FINALLY! A Good Visualization of Higher Dimensions

[phisgr]

I was hoping for a 3D cut of the 4D case before lowering to the 2D cut.

[theusaspiras]

This is actually the premise behind Principal Component Analysis (PCA), a popular dimensionality reduction technique which finds the largest variation amongst all dimensions and reconfigures the data as those axes. This can be understood as the long 'diagonals' of your data.

[carykh]

Wow, this is a great way to visualize high-dimensional space!

[TrimutiusToo]

It doesn't visualize higher dimensions, just finds a good 2D cut for a specific problem. Still extremely interesting

[meinbherpieg]

I never had anyone explain visualizations of higher dimensional objects with respect to their diagonals. Great job.

[FireyDeath]

Wish I could see the 3D demicube rotated and the tesseract realmically sliced for some more perspectives. And maybe some other cross-sections of the dekaract if we're feeling crazy enough

[imacds]

Not going to lie, the crazy artistic interpretation wasn't half bad either.

[VanVlearMusic]

Bae wake up a new visualization of higher dimensions just dropped two years ago!!

[stevethecatcouch]

Very nice. Your technique is more intuitive and satisfying than the spiky spheres in Matt Parker's Things to Make and Do in the Fourth Dimension.

[DamaKubu]

Really great visual!

The volume ratio between the red ball and blue balls peaks at Dimension 4 and then drops.

Diameter of red ball is sqrt(D) - 1

Thus red balls volume is proportional to (sqrt(D) -1)^3

The blue balls all have the same volume, but their number grows exponentially 2^D with dimension.

Kinda fun even if red ball grows without bound, its volume compared to blue balls quickly goes to zero.

In higher dimension space most volume is close to boundary.

[the]

I don't know if this necessarily helps me visualize the high-dimensional itself. But it does kind of solidify the understanding of the weird volume aspects that happens with higher dimensional geometry. Good visual 💯

[matematicke_morce]

Great video! And the title's right, this is the first video I've seen on this topic that actually helped me intuitively understand what's going on.

[Girasoleever]

I don't understand how increasing dimensions will only stretch the cube in one dimension.
Can we still call those "cubes"?

[RadicalCaveman]

This isn't a way of visualizing multiple dimensions, just a way of visualizing a 2-D slice of multiple dimensions. Still interesting.

[Henry.]

Most intuitive way to understand this puzzle, and it's better then a 3blue1brown video so good job

[Ykulvaarlck]

just so you know, the black frame until the first visualizatio0n made me think my playback was broken

[XMgamePlays]

This is a very good explanation.

Another way to "feel" the extra space. Could be looking at the space in the corners of the blue balls. In the 2d slices it not only grows bigger. But also that these grow in number with the power of 2 with each extra dimension.
4 in 2d.
8 in 3d.
16 in 4d.
1024 in 10d.

Correct me if I am wrong.

[davidwright]

The explanation of something so 'intuitively' impossible is direct, easy to follow and - 'obvious' - once it's been pointed out! Thanks.

[eryqeryq]

This is an amazing way to illustrate it... I never understand this until now!

[nahbruh]

with the way the title is worded, the thought would have never occurred to me. thanks for clarifying