there are a lot of new 3d objects in 4d

[arsenic]

In 3d there are much more 2d objects that flatlander don't understand
such as sphere's surface
most of flatlander think that anything that have area must have
line to cover them
but in fact sphere's surface is a 2d object that do not has line to cover them

I have found that some of 3d objects in 4d do not has surface but it has
volume

golme's volumic surface do not has any area to cover it
but it still has volume

[RQ]

planes are made of lines, if that's what you mean. Surely volume then is made of planes.

[Keiji]

Glomes would not have any planes. They would only have surface volume.

[RQ]

and what would those space be made out of ? planes, and those planes-lines, and those lines-points. That's all proof that you can plot them on the cartesian graph, and you can.

[Keiji]

the surface of a sphere has no lines, so the surface of a glome would have no planes.

[arsenic]

The way to erase surface from volume

First imagine a sphere then you imagine that you push a volume of
sphere in to marp dicrectional you will see that the volume live in
the forth dimension and the only surface to cover that volume is
an empty sphere .Next you imagine that there is another sphere that
it's volume have been pushed in to garp directional the only suface to cover it's volume is an empty sphere then you use two of those emphy sphere( which are the surface to cover those volume ) to joined together
after that those two of the empty sphere(which are the suface to cover those volume) will be one part of the volume and you will see the
volume that has no surface to cover them which is named empty glome

[pat]

First imagine a sphere then you imagine that you push a volume of sphere in to marp dicrectional you will see that the volume live in the forth dimension and the only surface to cover that volume is
an empty sphere.

I'm not following what this means. You are saying that I start with an ordinary ball (solid sphere) in three dimensions, correct? Then, I push it in a direction perpendicular to three-space, correct? What do you mean then by "surface to cover that volume"?

Let's talk coordinates. Let's say my original ball is the unit ball at the origin. That means that x² + y² + z² ≤ 1 and w = 0. Are you saying the "surface to cover that volume" is the surface x² + y² + z² = 1 with w = 0?

Now, let's translate it so that x² + y² + z² ≤ 1 and w = 5. Are you saying the "surface to cover that volume" is still x² + y² + z² = 1 with w = 0? or what do you mean by empty? Why isn't the "surface to cover that volume" x² + y² + z² = 1 with w = 5?

The same questions apply even if you mean a four-dimensional ball x² + y² + z² + w² ≤ 1 getting translated the same amount x² + y² + z² + (w - 5)² ≤ 1. Except then I'm not as clear on what you mean by "surface".

[BClaw]

Arsenic, are you talking about klein bottles and projective planes and stuff? The surfaces are 2-dimensional in those cases, but they enclose a 4-dimensional volume. The way a torus or a sphere have 2-dimensional surfaces (or maybe "skins") that enclose a 3-dimensional space. I'm interested in stuff like that too (of course, assuming that's what you are referring to.)

[Aale de Winkel]

Really confusing postings your guys have here of already discussed material.

The glome is the surface of the gongyl which you might call a filled glome which from its point is an empty glome.

volume without surface? Technically I see in equations such as:
x[sup]2[/sup] + y[sup]2[/sup] .... < R[sup]2[/sup]
the ball which doesn't include its surrounding sphere, the gongyl without its glome;

The glome;
x[sup]2[/sup] + y[sup]2[/sup] + z[sup]2[/sup] + w[sup]2[/sup] = R[sup]2[/sup]
for me is the surface of the gongyl:
x[sup]2[/sup] + y[sup]2[/sup] + z[sup]2[/sup] + w[sup]2[/sup] <= R[sup]2[/sup]
without its glome the shape:
x[sup]2[/sup] + y[sup]2[/sup] + z[sup]2[/sup] + w[sup]2[/sup] < R[sup]2[/sup]
ought to be called the "open gongyl", not to be confused with the "empty gongyl" which is the glome.
(without 'w' same thing for sphere versus ball, and if you also drop the 'z' in "Bionia" with repect to the circle versus the disc)

Bclaw, The Klein Bottle is the 4-dimensional analogon of the 3-dimensional Möbius Belt, it has but one surface enclosing the entire surrounding space.

http://home.wanadoo.nl/aaledewinkel/Enc ... hapes.html
.

[BClaw]

Bclaw, The Klein Bottle is the 4-dimensional analogon of the 3-dimensional Möbius Belt, it has but one surface enclosing the entire surrounding space.

I knew that! I just don't know how to *say* it.

[Aale de Winkel]

I didn't doubt that, it is just more or less opposed to the glome, you compared to, which has an inner and an outher "space". With the Klein Bottle this seperation does not exist since they are the same, or as said it "encloses all its surrounding space".
Not quite sure yet, but I do think you can view the Kleins Bottle as a rotation of the Möbius Belt, which is a tordated rotation of the line.
Möbius Belt: R[sub]180[/sub](line).
Klein Bottle: R( R[sub]180[/sub](line) ).
(see forementioned link, (probably the place I'll put your animations, iff space permits))
.

[BClaw]

... to the glome, you compared to, which has an inner and an outher "space". With the Klein Bottle this seperation does not exist since they are the same, or as said it "encloses all its surrounding space".

Wow! I never thought of that! A sphere or a glome encloses a space, so there is a space inside and a space outside, but a mobius strip or a klein bottle exists *in* a space, but does not *enclose* a space. That is so cool!