Seven Dimensions

Entry for the 2022 Summer of Maths Exhibition #SoME2.

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

[SuperSulc]

The algorithm thinks i'm much smarter than i really am.

[epsilonengineer]

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

[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

[tonaxysam]

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

[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.

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).

[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

[PopeGoliath]

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

[Pyotyrpyotyrpyotyr]

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

[trevormacintosh]

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

[MaxG]

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.

[kallekula]

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.

[stede]

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.

[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.

[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!

[JacenLP]

"Doing LA to it" sounds so violent yet graceful. I like it :D

[akeronan]

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.

[yash]

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)

[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.