How can I visualize 4D?

[Nick]

I have tried every website I can get my hands on but no matter what, I can't seem to visualize four dimensions. Does anyone have any links or explanations that can help me?

[moonlord]

Well, reading quickfur's pages several times helped me. Try to impose yourself that those things further away in the 4th dimension are 'elsewhere', so these strange things are normal afterall. It took me two months to properly imagine a tesseract... Just don't give up.

[thigle]

moonlord wrote:

Try to impose yourself that those things further away in the 4th dimension are 'elsewhere', so these strange things are normal afterall.

and he got the point.

another (simpler) exercise for me was 4-simplex (aka pentatope or 5-cell or whathaveyou), from sci.math forums, by Donald Davis, march1999:

*
_imagine a normal tetrahedron
_add 4 edges that run to the center from the vertices.
_imagine further that these 4 extra edges are somehow "elsewhere" the 3-D space occupied by the tetrahedron. maybe it'll help to imagine the 4 edges with a misty or hazy appearance at first.
_now, focus your attention on the 6 triangles between the 4 new edges. those triangles are "elsewhere", "beyond, behind..." the base tetrahedron, too.
_finally, focus on the 4 new tetrahedra that fill the gaps between the 6 new triangles. these too are "behind" the base, "elsewhere".
_now for the hardest part: between the base tetrahedron and the 4 "behind" tetrahedra, there's a single 4-dim'l void, which is the interior of the 5-cell.
_when you're beginning, you have to forcibly remind yourself, as you visualize this, that all the edges are the same length, all the triangles are normal equilateral triangles, and all of the 5 tetrahedral cells have the same size and shape.
_with practice, you may find that it's easier to "see" all these symmetries at the same time - that is, eventually, you stop seeing the object as a 3-d projection, and start seeing it in something like its 4-d shape.
*

btw, moonlord or anyone, what is meant by "properly imagine a tesseract" ? does it allow one to rotate it freely in 4space and project it freely form any position into 3space ?

[moonlord]

Well, not quite freely rotate it. I'd better say I am able to see the pyramid truncks that appear when scrolling a cube through the tessaract as real cubes. I keep struggling, though.

[thigle]

idea = unity of presences

to get an idea of a tesseract, one needs to be able to hold in one's awareness all the 8 3-cubes forming the hypersurface of tesseract. between these is a 4d chunk bounded by the tesseract.

what i don't understand, is the way the projection works from 4d to 3d ...

lets consider these analogies between seeing 3-cube(or looking at its 2-shadows in 2-plane, orthogonal to viewing axis) and 4-cube(or looking at its 3-shadows in 3-space):

in orthographic projection (from infinitely far):

3-cube, seen vertex first, which amounts to looking from viewpoint on 3-cube's diagonal, looks like hexagon with diagonals.
3-cube, seen face(2-cube = square) first, which amounts to looking at it from viewpoint at axis that passes through centre of the face and is orthogonal to it, looks like square.

between these 2 views of the 3-cube, one 'maximal' and one 'minimal', are all the other viewpoints with cube's image taking area between the hexagon and the square case.
it is like seeing 3 axies of 3-space: one extreme it looks like 3 axes crossing, with 2,3,6-fold rotational symetry, other extreme is just a cross, one axis becomes a point of intersection.

for 4d case, projecting the 4-cube into 3-space, one extreme is the rhombic cuboctahedron with 4 diagonals from vertices where 3 edges meet.
another extreme is a cube with its 4 diagonals.
(one is cell first, other is vertex first, which is which ?)

now between these 2 extremes are the other projections, like the familiar one where one 3-cube is within another 3-cube and their vertices are connected, which gives other 6 3-cubes.

out of the blue, i gotta go now (my girlfriend needs me ) but'll be back in the evening and completize my thoughts on this as well as pose some questions that might clarify what i don't get.

[Rkyeun]

When I first started imagining tetraspace, I used such diverse things as color, numerical value, and time.

Imagine a circle in a 2D graph. We will color it as follows: Starting from the center of the circle place a red dot. Draw concentric rings of color outwards towards the circumference. As you do this, change the color you use to draw so that the color approaches green. Do this more quickly as you progress, so there is quite a lot of green and not much red.
This is the bottom half of a sphere in a 3D graph, where color corresponds to the Z-axis. To draw the upper part of the sphere we trace the circles back in towards the center on "top" of that, staying green for quite some time but then rapidly turning blue near the end. All points except the outer circumference have two different colors. Two different Z-values. Were drawn twice at different times.

Imagine a sphere in a 3D graph. We will color it as follows: Starting from the center of the circle place a red dot. Draw concentric shells of color outwards towards the surface. As you do this, change the color you use to draw so that the color approaches green. Do this more quickly as you progress, so there is quite a lot of green and not much red.
This is the bottom half of a shape in a 4D graph, where color corresponds to the W-axis. To draw the "upper" part of the shape we trace the shells back in towards the center on "top" of that, staying green for quite some time but then rapidly turning blue near the end. All points except the outer surface have two different colors. Two different W-values. Were drawn twice at different times.

That shape is a kind of hypersphere. The glome perhaps? I'm not sure I know the right word for the 4D shape in which every point on its surface volume is the same fixed distance from its center.

I don't have to use color or transparency or ghostliness anymore. I can just directly in my head visualize four mutually perpendicular axis and the associated gridlines for graph paper. Five is tricky. For one brief instant I managed to visualize an infinite-dimensional space, but that gave me a headache for a week and I'm not eager to try again.

[Keiji]

Yes, it's called the glome.