approximation, via the Fibonacci sequence, to translate between miles and km. 13 miles ~ 21 km (actually 20.921470). My favorite approximation is π·E7 = 31415926.5... , which is a <1% error from the number of seconds in a year.
- saalweachter 2y ago... you all realize that phi is barely a better approximation than 8/5, right? 1.6 vs 1.609 (km in a mile) vs 1.618? (8/5)/(1 mile/1 km) = 0.9942; (1 mile/1 km)/phi = 0.9946. You're making things way harder on yourself for essentially no improvement in precision, especially when you're just rounding to the nearest whole number.
- eesmith 2y agoThat tells me you haven't memorized the first ten or twelve terms of the Fibonacci sequence.
- saalweachter 2y ago1 km = 5 furlong, with < 1% error.
- dr_dshiv 2y agoAlpha brainwaves are almost exactly 10hz, in humans and mice. The typical walking frequency (for humans) is almost exactly 2hz (2 steps per second). And the best selling popular music rhythm is 2hz (120bpm) [1]. Perhaps seconds were originally defined by the duration of a human pace (i.e. 2 steps). These are determined by the oscillations of central pattern generators in the spinal cord. One might suspect that these are further harmonically linked to alpha wave generators. In any case, 120bpm music would resonate and entrain intrinsic walking pattern generators—this resonance appears to make us more likely to move and dance. Or it’s just a coincidence. [1] https://www.frontiersin.org/journals/neurorobotics/articles/10.3389/fnbot.2022.850489/full https://www.frontiersin.org/journals/neurorobotics/articles/...
- jvanderbot 2y agoWell, a second is also a pretty good approximate resting heart rate (60 bpm)
- JadeNB 2y ago> Well, a second is also a pretty good approximate resting heart rate (60 bpm) I'm sorry to be the kind of person who feels compelled to make this comment, but you mean a Hertz, not a second.
- deleted 2y ago[deleted]
- saghm 2y agoReminds me of https://xkcd.com/687/ https://xkcd.com/687/
- panarky 2y agoAnother bad way to check for non-coincidences is to use a value like g which changes depending on your location. Pi is the same everywhere in the universe. g on Earth: 9.8 m/s² g on Earth's moon: 1.62 m/s² g on Mars: 3.71 m/s² g on Jupiter: 24.79 m/s² g on Pluto: 0.62 m/s² g on the Sun: 274 m/s² (Jupiter's estimate for g is at the cloud tops, and the Sun's is for the photosphere, as neither body has a solid surface.)
- TheRealPomax 2y agoFun fact: pi is both the same, and not the same, in all of those places, too. Because geometry. If you consider pi to just be a convenient name for a fixed numerical constant based on a particular identity found in Euclidean space, then yes: by definition it's the same everywhere because pi is just an alias for a very specific number. And that sentence already tells us it's not really a "universal" constant: it's a mathematical constant so it's only constant given some very particular preconditions. In this case, it's only our trusty 3.1415etc given the precondition that we're working in Euclidean space. If someone is doing math based on non-Euclidean spaces they're probably not working with the same pi. In fact, rather than merely being a different value, the pi they're working with might not even be constant, even if in formulae it cancels out as if it were. As one of those "I got called by the principal because my kid talked back to the teacher, except my kid was right": draw a circle on a sphere. That circle has a curved diameter that is bigger than if you drew it on a flat sheet of paper. The ratio of the circle circumference to its diameter is less than 3.1415etc, so is that a different pi? You bet it is: that's the pi associated with that particular non-Euclidean, closed 2D plane. So is pi the same everywhere in the universe? Ehhhhhhhh it depends entirely on who's using it =D
- mr_mitm 2y agoIn non-euclidean spaces, your definition of pi wouldn't even be a value. It's not well defined because the ratio of circumference to diameter of a circle is dependent on the size of the circle and the curvature inside the circle. It's probably true that it's only well defined in euclidean space. Your relaxed definition, which I have never seen before, is not very useful.
- antonvs 2y ago> What I’m saying is “this relationship vanishes when you change units, so it must not be a coincidence” is a bad way to check for non-coincidences in general. Yeah, it was a strange claim, which makes me think that the author may have had his conclusion in mind when writing this. I.e. what he meant to say may have been something more like: "The relationship vanishes when you change units, which suggests the possibility that the relationship is a function of the unit definitions... and therefore not a coincidence."
- ValentinA23 2y agoThere is relationship between the metric system and the French royal system. The units used in this system have a fibonacci-like relationship where unit n = unit n-1 + unit n-2. palm : 7,64 cm span : 12,36 cm handspan : 20 cm foot : 32,36 cm cubit: 52,36 cm cubit/foot =~ 1,618 =~ phi, el famoso Golden ratio. foot/handspan =~ phi too. And so on. From this it turns out that 1 meter = 1/5 of one handspan = 1/5 x cubit/phi^2 Another way to get at it is to define the cubit as π/6 meters (= 0.52359877559). From this we can tell that 1cbt = π/6m π meters = 6 cubits Source: https://martouf.ch/crac/index.php?title=Quine_des_b%C3%A2tisseurs https://martouf.ch/crac/index.php?title=Quine_des_b%C3%A2tis...
- KeplerBoy 2y agoIt does not? Pi has nothing to do with our arbitrary unit system.
- eigenket 2y agopi is always just pi, but g may be defined in terms of the meter.
- KeplerBoy 2y agosure, that's the entire point. heck, g is not even a constant, it just happens to measure to roughly 9.8 m/s² at most places around here.
- mikequinlan 2y agoPi is related to the circumference of a circle; the meter was originally defined as a portion of the circumference of the Earth, which can be approximated as a circle. "The meter was originally defined as one ten-millionth of the distance between the North Pole and the equator, along a line that passes through Paris."
- mistercow 2y agoBut that connection actually is a coincidence. From what I can tell, when they standardized the meter, they were specifically going for something close to half of a toise, which was the unit defined as two pendulum seconds. So they searched about for something that could be measured repeatably and land on something close to a power of ten multiple of their target unit. The relationship to a circle there doesn’t have anything to do with the pi^2 thing.
- mannykannot 2y agoIt was news to me, but that's what the article says, and it is supported by by Wikipedia, at least. [1] In addition, I feel the article glosses over the definition of the second. At the time, it was a subdivision of the rotational period of the earth (mostly, with about 1% contribution from the earth's orbital period, resulting in the sidereal and and solar days being slightly different.) Clearly, the Earth's rotational period can (and does) vary independently of the factors (mass and radius) determining the magnitude of g. The adoption of the current definition of the second in terms of cesium atom transitions looks like a parallel case of finding a standard that could be measured repeatably (with accuracy) and be close to the target unit - though it is, of course, a much more universal measure than is the meridional meter. [1] https://en.wikipedia.org/wiki/History_of_the_metre https://en.wikipedia.org/wiki/History_of_the_metre
- chrz 2y agoIm wondering is there connection or not? We use distance unit to get to π number, whatever the distance unit is right? We get π from circumference to diameter ratio, so however long the meter is the π in your distance unit is same ratio
- blablabla123 2y agoChanging units in Electrodynamics for instance comes with unexpected factors in formulas though, indeed containing π. (CGS <-> SI)
- HPsquared 2y agoIsn't that just the change between rad/s and Hz?
- elashri 2y agoIt is more involving [1] [1] https://phys.libretexts.org/Bookshelves/Electricity_and_Magnetism/Electricity_and_Magnetism_(Tatum)/16%3A_CGS_Electricity_and_Magnetism/16.03%3A_The_CGS_Electromagnetic_System https://phys.libretexts.org/Bookshelves/Electricity_and_Magn...
- setopt 2y agoIt’s more precisely the difference between “rationalized” and “unrationalized” units. You need a factor 4pi in either Gauss’ law or Coulomb’s law (because they are related by the area 4pi*r^2 of a sphere), and different unit systems picked different ones. It’s more akin to how you need a factor 2pi in either the forward or backward Fourier transform and different fields picked different conventions.
- vitus 2y ago> It’s more akin to how you need a factor 2pi in either the forward or backward Fourier transform and different fields picked different conventions. Some fields even use the unitary transform -- they split the difference and just throw in a 1/sqrt(2pi) in both directions. https://en.wikipedia.org/wiki/Fourier_transform#Angular_frequency_(%CF%89) https://en.wikipedia.org/wiki/Fourier_transform#Angular_freq...
- phkahler 2y agoDoesn't the relationship hold if we change units? It seems like it must. When I worked with electric water pumps I loved that power can be easily calculates from electrical, mechanical, and fluid measurements in the same way if you use the right units. VoltsAmps, torquerad/sec, pressure*flow_rate all give watts.
- deleted 2y ago[deleted]
- eigenket 2y agoNope, it completely vanishes in other units. If you do all your distance measurements in feet, for example, the value of pi is still about 3.14 but the acceleration due to gravity at the earth's surface is about 32 feet s^(-2). If you do your distance measurements in furlongs and your time measurements in hours then the acceleration due to gravity becomes about 630,000 furlongs per hour squared and pi (of course) doesn't change.
- ValentinA23 2y agoOnly because you're using metric seconds instead of "imperial seconds" (the time it takes for a 1 foot long pendulum to complete a full oscillation).
- eigenket 2y agoSure, if you change either of the units you can always change the other one to fix the equation again.
- koolala 2y agoBut does it work when you use the right Imperial technique?
- ValentinA23 2y agoIf I come up with my own measuring unit, let's call it the sneezle (whatever the actual length I assign to it) I will be able to also define a duration unit (say, the snifflebeat) based on the time it takes for a pendulum one sneezle long to complete a full oscillation, and vice versa I can define the sneezle by adjusting the length of a pendulum so that it oscillates in two snifflebeats. Here are the maths: T = 2π√(l/g) T/2π = √(l/g) (T/2π)^2=l/g g = l/(T/2π)^2 g = l/(T^2/4π^2) = 4π^2xl/T^2 Now substitue T with 2 and l with 1 an you get g = 4π^2x1/2^2 = π^2 It doesn't matter what the pair of units assigned to T and l are. However, they'll be interrelated. There is nothing arbitrary, and no coincidences behind g =~ π^2. It just requires to do some history of metrology and some basic maths/physics. If you want to discuss coincidences, may I suggest you to comment on this remark I made and which hasn't received any attention yet ? https://news.ycombinator.com/item?id=41209612 https://news.ycombinator.com/item?id=41209612
- glitchc 2y agoYour quibble seems nitpicky and unwarranted. What the author is saying is that the relationship becomes evident if we consider the units of m/s^2 for gravity. They just didn't quite say it like that.
- mistercow 2y agoObviously it’s nitpicky. That’s what a quibble is. But I don’t think it’s unwarranted. How you reason your way to a conclusion is at least as important a lesson as the conclusion itself. And in this case, the part I quoted is a bad lesson.
- lupire 2y agoThe "magic" doesn't disappear in "any" other units. Period = 2π√(length/g) So the "magic" holds in any units where the unit of time is the period of a pendulum with unit length.
- msteffen 2y agoIt’s really the best and only way to find non-coincidences involving the definition of units, though. All such non-coincidences will have this property
- mistercow 2y agoAll coincidences involving the definition of units will also have this property. Once you’ve narrowed to that specific domain, invariance to change of units is completely uninformative.
- koolala 2y agoReading this gave me a chill. Please take my temperature and compare it to the norm temperature of humanity.
- arcastroe 2y agoI'm surprised at the number of people disagreeing with your quibble. I had the exact same thought as you! If pi^2 were _exactly_ g, and the "magic" disappeared in different units, THEN we could say "so this is no coincidence" and we could conclude that it has to be related to the units themselves. But since pi^2 is only roughly equal to g, and the magic disappears in different units, I would likely attribute it to coincidence if I hadn't read the article.
- mjburgess 2y agoIt would be useful if people carried around some card with all the information that they understood on it, since opinions are largely symptoms of this. In almost all cases any apparent phenomenon specific to one system of measurement is clearly a coincidence, since reality is definable as that which is independent of measurement.
- twojacobtwo 2y ago> since reality is definable as that which is independent of measurement. In terms of quantum mechanics, would that mean the wave function is real until it collapses due to measurement? Or am I misunderstanding your use of measurement there? Something about that is sticking in my mind in an odd way, but I can't put my finger on exactly what it is - which is intriguing.
- mjburgess 2y agoMeasurement can change what is measured, but it doesnt change it from illusion to reality. I cannot measure santa clause into existence. But I can change the temperature of some water by measuring it with a very hot thermometer. That measurement changes what is measured is the norm in almost all cases, except in classical physics which describes highly simplified highly controlled experiments. The only 'unusual' thing about QM is its a case in physics where measurement necessarily changes the system, but this is extremely common in every other area. It is more unusual that in classical physics, measurement doesn't change the system.
- deleted 2y ago[deleted]
- john-aj 2y agoI agree. But if you remove the "so", there is no contradiction. It is possible the author used "so" not to mean "in other words", but simply as a relatively meaningless discourse marker.
- mistercow 2y agoHuh, interesting point. Writing unambiguously is ridiculously hard.
- anon457437 2y agoThe comma differentiates. The comma indicates a short pause and a certain intonation in speech (the period means a longer pause and a different intonation). If you say that sentence with and without a pause/comma, you'll see (hear) that the sentence is correct. Reading unambiguously is also hard.
- mistercow 2y agoThe problem with that is that writers are not consistent with comma usage either, particularly when it comes to informal writing, where prescriptive rules are out the window anyway. And I would argue that it would be a bit of a norm violation even in informal writing to introduce this new point at the end of a paragraph rather than starting a new one, which makes me think that that was not the author's intent.
- gklitz 2y agoWhat? The entire point is that it’s no coincidence in this unit set. Saying that changing units indicates a coincidence is like saying that if we see Trump suddenly driving a Tesla after Elon stated throwing money at him, that must be just a coincidence because if we change the car model to a ford then there would be nothing odd about it.
- throwawayk7h 2y agoyou can rule that heuristic out immediately because pi is unitless, surely?
- yunohn 2y agoI don’t agree with this. You could literally redefine any unit (as we have done so multiple times in the past) and end up with zero coincidences. All measurement metrics are “fake” - nothing is truly universal, and can easily be correlated with another human made measure eg Pi.
- mistercow 2y agoI seriously doubt you could define any system of units that has zero coincidences, even with significant computational effort. Some things in the real world are just going to happen to line up close to round numbers, or important mathematical constants, or powers or roots of mathematical constants, and then you’ll have some coincidences. There are just too many physical quantities we find significant, and too many ways to mix numbers together to make expressions that look notable.
- thayne 2y agoNot necessarily. One of the things I was taught when studying astronomy is that if you observe periodicity that is similar to a year or a day, that's probably not a coincidence, you probably failed to account for the earth's orbit or rotation.
- jvanderbot 2y agoThis is a good example, but actually this is exactly what GP was referring to. It is a coincidence that the thing you're observing is periodic with earth's rotation. Observing a similar thing from a satellite (allegorically the same as "changing bases") would remove the interesting periodicity. The earths rotation coincides with the phenomenon, so it's likely a coincidence.
- randallsquared 2y agoIn the example case, the earth's rotation is producing the apparent observation: it's the cause, not a separate phenomenon that happens to coincide, or that might be indicative of a deeper relationship. For something to be a coincidence, it must be otherwise unconnected causally, which is not the case if the reason you found a ~24 hour period is that you forgot to account for the earth's rotation.
- jvanderbot 2y agoI respectfully disagree (without attempting to say you're wrong!) about the definition of coincidence and the requirement of being non causally related. If I'm riding on a bus and the light poles going past line up with my music, that's a coincidence even though they are cause soley by the bus motion BPM matching an essentially random choice of song BPM.
- thayne 2y agoWhat you're describing is a coincidence. What I'm describing is an artifact in your data that is caused by the motion of the earth. To give a more concrete example, suppose you measuring the brightness of a trans-neptunian object, and observe that the brightness changes slightly with a period of about 1 year. You might think it has a non-uniform albedo, and a rotational period of one year, when in reality, it is just brighter when the earth is closer to it.
- billforsternz 2y agoAgreed. Irrespective of how the story is later developed, "So, this is no coincidence", is a baffling thing to put immediately after apparently demonstrating a coincidence!
- kazinator 2y agoIt is not unusual case. The heuristic you want is working. It's nothing more than a coincidence.
- mistercow 2y agoWhat are you disputing about the explanation given in the post? As far as I can tell, it’s basically accurate (although the pendulum unit was called the toise, and the meter seems to have targeted half a toise). If you accept that account, it’s not a coincidence.
- kazinator 2y agoI initially had an objection due to a misconception I was carrying. I see now that the pendulum formula is a pure relation between time and distance/length. It will apply regardless of the units used. For example if we measure time in fortnite and length and furlongs the formula will be the same. The gravitational acceleration of course will be in those units: furlongs per fortnights squared. Needless to say that will not be 9.81. Now the meter unit was chosen in relation to the second unit by the length of a pendulum that produced an integer period. So that choice/relation caused the gravitational acceleration g to take on such a value that its square root cancels out the π on the outside of the root. I was confused for a moment thinking that the definition of the kilogram would somehow be mixed up in this but of course it isn't. g doesn't incorporate mass; and of course pendulum swings are dependent only on length and not mass. There are all sorts of situations in which certain units either give us a nice constant inside the formula or eliminate it is entirely. For instance Ohm's law, V = IR. It's no coincidence that the constant there is 1. If we change resistance to some other unit without changing how we measure voltage and current we get V = cIR.
- ordu 2y ago> This is just an unusual case where that heuristic fails. I don't have this heuristic drilled into me, so I saw the point immediately. To be frank, I suspected the general direction of the answer after reading the headline, and this general direction, probably, can be expressed the best by pointing at the sensitivity of the approx. equation from the headline to the choice of units. So, I think, the reaction to this quote says more about the person reacting, then about this quote. If the person tends to look answers in a physics (a popular approach for techies), then this quote feels wrong. If the person thinks of physics as of an artificial creation filled with conventions and seeking answers in humans who created physics (it is rarer for techies and closer to a perspective of humanities and social sciences), then this quote is the answer, lacking just some details.
- mistercow 2y agoNo, like objectively, a dimensionless number lining up with a meaningful constant is more likely to be because of some underlying mathematical connection, and a dimensioned number lining up is more likely to be a coincidence. There are only a handful of ways for a unit’s heritage to have a connection to a local physical phenomenon like the post describes, and that’s what it takes to have a unit-dependent non-coincidence. That’s not dependent on your perspective. The thing that’s interesting in this case is that the meter’s connection to g is obscured by history, whereas most of the time a unit’s heritage is well known. Nobody is going to be surprised by constants coming out of amps, ohms, and volts, for example, because we know that those units are defined to have a clean relationship.
- ycui1986 2y agoI think what the author want to convey is that the metric system was designed based on the assumption that pi^2 = g. The assumption pi^2 = g is one of the source of the metric system (at least for the relationship between meter and second). The deviation was due to the size of earth being incorrectly measured by French in the original expedition.)
- deleted 2y ago[deleted]
- jdthedisciple 2y agoI think you are right and i had the exact same thought. I think people are misunderstanding you.