I"m wondering about this picture of a tesseract

[tryagain]

Picture:


I’ve been looking at this picture of a tesseract, and I wondered if there an error on it. Since everything from 2D is proportional to 3D just as 3 D is proportional to 4D,

If we open up a square, we get 4 lines:


If we open up a cube, we get four squares in a line and two squares sticking out:


So if we open up a 4-cube, shouldn’t we get one with 6 cubes in a line, and two cubes sticking out like this:?


or this?


I dont see why a tesseract opened up would look almostl ike a cube opened up, when a square opened up (a line) is completely different than a cube opened up

p.s. sry if the pics are small, just click on them if u cant see it


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[PWrong]

A tesseract has eight cubes in it. Your last two pictures have ten. :?
Nice pictures though

[bo198214]

Nice pictures though

yup

[wendy]

The top piccie is correct. The other two have extra cubes that ought not be there.

Orthogonal columns of 4 cubes in a line also work.

W

[Keiji]

erm, you are saying:

Square -> 4 lines
Cube -> 4 squares + 2 on the sides
Tesseract -> 6 cubes + 4 on the sides

how is that more logical than the tesseract having 4 cubes + 4 on the sides? It isn't.

Tesseract has 4 cubes + 4 on the sides like the picture shows.

[tryagain]

oh, i didn't see the extra two cubes on the front and back...

[PWrong]

Oh, I see now. So you think there should be six cubes in a line, plus two on the sides?

There are 11 distinct nets for a cube, and I'm not sure how many for the tesseract. However, I don't think yours is one of them.

[Keiji]

Two on the sides wouldn't work, as we have seen from various folding animations.

[Kevin_Ahab]

The way I see it, a n-dimensional hypercubes net is simply an n-1 dimensional hypercube with an n-1 dimensional hypercube touching each of its n-2 dimensional cells, and on one of those, another n-1 dimensional hypercube going further out. i.e.

Point: -n/a

Line: -unrepresnetnable in Pointspace

Square: -a line (n-1-d-hypercube) with a line from each of its points (n-2-d-hypercubes), and from one of those, another line

Cube: -a square with a square from each of its lines, and from one of those, another square

Tesseract: -a cube with a cube from each of its squares, and from one of those, another cube

This can be continued, but this forum is for the Fourth Dimension.

This is true becuase:

To find the number of n-spaces in an x-dimensional hypercube, simply: 1) double the number of n-spaces in the x-1-dimensional hypercube and 2) add this to the number of n-1-spaces in the x-1-dimensional hypercube.

[pat]

It's important to remember that there's more than one way to unfold a shape. The key part is how the pieces line up together. I personally, find the first picture you showed to be the easiest to see how they fit together.

For example, each of these is an unfolded cube:


But, the 4-cube can't be six 3-cubes in a row with two 3-cubes tacked on the sides. That would imply there is a run of six 3-cubes that you can go through where when you enter the 3-cube through face AA, you exit the cube through the face opposite AA. There are no such runs in the 4-cube.

Start in any 3-cube that makes up the surface of the 4-cube. Exit through a (2-cube) face and into another 3-cube. Exit that 3-cube through the (2-cube) face opposite the one you entered on. Lather. Rinse. Repeat. No matter which 3-cube you started in or which face of the 3-cube you exited through to start off, you will end up back in the original 3-cube on your fourth step.

[bo198214]

hey why not add all also the tesseract unfoldments?

[pat]

Because, I don't have that much time at the moment.

I should also note that the above may not be an exhaustive list of cube unfoldings. They were just all of the distinct 2-d ones that I came up with that do not involve adjacent squares with a cut between them.

[bo198214]

*nod*

[pat]

Here are some unfolded hypercubes. This is nowhere near exhaustive.

[pat]

Orthogonal columns of 4 cubes in a line also work.

Really? I can't get there.

In this picture, the lower green cube is between the blue cubes and between the magenta cubes. So, I can't find a way to make two orthogonal columns of 4 cubes since both columns have to share two cubes.



Ah, unless you meant this:


oops, I got the overlapping wrong on the top cube of the vertical column in this diagram

Which, of course, has to be what you meant... otherwise there would only have been seven cubes if the two columns of four shared a cube.

I got it now.

[bo198214]

Thanks pat, I was anyway thinking about programming a folding mode into 4d building blocks. Though I think that really interesting objects like the 3-torus looks rather awkward when surface is built by cubes ...

[pat]

There was an article in Volume 17, Issue 1 of the Journal of Recrational Mathematics (1984/85) titled "Unfolding the tesseract". The article is by Peter Turney. Here's the abstract:

The author shows that there are 261 distinct unfoldings of a tesseract (hollow hypercube) into order-8 polycubes in 3-space. This answers a question of M. Gardner [Scientific American 215 (1966), no. 5, 138-143].