The mystery of 0.8660254037844386467637231707529361834714026269051903140279034897...
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- čas přidán 12. 02. 2024
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OEIS sequence A010527 "Decimal expansion of sqrt(3)/2"
oeis.org/A010527
Do support the OEIS! oeisf.org/donate/
And here are a bunch of the digits:
0.8660254037844386467637231707529361834714026269051903140279034897
Wikipedia page on "ISO metric screw thread"
en.wikipedia.org/wiki/ISO_met...
Huge thanks to my Patreon supporters. They keep my triangle equlateral. / standupmaths
CORRECTIONS
- None yet, let me know if you spot anything!
Filming and editing by Alex Genn-Bash
Written and performed by Matt Parker
Produced by Nicole Jacobus
Music by Howard Carter
Design by Simon Wright and Adam Robinson
MATT PARKER: Stand-up Mathematician
Website: standupmaths.com/
US book: www.penguinrandomhouse.com/bo...
UK book: mathsgear.co.uk/collections/b...
Oh look at this: forms.gle/hWes6BgPji3L8RfH6 - Zábava
The creative desision to copy a real life object in editing so you can make it seem edited, only to interact with it physically, blew my mind it was so clever.
yup it was cool
Agreed
Reminds me of "the top 3 hardest things to say" by PixelzwithaZ
I finally know how it felt to be in an early theater and think I was going to be hit by a train
Yes.
Why are all these numbers the same?
...cos
This pun is a sin.
@@statesburgproductionsThat deserves a tanning.
Wait a sec...
That pun needs a cot to rest on.
Trigger warning
Another example of this number: the percentage of the speed of light that you need to travel at to have your time dilated by 2 (0.866c)
So if I set up a light clock, the distance between the mirrors is 1 light second, and I travel so that the distance light travels is double of that... then I have a right triangle with a leg of 1 and a hypotenuse of 2.
Spot the 30-60-90 triangle again
Oh yeah you’re right. I hadn’t thought about it that way before, I just was goofing around with an online calculator lol
@@nanamacapagal8342 wouldn't that be "other leg 2, hypotenuse root 3"?
@@lordchickenhawk root 3 is the other leg...
@@nanamacapagal8342 Ah yes, I'm trying to think all arse about
Reminds me of the video where Pi kept showing up in mysterious places and it was just because there was a circle hiding in there somewhere.
Reminds me of that 3blue1brown quote:
"If you see pi show up, there will always be a path somewhere in the massive interconnected web of mathematics leading you back to circles and geometry."
Which video are you talking about specifically? I'd love to see it. I love 3b1b's videos on unexpected pi (particularly the one that starts with two squares physically colliding with each other and the wall)
@@LetsGetIntoItMedia The Basel problem video (with pi^2/6).
There is another interesting case where pi of the parabola 2.29.. shows up in average distance between two points in a unit square. :)
@@LetsGetIntoItMediaI think it was the one about pi^2/6
Brady is probably a bit miffed this wasn't a Numberphile video. "0.8660254..." would've made a great title.
@@pekirt not in the modern numberphile but at the beginning and i think for a while, the channel was genuinely dedicated to individual numbers for each video, hence the name
Or Brady could make it next week….😂
@@pekirt Numberphile was pretty literal at first, I think sqrt(2) and the golden ratio are there originally. It was a counterpart to the other "collector" channels he did like sixty _symbols_ or the periodic table of _elements_ - so aiocafea is right, viewers just have to go back a bit. (fixed italics)
That being said I can see how Matt shows off the examples in a fun and engaging, and it's nice he has his own channel these days rather than having to work with meeting up with Brady every time he has something to publish!
Glad it ISN'T a Haran video because that saves everyone from his insufferable, nauseatingly hyperactive, artsy-fartsy, painfully forced hand-held, pseudo-action camera style he and that Riley guy just can't stop using.
This number looked strangely familiar and I realized where I know if from.. In graphic design, if you want to make a shape look isometric (skewing a square to make a side of a little isometric building icon, for example), I learned the formula shortcut in the Adobe Illustrator transform panel: Leave the width as is and change the height to 86.602%, then shear -30 degrees and rotate 30 degrees. I never understood where that really specific number came from, but it makes sense now after watching Matt explain the angle relationships. He is right, math is everywhere and it's all connected. So cool!
Fascinating. I will never look at bolts and icosahedrons the same way. Wait, I mean, from now on, I WILL look at them exactly the same way.
You forgot to raise an eyebrow.
It makes more sense to look at bolts and triangles the same way, but you do you.
"cos is cah"
I like the way a former maths teacher still uses the soh cah toa for remembering. 😀
"some old hippie, caught another hippie, well you know the rest"
in German we have
GAGA
HHAG
HHAG stands for "Hühnerhof AG" = chicken farm work group
The mnemonic I was taught uses the lower case cursive letters c, s, and t. Superimpose it on a right triangle, and use the edge you start with as the denominator and the next edge as the numerator. This had the added bonus of me tilting my head during exams, which was always good for a laugh.
I’m a current math teacher (teaching native American kids)! It sure makes telling that story a bit awkward, but they all know SOH-CAH-TOA!
@@lazykbys I tried to imagine what you're talking about from your post, but I'm totally lost. SohCahToa was my mainstay.
EDIT: It wasn't immediately obvious how to google that, but it came up from nailthemath at wordpress, and it does look about right. Sohcahtoa just had a nice ring to it tho, and it's never left my brain even for a second.
Saying that the cubic root of i would be √3/2, when in fact you mean that the _real part_ of one of the cubic roots of i doesn't feel like the kind of exact language I am used to from mathematics...
The icosahedron and √75 were also not exactly the same value as the others either, they're just related to them in a certain way. And yeah, it's not an exactly or precisely worded intro, instead it's mostly setup to let you imagine whatever relationships that might come to mind to try and think about, before coming in to tie things together with the triangle.
@@WindsorMason I can get behind the missing factor of 10. As he keeps on saying, they have the same digits and everything. But a complex number is not complete without the imaginary part. There ist more missing to the picture than just a factor 10 that is glossed over for the story and simpliciity.
@@kullen2042 He specifically mentions that in the video.
@@scottdebrestian9875well yes, but not in the introduction when he's saying these are all basically the same number.
I agree, it threw me off a bit.
@9:04 "We'll get that in post". Yes, you really nailed it.
Narrator: he did not, in fact, get that in post.
😂😂😂😂😂😂
I always enjoy the clever editing you throw in
Three-phase power at 120V per phase is ~208V. 120√3.
Also 230V single phase and ~400V 3-phase (outside of North America).
Yeah, not getting that in the video was a missed opportunity IMO.
Learning the mathematics involved in three phase electricity was, for me, when many mathematical concepts went from being strictly memorized mathematical facts/relationships to seeing connections and truly understanding them. And that understanding crosses so many boundaries as Matt displays in this video.
When Matt plucked that number placard out of the air in the intro I audibly went "what the f***".
the 30-60-90 triangle is one of those math things that shows up so much youd think math itself is picking favorites
1:13 ...impressive. Somebody's been watching VFX tutorials haha
For Matt's next trick, he'll find and grab the temperature icons that float above the UK when forecasters do the weather
@@purpleotteruk I remember seeing an Antiques Roadshow episode where someone brought in some weather magnets that were used by British TV forecasters until they were replaced with CGI and chromakey in the 80s (or 90s, I forget)
That must have taken so long to get right
He's come a long way since "GLORIA IN X-SQUARIS"
Surely, the duration of the video would be eastereggily chosen as 0.8860254... too.
And yes!
It is 0.8860254 units of 12½ minutes!
With a tiny Parker error.
The same triangle is why you can't tune perfect intervals across all twelve notes. It's called the Pythagorean pause. Since intervals represent a ratio between two frequencies, and there is an irrational number there, there is always a little fudge factor somewhere.
I mean you could if you don’t care about keeping rational values for the intervals
@@TheEternalVortex42 Then you don't have perfect intervals.
This is actually how we tune modern instruments. The difference is split between all the notes and there are no perfect intervals.
5:17 I love how you added your reaction to doing-things-the-hard-way-Matt’s dismay at learning we didn’t have to do things the hard way
man wtf he pulled that number from the overlay
I am making a game with tesselations of triangles, squares, and hexagons. Root 3 was turning up so often that i made a contant for it so that it wouldnt recalculate every time.
What did you name your constant?
@@zarzee8925 Now I wish I named it something cool to make this story better, but it is just called root3. Naming things is one the two hardest problems in programming.
GOOD JOB! I tell my students, “If something takes more than 5 seconds to write AND you need to write it more than twice, think about a substitution.”
@@zarzee8925Maybe it was 1.73205 or SQRT3.
@@zarzee8925 I wish I had named it something different to make the story better, but it is just called "root3"
2:53 The fact that Matt chooses to leave the 10 and the root (3) / 2 unsimplified is pretty comical. I bet some viewers were _begging_ for Matt to simplify it down into 5 root (3).
Haha I certainly was, until I realized he was trying to emphasize that the digits are indeed root(3)/2, just shifted over one place by the 10
Was about to write the same comment. Man that made me mad hahaha... Happy to see there are still sane fellas down here who dont want to just see the world sucumb to chaos
Yeah, until he got to the complex plane solutions for (3)root(i) had me wondering if he needed a doctor.
Apparently it's just a wobbly video about cos(pi/6).
But *NOT* simplifying it was the whole point... The entire video is about ``` sqrt(3) /2 ``` appearing in various places.
This is also the extra height of a layer of diameter 1 circles stacked atop another. I recognised the number immediately, since I had that figure well memorised from some designwork I did a while back.
circles don't have height
Rows in a hexagonal lattice@@mrosskne
@@mrosskne In think he means stacked like cannonballs are stacked, but in 2 dimensions, so a triangle with a circle at each point, each circle touching both others. i.e. the triangle has a sidelength of twice the radius.
I'm always impressed by the creative editing choices in these videos. like in that other video where past Matt, present matt and voiceover matt where having a conversation with each other. here the label graphic became a real entity, and that "aw" from Matt on the whiteboard. it's always fun to see
Your editing is always so slick!
2:44 the area of any triangle is 1/2 * base * height, not just a right-angled triangle, so the halving of the base to then double it again was a little excessive 😂
I saw 75 as 100 * 3/4 Instead, which also obviously make 10*cos(30deg) when you take the root.
Each new upload make my day. Such a great work. Love you Matt.
8:25
Really glad to hear this image of a cube with its diagonal dotted out is gonna make future appearances. Such an underrated concept and I'd love to see it used more.
This is so random, love it!
Well, when you picked the number "out of the screen" I kinda choked on the water I was drinking. Really cool effect; plus, as always, great video!
Unintentionally hilarious youtube ads
"That is precisely what "doing things the hard way" Matt is doing, right now"
Ad break cuts to someone opening a burger box and biting into a burger.
Took me a moment to realise that it wasn't just Matt doing an editing joke.
Yay for the Perth Wildcats t-shirt!
I had to scroll far too far to check if somebody else had noticed it! Not only the wildcats, but the proper 90s logo!
Reminiscing on the James Crawley Ricky Grace era
I love your videos, thanks for the great content!
It's so weird. As I was on my way to work today, I was just mentally figuring out the volume of a sphere that bounded a unit cube. A couple hours later, this video shows up in my feed.
I do the same kinda things of course, but reading this makes me realize why us nerds get made fun of.
Edit: for example while assisting in a classroom, I'd spend my time calculating my step distance on the tiles (standard is 1 foot each) and trying to make it the geometric root between feet and meters to make it easier to measure my pace in either system.
The simulation engineer assigned to Matt's videos had to fill in for the simulation engineer assigned to your thoughts, and just used copy+paste so they could just go home already 😅
I was driving to work and derived the difference in arrival time between two speeds in minutes.
Distance X Difference in speeds / Product of speeds X 60
Works for miles/mph and kilometers/kph and is really easy if one of the speeds is 60.
Man I was doing the same thing but I was wondering what the standard distance betweent the threads of ISO bolts was
I’m a math teacher and that kind of weirdness happens to me with startling regularity (at least weekly). My guess is that my subconscious is primed to find familiar things. We truly are pattern-finding geniuses!
Thanks for putting this out today Matt, I have a Maths mock tomorrow and this reminded me that I have to know about roots of complex numbers!
really great video production
Love how Matt interacts with the graphics in this video!
Very slick with the number out of thin air.
excellent video. Thank you!
What a masterpiece of a video! Thanks Matt!!!
Haven't checked out an episode of this in a while. This was cool. I'll be coming back more 🎉
for the bit on the root of 75, I found it easier to go the other way around (fewer steps, too!)
Start with (√3)/2
put the two inside: √3/2/2
= √3/4
= √.75
Great video!
Watched the trick in 4k frame by frame.
Very impressive.
Thank you for this video.
Far Out!
factorial(GREAT transition) 1:15
I didn't find pleasing the fact that you didn't refer, in the beginning, that it was the *real part* of the cube root of i Matt. factorial(But it's a great video)
I don't think it's a transition. I think it was there all along, just masked out by a black square in the overlay.
Didn't think I'd ever see a reverse factorial joke, but this is funny
Strongly agree about the real part - there's a difference between simplifying to aid understanding, and stating plainly wrong things
He DID say it was the real part of the solution. 6:30
@@Jimorian eventually, yes, but he at several points explicitly said that the decimal value WAS the cubed root of i, which is just nonsense
9:05 "little did Matt know, they did not, in fact, get it in post"
In my final practical exam becoming a maintanance technician (yes, in some countries you get proper training + exams), in one of the tasks we had to remake a cover for a bearing and of course with three equidistant holes ... it's been 15 years, but i'll never forget this number.
I'm a fitter and turner in South Australia. I also recognised 0.866025... (the cosine of 30 degrees) immediately for a similar reason.
I once had to figure out the area of hexagons because I was cutting and machining various hex bar stock into stackable hexagonal counterweights of various different values.
While I was working out how long to make each section to achieve each target weight I discovered that the "across the flats" measurement (of hex bar) squared and then multiplied by the cosine of 30 is equal to the area of the hexagon face. From there it was easy to figure out each length to cut given an SG of 7.8
It worked to within the accuracy of the scales I was given to test each weight. That saved screwing about with putting hex barstock in and out of the lathe trying to face off just the right amount of chips.
Of course, I'd come across cos 30 & sin 60 plenty of times before that but THAT was the job that burned it in permanently.
Bravo!! 👏🏼👏🏼😂 Absolutely hilarious video!
That editing was SMOOTH
I appreciate how much iteration that intro must have gone through to flow that nicely
I like your editing tricks, mister!
Did you know that: pi = e = 3
Also another shocking coincidence: pi² = g = 10
Found the engineer
As a programmer I don't care about pi but e is definitely 2
And (e**pi - pi) is 20
And 2^10 = 10^3
Do you folks know what = means?
my jaw visibly dropped when matt said “all six are the same number”! Great video!!
You inspired me to find sequence for the OEIS. I have two, though one is technically a subsequence
That cubed root of i seems pretty cheap, to say "oh it's 0.866... but that's just the real part"
the real and imaginary parts are in fact separate which is why they have to be written out as a+bi
@@joeyenniss9099 of course they are, which is exactly why saying 0.866... *is* the value is disingenuous. It's not. It's only part of it.
@@marshallc6215 bruh by that same logic all the numbers on the list are cheap because they are a+0i dumbass
Wow, someone is very annoying, huh?
@@marshallc6215 alright FINE, it's Re(nthroot{i}{3}). Happy now?
The graphic-pluck trick was so seamless I didn't even question it. It was only when you tried to put it back I was like, "Wait a minute..."
That quality planned editing is brilliant.
This better get a bunch of views. It’s soooo good
i'm always impressed by the lengths of matt's takes
The 30-60-90 is my favorite right triangle too. I once learned a way to use it to construct two-point perspective in drawing.
Finally, the Parker Constant!
Good idea, but it won’t work. Letting MP=0.866025403784 works great here in the US and in most places; however, it throws errors in Commonwealth countries. (Apparently there MP implies something incapable of remaining reliably constant.) SORRY MATT!
This is one of your best intros.
Sqrt(3) on 2 sounds awkward. It's sqrt(3) over 2. Or is this an Aussie (not the dog) thing?
Yes.
As an Aussie highschool maths teacher, we definitely say "on" unless it's complicated, then we say "all over", like "root 3 on 2" and "three times a^2 plus b^2 all over 2"
Oh this hits so close to home, or rather, work. I use this number a lot in my field of work. The standard for reducers in ventilation systems is 60 degrees so I always use the square root of 3 to determine the reducer length when I'm drawing them.
Thanks for breaking my brain with that 3D cube
Thank you! 😃
root 3 over 2 ...
Good grief, this ratio is everywhere. it is almost as universal as the golden ratio.
which is 1 plus root 5 on 2...
thats all very well, but what I found amazing is that you picked that graphic up out of thin air, seamlessly!!
This was fun. Good on ya
Nicely written and edited! Numbers like this are all over the place. While they don't have much recognition, videos like this bring them into more light.
With the internet, every number can have its fifteen minutes of fame.
I love this man!
Pretty sure I saw you in Waterloo tap a week or so ago. Did not want to bother you but could swear it was you. Made my day anyway
You had me, I was wondering for ages how the cube root of i could have a real solution!
nice edit!!
Why does the thumbnail look like a Bunnings ad?
(To non Australians, Bunnings is a hardware store. It uses that exact colour palette and style of line drawings in its advertising.)
Because my lowest prices are just the beginning.
I hope the future video featuring the cube graphic that is for a future video references the cube graphic as being made for a past video, but you're using it again 😂
Also the ratio of the distance between opposite sides of a hexagon and the distance between the opposite corners of a hexagon.
Holy intro batman, you pulled a real fast one on us there from all them being the same to the materialization of the number, I am in shock
Hi Matt, speaking of reoccurring numbers. Have a look at the Renard series or "preferred numbers". I think it is quite interesting to have some kind of "standard" numbers for everything in engineering, like pipe diameters, voltages. And the list goes on and on and on.
I remember when I found this years ago.
Sqrt 3 multiplied by the radius is the width of a circle halfway from the center to the edge.
I love your videos.
Stand up maths theme song - still one of my favourites! :D
Great video! ❤
0:42 Shouldn't you specify unit-cube or something? I was confused here.
I love the Stand up maths music... always brings me a smile :)
1:05 this reveal made me spit out my drink
McMaster-Carr has CAD models and drawings of all their bolts, should you need one in the future without paying some scummy stock photo site.
If you're referring to the diagram displayed at 3:27, based on the ep notes appears he used the illustration used on the wikipedia page for ISO metric nuts, which is released in the public domain per the wikimedia detail info, so presumably no scummy stock photo site involved...
Their mobile app is one of the best I’ve ever seen for a retailer. Some serious nerds working at McMaster-Carr it seems.
what made you think he was using a stock photo?
@@mrosskneThe part where he said he had to pay for the image and where if you google for "nut and bold drawing stock photo" the exact image he used appears on shutterstock for $50
That’s ok. He can afford it.
Wow that editing at the end was something else 😂😂
That visual effect made me gasp, rewind, play again, then show it to my wife who also gasped. Well done!
The pitch on a bolt, I was not surprised: I have been following a lot of channels from machinists, I know that the angle for the thread on a bolt is 60 deg.
Unless it's a Whitworth thread then it's 55 deg. Or a British Association thread, then it's 47.5 deg. Or Acme at 14.5 with trapeziums instead of triangles
The Pohms were really good at complicating matters when it came to arcane and opaque standards... though to be fair, while Acme was originally proposed at Worcester, it was the one in America, not the one in England
@@lordchickenhawk For some reason, I think I have only watched US machinists, and some Aussies, Austrian too, but no British one.
@@olivier2553 The English threads I mentioned are pretty much archaic now. That's why I wrote "were really good at" The American Acme thread doesn't really count because it isn't based on triangles.
My point was that 0.866... only applies to ISO and American threads because those standards arbitrarily chose a 60 deg included angle. There have been other types of threads over the years that used different depth to pitch ratios and even different shapes.
I'm an Aussie fitter and turner myself, so I've had to machine up all sorts of different threads over the years. A lot of our old heavy industrial machinery here dates back to our pre-metric (British) days.
I appreciate how long the number in the title goes for
Having done some programming with hexagon maps, I recognize this number as the ratio of a hexagon's edge-to-edge length with its corner-to-corner length.
It comes up when you're translating from axial coordinates to an (x, y) screen position.
Strong SEO with that title!
Great icosahedron Geogebra skills Matt. :)
I learn from the best.
Mathematical coincidence? 👀👀
(I made a video about those last week, what a coincidence!)
Given there are good (and known) reasons for these ones, this wouldn’t fall under your definition of mathematical coincidence (at least if I remember it correctly). Maybe we can call them quasi-coincidences or something like that? Initially surprising but actually very logical
the reveal in the beginning blew my mind
Low key genius editing with that number grab.
That intro was so tight. A masterpiece
you should edit together a youtube short with that goated intro, then at the end say "well actually i made a whole video discussing this topic"
5:50 Complex analysis became one of my favorite math classes ever when they showed us how to get roots geometrically on the complex plane. My mind was totally blown.
The old 1-2-v3 triangle. Flashback to high school Trig. class.