Yes - it doesn't give a very complete idea. Especially - you see the four dimensional object as a series of "slices" and have no appreciation of what it is like to rotate it or move it. Or experience it directly. Images like this can give a better idea, but more about that later.
In Flatland - Abbot has an example of his 2D creatures watching various 3D shapes as they pass through their realm.
So for instance if you watched a 3D sphere's intersection with a 2D plane - i.e. a 2D movie of a sphere - you'd see it start as a point, turn into a tiny circle, gradually get larger until it's a large circle, and then dwindle to a point again.
Here, actually they would see it as a line of course - though with binocular vision they could see that it is curved - and they could walk around it. Or go inside it and see it around them in all directions.
So - just as we can only see one side of a sphere but can infer that it is perfectly round on all sides, they could see one side of a circle and infer that it is round in 2D.
They might also be able to see the far side if it is semitransparent so you can look through it, and shaded.
All of this of course supposing that they have eyes and can see etc - quite a stretch to imagine physics that would permit that in 2D. But as a conceptual tool to use just as an analogy I think it helps.
So in a way they would be able to see a sphere in that way.
But that's not a good representation of a sphere.
A 2D creature of that sort, who enjoys movies about 3D geometrical shapes, might describe a familiar geometrical shape like this:
(Everybody knows this shape - but you may not recognize it from its 2D creature description).
Do you know what it is?
I'll scroll a few lines so you can puzzle it over if you don't know the answer
.
.
.
.
.
.
.
.
It's a cube.
from: how to cut a cube to reveal a hexagonal cross-section
However - we can get a reasonable idea about 4D shapes.
There are lots of great animations in the Wikipedia pages about them like this:
That's a 600-cell - 4D version of an icosahderon
A 120-cell - 4D version of a dodecahedron
A 24-cell It's self dual and has no analogue in 3D
A 16-cell - 4D version of an octahedron - exists in all dimensions
A Hypercube or 8 cell - 4D version of a cube
- exists in all dimensions
5-cell - 4D version of a Tetrahedron - exists in all dimensions
All of those figures, though they seem to change shape to us - are just doing a simple rotation in 4D. And - you have to think also - that what we see encloses a 4D interior which you most probably can't imagine - the 3D "cells" we see are, in 4D, of no thickness at all, as thin as the polygons that surface our 3D shapes.
And - this may make it even more baffling but is true: Though it seems to us that sometimes some of the faces are on the outside and some are on the inside - that's just because we can't see in 4D. In reality - all the vertices, edges, faces, and the entirety of all the 3D shapes we see as well - they are all on the very outside of the shape.
And none of them inside any of the others. It is just a perspective effect when sometimes some seem to intersect others or go inside others. Even the very centre of each tetrahedron in the 5 cell for instance, seen in 4D is on the outside of the 4D shape.
If you find that hard to imagine - well it's like the triangular faces of a tetrahedron. Every vertex, edge and the face, all the way to its centre, lies on the outside of the tetrahedron. The 3D tetrahedron is the space enclosed by everything we see.
Similarly in 4D everything we see here
is on the outside of the 5-cell - and the 5-cell is the space enclosed by everything we can see.
4D is the space richest in regular polytopes of all dimensions > 2., with six of them.
3D has one less, and all the higher dimensions have two less polyhedra (2D of course has infinitely many regular polygons).
You can learn a lot from studying these representations, and reasoning about them logically.
Also - in principle - you could learn to see in 4D directly. After all we see in 3D using binocular vision.
So, this is something I've wondered - if you put someone in a 4D virtual environment and show them 2D projections of it via binocular projection - and they may get some appreciation of 4D, especially if they have to interact, pick things up, solve puzzles etc.
You can also use tricks such as colouring to indicate the 4th dimension.
Some mathematicians have reported ability to be able to "think in 4D"
And one mathematician has gone further. For 30 years he has "lived in 4D" and has made an online course you can take to learn to think in 4D yourselves.
Dimensions
See also
Seeing in four dimensions
I've only just discovered this while writing this article :).
In Flatland - Abbot has an example of his 2D creatures watching various 3D shapes as they pass through their realm.
So for instance if you watched a 3D sphere's intersection with a 2D plane - i.e. a 2D movie of a sphere - you'd see it start as a point, turn into a tiny circle, gradually get larger until it's a large circle, and then dwindle to a point again.
Here, actually they would see it as a line of course - though with binocular vision they could see that it is curved - and they could walk around it. Or go inside it and see it around them in all directions.
So - just as we can only see one side of a sphere but can infer that it is perfectly round on all sides, they could see one side of a circle and infer that it is round in 2D.
They might also be able to see the far side if it is semitransparent so you can look through it, and shaded.
All of this of course supposing that they have eyes and can see etc - quite a stretch to imagine physics that would permit that in 2D. But as a conceptual tool to use just as an analogy I think it helps.
So in a way they would be able to see a sphere in that way.
But that's not a good representation of a sphere.
A 2D creature of that sort, who enjoys movies about 3D geometrical shapes, might describe a familiar geometrical shape like this:
(Everybody knows this shape - but you may not recognize it from its 2D creature description).
- starts as a point,
- turns into a triangle
- gets larger
- when quite large, the triangle gets truncated at each of its points, and starts to turn into an irregular hexagon
- At its largest it is a perfect hexagon
- Then shrinks and, turns irregular again as it diminishes
- Eventually it becomes a triangle and diminishes
- Finally turns to a point and vanishes.
Do you know what it is?
I'll scroll a few lines so you can puzzle it over if you don't know the answer
.
.
.
.
.
.
.
.
It's a cube.
However - we can get a reasonable idea about 4D shapes.
EXAMPLES OF 4D SHAPES
There are lots of great animations in the Wikipedia pages about them like this:
That's a 600-cell - 4D version of an icosahderon
A 120-cell - 4D version of a dodecahedron
A 24-cell It's self dual and has no analogue in 3D
A 16-cell - 4D version of an octahedron - exists in all dimensions
A Hypercube or 8 cell - 4D version of a cube
- exists in all dimensions
5-cell - 4D version of a Tetrahedron - exists in all dimensions
THESE ARE ALL SIMPLE ROTATIONS
All of those figures, though they seem to change shape to us - are just doing a simple rotation in 4D. And - you have to think also - that what we see encloses a 4D interior which you most probably can't imagine - the 3D "cells" we see are, in 4D, of no thickness at all, as thin as the polygons that surface our 3D shapes.
EVERYTHING YOU SEE IN THOSE DRAWINGS ARE ON THE OUTSIDE OF THE SHAPE IN 4D
And - this may make it even more baffling but is true: Though it seems to us that sometimes some of the faces are on the outside and some are on the inside - that's just because we can't see in 4D. In reality - all the vertices, edges, faces, and the entirety of all the 3D shapes we see as well - they are all on the very outside of the shape.
And none of them inside any of the others. It is just a perspective effect when sometimes some seem to intersect others or go inside others. Even the very centre of each tetrahedron in the 5 cell for instance, seen in 4D is on the outside of the 4D shape.
If you find that hard to imagine - well it's like the triangular faces of a tetrahedron. Every vertex, edge and the face, all the way to its centre, lies on the outside of the tetrahedron. The 3D tetrahedron is the space enclosed by everything we see.
Similarly in 4D everything we see here
4D IS THE DIMENSION RICHEST IN REGULAR POLYTOPES
4D is the space richest in regular polytopes of all dimensions > 2., with six of them.
3D has one less, and all the higher dimensions have two less polyhedra (2D of course has infinitely many regular polygons).
You can learn a lot from studying these representations, and reasoning about them logically.
Also - in principle - you could learn to see in 4D directly. After all we see in 3D using binocular vision.
So, this is something I've wondered - if you put someone in a 4D virtual environment and show them 2D projections of it via binocular projection - and they may get some appreciation of 4D, especially if they have to interact, pick things up, solve puzzles etc.
You can also use tricks such as colouring to indicate the 4th dimension.
Some mathematicians have reported ability to be able to "think in 4D"
And one mathematician has gone further. For 30 years he has "lived in 4D" and has made an online course you can take to learn to think in 4D yourselves.
Dimensions
See also
Seeing in four dimensions
I've only just discovered this while writing this article :).
About the Author
Robert Walker
Writer of articles on Mars and Space issues - Software Developer of Tune Smithy, Bounce Metronome etc.Studied at Wolfson College, Oxford
Lives in Isle of Mull
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