I find it to be the most frustrating and fascinating thing ever. I did a 
search for its objects: 
http://search.viewpoint.com/pl/websearch?vb=2&tn=&type=ONE&k=4th+dimension+objects

I don't know if some day would be possible but would be great to visit a 
4D world and meet 4D people :-D

I was thinking and maybe this is where you go when you die and ghosts 
are just what we can see from a 4D person. 4D makes my head go thinking 
pretty bizarre stuff, don't you?

Cheers.

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Saul Luizaga wrote:

> I don't know if some day would be possible but would be great to visit a 
> 4D world and meet 4D people :-D

I should point out that a rotation in 4D can leave your body in mirror 
image. If you go to 4D-land, turn around the "wrong" way, and then come 
back, your body will be inverted.

Why would you care? Well... certain biological molecules are chiral, 
so... good luck assimilating your food. :-P

(If 4D rotation seems weird, consider a 2D figure rotated through 3D 
space. For example, take the letter "d", draw it on a transparency, and 
then flip the transparency over. You now have the letter "b" - it's 
mirrored in 2D space, but in 3D space it's just a rotation!)

Let us not also forget that your body would have 0 thickness in 4D-land.

> 4D makes my head go thinking pretty bizarre stuff, don't you?

Perhaps.

The logic is fairly simple; the trouble starts when you try to 
"visualise" what it means.

For example, when you extrude an object, the following happens:

- Each point becomes two matching points.
- Each line becomes two matching lines.
- Each pair of matching points as a new line added between them.
- Each pair of matching lines has a new surface added between them.

Thus, clearly, when extruding an object into a new dimension:

- P = 2P
- L = 2L + P
- S = 2 + L

By that reconing, extruding a square (4 points, 4 lines) into a cube 
should yield

- 2*4 = 8 points
- 2*4 + 4 = 12 lines
- 2 + 4 = 6 surfaces

...which turns out to be correct.

Applying the exact same formulas, extruding a cube into a 4D hypercube 
should yield

- 8*2 = 16 points
- 2*12 + 8 = 32 lines
- 2 + 6 = 8 hypersurfaces
- I have no idea how many "normal" surfaces

For extra fun, look at a 2D projection of a rotating 3D cube. 
Unfortunately, this won't look like some 2D lines moving around, it will 
*look like* a true 3D cube - because our brains are designed this way. 
But now consider a 3D perspective projection of a rotating 4D 
hypercube... You can see exactly the same effects. It's just more 
mind-blowing. ;-)

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Invisible wrote:
> Saul Luizaga wrote:
> 
>> I don't know if some day would be possible but would be great to visit 
>> a 4D world and meet 4D people :-D
> 
> I should point out that a rotation in 4D can leave your body in mirror 
> image. If you go to 4D-land, turn around the "wrong" way, and then come 
> back, your body will be inverted.
> 
> Why would you care? Well... certain biological molecules are chiral, 
> so... good luck assimilating your food. :-P

Hahaha, yeah I've been reading... :-) didn't know about the 
bio-molecules tho.

> 
> (If 4D rotation seems weird, consider a 2D figure rotated through 3D 
> space. For example, take the letter "d", draw it on a transparency, and 
> then flip the transparency over. You now have the letter "b" - it's 
> mirrored in 2D space, but in 3D space it's just a rotation!)

Isn't this thing alone just cool? This reminds me the transformation 
between 3D rectangular and spherical coordinates, a circumference in the 
former is a line in the latter, fascinating...

> Let us not also forget that your body would have 0 thickness in 4D-land.

Imagine that, like a 2D being in a 3D world, great for a movie plot :-D

>> 4D makes my head go thinking pretty bizarre stuff, don't you?
> 
> Perhaps.
> 
> The logic is fairly simple;

agree

> the trouble starts when you try to 
> "visualise" what it means.

exactly, like how does a 2D being thinks about a 3D object or being, how 
do they "see" it, is just bizarre...

> For example, when you extrude an object, the following happens:
> 
> - Each point becomes two matching points.
> - Each line becomes two matching lines.
> - Each pair of matching points as a new line added between them.
> - Each pair of matching lines has a new surface added between them.
> 
> Thus, clearly, when extruding an object into a new dimension:
> 
> - P = 2P
> - L = 2L + P
> - S = 2 + L
> 
> By that reconing, extruding a square (4 points, 4 lines) into a cube 
> should yield
> 
> - 2*4 = 8 points
> - 2*4 + 4 = 12 lines
> - 2 + 4 = 6 surfaces
> 
> ....which turns out to be correct.

yeah, I was thinking for about 3 hours how to get to a 4D hypercube 
following the logic of building a 3D cube from a single line segment, 

dimension adds a new(n+1) dimension, the irony is you already must know 
what direction to rotate to in order to build the n+1-dimension object 
in that dimension, I was also noticing the increasing number of P,L and 
S, great brain exercise, also it reminded me of a possible explanation 
for the Bermuda Triangle as a "door" to a Wormhole into hyperspace, man 
this is the most pleasing way of meditation I have ever found :-); yoga? 
way stay meditating in a 3D world when you can imagine the awesomeness 
of a 4D world? :-D

> Applying the exact same formulas, extruding a cube into a 4D hypercube 
> should yield
> 
> - 8*2 = 16 points
> - 2*12 + 8 = 32 lines
> - 2 + 6 = 8 hypersurfaces
> - I have no idea how many "normal" surfaces
> 
> For extra fun, look at a 2D projection of a rotating 3D cube. 
> Unfortunately, this won't look like some 2D lines moving around, it will 
> *look like* a true 3D cube - because our brains are designed this way. 
> But now consider a 3D perspective projection of a rotating 4D 
> hypercube... You can see exactly the same effects. It's just more 
> mind-blowing. ;-)

In deed :-)
http://en.wikipedia.org/wiki/Fourth_dimension

Post a reply to this message

>> I should point out that a rotation in 4D can leave your body in mirror 
>> image. If you go to 4D-land, turn around the "wrong" way, and then 
>> come back, your body will be inverted.
>>
>> Why would you care? Well... certain biological molecules are chiral, 
>> so... good luck assimilating your food. :-P
> 
> Hahaha, yeah I've been reading... :-) didn't know about the 
> bio-molecules tho.

Remember Thalidomide?

One Thelidomide molecule is actually "safe", but it's mirror image 
causes... well, go Google it. (Unfortunately, it turns out that normal 
human metabolism can convert one molecule to the other, so it's not so 
"safe" after all...)

But there are lots and lots of biological molecules who's mirror images 
are at best inactive and at worst toxic.

> Isn't this thing alone just cool? This reminds me the transformation 
> between 3D rectangular and spherical coordinates, a circumference in the 
> former is a line in the latter, fascinating...

Have you read about non-Euclid geometry? You may find it interesting...

>> Let us not also forget that your body would have 0 thickness in 4D-land.
> 
> Imagine that, like a 2D being in a 3D world, great for a movie plot :-D

Assuming our hypothetical 4D world has 4D atoms and 4D gravity, I 
imagine your 0-thickness body would slip between the microscopic gaps 
between atoms...

>> The logic is fairly simple;
> 
> agree
> 
>> the trouble starts when you try to "visualise" what it means.
> 
> exactly, like how does a 2D being thinks about a 3D object or being, how 
> do they "see" it, is just bizarre...

And that's just about the best way for us 3D simpletons to imagine the 
4D world; try to imagine how you'd describe a cube or a torus to a 2D 
simpleton.

>> For example, when you extrude an object, the following happens:
>>
>> - Each point becomes two matching points.
>> - Each line becomes two matching lines.
>> - Each pair of matching points as a new line added between them.
>> - Each pair of matching lines has a new surface added between them.
>>
>> Thus, clearly, when extruding an object into a new dimension:
>>
>> - P = 2P
>> - L = 2L + P
>> - S = 2 + L
>>
>> By that reconing, extruding a square (4 points, 4 lines) into a cube 
>> should yield
>>
>> - 2*4 = 8 points
>> - 2*4 + 4 = 12 lines
>> - 2 + 4 = 6 surfaces
>>
>> ....which turns out to be correct.
> 
> yeah, I was thinking for about 3 hours how to get to a 4D hypercube 
> following the logic of building a 3D cube from a single line segment, 
> then a square and then a cube.

Start with 1 point.

Extrude. Now you have 2 points + 1 line.

Extrude. Now you have a square consisting of 4 points + 4 lines + 1 surface.

Extrude. Now you have a cube with 8 points + 12 lines + 6 surfaces + 1 
volume.

Extrude. Now you have a hypercube with... uh... 16 points, 32 lines, 24 
surfaces, 8 volumes and 1 hypervolume.

More fascinating, 4D space apparently has a regular hypergon with 400 
sides...

>> For extra fun, look at a 2D projection of a rotating 3D cube. 
>> Unfortunately, this won't look like some 2D lines moving around, it 
>> will *look like* a true 3D cube - because our brains are designed this 
>> way. But now consider a 3D perspective projection of a rotating 4D 
>> hypercube... You can see exactly the same effects. It's just more 
>> mind-blowing. ;-)
> 
> In deed :-)
> http://en.wikipedia.org/wiki/Fourth_dimension

Once you realise that a frustrum is a squished cube, and that you're 
looking *through* the 8th side, it's quite easy to find al 8 sides of a 
wireframe projection...

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On 10/20/2009 6:58 AM, Saul Luizaga wrote:

>> Let us not also forget that your body would have 0 thickness in 4D-land.
>
> Imagine that, like a 2D being in a 3D world, great for a movie plot :-D
>

Already the plot of a book: Flatland. ;)

-- 
~Mike

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Invisible wrote:
> Remember Thalidomide?

No idea about this man.

> One Thelidomide molecule is actually "safe", but it's mirror image 
> causes... well, go Google it. (Unfortunately, it turns out that normal 
> human metabolism can convert one molecule to the other, so it's not so 
> "safe" after all...)

oh, interesting, I'll dig into it later, thanks

> But there are lots and lots of biological molecules who's mirror images 
> are at best inactive and at worst toxic.

wow, maybe is not such a good idea without an anti-mirror protection :-)

> Have you read about non-Euclid geometry? You may find it interesting...

No, but I'll try to download a free book from wikipedia wikibooks, 
thanks :-)

> Assuming our hypothetical 4D world has 4D atoms and 4D gravity, I 
> imagine your 0-thickness body would slip between the microscopic gaps 
> between atoms...

wow, a 4D ghost :-D imagining is just great, someone has to do a movie 
out of this, is just too good to be just a Math tale. 4D gravity, that 
will be a problem since the simple "shadow" of a hypercube looks 
bizarre, imagine a true hypercube or more complex object or people, how 
gravity, being normal to a 4D surface by definition, affects the 4D 
matter, man! *blows mind*, if mathematicians help get this to the movies 
will break the Academy Oscar record set by Lord Of The Rings... assuming 
non-math-friendly-people don't go to sleep or exit the movie yelling for 
a refund...

> And that's just about the best way for us 3D simpletons to imagine the 
> 4D world; try to imagine how you'd describe a cube or a torus to a 2D 
> simpleton.

yeah, wow, assuming they understand our language, this is getting more 
and more a new Star Trek series: Voyager in the 4th Dimension.

> Start with 1 point.
> 
> Extrude. Now you have 2 points + 1 line.
> 
> Extrude. Now you have a square consisting of 4 points + 4 lines + 1 
> surface.
> 
> Extrude. Now you have a cube with 8 points + 12 lines + 6 surfaces + 1 
> volume.
> 
> Extrude. Now you have a hypercube with... uh... 16 points, 32 lines, 24 
> surfaces, 8 volumes and 1 hypervolume.
> 
> More fascinating, 4D space apparently has a regular hypergon with 400 
> sides...
> 

Thanks for this, I'd save this and the previous info for future 
reference. I have always liked Math over any other science, I'd be a 
mathematician if it were a Degree anywhere near my city; Math seldom 
bores me, all the opposite, is lots of fun :-)

> Once you realise that a frustrum is a squished cube, and that you're 
> looking *through* the 8th side, it's quite easy to find al 8 sides of a 
> wireframe projection...

OK, I'll take your word for it :-) and I'll start looking on this 
"frustratum" :-D to start with and then go through the 8 sides.

Thanks for all the info pal.

Cheers.

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Mike Raiford wrote:
> On 10/20/2009 6:58 AM, Saul Luizaga wrote:
> 
>>> Let us not also forget that your body would have 0 thickness in 4D-land.
>>
>> Imagine that, like a 2D being in a 3D world, great for a movie plot :-D
>>
> 
> Already the plot of a book: Flatland. ;)
> 
Oh! coincidence, wikipedia has it 
(http://en.wikisource.org/wiki/Flatland), thanks anyway :-)

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You seem to know an awful amount of Math, you have a Degree on it or 
something related to it?

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Saul Luizaga wrote:
> You seem to know an awful amount of Math, you have a Degree on it or 
> something related to it?

I have an Internet connection and too much free time. ;-)

Actually, I have an honors degree in Computer Science, which is closely 
related to mathematics. (However, this is only what it says on the sheet 
of paper; it was "really" a degree in Information Technology, which is 
quite different...)

When I did my maths GCSE, I only got a C. But that was back when I 
thought "maths" just meant doing endless runs of pen and paper 
calculations...

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have seen all 4 Cube movies, even the '70s one.

Cheers.

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