Half Dimension

[@docu73]

This is the best textbook math channel on KZbin. There're many math channels, some are just popular math like Numberphile, some talks about amazing stuffs at a more advance level like 3B1B and PBS Infinite Series, but there aren't 'interesting' (as far as I know) math channel that teach us textbook math. Until I find Dr Peyam. Thank you Dr. Peyam! Your character is what really make math exciting! It's been since primary school that I have this feeling of excitement when solving math problems, and now I feel like a kid again!

[@drpeyam]

Wow, thank you so much! This seriously made my day 🙂

[@penfriendz]

Can we have negative or complex dimensions?

[@PriXXifiaction]

Yes! For the ones interested check out these 3 papers/books: 1) DOI:10.1063/1.3625954 (negative Dim) 2) doi.org/10.1007/978-1-4614-2176-4 (complex Dim) 3) doi.org/10.1007/978-3-319-44706-3 (complex Dim)

[@la6mp]

Wow! I was surprised that I could understand and follow you to the end and still have a good time. Yesterday I would have thought of non-integer dimension as pure nonsense. You make strange things feel less strange :-) Please keep up the good work!

[@michaelzumpano7318]

I really love your videos! Great topics and great teaching.

[@frozenmoon998]

And people say 3D is complicated. I say this one gets the cake!

[@NuptialFailures]

At long last! Dr. Peyam, thank you ever so much. I’ve wanted this video for way too long now.

[@blackpenredpen]

That thumbnail!!!!!!!!

[@drpeyam]

Whoever made it is freaking awesome!!!

[@blackpenredpen]

Dr Peyam hahahahaa

[@Aviationlover-belugaxl]

Great video! In -1 dim space, the measure of something scaled down by 1/2 is 2x the thing’s original mass. When something is infinitely small,that thing has infinite measure. This makes sense b.c an infinitely small thing is a point, which is -1+1 dimensional, which would make the -1 dim measure infinite.

[@anegativecoconut4940]

If you scale me by 2 I will become 1/8 of my size.

[@mehdisi9194]

A very interesting subject in mathematics. Thank you so much

[@williamadams137]

WOAH!!! That reminds me with fractional derivative man 😁👍nice vid

[@TheMazyProduction]

13:29 “I think on the thumbnail I’ll put a better picture of this”. I like yours better.

[@anegativecoconut4940]

I live in in - 3 dimensional space.

[@shiina_mahiru_9067]

🤔 What about the dimension of Q? Would that also be true that if a given set A has dimension d, then A^n has dimension d^n?

[@benjamingrant5330]

I have a question. Is there any way to define a linear transformation from a whole-number-dimensional space onto a space with a fractional dimension? That would be super cool!

[@drpeyam]

Wow, I agree, that would be super cool! Not sure how to do it, though, but I think you can show that if T from R^n to R^n is an isomorphism and A is d dimensional, then T(A) is d dimensional

[@benjamingrant5330]

Dr Peyam awesome!! I guess a better question would be whether or not a fractional dimensional space would even be possible. Do you think it would be, and if so, can you think of an example?

[@giandomenicopanettieri5748]

What about complex fractional dimension?

[@yrcmurthy8323]

Alright thanks for uploading. It helps me a lot

[@plaustrarius]

Awesome!!! Been waiting for this video! Haha I first explored this sort of idea with the seirpinski triangle, koch triangle, and menger sponge.

[@pedrocusinato02]

reminds me of frac derivatives... more videos about it pls

[@drewpat9535]

Hi doctor, what do you mean by: X^d ? I'm Italian and wasn't able to grasp the meaning nor the spelling. Thanks a lot, your videos open my mind!!!!!

[@shelipemaktadir483]

love you, love your videos, love maths ~~ lots of love from London!

[@richardfredlund3802]

that was amazing! ... Dr Peyam I just finished watching your formula for sphere in N dimension before seeing this, which made me wonder does the formula hold for fractional dimensions?

[@drpeyam]

Sure!

[@richardfredlund3802]

wow that's so cool.

[@omkarpatilmaths]

Nice lecture Sir From Book you are referring to this hausdorff measure

[@drpeyam]

Evans and Gariepy

[@Л.С.Мото]

Hey Dr. Peyam... is it actually possible for you to make a video about the math you did in your phd work?

[@drpeyam]

I already did! Check out my 100th video special

[@kaandogan2470]

Are fractional dimensions used in Chaos Theorem? Btw great video , I have been wondering it for a while, thank you Dr. Peyam :)

[@drpeyam]

Pretty sure that yes

[@ibrahinmenriquez3108]

Dusty set... Love that terminology

[@mario1415]

Great! Do it for Brownian motion!

[@douglas_leimiceg]

22:09 don't say that negative values of x does not make sense XD half dimension did not make sense to me until today. btw, great video.

[@SKARTHIKSELVAN]

Thanks for your video.

[@ichigo_nyanko]

How come whenever I think I've thought of something new and interesting, and I search it on google soomebody already did it :( I at least figured out that in a number of dimenions 0 < d < 1 if you have a 2x2 cube and a radius 1 sphere(i.e. the sphere is touching all the edges of the cube, completely encaced by it), the sphere would have a bigger area than the cube, despite being inside of it. I thought that was neat and I couldn't find anything on google about it.

[@drpeyam]

Saturated market of ideas, unfortunately :( Happens to me too!

[@alwysrite]

interesting concept

[@dhunt6618]

Thanks! Go fractals!

[@dharmanshah1239]

Hey dr Peyam!! I want to add English captions for your videos.plz can we talk on email.

[@Uni-Coder]

If you think that it is hard to imagine 4D then first try to imagine 3.1D and after that 3.2D and so on :)

[@gustavocortico1681]

I don't understand. No matter how much you shrink the balls, it's impossible to tile R^2 without overlap. Whatever tiling you do will have an error proportional to the area you're covering, wouldn't it?

[@drpeyam]

We never said that there can’t be an overlap

[@gustavocortico1681]

Oh I think I see what you mean. It's more like a tiling where the net area equals the area of the object?

[@gourabghosh5574]

Hey bro your face does not seen to be symmetric in the picture of the video

[@Handelsbilanzdefizit]

Ok, arbitrary dimensions. When you multiply a [M x Pi] Matrix with a [Pi x N] Matrix, you'll get a [M x N] Matrix. But how do you write down a [M x Pi] Matrix? It's not possible (so far) ---> Linear Algebra fails! Harrr harrr harrr!!!!!!

[@drpeyam]

Maybe with tensors? :)

[@Handelsbilanzdefizit]

@@drpeyam The number of components/entries (a_ij) in a MxPi matrix, can not be an integer! So, you need an additional value that describes the "presence probability" of a_ij, like the wavefunction for electrons. a_ij is half there and half not there. Our world is binary and absolute. Something exists, or doesn't exist. Mathematical results are also absolute. There's nothing "fuzzy" between. Or you use fractal matricies with some weird shape, that is not rectangle any more. No idea how to sum or mulitplicate this. In my view, linear algebra is not developed enough.

[@sahilbaori9052]

Fractals are wierd...