Seven Dimensions

Entry for the 2022 Summer of Maths Exhibition #SoME2.
Spreadsheet: docs.google.com/spreadsheets/...
00:00 Intro
01:15 Pt. 1 - Abstract Spaces
04:19 Pt. 2 - Doing Linear Algebra To It
08:46 Pt. 3 - The Planck Units
12:01 Pt. 4 - Coherence
13:46 Conclusion
MUSIC
Crowander - Dreaming in a Dream
Crowander - Last Look
Sorry the audio's not very polished.

[@HBMmaster]

it's very fun seeing "the seven C's" in a more serious context like this. great explanation of these concepts

[@teovinokur9362]

woah it's the misali man

[@y.og.i]

hey it's that guy from the caramelldansen video

[@andyl.5998]

Hi, may I ask what "the seven C's" are? Thx~

[@U20E0]

@@y.og.i for me, it’s the guy from w

[@de_g0od]

Its him

[@tonaxysam]

Who would have thought that the 7 C's would have a sequel

[@brainandforce]

Don't you mean a Cquel?

[@paradox9551]

Never thought I'd see a jan misali reference here. Toki!

[@notwithouttext]

@@paradox9551 toki! sina toki ala toki kepeken toki pona? (hi! do you speak toki pona?)

[@paradox9551]

@@notwithouttext toki! mi ken toki e ona!

[@notwithouttext]

@@paradox9551 a!

[@lewismassie]

When I was struggling to memorise all the equations for my exams I realised if I could reverse engineer the positions of the equations from the units I wouldn't have to memorise the actual equations themselves. It wasn't until I had a casual chat in my university lab some years later that I found out it was called Dimensional Analysis. This obviously goes a lot deeper than my own brain could come up with (7-dimensional vectors was where you surpassed me) but this was still very interesting

[@felipevasconcelos6736]

When I was in High School, I used to participate in the national Physics Olympiad. There were always a few questions way beyond what I would have seen in physics class or on my own studies, but thanks to dimensional analysis and calculating areas on a graph there was usually enough time to answer a question from scratch.

[@chainemusique1792]

you still have to remember the constants

[@windywinend586]

@@chainemusique1792 no, constants are always given

[@felipevasconcelos6736]

@@chainemusique1792 constants are usually either given, or you can answer in terms of the constants. The first option allows people to “cheat” by doing dimensional analysis to answer questions they don’t really understand.

[@gizatsby]

Yeah this got me through my physics and chemistry classes back in school. The 1/2 in front of the kinetic energy equation got me a few times though haha

[@barigamb]

This entry is criminally underrated.

[@RoamingAdhocrat]

Every video published on this channel to date is criminally underrated.

[@1.4142]

@@RoamingAdhocrat This may be the best video he's made, I dare say.

[@RoamingAdhocrat]

@@1.4142 it's certainly in the top 10

[@gabrieltaylor4150]

It’s underrated but there isn’t really much you do about it. A lot of the topics and ideas he is talking about is things that most people don’t understand unless they have taken college classes on linear algebra or other similar higher level math classes to even understand what’s happening

[@Sergiuss555]

It's useless

[@UnitaryV]

The conversion matrix only handles matching dimensions across systems, but not the actual numerical value. However, what if we included the number 10 as an additional "unit"? It seems like that provides the last piece of the puzzle to perform full unit conversions, with the slight drawback that the resulting numbers would be expressed as non-integer powers of 10 (the speed of light becomes 10^8.477m^1s^-1 instead of the usual scientific notation form of 2.998*10^8m/s). Though a little odd at first, it's not wrong. In fact, it's a step up from the matrix at 10:05, which converts the speed of light c to m/s, with nothing indicating the value of 2.998*10^8. By adding an extra row at the bottom for the "unit" 10, containing 8.477 (the log of 2.998*10^8 in base 10) in the first column and appropriate values for the rest, the matrix becomes a bonafide unit converter that converts the numerical values too, instead of just matching the dimensions of the systems. Note that an extra column must also be added on the right for the unit 10, containing five 0's and a 1, so that we end up with a 6×6 invertible matrix. imgur.com/a/qMakuIY We can also choose to use any number greater than 1 other than 10, but that would change the values in the final row. For example, if we wanted to use e as our additional "unit" instead of 10, we would divide the entire final row (except the 1 in the corner) by log_10(e).

[@turun_ambartanen]

Very clever idea! For what it's worth, there is already a dimensionless number in the base unit system. I have no idea, why anyone would ever want to express results in multiples of 602214076000000000000000, but who am I to judge.

[@UnitaryV]

@@turun_ambartanen That's true of SI units, but not Planck units. Still, I'd be lying if I said the thought didn't cross my mind and give me a laugh. Edit: Now that I think about it some more, if we left N_A in the SI units instead of omitting it, and simply added 10 to the Planck units, that would be valid too. So the matrix can then convert from units of 10 to units of N_A, allowing you to express c as (N_A)^0.356(m)^1(s)^-1. Now that's what I call obfuscation, lmao.

[@felipevasconcelos6736]

@@UnitaryV choosing a number as arbitrary as 10 seems counter to the spirit of Planck units. Why not e, so the exponent is just the natural log?

[@UnitaryV]

@@felipevasconcelos6736 I agree, which is why I included the bit at the end. For the sake of pedagogy, I decided on using base 10 because log_10(x) can be approximated by a quick mental calculation. That way, you don't have to pull out a calculator to follow along with my explanation. For example, log_10(2.998*10^8)=log_10(2.998)+8. From this, you can be somewhat comfortable in accepting that 8.477 is log_10(2.998*10^8) without a calculator, since 8≤8.477

[@foogod4237]

@@UnitaryV Why not add 10 as the unit for the SI system and e as the unit for the Planck system? It would seem to parallel the differences of most of the other units in the different systems fairly well, IMHO... I actually came to the comments specifically hoping to find a thread about this stuff, because adding an 8th dimension to represent the actual quantity seemed like an immediately obvious next step the moment I saw where the video was going. You could then develop a single matrix to represent the complete conversion of any value in one measurement system to the corresponding value (with units) in another, essentially a complete _definition_ of any possible unit system using only math (and some other system as a reference point)...

[@WaluigiisthekingASmith]

I actually realized this a while back when I had a physics problem that forgot to give the mass of some object and, since there was a unit of mass in the answer but nothing involving mass was allowed in the answer it was unsolvable. In general this is a really good introduction to the idea of dimensional analysis. Dimensional analysis says that given some set of base quantities trying to derive some other quantity the answer is always the base quantities combined to get the one you want times some function of all of the dimensionless quantities

[@mathlitmusic3687]

How can we add a bit of mass with a bit of time, as in this "vector addition"? Wouldn't that contradict the "dimensional analysis" which says you can only add quantities with the same units? Or would you ignore dimensional analysis everywhere except when restricted to the "basis" lines? This kind of defeats the purpose of invoking dimensional analysis since that is only of any actual use when we multiply different quantities (like mass times time) not when we are simply adding the same quantity of different magnitude (like 1kg + 2.5 kg)?

[@v1298]

I love reading some advanced anecdote about math and "dimensional analysis" only to look at the profile picture and see Waluigi

[@Kalobi]

@@mathlitmusic3687 The vector addition in this vector space has nothing to do with the addition of physical values. The elements in the abstract vector space described in the video are things like "time" or "capacitance" or "length^4 divided by amount of substance", not "1s" or "3.5μF"

[@mathlitmusic3687]

@@Kalobi how can you get (length)^4 in this vector space? Since this vector space has the basis given by those SI units, which point/coordinate do you think will give you length^4?

[@Kalobi]

@@mathlitmusic3687 length^4 is 4*the length basis vector. Addition in this vector space corresponds to multiplication of physical quantities.

[@PopeGoliath]

This feels like a Part 1, Where part 2 goes on to define a new, mathematically optimal measurement system.

[@exciton9861]

Obviously that will be "The seven Cs"

[@brutusthebear9050]

There's no such thing. Measurement systems are context-dependent (which is why I defend US Customary, since it includes Metric and imperial, imperial units being better for things on a human scale, requiring less precision).

[@idontfeelsogood2063]

@@brutusthebear9050 "imperial units being better for things on a human scale, requiring less precision" cope x2 Why 99,9% of the world is using metric? Because it's better in everyday life. You just need to be raised and learn them from youth and you could measure weight, lenght and speed from your own sight/feeling. The thing is, you thinking imperial is better in everyday life is not because it is. It's because you've been raised and become accustomed to using it. Studies show 180° view on that = metric is better. That's why almost only USA is using it, they're medieval units.

[@brutusthebear9050]

@@idontfeelsogood2063 Alright. I'll humor you. Cut something into thirds using Metric. What is 1/3 of a meter? And then, cut something into thirds using Customary. What is 1/3 of a foot? A third of a meter is a repeating decimal, because Metric uses decimal. A third of a foot is 4 inches, because Customary doesn't use decimal. The reason most of the world uses metric isn't because it's inherently better. It's because it looks nice in decimal units and it's more precise. Customary works better on a human scale because it deals with division better. Units in Customary are usually base 12 or 16, which are more intuitive to divide. Metric is a system that was designed from the ground up to be a "rational"(istic) measuring system. Customary units are the result of actual human use. Also, Americans do learn Metric. Hell, we get taught more with Metric than Customary. If you actually did anything with your hands, you'd see why Customary is superior. But that would require actual effort.

[@idontfeelsogood2063]

@@brutusthebear9050 I do my "actual" effort everyday, as I'm engineer in production facility in Germany. But I won't discuss it any further, you seem based in imperial=better. No way it would be a civil discussion and I could convince you to the metric. You have been raised with Imperial and doing your best and apparently having success. This doesn't change my mind that metric>imperial. But your career is only limted to USA. Try traveling to Japan or Germany with using imperial. Not possible. Good luck bro.

[@Pyotyrpyotyrpyotyr]

I graduated university for engineering, and this video taught me linear algebra in a more intuitive way than university ever did.

[@pyropulseIXXI]

That is quite sad. This video just goes over stuff you could've learned by taking two minutes to read your textbook. I am sad that it takes these, admittedly awesome KZbin videos, to wake people up, when that information is already there if only you were self-motivated

[@teeletsetse445]

@@pyropulseIXXI You overestimate the textbook's power to explain.

[@pleaseenteranamelol711]

We all know that the real purpose of school is not to teach you, or encourage curiosity.

[@pyropulseIXXI]

@@pleaseenteranamelol711 Exactly

[@maya_unplugged]

Me(1): 🤩Oh nice, I’m going to learn something new. Me(2): 😳Reading your comment. Me(3): 😒ok, I’m out. Me(4): 💪🏼hmm. I’m not giving up so easy. Let’s give it a try. Me(5): 🤯ok, I’m out. Me(6) to myself: I TOLD YA.

[@MaxG628]

I really like using physics to motivate change of basis. It works a lot better than “I’m going to plot points in the plane using a system other than (1,0) and (0,1) because I hate myself”. At the same time I think I learned something about physics, too.

[@renhaiyoutube]

In quantum mechanics you encounter change of basis all the time, for example with spin and angular momentum

[@hanswoast7]

You might find the Fourier Transformation interesting then. It converts between a basis of X to 1/X, p.ex. from time to frequency. And it has wide application within physics and other sciences.

[@trevormacintosh3939]

Wow. That’s such a fascinating concept. I never would have thought of representing units as vectors.

[@pyropulseIXXI]

you can represent almost anything as a vector

[@easports2618]

@@pyropulseIXXI amount of sus moments in a childrens playground?

[@vrajeshpc]

@@easports2618 draw a vector towards the child's age on x axis and initially predicted age on y axis

[@epsilonengineer373]

First Astronaut: Wait, It's all Linear Algebra??? Second Astronaut with gun: Always has been.

[@TheMrSamusic]

As a math graduate, I thought excatly so. This is just a trivial video

[@b43xoit]

"What is the square root of an acre?" is a valid question, having a definite answer, and there might be times when it would be useful to know.

[@michaelleue7594]

Just to answer the question in case anyone cares to know, it's 66*sqrt(10) feet, or about 208 feet and 8.5 inches.

[@qwertystop]

Specifically, it answers the question of "what is the side length of a one-acre square of land". This is a less-trivial question than most other units of area would be, because the acre is in the odd position of being a unit of area defined in terms of two unequal side lengths (66 feet by 660 feet). This in turn is because square land parcels are not especially practical in pre-industrial farming: oxen pulling a plow are hard to turn, and 660 feet (a furlong, as in the length of a furrow) is about how far an ox can pull a plow before it needs to rest anyway. An acre is thus about how much plowing you can get done in one day with one ox, but if you got your land allotment (of one day's plowing) as a square, it'd have be smaller. Also, in distribution of a larger agricultural area to many serfs or tenants, it means more people can get a bit of riverbank, a bit of both the sunny and the shady side of the hill, and so on, and thus nobody is stuck only growing one kind of crop.

[@RoamingAdhocrat]

ok but if you have a one-acre field, what crop can you plant which would produce a square root perhaps if you inserted some kind of lattice of steel sheets, like a Kallax bookshelf on its side but much smaller, and planted one turnip into each cell…?

[@scrambledmandible]

@@RoamingAdhocrat I bet you were pining to get that one out :p

[@sponge1234ify]

@@RoamingAdhocrat I know this is unrelated, but i just wanna say thank you for giving me a proper name for those square racks/bookshelves. Now i can order one more properly in the future, and not have my books be in awkward Bantex files.

[@Libellisth]

What a remarkably concise way to convey a broader insight through this little practical exercise. It really clicked with me. Well done. You're a natural.

[@kallekula84]

You're very talented at conveying an idea in to a presentation like this and you should continue making more of these! Very interesting video and would love to see what's next on your channel!

[@Beashtman]

Great video, dimensional analysis can be a powerful tool in physics when trying to understand the meaning of an answer with bizzare combinations of units. Being able to see other ways of representing those units could provide some useful insight.

[@stede9304]

Fascinating. This didn’t make me think of vectors any differently. My math degree trained that out of me. It did allow me to see new & different representations of familiar concepts and units that gave an entirely new perspective on their relationships. And that is very cool.

[@jarroddt]

I watched this months ago and vaguley understood, having learnt 3-d vectors and matrix algebra. But now at university, having completed much of my way through the Linear Algebra course, its so cool to see these terms I've learnt come up in a video like this!

[@tima1639]

Exactly the same for me :D

[@akeron1an]

I think this was a brilliant video. It really makes you think about vectors in an entirely different way. To me the part about the determinant being 0 implies non-invertability made so much more sense explained through physics units than any previous explanation I had encountered.

[@mathlitmusic3687]

What is described here is merely change of variables (in a system of linear equations), nothing more.

[@johanngambolputty5351]

determinant basically gives you the change in volume elements, being 0 implies a volume can get mapped into a line or point (the result has no volume) and you can't uniquely unfold that back into the original arrangement (i.e. you can't invert that)

[@Pieter31]

For more insight on vectors, you should check out 3blue1brown's series "Essence of Linear Algebra"

[@General12th]

@@mathlitmusic3687 Are you implying that change of variables in linear algebra is not a brilliant subject?

[@IroAppe]

I have a shorter solution for 1:47: - Fill B - Transfer from B to A Now B=2. - Empty A - Transfer from B to A Now A=2. - Fill B - Transfer from B to A Now A=3 and B=4.

[@guidolongoni]

Yes, and as a bonus, less water is wasted (3 units instead of 5). I was thinking exactly the same thing.

[@cknox64]

Still need to empty A so you are left with B=4.

[@McShavey]

- Fill A exactly 1/2 full - Fill B exactly 1/2 full - Transfer A to B 😉

[@sodiboo]

​@@McShaveyThere are no markings to get it half full.

[@guilhermedamasceno343]

Just found out your channel and I want to congratulate how well you explain the essential, yet advanced concepts. I'm looking forward to see more of your content.

[@bencressman6110]

Phenominal video. As someone who dropped out of math before learning calculus and linear algebra, but who loves math, and learning… I can tell you that you made this extremely easy to understand. Engaging, and exciting 10/10

[@quarkup9296]

Really interesting and fascinating approach to unit systems. Great outside-of-the-box thinking to be able to use linear algebra in this context and, overall, great video

[@__8120]

This is definitely one of the most well presented SoME2 entries I've seen, good job!

[@CasparAddyman]

Nicely done, Kieran. And what a brilliant use of the Poincare quote!

[@bernardlaval6248]

First time in years YT algorithm works as I would have liked from the beginning. Great video. I've never thought of that. Thank you to have opened my mind today 👌

[@matthewgiallourakis7645]

I'd love to see a followup of this going over the Buckingham π theorem!

[@Korbad]

Wow, great video. The way you used the "SI basis" as was really great for visualizing and understanding linear algebra concepts. Thanks!

[@arttraynor5720]

Masterful Kieran , I've been noodling with SI units myself as a demonstration to students . I sensed there might be an abstract connection between them ( after much algebraic gymnastics ) but couldn't make the deft leap you have here . Thank you so much for this !

[@brunizzl]

The vectors representing a unit are actually used to represent units inside programming languages. This allows for example to automatically determine what unit the product of two variables with units has: just add their unit vectors.

[@9sven6]

This was surprisingly interesting! Good work

[@octo-pops]

Oh wow thank you so much for mentioning the seven Cs, they're amazing Great video! Always love seeing things like this and best of luck for your channel :)

[@PowerhouseCell]

Amazing!! I can't believe I just found your channel - as a video creator myself, I understand how much time this must have taken. Liked and subscribed 💛

[@Reydriel]

I knew about both Dimensional Analysis and Linear Algebra, but never thought to put the two together to GREATLY simplify the train of logic of converting between different systems of units. That visual of representing all possible dimensional units as a vector of their powers blew my mind lol

[@Yossus]

Dimensional Analysis was my favourite part of my physics degree, but I haven't had to do a lot of linear algebra since then. This was basically a solid 15 minutes of me sitting there mouthing "that's so coooooool" over and over

[@ghkthILAY]

thats a really refreshing point of view, thank you for this video! i had some problems with the concept of natural units (specifically in QM combined ED) but seeing it as a change of basis really helps.

[@user-tn4cl5dz1v]

I have never seen dimension analysis in a effective way like this!!!You are such a genius!!!

[@strangeWaters]

The things you're working with in this video are usually called "tensors". If a vector is a linear combination of unknowns (1x + 3y + .5z), a tensor is a linear combination of *products* of unknowns (3xy + 5z + 7x^2). (We don't usually think of adding things with different units, but it's just a way to keep track of multiple things at once.) What you're doing here is sort of taking the "logarithm" of basis tensors to get vectors ("log" xy²z = x + 2y + z). I bet there's a formal name for this operation, but idk what it is. As you've clearly shown, after taking the log, the result is a vector space. Great video :)

[@redsix5165]

Where can I learn more about tensors the the description of a log…your comment is very understandable but I am not quite there.

[@bayleev7494]

is this right? to use all operations in the tensor algebra, this kind of assumes that any two units can be added together as well as multiplied, which isn't really true (for example mass + time doesn't really make sense).

[@alxjones]

There are a few issues with this. (1) The multiplication is commutative here, which is not typical for general tensors. We could call it a symmetric (tensor) algebra, if it weren't for... (2) Tensors have a concept of addition, scalar multiplication, and tensor multiplication. Your proposal is that products of dimensions are tensor products, and so the compatible addition here would allow for addition of terms with different units. For example, mass + length would be a valid tensor in this system. That 'log' you take note of is actually the isomorphism between the space of dimensions (with multiplication and exponentiation) and Z^7 (with addition and component-wise scaling), both considered as vector spaces over the field of integers Z. Any isomorphism F between these spaces must satisfy F(x y^k) = F(x) + k F(y) It's common to see something that looks a lot like exp and/or log when looking at morphisms in algebra, but they are just examples of a more general concept.

[@easy_riders]

This is an incredible video! Thank you!

[@CheesePumpkin2137]

As a person simply interested in mathematics, your video is extremly helpful and shows the utilisation of matrices in a very intuitive way. Good job

[@nbtbn]

Awesome video, and super fascinating to see systems of measurement as sets! As a chemical engineering student who does unit conversions all the time without questioning their existence, this got me genuinely excited and curious, and has permanently changed the way I think about units.

[@Seltyk]

This is by far the best SoME2 video I've seen yet. Add in Uniit's comment about the extra column for the number 10, and you've got some delicious linear algebra on your hands.

[@ReptillianStrike]

what is SoME2?

[@Seltyk]

@@ReptillianStrike 2nd annual Summer of Math Exposition

[@ReptillianStrike]

@@Seltyk Summer of math? What's that?

[@Seltyk]

@@ReptillianStrike yearly competition among youtube creators hosted by 3blue1brown to make math explainer videos

[@ReptillianStrike]

@@Seltyk ah ok thank you! I was completely out of the loop on this, but still got these videos in my recommended when it was going on lol

[@dechair3113]

What an interesting application of linear algebra!

[@nuparuchi]

Lovely video, very clear and concise. Plus, the style and your delivery are quite pleasant. Really great watch

[@paxdriver]

This channel is absolutely amazing! Great work dude

[@DavidARowland]

Playing with this recently I became aware that energy and torque have the same units: Length times force. L*F or M*L*T(-2).

[@LPT4]

wouldn't L be squared because F is derived from M*L*T^(-2) [m/s^2]

[@avnishbadoni1393]

add a dimension of angle to the other basis. And you'd see the difference.

[@qdrtytre]

@@avnishbadoni1393 Angles are dimensionless. Well, that's the party line anyway.

[@avnishbadoni1393]

@@qdrtytre To those who say this, you can ask if it wouldn't take any force or energy to rotate a 2 tonne wheel on it's axis without changing its x, y or z coordinates. 😁😉

[@triffid0hunter]

Energy is length linear-multiply force, while torque is length 3d-cross-product force - so energy is a real number (ie scalar) while torque is a 3-vector. So yeah, the _units_ may be the same, but energy and torque are still different.

[@MCLooyverse]

I had this idea a while ago, but never did much with it. It's nice to see it explored. Also, every time it comes up, I feel compelled to ridicule the idea of the mole as a unit. It's just a number.

[@felipevasconcelos6736]

It’s a bit silly, which’s why in the Seven C’s the unit of “amount of matter” is just “a hundred”. I also think that it’s kind of weird luminous intensity has its own unit.

[@matj12]

SI defines some counts: kilo, mega and others. Mole (~6×10^23) could be one of them. That would allow that it would be used as a prefix to multiply a unit. The number is similar to yotta (the highest count named by SI) (10^24) in orders of magnitude, and yotta is very rarely used, so mole as a prefix too would be very rarely used. But there are cases where that would be convenient; a moleohm would be a realistic resistance of an insulator. I found that the resistivity of Teflon is around 10^24 ohmmeters. In the other direction, mole is used usually only in chemistry. Chemists could completely ignore mole and express amounts of particles in yottas, which wouldn't change the numbers much because mole and yotta are similar.

[@matj12]

@@felipevasconcelos6736 That luminous intensity has its own unit is not strange. It's independent of other SI units. That the unit is in SI is strange. I expect that SI units are for objective measures. Luminous intensity denotes how bright some light seems to an average human, which is quite subjective IMO.

[@inciaradible7144]

Very fun video; playing around with unit systems can be incredibly useful to see what it is that units really represent. Additionally, I think this is a great lesson in how to apply these fundamental concepts of linear algebra.

[@ThomasValadez-tv]

This is such an elegant video. I feel like this more than anything else has helped me understand the math and theory behind physics.

[@ebentually]

What always bother me about the 7 base SI units is why amount of substance is considered a base unit, as that is just a contant to deal with quantities like a dozen or a mega (the prefix to refer to million) may there is a fatal flaw i'm overseeing, so if anyone can explain what this flaw is it would be really great

[@lunam7249]

the scientific world confused "quaLity" with "quaNtity".....quaLity = something stuff that different from something else..... quaNtity = the NUMBER of stuff....so time, mass, length are really Qualities.... 12, 3.44657, 287335546.3736 are QUANTITYS...

[@Sandromatic]

I think a version of this that included scaling factors somehow to allow converting between units and not just different basis systems would be much more useful. Like, the main problem is that you can't just convert between SI and 7C or Plank units because they don't map to the same values. If the scalar value of these vectors *did* map properly then that'd be more useful but the exponents-as-vectors approach is just missing a fundamental part of unit conversion. It will tell you what units you expect to have in your result, yes, but it won't tell you what scaling factor you will have to use in order to actually convert the quantity.

[@pyropulseIXXI]

This is just a dimensional analysis conversion

[@cmyk8964]

You may be able to derive that with even more linear algebra, but I don’t know hew.

[@bensonprice4027]

What a great example of adding depth to two seemingly unrelated topics! This has helped expand my understanding of dimensional analysis and linear algebra.

[@foo0815]

I always found the inclusion of mol as a physical unit questionable. It's just a number without physical meaning.

[@RuyVuusen]

Not really, just like a meter is practically a random length in one dimension, it, too, is a practically random quantity of amount of subtance; both still hold a lot of meaning. What reduces their random nature is that they are derived from universal constants or agreed upon numbers and thus are not subject to change-unlike only defining your mesurements in non-constant concepts such as the human foot or the length of day (both of which evidently can work, but have to be standardized, aka separated from their original definition). In the end, units of measurement are merely a human convention, and for that reason they may as well be random, as long as they are constant and useful to their purpose (which moles are).

[@taimunozhan]

@@RuyVuusen The problem with mol is not that its value is arbitrary (which all units ultimately are, natural units included) but that it really doesn't express any physical quantity that would even require units to be measured. Meters, feet, or whatever crazy length unit one might conjure will too have an arbitrary value but they will reference the physical concept of length; the number is coupled with a certain physical feature. For mol, there's only a number; it has more in common with the prefixes like kilo- and micro- than with any of the proper units. There are far better candidates for a linearly independent seventh dimension. Angle is often brought up in this context, with the radian sometimes being mentioned as a base SI unit. I believe there is a fairly good case for information (measured in bits, bytes or other quirkier units like nats) to be treated as another dimension to incorporate into a metric system as well.

[@Anonymous-df8it]

@@taimunozhan Don't obsess over inaccuracies. He's probably American lol

[@sebastiandierks7919]

Nice video :) I was honestly very sceptical when I saw the thumbnail as in relativity, mass is explicitly NOT a vector, but a Lorentz scalar, the norm of the energy-momentum 4-vector. I also thought about mass distributions, where mass would however still be a scalar field on spacetime. I then thought about the inertia tensor of rigid bodies, but then "mass" would be a second order tensor, not first order. Anyway, I had to click to find out what the video was about and would not have guessed a video on unit systems! I also have a question as I'm not familiar with coherence of unit systems: So the SI-system would then actually not be coherent right? As the mol and candela are redundant? mol measures the amount of substance, which can be expressed as the amount of atoms/molecules, which is a dimensionless number, which is equivalent to the 0-vector. So (0,0,0,0,0,0,0) and (0,0,0,0,0,1,0) would represent the same unit? Or is that wrong, as in this argument I considered the Avogadro constant to be a known constant, similarly to how you assume to know the speed of light, Planck's constant and the gravitational constant to be known and fixed in Planck units? Also what about natural units with c=hbar=1, where length and time (for example) have both the same unit of GeV^-1? What does that mean in the context of this video? I think one actually only needs 1 unit and set a bunch of natural constants to 1 (which is however like picking another unit maybe?). E.g. pick the second as your one basic unit of time and express length = speed of light * time, i.e. express length in (light)seconds etc. All physical quantities can then just be measured in powers of the second. Otherwise, who says there should be 7, or 5, or whatever arbitrary amount of basic units?

[@MrAlRats]

The mass, energy, and momentum of any physical system are related to each other by the formula m² = E² - p² (in any system of units where the speed of light is dimensionless). So, they can be measured using the same units. The energy of a system is also proportional to the frequency associated with the wave nature of the system, so all these quantities can be measured using units of the frequency. There should only be two base units: The second, s, and the electronic charge, e. Lengths and time intervals should be measured in seconds. Mass, energy, frequency, momentum, acceleration, and temperature should be measured in units of the reciprocal of the second, s⁻¹. Pressure and density should be measured in units of s⁻⁴. Speed, entropy, and angular momentum should be dimensionless. Capacitance should be measured in units of e²s. Voltage should be measured in units of e⁻¹s⁻¹. The electric current in units of es⁻¹. All the fundamental constants disappear in this system of units. The size of the second is arbitrary and so can be adjusted for convenience.

[@98danielray]

it is not the zero vector, no

[@sebastiandierks7919]

@@MrAlRats Are you sure you could not also relate time and charge by an equation/an experimental setup and measure charge in a certain power of seconds? Why use two base units? It's just as arbitrary as 5 or 7 in the video. Also, you could set e=1, as you set c, hbar, k_B, G = 1 in other unit systems. Although setting a natural constant to 1 is the same as picking a dimensionful constant, you again choose a unit to measure in, it's just not introducing an additional physical dimension.

[@MrAlRats]

@@sebastiandierks7919 It's the discovery of relationships between different quantities due to various developments in the history of physics (such as statistical mechanics, relativity, quantum mechanics) that has allowed the number of base units to be reduced to just two. The best we can currently do is to devise a system of units with two dimensions (Time [T] and Electric charge [Q] ), with one base unit associated with each dimension - the second,s, and the electronic charge, e. All other measurement units can be expressed as some integer powers of these two base units multiplied together. Until some deeper connection is known between these quantities I think we will need at least two base units. Perhaps we'll have to wait for a theory of quantum gravity or theory of everything and then maybe everything could be measured in qubits of information or something.

[@adamrezabek9469]

Yeah, mol is wird. It should not be an unit. And if we want to treat it like a unit, we have to forget that it's actually just shortcut for writing 6,022E23 and treat it as a unit.

[@agranero6]

I read an article about that about 30 years ago (Natural Units via Linear Algebra, American Journal of Physics). Basically, this works because a vector space can be made with the vector set being the set of real numbers, the scalar set being the set o real numbers too (nothing forbids that) and the multiplication of a vector by a scalar is the operation of elevating the vector "number" to the scalar "number". All vector space conditions hold (as the exponentiation is to the multiplication what multiplication is to addition in terms of distributivity). The only condition is that the base conversion matrix is nonsingular. For the same reason if you take natural numbers instead the prime numbers form a base of natural numbers which is a vector space way to express the fundamental theorem of arithmetic.

[@darkside3ng]

It's amazing to know that there is someone who knows all these things and can also give such an amazing presentation!!!!

[@guidosalescalvano9862]

I wish you had derived the eigenvectors of unit space. I.e. what is a coherent system for expressing all of physics?

[@aboprivatkanal2493]

What

[@zokalyx]

what do you mean? the 7 base SI units are coherent and can be used for all magnitudes in physics.

[@guidosalescalvano9862]

@@zokalyx But are they the minimal vectors to span unit space? i.e. are they orthogonal?

[@MrAlRats]

The mass, energy, and momentum of any physical system are related to each other by the formula m² = E² - p² (in any system of units where the speed of light is dimensionless). So, they can be measured using the same units. The energy of a system is also proportional to the frequency associated with the wave nature of the system, so all these quantities can be measured using units of the frequency. There should only be two base units: The second, s, and the electronic charge, e. Lengths and time intervals should be measured in seconds. Mass, energy, frequency, momentum, acceleration, and temperature should be measured in units of the reciprocal of the second, s⁻¹. Pressure and density should be measured in units of s⁻⁴. Speed, entropy, and angular momentum should be dimensionless. Capacitance should be measured in units of e²s. Voltage should be measured in units of e⁻¹s⁻¹. The electric current in units of es⁻¹. All the fundamental constants disappear in this system of units. The size of the second is arbitrary and so can be adjusted for convenience.

[@robheusd]

Why is luminous intensity a fundamental unit, isn't it expressable as amount of energy per second per area (square length)?

[@KieranBorovac]

Intensity is indeed measured in W/m^2, but 'luminous intensity' is not technically the same thing - it's a special unit that measures brightness as perceived by human eyes, which is more complicated than just 'radiant power per unit area' because vision is complicated. (I recommend searching 'photometry' for a more detailed explanation.)

[@enderyu]

@@KieranBorovac but including the mole is still a bad idea, right? Its just a pure number, so you can represent 1 m as (6x10^23)^-1 m*mol or even (6x10^23)^-2 m*mol^2

[@zokalyx]

@@enderyu I kinda feel the same thing, but at the same time, a mole is a really relevant number in chemistry that we would benefit a lot from knowing precisely.

[@vorpal22]

Fascinating video. I had never thought of using linear algebra and change of bases units to go from one set of units to another! Now it seems so natural and obvious.

[@scaredyfish]

I’ve often thought how handy it would be to have a spreadsheet that is aware of units and can convert between them at will. I think this concept could be quite useful in implementing something like that.

[@elliotderbyshire5859]

This is a cool video

[@BleachWizz]

yes, very indeed

[@M_1024]

mol is not a unit, mol is a number, so in Planck units it will be 1

[@brendawilliams8062]

Amy Noether

[@robinwang6399]

This is very interesting, I have never thought of units this way, nor applied vectors in ways other than the usual physical computations. You video have opened a door to me in representing things in maths, making it easier for me to try to understand the world around me.

[@dhruv4028]

This guy has one video and it is an absolute banger. Waiting for more content from you

[@ClementinesmWTF]

I’m so happy someone has put into words and good visuals what I’ve always thought. And I’m even happier you used janMisali in your examples. This has always been something that bothered me in the field of metrology and SI (especially when it comes to SI “supremacists”). There truly is no “true” measurement system and all systems can be equally expressed as all others. Sure, some might have some other nice properties (eg base-10 or human-scale-ness), but even those are arbitrary to some extent unless you’re using natural systems. And even then, metric’s base-10ness isn’t even that good from a mathematical standpoint…2x5? It’s cool to have a standardized system, but for people who trash on Customary, they shouldn’t for a second think their system is any less arbitrary. *meters are based on the distance from the North Pole to the equator thru Paris, seconds are based on Cesium atoms, and temperature isn’t even based on an atom, but a molecule (H2O), and even then the definition isn’t 0° like most people think it is (don’t even get me started on relative vs absolute temps). The SI definitions have changed over time to become less subjective, sure, but the current definitions are just “more exact subjectiveness” when it comes to that. There’s more than just those, but it makes the point and I still love this video so much for showing how subjective most metrological systems are

[@mathlitmusic3687]

SI isn't a superior system of units, in the sense that any "system of unit" is equivalent to it- it's just more convenient for our use. The great thing about it is suitable for science, because it is designed according to the decimal system (kilometers, kilograms, kiloJoules, nanometer, nanojoules, nanogram etc are easily understood by knowing what kilo or nano means as 10^x) which is way better than the stone age measurement systems like yards, feet, inches, score, stones, etc, which were designed by primitive people for a largely primitive, non-scientific world.

[@KnTenshi2]

@@mathlitmusic3687 Those are only intuitive if you already know (or are taught) that kilo means 10^3 and nano means 10^-9. What about Lahk? Or a Pak?

[@mathlitmusic3687]

@@KnTenshi2 Once you know what "kilo" means then you can use it for any quantity that's the difference- kilolitres, kilometres, kilometres, kilojoules, etc any quantity can have a kilo of that. But other archaic systems are quantity specific- like inch or feet has no meaning when we are talking about mass. That's the essential difference. Of course, another convenience is that it's always 10^x which is easier in conversions, than the 12 inches = 1 feet , or 1 score = 20 years or whatever..

[@felipevasconcelos6736]

@@KnTenshi2 that’s how numbers in general work. No one is born knowing that “thousand” means 10^3, so we’re taught that. Learning numbers is so easy a child can do it, though, so everyone learning a new (very limited) set of numbers isn’t a big deal. It’s “intuitive” because, if you know units or length, that knowledge is immediately transferable to units of mass, for example.

[@felipevasconcelos6736]

Seconds aren’t really based on Cesium atoms. They were redefined that way, but only to match the earlier definition as well as possible, and the earlier definition was that one day had 24*60*60 seconds, for no reason other than the the Babylonians liked 60.

[@oceannuclear]

This is ridiculous... I love it. I'd think people who think about units in such depth and people who are familiar with the axioms of vector spaces are mutually exclusive. You have proved me wrong. Also I love your outro music and concluding remark + quote. It makes the video beautiful :)

[@Zaneclodon]

great video! i love thinking about units and this video brought that together with linear algebra to show how to think about them in a whole new way! minor correction at 11:11: the vector displayed for capacitance in Planck units is incorrect, but the spreadsheet correctly lists it as (-1.5, 0.5, 0.5, 1, 0)ᵀ.

[@broken_radar]

This is an amazing video. This explains me things I have been thinking about and I think there are larger implications for this. Thank you!

[@yash1152]

1:14 3-5L jar problem as coordinates on a grid 1:58 pendulum's state space - i've seen that several times before 4:47 we now need to make sure that basic ops of LA are meaningfull 5:13 axioms of linear algebra and their corresponding meaning 7:09 change of basis in square matrix (with annotations of each vector)

[@rainbowskyrunner]

Thank you for producing this and sharing you thoughts so kindly and concisely. You are awesome 👏🏾 😎

[@aidenm.893]

*Super* straightforward. Thank you. I don't even have a degree, but I follow you the whole way.

[@BleachWizz]

I like this way of thinking, it feels so intuitive

[@PlaneShaper2]

What an elegant video on this topic! One of my physics professors was absolutely a fan of using Energy as one of his base units. We were allowed exactly one page of notes and the text book during exams in my first year college physics course two decades ago, and a representation of this constituted my notes. Really, Part 1 of this video ought to be shown to every high school physics student, part 2 to every first year physics college student, and parts 3 and 4 to every second year student.

[@tima1639]

loved this! On Wednesday i am writing my linear algebra I exam and its so refreshing to understand these concepts of linear algebra, which i would have not understood before So insightful, thank you!

[@leftyrighter8662]

Thanks for the simplified explanation. I always wondered what matrix multiplication would do in a real scenario.

[@salmagamal5676]

This is incredible. Please keep going I would love to see more of your videos

[@lonestarr1490]

_That_ was a debut video? Damn... Talking about "start as you mean to go on". I'll definitely stay tuned for more.

[@brucedickson6019]

This is fascinating. I've used Nastran since the early 90's. One of the impediments to effective finite element analysis is learning to pick a set of units that inherently consistent. Ideally, the units should be selected so that quantities describing your model are in he range [0.1, 10] . You're then faced with the challenge of interpreting the results correctly. This idea makes the process easy.

[@etdr]

This video is just breathtaking. Just a masterwork of thinking outside the bun

[@evanmarshall322]

I’m honestly super disappointed that this is the only video on the channel. Definitely subscribing AND (for once) ringing the bell.

[@mt2aod]

Amazing, thought-provoking video. This very relevant to my field of study 'changing basis units in general relativity'. It's enjoyable to see your perspective.

[@Maazin5]

This is really cool. I struggled with Linear Algebra in school and this video connected a lot of gaps I had about my understanding 👍🏾

[@loganfisher3138]

One note regarding Planck units: we can reasonably consider the Planck version of candelas to just be in units of power, and we just drop "mol" and treat it as a pure number instead. If you want to try something fun: figure out how units of information are represented in Planck units. Hint: Bekenstein bound.

[@nevyn]

I love linear algebra, and seeing it applied in this novel fashion is just so delightful that I'm giggling as I'm watching it!

[@gumbilicious1]

This is an incredibly brilliant and enlightening observation. I don’t know if this is an original idea or not, but thank you presenting it

[@Ayeloo]

Despite me hating physics and mostly watching videos like these fo the entertainment value, and having no expectations coming in for such an abstract topic, I ended up coming out with a much more meaningful understanding of linear algebra from this than I did from the 4 months I spent on a college course for it, so thanks :) Great video

[@holylampposts]

My favorite definition of vector that I've heard is, "a vector is an element of a vector space." In my undergrad, I spent a lot of time scratching my head, wondering what exactly a vector is. I asked a grad student friend of mine, and he provided that definition to me. Suddenly, the abstract thinking I needed to understand linear algebra was unlocked to me and I realized that I don't have to physically interpret the math I was studying; I just needed to understand the rules and definitions I was playing with. "If you've been conditioned to think..." Your conclusion takes me back to that moment. I hope it unlocks other students' minds the way my mind was released all those years ago.

[@amkessel2014]

Excellent video! Clear and cogent. Looking forward to future content.

[@LuckaPow]

Watching this video made me learn so much and gave me a reason for why i took linear algebra

[@nimnim5149]

Really fun to watch and very informative... I'm sure this is going to be in one of top picks by Grant Sanderson .

[@lukisprieston477]

I was in physics class earlier today wondering why we call a series of unit conversions via multiplication "dimensional analysis." Then I got to the part in the video where you flip the list of exponents into a vector and I literally shouted "Aha! I get it now!" Great video

[@ZakaiOlsen]

Had no idea what the title meant until the Abstract Spaces slide came up and it finally clicked. Great explanation! I wrote a section on the algebra for dimensional analysis in my dissertation. Another neat trick for a non-coherence system of quantities is that the algebra gives you the dimensionless numbers you might be interested in during an experiment for free. For example, consider the fluid flow past a sphere, where we’re interested in the drag force exerted on the object in the flow. Now assume for our experiment that we’re interested in the drag force, flow speed, sphere diameter, fluid density, and fluid viscosity. These can be written in base units of mass, length, time. The 3x5 matrix formed by our quantities is non-coherent, but the null vectors form the dimensionless force and Reynolds number. This means the physics of the flow past the sphere boil down to an equation with two variables : dimensionless force = f(Reynolds number), so we only have two parameters to worry about instead of 5. For larger multiphysics systems you can automate the derivation of dimensionless groups using the same algebra, but that reduction in model parameters hits a limit after you’ve reached the number of bass units used (I.e., for an n-parameter system with b base units you reduce down only as far as n-b parameters).

[@nikolajgylling4651]

This remains one of the best videos I have seen on KZbin!

[@generalaswalter5394]

SoMe is one of the best things than happened to education industry, there are so many new channels with videos marching the quality of channels with a hired crew, so interesting.

[@cassiopeiasfire6457]

The conclusion is one of the best descriptions of what math *is* that I've heard.

[@pra.]

Applying linear algebra to S/I units is so cool. This made me happy

[@gigs1890]

How is this the only video on your channel? Loved it, looking forward to the next one

[@andrewgr144]

This is so well done and entertaining, I'm sad you haven't made anything else of a similar nature.

[@Kinqsly]

I’ve been putting this video off for awhile, thanks again! Great work. 12:51

[@haraldhuber3734]

Dimensionsanalyse! Sehr schön gemacht, vor allem die Transformations-matrix aus der 8. Minute bereitet mir von nun an bestimmt noch häufiger freude :D Danke vielmals, und beste Grüße aus Wien. Peace, Love and Tschaka laka la!

[@rktiwa]

It's deeper than it seems. It gives wings to your imagination. Thanks