egg|nomz|egg changed the topic of #kspacademia to: https://gist.github.com/pdn4kd/164b9b85435d87afbec0c3a7e69d3e6d | Dogs are cats. Spiders are cat interferometers. | Космизм сегодня! | Document well, for tomorrow you may get mauled by a ネコバス. | <UmbralRaptor> egg|nomz|egg: generally if your eyes are dewing over, that's not the weather. | <ferram4> I shall beat my problems to death with an engineer.
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<kmath> <johnregehr> @TerminateThread I definitely need a better "it's going to get squashed anyway" commit msg
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<UmbralRaptop> TFW walking 3 miles is faster than waiting for the bus.
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<bofh> UmbralRaptop: been there done that
<UmbralRaptop> Bonus points if it's raining.
<bofh> UmbralRaptop: been there done that
<bofh> :P
* UmbralRaptop awards bofh the requisite number of bonus points.
<UmbralRaptop> Anyway, ChExo was fun, aside from transportation issues.
<UmbralRaptop> Also, apparently Uranus has a weirdly active radiation belts (2nd only to Jupiter), while Neptune's are notably quiescent?
<bofh> Wait, *what*?
<bofh> via what proposed mechanism?
<bofh> (mostly curious about the Uranian ones, Neptune being quiescent kinda is unsurprising).
<UmbralRaptop> Mechanism unknown, the speaker just said that's what has been measured, and a reason for ice giant orbiters.
<tawny> quiescent here means not as radioactive, or something else?
<UmbralRaptop> Sorta. Lack of particles, especially at higher energies.
<bofh> I mean the reasons for ice giant orbiters are { gigantic scroll that runs a km in length and is written in 12-point Courier New }, tbqh.
<iximeow> this is accurate
<UmbralRaptop> Yep.
* UmbralRaptop does hope that they'll be a high priority on the 2020 decadal survey.
<bofh> Same.
<bofh> Like, we only have so many launch windows for those since we need at least Jupiter grav assists.
<bofh> Ideally J+S.
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<B787_300> UmbralRaptop: if you are so concerned about photons why don't you use a really narrow FOV and a pmt?
<bofh> seconding the PMT suggestion
<UmbralRaptop> PMT?
<B787_300> Photo multiplier tube
<UmbralRaptop> Ah. Because that's solving the wrong problem. The targets are decently bright, even spread out over an R~100k
<UmbralRaptop> I just need absurd SNR over a night to get a precision in radial velocity slower than most people walk.
<UmbralRaptop> "just"
<B787_300> Right but you are trying to detect dips in brightness?
<UmbralRaptop> minute red/blue shifts.
<B787_300> ...
<UmbralRaptop> Radial velocity method, not transit method. >_>
<B787_300> Oh you poor soul
<B787_300> No wonder a 3.2 is too small for you
<bofh> Oh, yeah, it all makes sense now.
<B787_300> Hrm I wonder if you could get creative and use 3+ PMTs with narrow band pass filters on them
<B787_300> Although putting a splitter block and filter is putting something that might absorb the light
<xShadowx|2> any math geniuses? not even sure what to google for this, lets pretend there is a solar system, and a planet that revolves around its star at some speed, and theres a ship at some point in the system, and can move at some speed, what kind of math is used to calculate the intercept point between the ship and the planet? i know how intercept works with 2 straight lines, but planet moves in circle
<xShadowx|2> oh and ignore gravity effects :|
<B787_300> UmbralRaptop: also how were you going to detect the shift with a monochromatic sensor?
<iximeow> parametric equations \o/
<xShadowx|2> just straight line intercepting circle path
<xShadowx|2> thats what i should google? :D
<B787_300> xShadowx|2: uuum well as you said ignore gravity the ship ain't moving unless it thrusts
<B787_300> And can the ship move fast enough that you can assume the planet doers move?
<UmbralRaptop> B787_300: dozens of orders from an echelle grating.
<iximeow> it's probably not the ideal way, but the way i've done that in the past is phrasing the motion of all objects as functions of t, solving X and Y independently for t, and if any t is the same for both, they intersect
<iximeow> (extending that to also Z if you're worried about three dimensional motion, which uh.. at the time i was not)
<B787_300> xShadowx|2: I mean iximeow is right in the construct given to us. In actual space flight you need ephemerides and isp and wonderful conic sections
<xShadowx|2> hoping theres some formula using X,Y (ship coords, assuming star is 0,0), R (radius of planet orbit), S1 (speed of planet), S2 (speed of ship) to just say "ok heres the closest point" :3
<iximeow> B787_300: yeah given the way it's posed i figure it's a sufficient model
<B787_300> UmbralRaptop: what is the QE of your detector? And could you get the Dark/read current any lower on it?
<UmbralRaptop> I suspect that given initial positions / velocities, there's an analytic solution.
<B787_300> Because if you need SNR and the S is locked you need to reduce the N
<B787_300> So can you get cyrogens to cool your detector?
<UmbralRaptop> B787_300: decently high, though overall system efficiency is <5%. Read noise is smallish, dark current is assumed to be 4 electrons/hour/pixel.
<iximeow> ah, read noise measured in single electroncs/hr
* iximeow needs to astro better
<B787_300> I don't want to know how much that detector cost then...
<UmbralRaptop> hah
<B787_300> Like the I am used to seeing ~5e/sec/pix
<UmbralRaptop> (Yes, it's cooled. -100 C, IIRC)
<B787_300> And that isn't a cheap sensor
<B787_300> iximeow: so you also astro some too?
<iximeow> B787_300: not my best image, but https://pbs.twimg.com/media/DbxdNegUwAAukRt.jpg
<iximeow> i'm slowly trying to figure out how to align my 11" sct as i know it's still a little off
<UmbralRaptop> Teledyne's price for their Hawaii line of sensors is "call us" o_O
<iximeow> lol
<B787_300> UmbralRaptop: here is a thought... do you have a coude room and optics table to use? Because then you might be able to set up so that the grating is on one side of a table the you have multiple detectors that are close together physically on the other end so you are using the grating to help separate out really close color shifts
<B787_300> iximeow: what brand is your SCT?
<iximeow> B787_300: celestron. i got it "used/for parts" because the previous owner uh... didn't mount the secondary properly, it fell, chipped the primary, it's a whole thing
<iximeow> i've not yet convinced myself the primary is seriously damaged (that picture ain't bad) but..
<B787_300> Ah... all my recent experience is with meade or takahashi
<B787_300> But I got to use a 32 grand sensor on a 16" ... So yeah the sensor was worth more than the scope
<UmbralRaptop> iximeow: damaged secondary on an SCT? I'm slightly impressed. O_o
<iximeow> UmbralRaptop: another source of problems i've not yet addressed is the uh
<iximeow> smudge, on the secondary
<UmbralRaptop> B787_300: We have an optics table, and will eventually have a clean room, though I'm not as close to the hardware as you might think.
<iximeow> i think the previous owner fished the secondary out of the telescope and got a thumb on the mirror or.. something
* iximeow cringes at the thought
<UmbralRaptop> iximeow: as long as it; s not a blood spatter, or something…
<B787_300> I am just thinking of ways to increase S and as PMTs are a bit large you need the angular separation
* iximeow considers trying to fiddle with alignment again tonight
<kmath> <AmyZenunim> Due to European data protection regulations, you are required to meow back at any cat who meows at you, effective May 25th, 2018.
* iximeow meows at egg
<egg|zzz|egg> meow
<egg|zzz|egg> !wpn bofh
* Qboid gives bofh a DEC nickel
* iximeow looks outside
<egg|zzz|egg> UmbralRaptop: should I get a cat or a reflector
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<iximeow> hm. this programmer makes slightly more sense
<iximeow> time to sleep
<kmath> <ChemPlayer1> A bit of sleuthing indicates that we were finally killed off for 1) chocolate cake, 2) Grignard reaction, and 3) ph… https://t.co/SUjYEsFCUY
<egg|zzz|egg> !wpn whitequark
* Qboid gives whitequark a giant double/goto hybrid
<egg|zzz|egg> bofh: UmbralRaptop: whitequark: so apparently there are moar Spitzer observations, https://arxiv.org/abs/1801.02554, so I guess I should update https://twitter.com/eggleroy/status/948349779389636609
<kmath> <eggleroy> Transit observations behind the TRAPPIST-1 papers [1,2]. Spitzer time red [3], VLT time blue [4], K2 campaign 12 ye… https://t.co/A0htiu4NU4
<egg|zzz|egg> ... did they typeset Spitzer in mathmode
<egg|zzz|egg> 21st february to 26th march, so a fair bit of overlap with the K2 observations
<bofh> oh nice
<kmath> <larplover69> D&PhD, a tabletop in which you have only one very high stat and have to justify how it relates to anything you want to do
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<egg|zzz|egg> bofh: what kind of coordinate-free animal is the moment of inertia
<egg|zzz|egg> bofh: well, a bilinear form, but what is this weird bracket squared really https://en.wikipedia.org/wiki/Moment_of_inertia#Motion_in_space_of_a_rigid_body,_and_the_inertia_matrix
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<bofh> looks like a skew-symmetric matrix constructed componentwise and then squared?
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<bofh> which honestly just looks like a way of avoiding exterior products???
<UmbralRaptop> Finally an achievable goal. https://twitter.com/neuromusic/status/997335265768230912
<kmath> <neuromusic> By 35, you should have upgraded to Python 3.
<egg|zzz|egg> bofh: squared means the result is symmetric tho
<egg|zzz|egg> bofh: and casting a vector to a bivector is orientation dependent
<egg|zzz|egg> bofh: also wtf is squaring a bivector
<bofh> ...that's a good point, what the heck.
* bofh stares at the page again
<bofh> UmbralRaptop: I'unno, https://twitter.com/lexaloffle/status/997557186334048256 was extremely easy to achieve imho.
<kmath> <lexaloffle> By 35, you should have started more projects than you can hope to complete in your lifetime.
<UmbralRaptop> heh
<egg|zzz|egg> bofh: physicists like reinterpret_casting things in coordinates
<egg|zzz|egg> bofh: which violates the strict aliasing rule^W^W^W coordinate-free invariants
<bofh> THAT'S THE BEST WAY OF SUMMARIZING IT I'VE EVER HEARD THANK YOU
<bofh> okay the more I stare at these expressions on this wikipedia page the less sense it makes.
<bofh> like it seems to not actually *be* a bivector? but that doesn't make sense physically then
<egg|zzz|egg> bofh: I obviously isn't a bivector, it's symmetric
<egg|zzz|egg> and probably positive definite
<egg|zzz|egg> it's a bilinear form
<egg|zzz|egg> but how do you get from a vector to a bilinear form via a sinful skew-symmetric matrix
<bofh> it's positive semidefinite via construction, at least
<bofh> ¯\_(ツ)_/¯ my linalg prof would've docked marks for this construction in all honesty.
<bofh> I seriously want to rewrite this section now but first I need to figure out why this construction is sensible (for some weak value of "sensible").
<SnoopJeDi> !wpn -add:adj sinful
<Qboid> SnoopJeDi: Adjective added!
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<egg|zzz|egg> bofh: blargh I really don't see what this [.]^2 is doing >_>
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<egg|zzz|egg> bofh: hm so L = r ∧ p = r ∧ mωr which is linear in ω =: Iω (where I is then also a bilinear form on bivectors via the inner product), but what operation is r ↦ (ω ↦ r ∧ m ω r)
<egg|zzz|egg> (wlog ignore the m)
<bofh> hrm
<bofh> I'm not seeing wtf the [.] construction makes sense to do, honestly.
<egg|zzz|egg> oh that one certainly makes no sense
<egg|zzz|egg> but [.]^2 might be somethnig
<egg|zzz|egg> let ω be pure, ω = v ∧ w, then that is r ∧ (v ∧ w).r where . is "bivector acting as an alternating form"
* egg|zzz|egg stares blankly
* bofh blinks
<egg|zzz|egg> bofh: okay what if we start from the insanity, then it's Iω = (*r)((*r)(*ω)) where * is the Hodge star, how do we get of all those stars now Ꙩ_ꙩ
<egg|zzz|egg> wait no there's probably another star here, Iω = *((*r)((*r)(*ω)))
<bofh> sec, let me try and remember how Hodge * works
<egg|zzz|egg> it's the [silly bracket]
<egg|zzz|egg> given an orientation the [silly bracket] is sensible
<bofh> okay yes there's definetly a fourth one, and I *think* that lets us cancel
<bofh> ohh. right, b/c the construction fundamentally requires a choice of coordinate
<egg|zzz|egg> bofh: just of orientation (I mean, it's in coordinates, but the construction can be rewritten *((*r)((*r)(*ω))) which only summons the Hodge star)
<egg|zzz|egg> now how do we banish the Hodge star
<egg|zzz|egg> bofh: maybe ncatlab has something interesting to say https://ncatlab.org/nlab/show/moment+of+inertia
<SnoopJeDi> !wpn -add:adj Hodge
<Qboid> SnoopJeDi: Adjective already added!
<SnoopJeDi> good, good
<egg|zzz|egg> bofh: blarg no, ncatlab doesn't say anything we haven't already
<egg|zzz|egg> oh there's the Lie approach
<bofh> hm. *can* we banish the Hodge* entirely, even?
<egg|zzz|egg> bofh: I certainly hope so, if rigid body dynamics depend on orientation I'll be annoyed
<egg|zzz|egg> so will the white rabbit
<bofh> like I feel like the innermost (*r)(*ω) should be simplifiable to an equivalent expression with the * outside and a wedge inside, but... actually nevermind. hm.
<egg|zzz|egg> so *r is a bivector, *ω is a vector, this is a bivector acting on a vector so a vector
<egg|zzz|egg> does it have something to do with ωr
<bofh> sec, trying to make sense of bivector ⇔ pseudovector isomorphism https://en.wikipedia.org/wiki/Pseudovector#Geometric_algebra
<egg|zzz|egg> bofh: I think they're the same thing, except pseudovector is the careful physicist word?
<SnoopJeDi> "careful" :P
<egg|zzz|egg> well the careless one will call it all vectors
<egg|zzz|egg> bofh: there's the so(V) = V ∧ V isomorphism that I'm using extensively
<bofh> yeah, like they're clearly isomorphic thanks to Hodge duality
<egg|zzz|egg> bofh: okay (*r)(*ω) = rω afaict
<egg|zzz|egg> bofh: yeah and then it's going to be obvious that this is r ↦ (ω ↦ r ∧ m ω r)
<egg|zzz|egg> bofh: so the insane construction is correct
<egg|zzz|egg> bofh: but what *is* r ↦ (ω ↦ r ∧ m ω r) anyway
<bofh> okay yeah in that case we ARE good, but... yeah.
<bofh> uh what is m in (ω ↦ r ∧ m ω r)?
<egg|zzz|egg> bofh: so, in coordinates, this thing is the tensor product of r with itself except on the diagonal, where instead of x^2 you have -y^2-z^2
<egg|zzz|egg> (etc.)
<bofh> *nods*
<bofh> hm.
<egg|zzz|egg> bofh: so that's, uh. r⊗r-⟨r,r⟩𝟙 ?!
<egg|zzz|egg> bofh: but, why
* UmbralRaptop abuses the school network to keep software on emergency backup computers up to date.
<B787_300> UmbralRaptop: https://ieeexplore.ieee.org/document/6872788/?reload=true so they make SPD arrays that might be able to be used with a grating to increase your detection rate
<bofh> egg|zzz|egg: hrm. I *think* so
<bofh> but indeed, why the heck
<bofh> this seems... weirdly more complicated than it needs to be?
<UmbralRaptop> B787_300: er, my problem is less detection than needing huge amounts of data to get the requisite precision. eg: Altair is 1) the brightest star in my target list, and 2) basically impossible to get useful data on because it has so few absorption lines.
<B787_300> UmbralRaptop: but if you can use less integration time to get a lot of data
<B787_300> as you mentioned you were needing silly long integration times
<B787_300> and if you are detecting more efficiently then you are getting more data for a shorter time
<bofh> https://pbs.twimg.com/media/Ddj_rcPW0AANb2A.jpg:orig oh hey it's my breakfast cereal
<egg|zzz|egg> !wpn rqou
* Qboid gives rqou a baka code
<rqou> lolol
<egg|zzz|egg> rqou: btw I need to try to find a closed-form for the rqou numbers
<rqou> meh i don't care enough :P
<rqou> i only needed that one number's order of magnitude :P :P
<egg|zzz|egg> yes, but still, I should find a closed-form
<egg|zzz|egg> it's fun
<egg|zzz|egg> enumerative combinatorics is fun
* egg|zzz|egg did his MSc. thesis in that
<egg|zzz|egg> I mean the goal was some sort of eggstremal combinatorics but I only found a very mildly incremental result
<egg|zzz|egg> bofh: so, being more careful with the musical isomorphisms, Iω = (r⊗r♭)ω-⟨r,r⟩ ω?? but why what how
<bofh> egg|zzz|egg: I, uh, wtf. I think you have to get that result but I'm not seeing how to get it.
<egg|zzz|egg> bofh: I'm mostly unsure what the fuck that operation is
<egg|zzz|egg> tensor-minus-inner-times-the-identity-wat
<egg|zzz|egg> bofh: and as a form, vIw = ⟨v,r⟩⟨r,w⟩-⟨r,r⟩ ⟨v,w⟩?
<egg|zzz|egg> bofh: wait is a bilinear form on bivectors the same as a bilinear form on vectors, or do you need an orientation
<bofh> I... that's a good question, actually. I THINK it should be the same for both vectors and bivectors, but now I'm not at all sure.
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<egg|zzz|egg> bofh: we have an inner product so I'd say yes, bilinear forms are linear maps and linear maps induce linear maps on multivectors?
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<bofh> egg|zzz|egg: oh, good point. and yeah, writing it out in terms of inner products and linear maps, yes, it better be the same.
<egg|zzz|egg> bofh: okay so now we can ignore the bivectors and we just have the bilinear form I(r) with I(v,w) = ⟨v,r⟩⟨r,w⟩-⟨r,r⟩ ⟨v,w⟩
<egg|zzz|egg> bofh: I guess principia can just have a SymmetricBilinearForm type :-p
<bofh> that makes things slightly clearer I guess?
<egg|zzz|egg> bofh: and then you have the inner product form, given a vector its tensor product with itself is a symmetric bilinear form, symmetric bilinear forms form a vector space, and that's sufficient to construct I
<egg|zzz|egg> bofh: of course you can generalize to the symmetric product (v⊗w+w⊗v)/2
<egg|zzz|egg> but I don't think I actually need it
<egg|zzz|egg> apparently that's known as v⊙w? https://en.wikipedia.org/wiki/Symmetric_tensor#Symmetric_product
<bofh> Huh.
<bofh> Okay I *have* seen that notation awhile back
<egg|zzz|egg> bofh: meanwhile in openfpga the 2 norm was beaten back with a cat :-p
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<bofh> egg|nomz|egg: GOOD
<kmath> <qntm> Hypothetically speaking would it be possible for someone to punch out the window on an Apollo command module? I nee… https://t.co/5XWeyiKoZl
* egg|nomz|egg should build a pinyin to viet consonants transliterator
<B787_300> UmbralRaptop: i mean hypothetically yes... their hand might not be a hand afterword as i am pretty sure you cant force it out but you might be able to cause fatigue in the bolts/support stucture by applying a repetitive force to it
<UmbralRaptop> Ow
<bofh> egg|nomz|egg: that seems useful
<egg|nomz|egg> bofh: I mean mostly if you can pronounce (some dialect of) viet but not standard pinyin :-p
<egg|nomz|egg> bofh: although viet has no such thing as pinyin q or ch, only pinyin j or zh
<egg|nomz|egg> cc rqou for chinese phonology
<B787_300> UmbralRaptop: yeah... i mean you have to think that: 1. it is a wedge design with the narrow part on the outside to make it so that a blowout would be hard as hell (like how Airline windows are designed) and 2. it is a designed for a 14.7 psi pressure differential
<UmbralRaptop> B787_300: given that this is for a silly SF story, I wonder if that means someone punching in on a depressurized capsule is plausible. >_>
<B787_300> nope
<B787_300> as then you have the same support structure
<bofh> egg|nomz|egg: I mean such a mapping obviously need not be bijective, I guess you pick which of injectivity or surjectivity will fail in your mapping.
<egg|nomz|egg> bofh: pretty sure on consonants there are just more in zh-cmn
<egg|nomz|egg> bofh: fewer tones but they're afaict different so I have no idea how to pronounce them
<egg|nomz|egg> rqou: halp i cannot into zh
<rqou> er, what?
<rqou> i'm not very good either
<egg|nomz|egg> rqou: well, vi has [ʈʂ] and [tɕ] but none of the [ʰ] stuff so that confuses me; also I don't think [ɕ] is a thing ([ʂ] is though)
<egg|nomz|egg> bofh: the reference epoch in all TRAPPIST-1 papers seems to systematically be the first observed transit, to the extent that this now appears to be unstated
<egg|nomz|egg> bofh: clearly we should date everything in reference to the first observation of the transit of TRAPPIST-1 c
<bofh> egg|nomz|egg: how to convert between that and J2000?
<egg|nomz|egg> bofh: BJD_TDB 2457282.80570
<egg|nomz|egg> uncertainty 0.00140
<bofh> "uncertainty"?
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<SnoopJeDi> bofh, Ellied I believe this is your genre as well as mine: https://www.youtube.com/watch?v=0aFhzGEBQlk
<kmath> YouTube - Trinitron: Sony's Once Unbeatable Product
<whitequark> !wpn egg
* Qboid gives egg a Schläfli fractal
<bofh> !wpn egg
* Qboid gives egg a segfault
<iximeow> 0x41414141
<Majiir> .hat
<Majiir> !wpn egg|nomz|egg
* Qboid gives egg|nomz|egg a gzip'd Gaelic stack
<UmbralRaptop> !wpn Majiir
* Qboid gives Majiir a restricted summation-like thermistor
<UmbralRaptop> So, Twitter sent an email about how I could watch the royal wedding on their site. ☭☭☭
<kmath> <sigfig> distinctly unclear if i just have a high probability of encountering strange events or if i do something to generate them
<bofh> the local ClearChannel top40 station just inexplicably out of nowhere played Amish Paradise and then went right back to regular programming, with *zero* mention of it at all.
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<Majiir> !wpn UmbralRaptop
* Qboid gives UmbralRaptop a Vulcan McParseface
* UmbralRaptop coats Vulcan McRubberface
<bofh> !wpn Majiir
* Qboid gives Majiir a sigma branch
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<egg> bofh: uncertainty in the mid-transit time
<egg> bofh: also, with the resonance you could probably define a whole calendrical system based on the transits!
<egg> whitequark: are there any new котяpics?
<bofh> oh my gods you're right and it's really tempting to make a trappist-1 calendrical system now :P
<egg> bofh: let's do it! :-p
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<egg> let me dig up a recent paper
<egg> bofh: btw did you see that thing on cosmic censorship I linked yesterday
<Majiir> egg, hello
<egg> o/ Majiir
<egg> !wpn Majiir
* Qboid gives Majiir a polarized prediction which strongly resembles a minion
<iximeow> !wpn Majiir
* Qboid gives Majiir a rotary dust which strongly resembles a div
<Majiir> Are you still at Google, egg?
<egg> yeah, still getting paid to stare at number formatting at youtube
<Majiir> I always knew you had a future in numerics
<egg> also date formatting
<egg> also pluralization
<Majiir> Are you happy?
<egg> also teaching UX writers about the grammars of 80 languages
<bofh> egg: yea
<egg> Majiir: I get to write emails in latin at work, so that's fun :-p
<bofh> I'm not a black hole person so I'm not sure I properly appreciate it
<Majiir> egg, that does sound fun
<Majiir> Maybe I'll start opening with _salve_ from now on
<bofh> Majiir: egg: quidquid Latine dictum sit, altum videtur. :P
<egg> (context: I didn't know enough russian to write examples of declension of names making translation problematic, so I conjured up some latin UI strings instead)
<egg> bofh: I think it didn't :-p
<bofh> egg: touché :P
<egg> bofh: hm the resonances are far from eggsact, so that would probably be lunisolar-but-much-worse
<egg> the 8/5 is pretty close
<bofh> can you really call them resonances in that case?
<bofh> I mean it's a system with seven planets, even picking a subset of them you're going to wind up with something much worse than lunisolar
<egg> iximeow: cat!
<iximeow> i really can't tell what shape or orientation she currently has through all the fur, but
<egg> is the cat orientable
<bofh> are cats, in general, orientable?
<iximeow> i suppose not
<bofh> !wpn -add:adj Möbius
<Qboid> bofh: Adjective added!
<egg> bofh: so the latest periods are 1.51087637 2.42180746 4.049959 6.099043 9.205585 12.354473 18.767953, see arXiv:1801.02554v1
<egg> hm, Qboid expands github issues and pull requests, but not arXiv abstracts
<egg> Thomas: feature request!
<bofh> middle one looks like 81/20, which is a really odd long-period resonance.