the impossible cube and the hypercube are different examples, right?

]]>I have had a loooong think and come up with this:

Firstly, I'll examine an impossible cube. Then try to repeat this with a 4-D hypercube.

Why a cube? Just that it's easier to write the coordinates of the corners.

Let's start with a 0-D point. This has 1 vertex only.

In 1-D I'll consider a line. This has 2 vertices and 1 line. Each vertex has 1 line leading from it.

In 2-D I'll consider a square. This has 4 vertices, and 4 lines. Each vertex has 2 lines leading from it. These lines are at right angles.

In 3-D I'll consider a cube. This has 8 vertices, and 12 lines. Each vertex has 3 lines leading from it. Pairs of these lines are at right angles. There are 6 square faces.

In 4-D I'll consider a hypercube. By continuing the pattern this will have 16 vertices and 32 lines. Each vertex has 4 lines leading from it. Pairs of these lines are at right angles. There are 8 cubes.

Here's an impossible cube:

https://i.imgur.com/3kkOsTY.gif

The line HD is in front of the line EF. This is impossible. By choosing coordinates for the position of the viewer such as (1.5, -1.5, 1.5) you can compute the distance to EF and to HD and hence show that HD is further away from the viewer.

To show a hypercube on a 2-D surface is not easy. The usual way to to draw one cube inside another and then join corresponding vertices. Here's my attempt:

https://i.imgur.com/b5ydZpR.gif

I have tried to show the cube ADHEILPM in front of everything else. Best I could manage so far.

Bob

What program do you use to draw these geometric shapes?

]]>Certain 2d objects, such as the penrose triangle, are possible in 2d but not in 3d. My question is, are certain 3d objects impossible in 4d, and what are they?

There are no nontrivial knots in 4 dimensions. so any way you twist a piece of string and glue it in 4 dimensions, it can be unknoted.

]]>Firstly, I'll examine an impossible cube. Then try to repeat this with a 4-D hypercube.

Why a cube? Just that it's easier to write the coordinates of the corners.

Let's start with a 0-D point. This has 1 vertex only.

In 1-D I'll consider a line. This has 2 vertices and 1 line. Each vertex has 1 line leading from it.

In 2-D I'll consider a square. This has 4 vertices, and 4 lines. Each vertex has 2 lines leading from it. These lines are at right angles.

In 3-D I'll consider a cube. This has 8 vertices, and 12 lines. Each vertex has 3 lines leading from it. Pairs of these lines are at right angles. There are 6 square faces.

In 4-D I'll consider a hypercube. By continuing the pattern this will have 16 vertices and 32 lines. Each vertex has 4 lines leading from it. Pairs of these lines are at right angles. There are 8 cubes.

Here's an impossible cube:

The line HD is in front of the line EF. This is impossible. By choosing coordinates for the position of the viewer such as (1.5, -1.5, 1.5) you can compute the distance to EF and to HD and hence show that HD is further away from the viewer.

To show a hypercube on a 2-D surface is not easy. The usual way to to draw one cube inside another and then join corresponding vertices. Here's my attempt:

I have tried to show the cube ADHEILPM in front of everything else. Best I could manage so far.

Bob

]]>The same happens if you look at a painting. There may be a small amount of relief on the surface but not enough to tell us it's a picture of the 3-D world. Again, our brains do that.

Illusions like the Penrose Triangle (for more look here https://www.illusionsindex.org/illusions rely on the way our brains work. We're so used to seeing solid objects that we'd rather believe an impossible object exists than see it as just a clever bit of 2-D drawing.

Because our retina only gets a 'flat image', all 3-D world pictures are a construct of our brain. So how can we perceive a 3-D object in all its 3 dimensions in the way you want.

Nice question though and I shall continue to ponder it. If I come up with anything, I'll let you know.

Bob

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