Impossible 3D shapes=Possible 4D figures?

[Muzozavr]

Which using 4D only to turn in different view of 3D in which it CAN'T normally?

[houserichichi]

The Klein bottle is a popular example of a four dimensional shape that can't be seen "properly" in only 3D.

[Muzozavr]

No, I don't mean the true 4D figures, like Klein Bottle.
You know Penrouse's triangle, yeah? And "Devil's Fork"? I thought that maybe the figures like that are actually 4D...

[bo198214]

As far as I can see Neckar Cube, Penrose Triangle and Devil's Fork are optical illusions imposed by loosing one dimension in the projection 3d to 2d and have nothing to do with 4d. But if any one can draw a connection ...

For example what happens if we create similiar illusions for tetronians, i.e. with the 4d to 3d projection? Probably we poor trionians wouldnt even notice anything strange about it .... :?

[PWrong]

You could try making an impossible hypercube, i.e. a projection so that a tetronian can't tell which cube is supposed to be the "top".

Think about how Fred would make a projection of a cube. He wants one line to look like it's passing "underneath" another line, so he makes a gap in it. He actually needs 14 pieces in total, not twelve. Also, he'd have to put together the inside first, because once it's finished, all he'll see is a hexagon.

Here's the steps he would take to make a normal (possible) cube. To make an impossible cube, you simply put one gap in the wrong place.
It has two squares and four "legs". Note that the bottom left square is supposed to be in front of the other.

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1. Make the bottom and left sides of the front square. (2 pieces)

|
|
|
|______

2. The inside of the front square, which is a small part of the back square, and one "leg". (3 pieces)

|  |
|  |__
| /   
|/_____


3. Put the rest of the higher square on, hiding the inside. (2 pieces)
______
|  |  |
|  |__|
| /   |
|/____|


4. Add the three extra legs, and complete the bottom and left sides of the back square. (5 pieces)
   
  /|    /
/__|___/
|  |  | 
|  |__|__
| /   |  /
|/____|/

5. Put the top and right sides of the back square on. (2 pieces)
   _____
  /|    /|
/__|___/ |
|  |  |  |
|  |__|__|
| /   |  /
|/____|/


This actually looks more like an impossible cube, because of the gap in the underscores. But the construction shows that there has to be a gap in two sides of the back square, so it would look normal.

It would be quite difficult to make an impossible hypercube. It would have to be the 2D form of the hypercube. The wireframe model doesn't intersect anywhere, and the solid one intersects nearly everywhere. You'd probably make it out of paper, although something see-through would better. I might try making one next year before uni starts (unless I keep my new year's resolution to get a life).

For a normal (possible) hypercube, you need 33 seperate pieces to make the 24 faces. 9 large squares, 3 small squares, 3 L shapes (i.e. a large square with the small square cut out), 6 parallelograms, 6 more parallelograms with a triangle cut off, and the 6 cut off triangles.

The parallelograms are for the projected faces of the hypercube. The cut parallelograms look like this:

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/__|


To make the hypercube, we could use the same five steps as above, although each step is a bit more complicated. It would be easier to draw up some nets and fold them up, then glue them together to make the whole thing.

[bo198214]

I dont know if it have to be so difficult.
As you said whats before and whats behind in the 3d->2d projection is marked (for us trionians) by making a gap in the line that is behind.

For duonians its difficult to see through lines so they usually would use points to draw a picture, when want to see the backside.
For trionians its difficult to see through planes thatswhy we use lines to draw a cube of which we want to see its backside.
For tetroninas its difficult to see through 3d-shapes so they use 2d-shapes if they want to see the backside of a hypercube.

So a hyper-Necker-cube would be drawn with 2d-shapes by a tetronian.
And more essentially if we draw a hypercube with lines there is no permeation of the lines, but of 2d-shapes (i.e. parallelograms) there is.
Its really difficult to still draw this, but if you draw hypercube (in cavalier-perspective as the necker-cube is) including the hidden (3d-)faces and now contemplate the 2d-facets, you realize that some of them permeate another (without hidden faces there is no permeation of any 2d-facets).
A tetronian would draw the 2d-facet behind as being broken/slitted (for two permeating 2d-facets).

There are 2 inner points in the 3d projection of a hypercube.
One point is connected to all hidden (3d-)faces - call this point h - the other to all visible faces - call this point v.
4 edges are connected each to each (inner) point.
6 2d-facets are connected to each (inner) point.
Two 2d-facets can only permeate each other if each is connected to an (different) inner point.
As far as I can see those two 2d-facets permeate each other exactly if they have no unit direction in common.
If we label the 2d-facets connected to either h or v by the spanning unit axes this would give:
h12 permeats v34
h13 permeats v24
h14 permeats v23
h23 permeats v14
h24 permeats v13
h34 permeats v12
All 2d-facets connected to h, would be drawn slitted, because they are 4d-behind the ones connected to v.
If we would draw all v-2d-facets slitted and the h-2d-facets unbroken, this would give another valid projection of a hypercube. (So if the 2d-facets would be drawn unslitted the tetronian could/would mentally switch between these two projections ....)
And a hyper-Necker-cube would be any different slitting distribution ...

[pat]

No, I don't mean the true 4D figures, like Klein Bottle.
You know Penrouse's triangle, yeah? And "Devil's Fork"? I thought that maybe the figures like that are actually 4D...

The mathematician John H. Conway has on his desk a gift from Roger Penrose. The gift is a cube that, when viewed by a 4-dimensional creature (with 3-d retinas) from the right spot is an impossible 4-d shape. The article that I read this in gave no further details.

[4D guy]

Which using 4D only to turn in different view of 3D in which it CAN'T normally?

Yes, it can. If a 2D universe was "wrapped" around a sphere the universe would seem infinite. So if a 3D universe was "wrapped" around a glome (hypersphere(4D sphere)) it would seem infinite too.

[mjjirachi]

what is a devils fork?

[PWrong]

what is a devils fork?

http://mathworld.wolfram.com/ImpossibleFork.html

[bo198214]

though I already gave this link