Actually used this in KSP without realizing it was a thing. Was curious if I could line the boosters to perfectly balance without even spacing. Found the 7 on 12 configuration when thinking about it while waiting for a sandwich to toast.
Love this. I've been sticking 5 and 7 tubes in 12-rotor centrifuges for decades. I've always wondered about working out the mathematical possibilities.
The balancing only really matters for small centrifuges though. Most of the ones with large number of slots have fixed wells inside the rotor to the point where the weight of one tube becomes negligible, so they'll run perfectly fine with a single tube in an unbalanced configuration.
The catch is - almost all lab centrifuges are equipped with so called auto balancing unit. The rotor just auto-balance himself no matter how you load your cr*p into it :-)
I do a lot of repairs for centrifuges where I work. I've seen that a lot. The worst ones are when the rotor jams so bad that it prevents the door from opening and you have to bust the hinge to be able to un-wedge the swinging test tube holders and pull everything apart.
I always loved adding configurations because of how unbalanced they looked. People would always be like "you're gonna wreck the centrifuge!" only to hear it whir quietly and peacefully 😂
Brady, you should follow up with her and ask if the same balancing principle applies with a sphere and quaternions like it does with the centrifuge (disk) and complex numbers.
Quaternions are made of 3 imaginary and 1 real numbers. So its kind of like have an extra dimension compared with free space. I'm not sure if it can be done in a short video.
duggydo I think quaternions are 4 dimensional so they would be on a hyper sphere. From a little bit of research I did I think quarternions when brought to a power the argument doesn't add nicley like regular complex number, so the problem will be more complicated and probably better understood with linear algebra
I don't think that would be the case because a quarernion squared doesn't just multiply its angle from the real axis by two like imaginary numbers do. I might be wrong
@@MrMctastics I do agree that linear algebra might be easier than using quaternions. It might be possible with quaternions, but one might have to take into account for the reduction in dimension.
Rajesh Thomas quaternions are used almost exclusively in 3d graphics computation. You need the extra degree of freedom for it to work. That’s why I’m curious if it applies in this scenario when extending to balancing in 3D.
Atlas rocket doesnt give a sh*t tho, lol. Because of the fuel lines they have to put boosters where they will fit. Thus the uneven boosters cause the rocket to slip sideways. It's pretty crazy to watch.
Chemistry is very similar to physics in lots of ways - it is pretty much in-depth nuclear physics. Chemistry was even one of the prerequisites for my Engineering Physics program.
Jesse H. Betas pretending to be alphas in order to impress some imaginary female on the internet, when we're in a comment thread of a video talking about centrifuges from a statistical standpoint not many women would be reading this. Or to be a bit better worded, the ratio of men : women that would see this thread is fairly large
When I was a biology grad student I would use this trick to balance odd numbers of tubes in the centrifuge. We had one with 24 slots, so I could do every number except 1 and 23. I even tried balancing tubes with uneven volumes by putting the two less full ones a little closer to each other and the fuller ones further away.
@@Son96601 Still, this doesn't mean the configuration is balanced in the remaining direction. That is, the center of mass is in the line you've drawn but not necessarily in the middle.
Cuntslaw, but reflectional symmetry is *not* a sufficient condition for valid tube configurations (I'm sure you can easily find a counter-example). Rotational symmetry is.
@@jasondoe2596 the 7 configuration has no rotational symmetry. you can rotate it as many times and by how ever much you want and it won't look the same. What are you even talking about?
Another big advantage to using complex numbers, is that you can look at this problem in the frequency domain. You can take the DFT (discrete fourier transform) of a placement vector and inspect the first harmonic (in the second, and last positions of the DFT) to see if the configuration is balanced or not. A balanced condition, is where the first harmonic is zero. This can also give you a quantitative value for how "unbalanced" a configuration is. p = [1 0 0 1 1 1 0 0 1 1 0 1]'; % position vector, 1=> test tube, 0=> no test tube >> fft(p) ans = 7.00000 + 0.00000i 0.00000 + 0.00000i % first harmonic term -1.00000 + 1.73205i 3.00000 + 0.00000i 1.00000 + 1.73205i 0.00000 + 0.00000i -1.00000 + 0.00000i 0.00000 - 0.00000i 1.00000 - 1.73205i 3.00000 - 0.00000i -1.00000 - 1.73205i 0.00000 - 0.00000i % also first harmonic term (conjugate)
One laboratory teaching assistant that I had insisted that 3 tubes could be balanced in an 8 hole rotor, and nothing that I or any of the other students could dissuade him. Luckily, (1) the centrifuge was robust enough to work anyway; and (2) the students were, to a person, smart enough to figure out the TA was wrong (meaning the TA's misapprehension was not passed on to others).
It gets weird once you have 4 or more prime factors. Then you can have negative contributions! For example, for N=210, you can combine 3 dots in a triangle with 5 dots in a pentagon and 7 dots in an upside-down heptagon. This looks like there are too many dots at the "top" (where the triangle and pentagon overlap), but then you can take away 2 dots at the top and bottom to maintain balance. The resulting pattern has 13 dots in a weird pattern that is not just built from adding symmetric prime-factor sets of dots.
Actually, it's easier with just 3 prime factors. That allows N=30. I'll use N=60 because then the dots correspond to minute marks on a clock, which is intuitive for many. To the set of balanced dots: Add an upside-down triangle: 10, 30, 50 Add a pentagon: 0, 12, 24, 36, 48 Subtract opposites: -0, -30 The resulting pattern is 10, 12, 24, 36, 48, 50 and balances without any dots that are opposite, form a complete triangle, or form a complete pentagon.
Notably, all numbers divisible by 6, can be worked out in any configuration aside from 1 and k-1. We know this because all even numbers can be created with some number of pairs. And we know that if a number works with 2 and 3, then you could create any number by adding enough 2s, then a single three. So with 210, we know everything works. Regardless of how weird it looks.
This is interesting. I wonder, though, is it possible to have a balanced set up not built by balanced polygons (in the way that it is done here with negative contributions) whose complement is also not built by balanced polygons (i.e. on that requires a negative contribution) ? An easy example of what I mean (very similar to what Jeff gives), for N = 30, the triangle (0,10,20) together with the pentagon (3,9,15,21,27), but removing the pair (0,15) gives the balanced set (3,9,10,20,21,27) which cannot be built from balanced polygons without negative contributions. However, its complement in the 30 holed centrifuge can by built up from the pentagon (0,6,12,18,24), the triangle (5,15,25) and then all the remaining opposite holes can be occupied by pairs, so this configuration does not require any negative contributions. Does there exist a configuration such that it and its complement can only be arrived at by including some negative contributions? I feel that it is impossible.
It brings back so many memories of myself standing in front of the centrifuge doing biology experiments. We always kept tubes of different weight with just water in it to help us balance.
In practice (low tech) you fill the necessary other tubes with water. Another solution (high tech) there are self-balancing centrifuges that counterweight in their rotationmechanism, so it doesnt matter what you put in.
@@Robocop-qe7le I don't think a cell biologist is going to be building a self-balancing rotation mechanism any time soon or running the math for it to make sure it can counterbalance any number of tubes in sat in any formation. But that's just me.
And you don’t need maths to do that. Just make a mechanical negative feedback loop that moves the counterbalance until the top is horizontal. With some damping, of course, otherwise you will need to analyse the stability of the loop - which does need maths. An oscillating feedback loop would be fun!
Balancing the 7 tubes in the n=12 configuration makes perfect sense, actually! The side with 2+0+2 tubes balances out the side with 3 because the weight of the 4 tubes (in particular, the 2 inner tubes to a greater extent than the outer 2, but anyway) compensate for the missing tube between them which of course is occupied on the opposite side.
I’m a drummer and I’m fascinated because this is practical to tuning drums: I had a drum with 8 lugs, and 3 of the screws fell out and went missing. I realized one day I could tune it better by removing the fifth screw and leaving four in balance. Not only that, I used your equation at the end to answer a hypothetical I came up with: could you properly tune a 5 lug Gretch drum with only 3 screws? And the answer is no, because 5 doesn’t have any prime factors! Easy as pie! This does make me want to get a 12 lug snare drum and see if there’s a difference in sound tuning up with only 7 screws tho... fascinating!
Ah this was great. I was finding it fairly intuitive and pleasing, but didn't see the relation to complex numbers coming out of it at all until it was mentioned. I love when suddenly something just nicely shifts into a different kind of maths.
Molecular biologist here, it totally blew my mind the first time i learned the balancing trick without having to put an extra tube of water! E.g. putting 7 tubes into a centrifuge with 12 slots. Funny how a lot of the time these configurations look totally crazy and asymmetric but are still balanced, haha.
Extremely interesting math. But, if pragmatism and efficiency in adding test tubes is the only concern, it's easiest to simply look for any two open slots across from each other, add two test tubes at a time, and stop when you run out of empty opposing slots.
I mean that's great until you need to spin an odd number of tubes. Even though it's perfectly viable to spin 3 tubes in a 6 tube centrifuge, your method wouldn't work.
Totally irrelevant but I feel like this centrifuge wouldn't work because the test tubes are vertical. Am I wrong ? Edit : 1. By irrelevant I mean it's irrelevant to the math and symmetry problem. 2. By wouldn't work I mean it would be really inefficient compared to hinged tubes that are able to angle outwards.
@The Porcupine In principle, you are wrong, but of course the kind of technique and rotor you'd like to use, depends on what you want to centrifuge and for what reason. There are different kinds of rotors, used for different things. In this case, precipitate (assuming that there is any, and that we are trying to separate it) would end on the outwards side of test tube, instead of the "bottom".
This is really helpful as a biochemist. Especially knowing that I can add a balanced configuration to another balanced configuration and still have a balanced centrifuge.
As a chemist, I immediately see the answer to this mathematical problem isn't a mathematical answer. You just add an extra equally weighted tube when it's unbalanced. Sod the maths ;)
I was thinking that as the solids separate out, the centrifuge would get unbalanced as the weight distribution of the samples would change but the dummies wouldn't. But i guess not
This is a particularly interesting video because of that Mandelbrot set twist. "Suddendly a wild Mandelbrot set appears". I mean, after watching Numberphile for so many years I have more or less become used to all kinds of crazy connections happening in mathematics, but this is a real surprise, a very welcome one. Will show this video to coworkers tomorrow.
Two things: 1. This takes me back a few years to working in a Medical Laboratory when I would deliberately seek new ways to balance the centrifuge and freak out my colleagues with 40 years experience by using unorthodox (but balanced) patterns 2. The maths at the end with the complex numbers was the first time I could actually see a practical application that I understood for complex numbers (and I did second year university calculus for fun while studying medical laboratory science - however complex numbers were always too abstract and theoretical to me.)
I already read Dr Baker's blog about this but nice to see it outlined here with Dr Krieger. Yes, Holly has a brilliant mind and is also impossibly cute! Such an attractive combination.
Luckily the centrifuge rotor makers are well aware of this, the number of holes is almost always divisible by 2 and 3 at least for convenience. For example, if you only wanted to stably spin something fast: a prime number of slots is most stable, with fewer vibrational modes, this is why ceiling fans have 5 blades.
That's funny because someone above said the manufacturer had no idea and their documentation etc only allowed for simple balancing. They were apparently pretty shocked people were balancing 12 with 5
I think Holly is the best person that appears on Numberphile tbh, really nice and talks in an interesting way, plus I love complex numbers and things pertaining to them. Not that I don't like others - James, Matt, Zvezda, Tadashi... - they are amazing, but I am always excited for a video with Holly
@@linyenchin6773 "mathematics" is the long form, correct? Not mathematic. American and Canadian English prefers shortening it to "math", and British and some others to "maths".
This is awesome, as molecular biologist I have been doing centrifuge balancing math almost on a daily basis for the past 7 years and i always found it very interesting. A good follow up on this would be to investigate what would be the ideal centrifuge, which supports the most possible configurations as a fraction of seats. I ve worked with 6, 12, 14, 20 and 24 seats and suspect 24 is ideal but i dont have the prove for it.
Doesn't seem very dodgy to me. If you tie a string to the top and bottom of a water bottle and swing it above your head, the water bottle could be made to stand "upright" according to how it would sit on a table but the water would still flow to the outside of the bottle. That's literally the point of a centrifuge.
Here's a thought: The guy at the county fair running the "Tilt-A-Whirl" (which is a giant centrifuge for people instead of test tubes), you know, the guy with the greasy clothing, smoking the smelliest cigar known to man, and can barely speak more than a three word sentence, usually named Bob or Maynard or something ... knows how to do this instinctively, and does it every two minutes for eight hours a day. Not only that but Maynard has to do it with "test tubes" of different sizes and weights!
Maybe a modern theme park ride might have some sort of self adjusting counter-balancing system, but I'm talking about my memories of the ride, back in the 60's at the state fair. Although the "Tilt-a-whirl" and "Octopus" rides are similar, I actually was thinking of the "Roundup" ride. Never the less, these rides travel around the country on trailers and set up in a few hours and I'm doubtful that they have such complex mechanisms as self-adjusting counter balancing systems. Having worked in a theme park in my college days, I know that the safe operation of most rides, even the permanent ones in theme parks, are largely the responsibility of the operator and his ability to understand the ride's operation and to be able to asses the people he's putting on the ride. Weight distribution is a big factor on any spinning ride - with the exception of a merry-go-round.
As a recent graduate with a biochem phd.. i certainly have a lot of experience with centrifuges.. perhaps the highlight of my day some days was finding new and fun centrifuges configurations
Mathematics is not a science. It is something of a foundation for the other sciences to build on. Science deals with real measurable stuff and works empirically. Unlike maths, which is absolute.
@@Banzybanz (Look up Go(e)dels' work.... not so absolute anymore since he came long, en also Heisenberg puts deteminism or absoluteness off a bit.. and lets not forget Noether either). Mathematics is certainly inspired by science and vice versa. Rigidity in thinking in this way is not useful. And however a system is (psycically) arranged, then comes along a spacetime expansion contraction and poof, not so rigid after all. Even all stuff we depend on such as sum of internal angle of a triangle sums to 180 degrees... not longer the case for a short time. And science and mathematics became a noun, but it just the act of thinking with a feedback loop and as much as possible to prevent cognitive dissonace, and adapt with new insight or data or facts. Call that the scientific method, or the mathematical thinking, or science, I do not care. The above parts is namely is the boring part, sure it should be done 'well' , but the part that is really motivating: are the fun parts/play/creativity/discovery/fullfillment and yes even findind out stuff does not works you hoped or thought gives direction and can feel ok (although it feels horrible sometimes i must admit, and then also still having to write stuff down and even put it in paper form, to prevent others to waste time is not fun but usefull nonetheless, and in general to counter the bias to only publish the sucess stories). cheers, the whole field of human thinking and doing and adapting is at least interesting, although all that is, is prety.
Well, if you’re dealing with a real-life centrifuge, you’re more likely to just add some extra dummy tubes to balance things out than futz with this. The question is, for a centrifuge with N slots, how many extra tubes do you need to make sure it will always balance? For example, if you had 15 slots, you could only balance it on 3, 5, 6, 9, 10, 12 and 15. There are at most 2 consecutive numbers that are unbalanced (1-2, 7-8, 13-14), so you need 2 extra tubes to make sure you can always balance it (worst cases being if you have 1, 7 or 13). One part of the solution is that at most you need p-1, where p is the smallest prime factor of N. This is because, if p is a prime factor of N, every multiple of p can easily be balanced (split the N slots into groups of p evenly distributed ones, like the 6 slots being split into 3 pairs at the start, and fill the appropriate number of these groups), and at most p-1 are needed to make any number into a multiple of p, so you can never need more than that. The question is, is this number alway necessary? Are there numbers N where less than p-1 extra tubes are sufficient? And if so, when does this happen? Edit: Just realized p-1 is necessary because 1 will always be the worst case; nothing below p will balance.
Interesting, I've never thought of this in any formal mathematical way, but I did have fun balancing 17 tubes in a 24-well centrifuge last week. I used the n-k trick and balanced the 7 empty spaces because it was quicker.
I like these types of videos because because they teach me how to do maths instead of why maths . thank you so much ma'am. Please continue to teach me maths.
I did this using vectors (center of mass method) to see if this balances. The angle is correct but 3 test tubes does not balance 4 test tubes in the given angles. It is quite close though, its √8 test tubes which balance 4 test tubes (√8 is approximately 2.83 which is quite close to 3). But as this is a Mathematics video I don't think approximations are involved. Did anybody else notice this or am I wrong here?
Did you consider that the 4 test tubes also balance themselves out just a little bit? This arrangement has to be correct, as the three component arrangements are balanced, and putting two balanced arrangements together leaves us with a balanced arrangement.
Your calculation must be wrong. Assuming you are referring to the example with 7 test tubes, partition your angle computation like Dr. Krieger does. The first 3 test tubes balance each other out, because v1 + v2 + v3 = 0. The two sets of two also balance each other out, because v4 + v5 = v6 + v7 = 0.
Yes speaking completely in terms of vectors, they do balance themselves out. But putting mass in to the equation, and assuming each test tube is a mass of m, we have a mass of 2m and another mass of 2m perpendicular to each other. Their resultant is given by the Pythagoras theorem which turns out to be √8m. However in this example, the balancing mass is 3m which isn't right.
An imaginary coordinate (z) is just a point on a place, and all planes can be broken down into cartesian coordinate (a + bi). A Z is just a point on a plane, a+bi describes where that point is.
I think they should've spent more time on the complex number implementation of the problem, though that would've made the video twice as long. Letting z stand for a complex number, the six tube configuration was represented by the solutions to z⁶ - 1 = 0. Deciding if some subset of those solutions sum to 0 is solving the centrifuge problem. I don't quite remember enough abstract algebra to know where to go next, but that's the link.
The imaginary notation is equivalent to vector addition in two dimensions. The reals add to the reals, and the imaginaries add to the imaginaries. The physicists will just sum the x and y components independently, and see they sum to zero. If you allow different distances from the center, then you multiply the masses times the distances from the center [ taken at (0,0) usually ]] and add them up. This is kind of like adding linearly independent solutions of a homogeneous equation. It works in arbitrary dimensions. The easiest way to handle arbitrary dimensions is to imagine a table with as many columns as you have indepependent dimensions (or new variables). In theories that need lots of indendent variables to cover the full range of phenomena, the table of properties at every point in space and time gets larger and larger. Our time and space table has x,y,z,t columns, then for every point in space and time (for every row in the table) a fairly large number of vector and scalar fields, each with their own columns. It is rare to have a full table with every column filled. Physics and most quantitative disciplines now have many holes, and things are not balanced overall. Also much duplication and overlapping. Many columns and cells in the rows have too many very smart people fighting over tiny scraps. While vast areas of the table have only rough entries for position and time. I really like your insight. I tried to lock this into my memory so anytime I see anything, I will try to visualize its balanced and transitional states. We do this routinely with solutions of the Schrodinger equation for chemicals, nuclei, for the gravitational potential field of earth and masses, and for the magnetic potential field of the earth and masses, for quantum and acoustic wave solutions for solids, plasmas and anything. Now in x-ray crystallography, there are many interesting symmetry tools. And physicist love to get together, titter and tease each other about symmetry. I like your introduction of the notion that prime numbers might have something to say about balancing. I was reading the other day about resonance stable states for orbits of the planets, with certain exact integer ratios for stable states -- but with chaotic transitions common. The resonances and states of spectroscopy should all be tied to the states of the particles or molecules or bodies involved. The constraint equations for balance of forces, momentum conservation, energy conservation, and endless constraints in industrial and engineering practice - all come down to this notion of balance. Thing adding to zero, or close to zero. The columns of the univeral accounting sheet coming out to the exact penny. You really are doing a great global public service. Thank you,
Sure, not having the "proper" number of test tubes won't stop you in practice, but for a large centrifuge it can (presumably) be tedious. So the problem actually becomes; _what's the minimum number of dummy tubes that you need?_ So it still boils down to the same mathematical problem, even if you're being practical :)
@@jasondoe2596 but for a large centrifuge (that presumably has an amount of tubes that can be divided by several numbers, for example 60) there will be a lot of options to balance it in this way without using any dummys. In fact every number (exept for 1) can be written out as some combination of 2's and 3's, and these larger centrifuges will probably have a lot of ways to add pairs of 2 and 3 together, thus no dummys will be needed.
Depend's on the accuracy needed and how sensitive the equipment is. Water doesn't have the same density and as such you can get the weight balanced or the momentum, but not both. For most cases water is fine, but spacing is a far superior option for quality in sensitive scenarios.
When we were using ultra-centrifuges (rotating your tubes at 20k-30k rpm) then OF COURSE we had to fine tune weight and counter weight in a balance before pressing the start button. Otherwise you just create some sort of ballistic projectile und co-workers wont like you very much in the foreseeable future....
6:50 if you're having problems understanding the mathematical interpretation of this, remember that you can treat the sum of complex numbers as if it was the standard vector sum in R^2. Every number (a,b) is represented by an arrow that starts in (0,0) and ends in (a,b); you sum two vectors component-per-component or use the graphic head-to-tail method (google it, very easy to understand). The important thing is that the sum of two vectors lying on the same line, of opposite direction but equal length is the null vector (0,0) That's why you can add up stable configurations and obtain a new stablr configuration: if a+b=0 (vector sum), and c+d=0, then a+b+c+d=0+0=0. The next claim is sightly more complicated: when a vector z, forming an angle of 1/nth of the circumference relative to the x-axis, is squared, cubed, ..., Raised to the power of k, the resulting vector z^2, z^3, ..., z^k forms an angle of k/nth of the circumference relative to the x-axis. The reason is the way the product between vectors is defined. Long story short, when a vector (a,b) is raised to the power of k, the angle it forms relative to the x-axis gets multiplied by a factor k, and its length is equal to the original length raised to the k. If the vectors live on the unitary circumference, their length is one and only the angle they form with the x-axis is affected by the exponentiation.
I found this really interesting as a physicist, I instantly thought of balancing forces to keep the CoM of the origin... But then wouldn't have ever thought of doing powers of complex numbers in a million years.
@@Errenium in this case a number x is a linear combination of the numbers of set B if x can be written as the sum of elements of B multiplied by a constant. For n=6 k=1 has no prime factors k=2 2×3 is 6, works k=3 3×2 is 6, works k=2×2 2×3 is 6, works k=5 6 can not be written as a product of 5 and an integer. And as far as I can see, you just take the prime factors of k and if n can be written as a linear combination of the prime factors (so the sum of every prime factor multiplied by a positive integer), then it works. I think there's no need for an explanation using the complex plane.
First of all - I love this video! :) So I'm not trying to bash it by saying that: In reality, you would simply fill an empty test tube with water to match the mass of the sample test tube (if, for instance, you only had a single sample) in order to balance the centrifuge.
Just not enough holes yet to provide counterexample. With 24 holes you can arrange 7 tubes without any symmetry. (Another question: what are the minimum numbers of holes and tubes needed for this?)
@Gaston Fontenla for any balanced configuration you will ALWAYS have at least one line of symmetry. It's more of a physics explanation but it's always true. Otherwise, it by definition isn't balanced. :)
Isn't there a non-symmetric arrangement of 7 even in 12? 0,2,4,5,8,9,10. I was going to say the answer to your question is 5 tubes in 12 holes, but then realized that was also a solution for 7 since the empty spots must also be non-symmetric.
I had a.centrifuge problem once. In Organic Chemistry. My older than dirt test tube blew up in the older than dirt centrifuge. Professor said it was okay, accidents happen. He also gave me zero for the day. Thanks.
Here's an example of balancing a centrifuge that has an *even* number of slots by adding an *odd* number of test tubes with *none* of the test tubes being directly opposite any other: Imagine a centrifuge with *ten* slots. If you then picture those slots as the vertices of two overlapping pentagons, it's easy to see that if you place *five* test tubes in every other slot, they will be at the vertices of an imagined pentagon. Further, the centrifuge must be balanced. There are intuitive geometric proofs (ways to easily visualize) that it must be balanced that don't require doing any algebraic calculation.
the complex number z represent a vector, the powers of z represent the same magnitude of z, but different directions. so when you sum all the z and its powers, you are essentially doing vector addition. if the sum is 0, then it means you haven't moved from the origin. in this case, the origin represent perfect balance, and anything else means an imbalance. it's the same as solving for the center of mass/gravity of an object.
Assume equal point masses on the complex plane. Then the center of mass is simply the mean position of the masses. And this mean is zero if and only if the sum is zero. Finally, because of how a centrifuge works, each point mass can only be placed at one of the complex roots of unity. So if you have 7 masses and 12 slots, as in the video, the question becomes: can you add together 7 of the 12 twelfth roots of unity so that their sum is zero?
To balance effectively the average position of the mass must be the center, because in that case the centerfuge motor isnt having force pulling it in a specific direction, because any outward movement in one direction is counteracted by movement in the opposite direction. The complex numbers work, because mathemeticians have found very useful properties of complex numbers (a+bi where i = square root of -1). You have to think of a as an x coordinate, and b as a y coordinate on a 2d graph. Multiplying 2 complex numbers gives a result where the angles of the 2 inputs are added, which is why the powers are evenly spaced around the unit circle. All the addition is doing is calculating the center of mass of the beakers (assuming equal mass). Without complex numbers, you can think of balancing on a seesaw with one end being -1, and the other +1. To balance the seesaw, weight on the -1 end must be equal to the weight on the +1 end so that (weight1 - weight2) =0 The specific example works because z^3 = -1, so (z +z^2 + z^4 +z ^5) = (z + z^2) + z^3 (z + z^2) = (z + z^2) - (z + z^2) = 0. Hope this helps.
Honestly, at this point I'm just surprised no one has mentioned polar representation, as in, every complex number can be caracterized by its distance to the origin (its module, 1 in this case) and an angle between 0 and 2pi (its argument). That's probably the most natural way to think about the roots of unity, by saying their arguments can be written as 2kpi/n where n is a natural number and k is in |[0,n-1]|.
You have a set of complex numbers on the unit circle (~centrifuge model with all the possible positions of the tubes, though we are only interested in equally spaced ones that can be expressed in the form of z^n with the N-th root of unity formula). If you consider a set of K points from N equally spaced points around the circle (all the holes of the centrifuge), then the center of mass of these K points (which are the tubes) is the average of the K complex points. If average of the K complex points is 0 (the sum is also 0) it means that the center of mass of the tubes is the center of the circle (=of centrifuge), thus the configuration is balanced because it is part the axis around which the centrifuge spins.
Similar principle applies to cartons of eggs, too. If there are four eggs left in the carton, they should not all be in one end, catching the next person to pick it up off guard. They should be spaced evenly away from the center, to balance the carton.
This is actually an important problem for space rockets. How many, and which engines can be activated. Think of Falcon 9: 9 engines; 1 central, 8 in a circle. It must vary thrust on landing, so it lights up only a certain number of engines for braking. How many? Which ones? 1: 1 central. 2: 2 on opposite sides. 3: as 1+2. 4: every other from the outer set. 5: 4+1. 6: 9-3. 7: 9-2. 8: 9-1. And of course 9, all. If it didn't have the central engine, all the odd combinations would be impossible, unbalancing the rocket.
Great example, as it's not just the masses of some funny number of parts making trouble due to imbalance, but active thrusters applying serious torques to the whole. If a rocket like that goes hayware, it's not just a matter of annoying vibrations!
If you have self balancing centrifuge this isn't so much of a problem; Though it does still help to have some dummy tubes on hand.
Overlap of my two favorite channel
I love it when one channel I watch comments on another video by a channel I also watch.
kkkkkk I really like finding Cody's Lab comments in random videos! Mumbo, Numberphile, and the list continues to grow
I'm more surprised than i should be that cody watches numberphile.
Where can I buy self-balancing complex numbers?
This principle can also be used to attach side boosters to rockets with zero torque in KSP.
Jebediah approved it
Actually used this in KSP without realizing it was a thing. Was curious if I could line the boosters to perfectly balance without even spacing. Found the 7 on 12 configuration when thinking about it while waiting for a sandwich to toast.
ohhhhh
Finally an actual and important use…
@@robinx1615 Unfortunately, there will still be considerable torque due to Jeb fidgeting about in his seat.
Loved you in "Arrival".
No
Did Holly Krieger do a cameo in Arrival? Because that would be pretty awesome. Or are you confusing Holly Krieger with Amy Adams?
@@TevelDrinkwater Who is Amy Adams?
i was going through all the comments just to find this one, thank you.
@@TevelDrinkwater you don't understand jokes, do you?
Love this. I've been sticking 5 and 7 tubes in 12-rotor centrifuges for decades. I've always wondered about working out the mathematical possibilities.
Every time she smiles my heart skips a beat
stop being so hetero
I think you may have a heart condition. You should probably get that checked out.
nerd
...yeah, until you turn on the sound and realize that every second word is 'right'.
add the 'ok's and there aren't much words left...
@@JaykTheJackal I don't. Can you explain why?
Fun fact : for 30 spots, you can do it for all numbers of tubes except 1 and 29.
The balancing only really matters for small centrifuges though. Most of the ones with large number of slots have fixed wells inside the rotor to the point where the weight of one tube becomes negligible, so they'll run perfectly fine with a single tube in an unbalanced configuration.
I don't know if that's true. But I'm too lazy to check, so...
But they will overlap so no
Noice
Yes in fact for N any multiple of 6, you can always have any value of k except N -1 and 1.
In biochemistry, we just throw in an extra tube when there's a hard number.
Yes! But I did the 7 tube configuration several times. Not my innovations though, saw a senior do it, then I just played with balancing.
Yeah, easier to carry some dummy tubes than deal with all this nonsense.
@Bill Why call it the centrifuge problem when it has nothing to do with centrifuges then?
First thing that comes to mind, when you "can not balance it", actually =)
@Bill I loled
So much joy in one person, this fills my heart
😍
And then there is this one guy in the lab, who puts 2 tubes in next to one another. >:(
yeah that is a class one nobendery
The catch is - almost all lab centrifuges are equipped with so called auto balancing unit. The rotor just auto-balance himself no matter how you load your cr*p into it :-)
Some people just want to be anarchists.
@@shurmurray last eppendorf centrifuge i used certainly didnt have that
I do a lot of repairs for centrifuges where I work. I've seen that a lot. The worst ones are when the rotor jams so bad that it prevents the door from opening and you have to bust the hinge to be able to un-wedge the swinging test tube holders and pull everything apart.
Her joyful delivery of a math problem is a thing of beauty.
That laugh could disarm a nuclear bomb :)
Attlanttizz couldn't have said it better myself
I always liked Anne Hathaway's laughing in Interstellar. Dr Holly Krieger sounds like same.
Or set it off !
@@countingfloats Oh, the world would be in shambles then :)
yikes
I always loved adding configurations because of how unbalanced they looked. People would always be like "you're gonna wreck the centrifuge!" only to hear it whir quietly and peacefully 😂
Brady, you should follow up with her and ask if the same balancing principle applies with a sphere and quaternions like it does with the centrifuge (disk) and complex numbers.
Quaternions are made of 3 imaginary and 1 real numbers. So its kind of like have an extra dimension compared with free space. I'm not sure if it can be done in a short video.
duggydo I think quaternions are 4 dimensional so they would be on a hyper sphere. From a little bit of research I did I think quarternions when brought to a power the argument doesn't add nicley like regular complex number, so the problem will be more complicated and probably better understood with linear algebra
I don't think that would be the case because a quarernion squared doesn't just multiply its angle from the real axis by two like imaginary numbers do. I might be wrong
@@MrMctastics I do agree that linear algebra might be easier than using quaternions. It might be possible with quaternions, but one might have to take into account for the reduction in dimension.
Rajesh Thomas quaternions are used almost exclusively in 3d graphics computation. You need the extra degree of freedom for it to work. That’s why I’m curious if it applies in this scenario when extending to balancing in 3D.
Reminds me of balancing booster stages in KSP. Same rules apply.
YES!
EUREKA
kinda. This only works if your boosters have same "mass" (leverLength*(TWR-(Mass)))
@@sergey1519 Yeah true, again it's the same as for test tubes.
Atlas rocket doesnt give a sh*t tho, lol. Because of the fuel lines they have to put boosters where they will fit. Thus the uneven boosters cause the rocket to slip sideways. It's pretty crazy to watch.
Dr Holly is a great guest on this channel, love her enthusiasm :D
People: You cannot talk about imaginary numbers in a centrifuge problem video
Mathematicians: Hold my beer
Hold my brown paper
THIS is an underrated comment. Thank you.
Hold my test tube
josepmb19 People: you cannot get people to care about imaginary numbers in a centrifuge problem.
Mathematicians: True
Hold my blue pen
I never thought I would enjoy a video that smells of chemistry so much! Well done, Brady and Dr. Holly. :)
Chemistry is very similar to physics in lots of ways - it is pretty much in-depth nuclear physics.
Chemistry was even one of the prerequisites for my Engineering Physics program.
I love how the animators actually put effort in to tilt the fluid toward the edge when it spins like at 0:35
She has such a beautiful laugh ^_^
so true! i actually replayed 5:59
Hmm... actually it really annoyed me. Felt too fake.
@@zarboov88 could barely watch the video tbh
I shouldn't've clicked into this thread. Dunno what I expected.
Jesse H. Betas pretending to be alphas in order to impress some imaginary female on the internet, when we're in a comment thread of a video talking about centrifuges from a statistical standpoint not many women would be reading this. Or to be a bit better worded, the ratio of men : women that would see this thread is fairly large
When I was a biology grad student I would use this trick to balance odd numbers of tubes in the centrifuge. We had one with 24 slots, so I could do every number except 1 and 23. I even tried balancing tubes with uneven volumes by putting the two less full ones a little closer to each other and the fuller ones further away.
my mind is blown with that 7 balanced config...
I would have expected it to be a Parker configuration.
it shouldn't be blown. If you draw a line down the tube at the centre, you can see the configuration is symmetrical.
@@Son96601 Still, this doesn't mean the configuration is balanced in the remaining direction. That is, the center of mass is in the line you've drawn but not necessarily in the middle.
Cuntslaw, but reflectional symmetry is *not* a sufficient condition for valid tube configurations (I'm sure you can easily find a counter-example).
Rotational symmetry is.
@@jasondoe2596 the 7 configuration has no rotational symmetry. you can rotate it as many times and by how ever much you want and it won't look the same. What are you even talking about?
"So we're gonna do chemistry today..."
😍
"... just kidding, I can't do any chemistry!"
😭
7:21 - eeeeerm…OK 😂️
Miha Rekar it seemed more like an "ohhh ok" but eh we all hear differently
Another big advantage to using complex numbers, is that you can look at this problem in the frequency domain. You can take the DFT (discrete fourier transform) of a placement vector and inspect the first harmonic (in the second, and last positions of the DFT) to see if the configuration is balanced or not. A balanced condition, is where the first harmonic is zero. This can also give you a quantitative value for how "unbalanced" a configuration is.
p = [1 0 0 1 1 1 0 0 1 1 0 1]'; % position vector, 1=> test tube, 0=> no test tube
>> fft(p)
ans =
7.00000 + 0.00000i
0.00000 + 0.00000i % first harmonic term
-1.00000 + 1.73205i
3.00000 + 0.00000i
1.00000 + 1.73205i
0.00000 + 0.00000i
-1.00000 + 0.00000i
0.00000 - 0.00000i
1.00000 - 1.73205i
3.00000 - 0.00000i
-1.00000 - 1.73205i
0.00000 - 0.00000i % also first harmonic term (conjugate)
It was more like "did you say something ? I dozed off"
@@jessstuart7495 eeeeerm .. OK
@@goretrade,
Electrical Engineer.
One laboratory teaching assistant that I had insisted that 3 tubes could be balanced in an 8 hole rotor, and nothing that I or any of the other students could dissuade him. Luckily, (1) the centrifuge was robust enough to work anyway; and (2) the students were, to a person, smart enough to figure out the TA was wrong (meaning the TA's misapprehension was not passed on to others).
Sooo..my blood test results will be ready when..?
Right after they work out the math. Are you still with us?
D O N O T I N T E R R U P T T H E M A T H
It gets weird once you have 4 or more prime factors. Then you can have negative contributions! For example, for N=210, you can combine 3 dots in a triangle with 5 dots in a pentagon and 7 dots in an upside-down heptagon. This looks like there are too many dots at the "top" (where the triangle and pentagon overlap), but then you can take away 2 dots at the top and bottom to maintain balance. The resulting pattern has 13 dots in a weird pattern that is not just built from adding symmetric prime-factor sets of dots.
Actually, it's easier with just 3 prime factors. That allows N=30. I'll use N=60 because then the dots correspond to minute marks on a clock, which is intuitive for many.
To the set of balanced dots:
Add an upside-down triangle: 10, 30, 50
Add a pentagon: 0, 12, 24, 36, 48
Subtract opposites: -0, -30
The resulting pattern is 10, 12, 24, 36, 48, 50 and balances without any dots that are opposite, form a complete triangle, or form a complete pentagon.
Jeff S, haha, that's awesome!
Great example, too.
The question is whether any of those arrangements let you have a number of tubes you couldn't get just by addition of sets.
Notably, all numbers divisible by 6, can be worked out in any configuration aside from 1 and k-1. We know this because all even numbers can be created with some number of pairs. And we know that if a number works with 2 and 3, then you could create any number by adding enough 2s, then a single three.
So with 210, we know everything works. Regardless of how weird it looks.
This is interesting. I wonder, though, is it possible to have a balanced set up not built by balanced polygons (in the way that it is done here with negative contributions) whose complement is also not built by balanced polygons (i.e. on that requires a negative contribution) ?
An easy example of what I mean (very similar to what Jeff gives), for N = 30, the triangle (0,10,20) together with the pentagon (3,9,15,21,27), but removing the pair (0,15) gives the balanced set (3,9,10,20,21,27) which cannot be built from balanced polygons without negative contributions.
However, its complement in the 30 holed centrifuge can by built up from the pentagon (0,6,12,18,24), the triangle (5,15,25) and then all the remaining opposite holes can be occupied by pairs, so this configuration does not require any negative contributions.
Does there exist a configuration such that it and its complement can only be arrived at by including some negative contributions? I feel that it is impossible.
3:54 LOL, that playful animation xD
PS. Great video!
Why do you people leave so many blank lines, that the final line of text gets hidden?
@@dyhrbergdk9541 - I guess it's to give you a little... surprise.
Dyhrberg DK, yeah, sorry, I should have reserved that trick for something more witty than that :P
It *is* a great video though!
"Let's do an example with 12 spots."
*Shows circle with 12 holes*
"Let's pretend these are all evenly spaced"
*Corrects the spacing*
Old phone rotary disc, if * and # were ever added in the analog age, or if it came from a duidecimal age
It brings back so many memories of myself standing in front of the centrifuge doing biology experiments. We always kept tubes of different weight with just water in it to help us balance.
In practice (low tech) you fill the necessary other tubes with water.
Another solution (high tech) there are self-balancing centrifuges that counterweight in their rotationmechanism, so it doesnt matter what you put in.
Which were engineered doing this kind of math ;)
not really, is just cell biologist's common sense. The worst thing you want to have is a centrifuge going nuts in the lab at 12000 rpm.
@@Robocop-qe7le I don't think a cell biologist is going to be building a self-balancing rotation mechanism any time soon or running the math for it to make sure it can counterbalance any number of tubes in sat in any formation. But that's just me.
And you don’t need maths to do that. Just make a mechanical negative feedback loop that moves the counterbalance until the top is horizontal. With some damping, of course, otherwise you will need to analyse the stability of the loop - which does need maths. An oscillating feedback loop would be fun!
@@glasgowbrian1469 that seems rather.. mathematical...
This is one of the most simplistically beautiful results i've seen in a while!!! Makes me appreciate having taken modern algebra
I can’t watch, listen and understand. If I watch I don’t listen, but if I just listen I can understand. Thanks Holly.
Interesting. Yet, very very simple. But it's always a delight to see Holly Krieger, thanks for sharing!
beautiful connection to complex numbers and roots of unity!
Man she's great. Very natural thinking process that leads you to the equations, meaning I end up understanding them better in the end.
I feel balanced while listening to you
I could listen to Dr. Krieger all day and not get bored
Thank you Dr. K, and you, too, Brady.
Woke up this morning, not expecting any problems.....
Centrifuge problem.
Perfect.
Balancing the 7 tubes in the n=12 configuration makes perfect sense, actually! The side with 2+0+2 tubes balances out the side with 3 because the weight of the 4 tubes (in particular, the 2 inner tubes to a greater extent than the outer 2, but anyway) compensate for the missing tube between them which of course is occupied on the opposite side.
biologists’ use of centrifuges completely ignored. As a chemist, this pleases me. Subscribed.
I can listen to holly talk for hours
I’m a drummer and I’m fascinated because this is practical to tuning drums:
I had a drum with 8 lugs, and 3 of the screws fell out and went missing. I realized one day I could tune it better by removing the fifth screw and leaving four in balance.
Not only that, I used your equation at the end to answer a hypothetical I came up with: could you properly tune a 5 lug Gretch drum with only 3 screws? And the answer is no, because 5 doesn’t have any prime factors! Easy as pie!
This does make me want to get a 12 lug snare drum and see if there’s a difference in sound tuning up with only 7 screws tho... fascinating!
Me: wakes up in 3 am
Me: Tf should I do?
Me: Watches math
Ah this was great. I was finding it fairly intuitive and pleasing, but didn't see the relation to complex numbers coming out of it at all until it was mentioned. I love when suddenly something just nicely shifts into a different kind of maths.
“Ok so here is why I am wrong”
Summary of my academic career right there.
Well, knowing _why_ you're wrong is a valid achievement in academia. Most people don't even realize _that_ they're wrong.
Molecular biologist here, it totally blew my mind the first time i learned the balancing trick without having to put an extra tube of water! E.g. putting 7 tubes into a centrifuge with 12 slots. Funny how a lot of the time these configurations look totally crazy and asymmetric but are still balanced, haha.
Extremely interesting math. But, if pragmatism and efficiency in adding test tubes is the only concern, it's easiest to simply look for any two open slots across from each other, add two test tubes at a time, and stop when you run out of empty opposing slots.
I mean that's great until you need to spin an odd number of tubes. Even though it's perfectly viable to spin 3 tubes in a 6 tube centrifuge, your method wouldn't work.
Just start with the largest prime factor of slots total (e.g. 3) and continue while possible. :)
@@andrewb378 or have a centrifuge with an odd number of holes
I never thought that the unit circle that I learned in high school would come back as a solution to this problem. MIND BLOWN!!
Totally irrelevant but I feel like this centrifuge wouldn't work because the test tubes are vertical. Am I wrong ?
Edit : 1. By irrelevant I mean it's irrelevant to the math and symmetry problem.
2. By wouldn't work I mean it would be really inefficient compared to hinged tubes that are able to angle outwards.
Correct. A real centrifuge allows the bases of the tubes to swing outwards so the heavier contents are separated to the bottom.
How a centrifuge works should be relevant. And this is also why you can't add an extra hole in the center just to satisfy k=1 and k=n-1, for example.
Yes... Right, center of mass, but then we need a physicist's input.
hftf Indeed that would just completely defeat the point of having a centrifuge at all, not being vertical wouldn’t help a whole lot.
@The Porcupine In principle, you are wrong, but of course the kind of technique and rotor you'd like to use, depends on what you want to centrifuge and for what reason. There are different kinds of rotors, used for different things.
In this case, precipitate (assuming that there is any, and that we are trying to separate it) would end on the outwards side of test tube, instead of the "bottom".
This is really helpful as a biochemist. Especially knowing that I can add a balanced configuration to another balanced configuration and still have a balanced centrifuge.
And you don't need to have ALL tubes of equal weight ... just the ones in each balanced sub-configuration.
As a chemist, I immediately see the answer to this mathematical problem isn't a mathematical answer.
You just add an extra equally weighted tube when it's unbalanced.
Sod the maths ;)
The math solution is to help you figure out where to add the extra weight, or the minimum extra weight required.
Smart. Using your own blood right?
Or just add an extra copy of one of the samples being tested. Works as verification.
Wow, that's smart. Didn't even think of it. :)
I was thinking that as the solids separate out, the centrifuge would get unbalanced as the weight distribution of the samples would change but the dummies wouldn't.
But i guess not
This is a particularly interesting video because of that Mandelbrot set twist.
"Suddendly a wild Mandelbrot set appears".
I mean, after watching Numberphile for so many years I have more or less become used to all kinds of crazy connections happening in mathematics, but this is a real surprise, a very welcome one.
Will show this video to coworkers tomorrow.
Two things:
1. This takes me back a few years to working in a Medical Laboratory when I would deliberately seek new ways to balance the centrifuge and freak out my colleagues with 40 years experience by using unorthodox (but balanced) patterns
2. The maths at the end with the complex numbers was the first time I could actually see a practical application that I understood for complex numbers (and I did second year university calculus for fun while studying medical laboratory science - however complex numbers were always too abstract and theoretical to me.)
Complex numbers apply to the analysis of circuits with alternating current, also to quantum mechanics.
Dr holly keep making these videos. I could watch this all day 😍
Haha, I've always thought about a similar problem when taking eggs out of the box.
You take them from the far end of the box, so the center of mass doesn't end up too far from your fingers.
Holly Krieger and comments on Holly Krieger are perfectly balanced as all things should be.
Every time you make a video in someone's office, I end up being jealous of their collection of yellow Springer GTM books. Lol
I already read Dr Baker's blog about this but nice to see it outlined here with Dr Krieger. Yes, Holly has a brilliant mind and is also impossibly cute! Such an attractive combination.
Luckily the centrifuge rotor makers are well aware of this, the number of holes is almost always divisible by 2 and 3 at least for convenience. For example, if you only wanted to stably spin something fast: a prime number of slots is most stable, with fewer vibrational modes, this is why ceiling fans have 5 blades.
That's funny because someone above said the manufacturer had no idea and their documentation etc only allowed for simple balancing. They were apparently pretty shocked people were balancing 12 with 5
I think Holly is the best person that appears on Numberphile tbh, really nice and talks in an interesting way, plus I love complex numbers and things pertaining to them. Not that I don't like others - James, Matt, Zvezda, Tadashi... - they are amazing, but I am always excited for a video with Holly
I love her laugh. Shows how much she loves maths
math*
That "s" is overkill~redundant.
Nodody says "logics" so why would you type or even think "maths"?
Lin Yen Chin he's probably British or distantly related. They say maths. Americans say math.
@@jamesjarrait2231 It bothered me too when I studied. But both sides are quite consistent about it
@@linyenchin6773 "mathematics" is the long form, correct? Not mathematic. American and Canadian English prefers shortening it to "math", and British and some others to "maths".
This is awesome, as molecular biologist I have been doing centrifuge balancing math almost on a daily basis for the past 7 years and i always found it very interesting. A good follow up on this would be to investigate what would be the ideal centrifuge, which supports the most possible configurations as a fraction of seats. I ve worked with 6, 12, 14, 20 and 24 seats and suspect 24 is ideal but i dont have the prove for it.
Multiples of 6 support all but 1 tube empty or full.
Came to the comments section just for the pedantry over the dodgy graphic that doesn't tilt the tubes, but does tilt the fluid.
Fluids behave differently than solids. This is what you'd expect if the test tubes were securely placed.
Doesn't seem very dodgy to me. If you tie a string to the top and bottom of a water bottle and swing it above your head, the water bottle could be made to stand "upright" according to how it would sit on a table but the water would still flow to the outside of the bottle. That's literally the point of a centrifuge.
Only if you assume the plate is infinitely thin.
@@buttsufancypantsu1644
You HAVE to be kidding, right?
@@ThunderChunky101 Why do you think I'm kidding?
I wish my maths teachers at Uni would have spontaneously laughed half as much as Dr Krieger !
Here's a thought: The guy at the county fair running the "Tilt-A-Whirl" (which is a giant centrifuge for people instead of test tubes), you know, the guy with the greasy clothing, smoking the smelliest cigar known to man, and can barely speak more than a three word sentence, usually named Bob or Maynard or something ... knows how to do this instinctively, and does it every two minutes for eight hours a day. Not only that but Maynard has to do it with "test tubes" of different sizes and weights!
Maybe a modern theme park ride might have some sort of self adjusting counter-balancing system, but I'm talking about my memories of the ride, back in the 60's at the state fair. Although the "Tilt-a-whirl" and "Octopus" rides are similar, I actually was thinking of the "Roundup" ride. Never the less, these rides travel around the country on trailers and set up in a few hours and I'm doubtful that they have such complex mechanisms as self-adjusting counter balancing systems. Having worked in a theme park in my college days, I know that the safe operation of most rides, even the permanent ones in theme parks, are largely the responsibility of the operator and his ability to understand the ride's operation and to be able to asses the people he's putting on the ride. Weight distribution is a big factor on any spinning ride - with the exception of a merry-go-round.
You're doing a big job for Maths, keep them coming
Holly is both super smart and adorable.
That's neat. I really enjoyed studying for the GRE and applying all of these tricks.
I like Holly Krieger.
: D
Who doesn't??
As a recent graduate with a biochem phd.. i certainly have a lot of experience with centrifuges.. perhaps the highlight of my day some days was finding new and fun centrifuges configurations
Lord Kelvin: In science, there is only physics. All the rest is stamp collecting.
Mathematician: Hold my beer.
Apparently that quote was from Ernest Rutherford, and apparently he won the Nobel prize in chemistry later that year.
Mathematics is not a science. It is something of a foundation for the other sciences to build on. Science deals with real measurable stuff and works empirically. Unlike maths, which is absolute.
@@Banzybanz (Look up Go(e)dels' work.... not so absolute anymore since he came long, en also Heisenberg puts deteminism or absoluteness off a bit.. and lets not forget Noether either).
Mathematics is certainly inspired by science and vice versa. Rigidity in thinking in this way is not useful. And however a system is (psycically) arranged, then comes along a spacetime expansion contraction and poof, not so rigid after all. Even all stuff we depend on such as sum of internal angle of a triangle sums to 180 degrees... not longer the case for a short time.
And science and mathematics became a noun, but it just the act of thinking with a feedback loop and as much as possible to prevent cognitive dissonace, and adapt with new insight or data or facts.
Call that the scientific method, or the mathematical thinking, or science, I do not care.
The above parts is namely is the boring part, sure it should be done 'well' , but the part that is really motivating: are the fun parts/play/creativity/discovery/fullfillment and yes even findind out stuff does not works you hoped or thought gives direction and can feel ok (although it feels horrible sometimes i must admit, and then also still having to write stuff down and even put it in paper form, to prevent others to waste time is not fun but usefull nonetheless, and in general to counter the bias to only publish the sucess stories).
cheers, the whole field of human thinking and doing and adapting is at least interesting, although all that is, is prety.
Well, if you’re dealing with a real-life centrifuge, you’re more likely to just add some extra dummy tubes to balance things out than futz with this. The question is, for a centrifuge with N slots, how many extra tubes do you need to make sure it will always balance?
For example, if you had 15 slots, you could only balance it on 3, 5, 6, 9, 10, 12 and 15. There are at most 2 consecutive numbers that are unbalanced (1-2, 7-8, 13-14), so you need 2 extra tubes to make sure you can always balance it (worst cases being if you have 1, 7 or 13).
One part of the solution is that at most you need p-1, where p is the smallest prime factor of N. This is because, if p is a prime factor of N, every multiple of p can easily be balanced (split the N slots into groups of p evenly distributed ones, like the 6 slots being split into 3 pairs at the start, and fill the appropriate number of these groups), and at most p-1 are needed to make any number into a multiple of p, so you can never need more than that. The question is, is this number alway necessary? Are there numbers N where less than p-1 extra tubes are sufficient? And if so, when does this happen?
Edit: Just realized p-1 is necessary because 1 will always be the worst case; nothing below p will balance.
I'm a simple man. I see brains, I hit like.
Interesting, I've never thought of this in any formal mathematical way, but I did have fun balancing 17 tubes in a 24-well centrifuge last week. I used the n-k trick and balanced the 7 empty spaces because it was quicker.
I was born in an unbalanced test tube...
dont worry, they threw in a dummy tube to counter act it.... but which did they pick in the end????
I like these types of videos because because they teach me how to do maths instead of why maths . thank you so much ma'am. Please continue to teach me maths.
I did this using vectors (center of mass method) to see if this balances. The angle is correct but 3 test tubes does not balance 4 test tubes in the given angles. It is quite close though, its √8 test tubes which balance 4 test tubes (√8 is approximately 2.83 which is quite close to 3). But as this is a Mathematics video I don't think approximations are involved. Did anybody else notice this or am I wrong here?
Did you consider that the 4 test tubes also balance themselves out just a little bit? This arrangement has to be correct, as the three component arrangements are balanced, and putting two balanced arrangements together leaves us with a balanced arrangement.
Your calculation must be wrong. Assuming you are referring to the example with 7 test tubes, partition your angle computation like Dr. Krieger does. The first 3 test tubes balance each other out, because v1 + v2 + v3 = 0. The two sets of two also balance each other out, because v4 + v5 = v6 + v7 = 0.
Yes speaking completely in terms of vectors, they do balance themselves out. But putting mass in to the equation, and assuming each test tube is a mass of m, we have a mass of 2m and another mass of 2m perpendicular to each other. Their resultant is given by the Pythagoras theorem which turns out to be √8m. However in this example, the balancing mass is 3m which isn't right.
@@iyaz4004 how long did you scroll to find this
Mass doesn't matter, as long as all the test tubes weigh the same.
Okay, I am officially blown away by this one - more than I usually am on Numberphile! :-)
Bit confused, we're adding a+bi then we're on Z stuff. That bit wasn't very clear to me.
Rob Read Z is just shorthand for a+bi.
An imaginary coordinate (z) is just a point on a place, and all planes can be broken down into cartesian coordinate (a + bi). A Z is just a point on a plane, a+bi describes where that point is.
No high school math?
I think they should've spent more time on the complex number implementation of the problem, though that would've made the video twice as long.
Letting z stand for a complex number, the six tube configuration was represented by the solutions to z⁶ - 1 = 0. Deciding if some subset of those solutions sum to 0 is solving the centrifuge problem. I don't quite remember enough abstract algebra to know where to go next, but that's the link.
@@tfos993 Don't be mean. Most high school curricula barely cover this.
The imaginary notation is equivalent to vector addition in two dimensions. The reals add to the reals, and the imaginaries add to the imaginaries. The physicists will just sum the x and y components independently, and see they sum to zero. If you allow different distances from the center, then you multiply the masses times the distances from the center [ taken at (0,0) usually ]] and add them up.
This is kind of like adding linearly independent solutions of a homogeneous equation. It works in arbitrary dimensions. The easiest way to handle arbitrary dimensions is to imagine a table with as many columns as you have indepependent dimensions (or new variables). In theories that need lots of indendent variables to cover the full range of phenomena, the table of properties at every point in space and time gets larger and larger. Our time and space table has x,y,z,t columns, then for every point in space and time (for every row in the table) a fairly large number of vector and scalar fields, each with their own columns. It is rare to have a full table with every column filled. Physics and most quantitative disciplines now have many holes, and things are not balanced overall. Also much duplication and overlapping. Many columns and cells in the rows have too many very smart people fighting over tiny scraps. While vast areas of the table have only rough entries for position and time.
I really like your insight. I tried to lock this into my memory so anytime I see anything, I will try to visualize its balanced and transitional states. We do this routinely with solutions of the Schrodinger equation for chemicals, nuclei, for the gravitational potential field of earth and masses, and for the magnetic potential field of the earth and masses, for quantum and acoustic wave solutions for solids, plasmas and anything. Now in x-ray crystallography, there are many interesting symmetry tools. And physicist love to get together, titter and tease each other about symmetry.
I like your introduction of the notion that prime numbers might have something to say about balancing. I was reading the other day about resonance stable states for orbits of the planets, with certain exact integer ratios for stable states -- but with chaotic transitions common. The resonances and states of spectroscopy should all be tied to the states of the particles or molecules or bodies involved. The constraint equations for balance of forces, momentum conservation, energy conservation, and endless constraints in industrial and engineering practice - all come down to this notion of balance. Thing adding to zero, or close to zero. The columns of the univeral accounting sheet coming out to the exact penny.
You really are doing a great global public service. Thank you,
or... you just add a dummy tube filled with water. but nice video, ill keep it in mind next time i centrifuge something
Sure, not having the "proper" number of test tubes won't stop you in practice, but for a large centrifuge it can (presumably) be tedious. So the problem actually becomes; _what's the minimum number of dummy tubes that you need?_
So it still boils down to the same mathematical problem, even if you're being practical :)
@@jasondoe2596 but for a large centrifuge (that presumably has an amount of tubes that can be divided by several numbers, for example 60) there will be a lot of options to balance it in this way without using any dummys. In fact every number (exept for 1) can be written out as some combination of 2's and 3's, and these larger centrifuges will probably have a lot of ways to add pairs of 2 and 3 together, thus no dummys will be needed.
Dalitas D applying an engineering solution to a maths problem :)
Depend's on the accuracy needed and how sensitive the equipment is. Water doesn't have the same density and as such you can get the weight balanced or the momentum, but not both. For most cases water is fine, but spacing is a far superior option for quality in sensitive scenarios.
When we were using ultra-centrifuges (rotating your tubes at 20k-30k rpm) then OF COURSE we had to fine tune weight and counter weight in a balance before pressing the start button. Otherwise you just create some sort of ballistic projectile und co-workers wont like you very much in the foreseeable future....
6:50 if you're having problems understanding the mathematical interpretation of this, remember that you can treat the sum of complex numbers as if it was the standard vector sum in R^2. Every number (a,b) is represented by an arrow that starts in (0,0) and ends in (a,b); you sum two vectors component-per-component or use the graphic head-to-tail method (google it, very easy to understand). The important thing is that the sum of two vectors lying on the same line, of opposite direction but equal length is the null vector (0,0)
That's why you can add up stable configurations and obtain a new stablr configuration: if a+b=0 (vector sum), and c+d=0, then a+b+c+d=0+0=0.
The next claim is sightly more complicated: when a vector z, forming an angle of 1/nth of the circumference relative to the x-axis, is squared, cubed, ..., Raised to the power of k, the resulting vector z^2, z^3, ..., z^k forms an angle of k/nth of the circumference relative to the x-axis.
The reason is the way the product between vectors is defined. Long story short, when a vector (a,b) is raised to the power of k, the angle it forms relative to the x-axis gets multiplied by a factor k, and its length is equal to the original length raised to the k. If the vectors live on the unitary circumference, their length is one and only the angle they form with the x-axis is affected by the exponentiation.
Do I smell the mind of a pasta laden mechanical engineer...
How can I apply this knowledge so my washing machine doesn't try to escape out the back door when I do laundry?
I found this really interesting as a physicist, I instantly thought of balancing forces to keep the CoM of the origin... But then wouldn't have ever thought of doing powers of complex numbers in a million years.
So it's a linear combination of the prime factors. You can leave out the complex plane and just do high-school algebra here, right?
@@Errenium can you give a counterexample?
@@Errenium can you give an example n and k where n is a linear combination of the factors of k and there's still no arrangement?
@@Errenium in this case a number x is a linear combination of the numbers of set B if x can be written as the sum of elements of B multiplied by a constant.
For n=6
k=1 has no prime factors
k=2 2×3 is 6, works
k=3 3×2 is 6, works
k=2×2 2×3 is 6, works
k=5 6 can not be written as a product of 5 and an integer.
And as far as I can see, you just take the prime factors of k and if n can be written as a linear combination of the prime factors (so the sum of every prime factor multiplied by a positive integer), then it works.
I think there's no need for an explanation using the complex plane.
@@Errenium can a prime n centrifuge ever be balanced without being empty or full?
@@MarioWenzel There’s actually one in the video! n=6, k=5
You can do three, and you can do two, but you can’t do 3 and 2 without getting an overlap.
Interesting how the 'sum of prime factors' condition satisfies the non-collision constraint. Great video
Don't forget about us cell culture biologists...….
You got stuck in the supernatant, sorry.
That's just really, really complex chemistry that nobody fully understands yet, so you're kinda(?) covered.
Yes, and don't forget molecular biology: that's like wizardry but nobody understands it.
0:10 was anybody else bothered by the fact that the tubes should be tilted when rotating?
@@dragoncurveenthusiast Not all of them do!
First of all - I love this video! :) So I'm not trying to bash it by saying that:
In reality, you would simply fill an empty test tube with water to match the mass of the sample test tube (if, for instance, you only had a single sample) in order to balance the centrifuge.
I observed that there was at least one line of symmetry in each of those balanced configurations.
You need that line of symmetry for symmetric distribution of weight
Just not enough holes yet to provide counterexample. With 24 holes you can arrange 7 tubes without any symmetry. (Another question: what are the minimum numbers of holes and tubes needed for this?)
Gaston Fontenla actually there is assuming the holes are spread even, your line of symmetry would lie down the center of one of the holes
@Gaston Fontenla for any balanced configuration you will ALWAYS have at least one line of symmetry. It's more of a physics explanation but it's always true. Otherwise, it by definition isn't balanced. :)
Isn't there a non-symmetric arrangement of 7 even in 12? 0,2,4,5,8,9,10. I was going to say the answer to your question is 5 tubes in 12 holes, but then realized that was also a solution for 7 since the empty spots must also be non-symmetric.
I had a.centrifuge problem once. In Organic Chemistry. My older than dirt test tube blew up in the older than dirt centrifuge. Professor said it was okay, accidents happen. He also gave me zero for the day. Thanks.
Radiator fans are similarly designed to prevent harmonic resonance.
Is there an award for most charming mathematician?
You can use this to balance wire coils on a motor too!
Here's an example of balancing a centrifuge that has an *even* number of slots by adding an *odd* number of test tubes with *none* of the test tubes being directly opposite any other: Imagine a centrifuge with *ten* slots. If you then picture those slots as the vertices of two overlapping pentagons, it's easy to see that if you place *five* test tubes in every other slot, they will be at the vertices of an imagined pentagon. Further, the centrifuge must be balanced. There are intuitive geometric proofs (ways to easily visualize) that it must be balanced that don't require doing any algebraic calculation.
Sure I'm in love of that enthusiasm for math and that smile
Here's another factor in why math is a beautiful science. We need more such mathematicians :)
This all made total sense
Until Dr. Krieger jumped into complex numbers and z^n. How does that correspond? What is going on? halp
the complex number z represent a vector, the powers of z represent the same magnitude of z, but different directions. so when you sum all the z and its powers, you are essentially doing vector addition. if the sum is 0, then it means you haven't moved from the origin. in this case, the origin represent perfect balance, and anything else means an imbalance. it's the same as solving for the center of mass/gravity of an object.
Assume equal point masses on the complex plane. Then the center of mass is simply the mean position of the masses. And this mean is zero if and only if the sum is zero. Finally, because of how a centrifuge works, each point mass can only be placed at one of the complex roots of unity.
So if you have 7 masses and 12 slots, as in the video, the question becomes: can you add together 7 of the 12 twelfth roots of unity so that their sum is zero?
To balance effectively the average position of the mass must be the center, because in that case the centerfuge motor isnt having force pulling it in a specific direction, because any outward movement in one direction is counteracted by movement in the opposite direction.
The complex numbers work, because mathemeticians have found very useful properties of complex numbers (a+bi where i = square root of -1). You have to think of a as an x coordinate, and b as a y coordinate on a 2d graph. Multiplying 2 complex numbers gives a result where the angles of the 2 inputs are added, which is why the powers are evenly spaced around the unit circle.
All the addition is doing is calculating the center of mass of the beakers (assuming equal mass). Without complex numbers, you can think of balancing on a seesaw with one end being -1, and the other +1. To balance the seesaw, weight on the -1 end must be equal to the weight on the +1 end so that (weight1 - weight2) =0
The specific example works because z^3 = -1, so (z +z^2 + z^4 +z ^5) = (z + z^2) + z^3 (z + z^2) = (z + z^2) - (z + z^2) = 0.
Hope this helps.
Honestly, at this point I'm just surprised no one has mentioned polar representation, as in, every complex number can be caracterized by its distance to the origin (its module, 1 in this case) and an angle between 0 and 2pi (its argument). That's probably the most natural way to think about the roots of unity, by saying their arguments can be written as 2kpi/n where n is a natural number and k is in |[0,n-1]|.
You have a set of complex numbers on the unit circle (~centrifuge model with all the possible positions of the tubes, though we are only interested in equally spaced ones that can be expressed in the form of z^n with the N-th root of unity formula). If you consider a set of K points from N equally spaced points around the circle (all the holes of the centrifuge), then the center of mass of these K points (which are the tubes) is the average of the K complex points. If average of the K complex points is 0 (the sum is also 0) it means that the center of mass of the tubes is the center of the circle (=of centrifuge), thus the configuration is balanced because it is part the axis around which the centrifuge spins.
Similar principle applies to cartons of eggs, too. If there are four eggs left in the carton, they should not all be in one end, catching the next person to pick it up off guard. They should be spaced evenly away from the center, to balance the carton.
It all began with that laugh. That damn laugh.
This is actually an important problem for space rockets. How many, and which engines can be activated. Think of Falcon 9: 9 engines; 1 central, 8 in a circle. It must vary thrust on landing, so it lights up only a certain number of engines for braking. How many? Which ones? 1: 1 central. 2: 2 on opposite sides. 3: as 1+2. 4: every other from the outer set. 5: 4+1. 6: 9-3. 7: 9-2. 8: 9-1. And of course 9, all. If it didn't have the central engine, all the odd combinations would be impossible, unbalancing the rocket.
Great example, as it's not just the masses of some funny number of parts making trouble due to imbalance, but active thrusters applying serious torques to the whole. If a rocket like that goes hayware, it's not just a matter of annoying vibrations!