Could an $n$-dimensional creature scan an $(n-1)$-dimensional QR code without any problem?

Question

I was thinking about "How much data could be squeezed into the $2$D QR code", then it stroke me, that if it were a $3$D cube, probably inner voxels would be hidden from the outside observer/scanner. So there will be little usefulness of such codes.

The questions, that I find hard to answer are:

If I was an $n$-dimensional creature, would I be able to scan a $n-1$-dimensional "QR"-code without any problem, as we do with $2$D version?

or rather,

would I still be struggling to read even $3$ dimensional version?

Answer 1

I think you would be able to do so. For a 2D creature, seeing the inside of a square from outside is impossible, for us it's trivial since we can look at it from "above" (by moving to a plane parallel to the square). Similarly, a $n$-dimensional creature could move along the $n$-th axis to a $(n-1)$-dimensional plane parallel to the $(n-1)$-dimensional qr code. Maybe this non-formal answer could be better suited for a comment but I can't leave them yet.

Answer 2

I'll discuss a more general topic: that of $n$-dimensional cameras. What is a camera? It's an operator that projects a space onto a hyperplane.

Consider a camera obscura in $\Bbb R^3$. Since we are doing math, imagine it is ideal, i.e. the pinhole is a point. If we put the pinhole at the origin, then this camera takes lines passing through the origin and transforms them into dots on a plane, i.e. it sends the projective space $\Bbb{PR}^2$ onto $\Bbb R^2$, mapping the line $[x:y:z]$ to $(\frac xz,\frac yz)$ (if $z\neq 0$). In particular, it sends a point $(x,y,z)$ of $\Bbb R^3$, $z\neq0$, to $(\frac xz,\frac yz)$.

We can easily generalize this to an arbitrary dimension: a camera (or scanner) in and $n$-dimensional world is a map from $\Bbb{PR}^{n-1}$ to $\Bbb R^{n-1}$, which we can see as a map from $\Bbb R^n\setminus H$ to $\Bbb R^{n-1}$, where $H=\{x_n=0\}$.

So even if it's hard to visualize (impossible, even), it is mathematically possible to say that there is no issue scanning $(n-1)$-dimensional QR codes in an $n$-dimensional world.

Answer 3

if you read 3d QR Code with 3D Device (or 3D eyes), then yes, you'll still be struggling.

BUT if you are a higher dimension creature and use higher dimensions eyes and device, the perspective will be different. the what so called 'inner voxels' from a 3D Cube will also visible.

as if we are in 3d world able to see the whole 2d shape. but of course, that won't be the case if you are a 2D creature.

lets take an example. You are a 2D creature (e.g. small circle) beside a 2D object (e.g. big square). as a small circle (2D creature), your perspective toward the big square is limited to what's available in front of you. the back side of the big square is hidden.

BUT as a human (3D creature) who see the picture, sure you'll see differently. You will be able to see the WHOLE big square.

CMIIW