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.
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.
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 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 😂
It's _always_ better when a scientist is down-to-earth and has a sense of humour, IMO. I never enjoyed those super-serious professors who always look grave and never smile or anything.
@@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?
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.
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).
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.
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.
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)
When I was a graduate student back in the 70s I realized that I should be able to balance 5 tubes in a 12 slot centrifuge rotor. Nobody in the lab believed me. Then I asked the Beckman service rep and he didn't know. So I tried it and the rotor spun just fine.
Did you test an unbalanced configuration to show that it makes a difference? I mean, as a grad student, you should have been able to afford to buy a replacement in case of disaster, uh, right? :P
Perhaps I'm being too simple minded, but for twelve slots I imagine a clock dial. I can balance 2 by using twelve o'clock and six o'clock. . I can balance 3 by using one, five and nine o'clock. No conflict for occupancy there, so I can have both, giving 5 tubes.
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
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.
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.
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.
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.
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!
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!
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.
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.
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.
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.
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.)
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.
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.
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!
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
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.
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.
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
correct me if i'm wrong but you can always balance if: a) two tubes creates line going trough center of centrifuge b) you can create a regular polygon from tubes (same side lengths, same angles) c) any combination of above (assuming no overlaps)
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".
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.
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
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....
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,
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.
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 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 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.
@@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.
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.
In the 1990s we surprised our colleagues at the University of Goettingen with this experiment. We did this with real samples in a real Eppendorf centrifuge running at high speed with n=12 and 5 test tubes. Most people could not believe their eyes... It was fun. And you could save some time if you had only 5 samples tubes and n= 12. And seven samples also worked fine but also looked quite weird.
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.
This is fun. Chemistry has a lot to do with symmetry and this can also be solved by looking at the symmetry operations found (and combine them). For example a mirror plane through 2 holes and the center means 2+n*2 tubes are possible. n-fold symmetry axis means n are possible and so on.
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.
1. You're assuming that all the tubes have the same mass & the same mass distribution. If you're allowed to fill them differently, other patterns will likely become possible. 2. The balancing you're doing is "static balancing." But for the centrifuge to actually spin without going into conniptions, requires that it also be dynamically balanced. That is, its moment of inertia tensor must have one of its stable principal axes along the direction of the spin axis. I suspect you're OK on that score, using the addition-of-groups principle you've described, but I'm not completely certain. When there's a high degree of symmetry, it will work; but for the 7-of-12 pattern you present, it might not. Regardless of all that, the connection to the sums-of-complex-nth-roots-of-unity problem, is inherently very interesting. Thanks!! Fred
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.
Maybe the best math video I've ever seen after turning out a sphere. The only question I have, is this problem have relation to Music Theory? In an octave from 12 semitones, any tonality have 7 notes working and 5 missing, that is what I reminded.
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