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.

Post a reply to this message

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. ;-)

Post a reply to this message

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, 
then a square and then a cube, the way a 90º rotations on the n-1 
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...

Post a reply to this message

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

Post a reply to this message

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.

Post a reply to this message

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 :-)

Post a reply to this message

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

Post a reply to this message

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...

Post a reply to this message

Just wondering, have you or anybody seen: Cube²: Hypercube, movie? I 
have seen all 4 Cube movies, even the '70s one.

Cheers.

Post a reply to this message

Saul Luizaga wrote:
> Just wondering, have you or anybody seen: Cube²: Hypercube, movie?
 

I have. I liked Cube better. I didn't know there were more.

-- 
   Darren New, San Diego CA, USA (PST)
   I ordered stamps from Zazzle that read "Place Stamp Here".

Post a reply to this message

Darren New wrote:
> I have. I liked Cube better. I didn't know there were more.

I liked Cube²:Hypercube better because of the wonderful weirdness of 
waking up inside a 3d Cube which was part of a 4D cube, all the 
fascinating theories and convoluted plot and the non-sense ending was 
just classy for me, I only wish I had know more Math and understand was 
was the cylinder that psychiatrists brought back, but for me was the 
best anyway :-)

Ironically Cube 0, was the fourth movie and explain how the first Cube 
was build, is very interesting and creepy like the res of them. The 
first one was: Man in a Cube, literally; was a UK idea originally at the 
seems, there was a German version of Man in a Cube too, I don't 
understand much German so I didn't saw it completely.

Post a reply to this message

Saul Luizaga wrote:
> 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

It's good to know that some people are taking the idea of a fourth 
spatial dimension seriously. A lot of people argue that time is the 
"true" 4th dimension, but that idea just doesn't sit well with me, even 
though I don't have a lot of knowledge in this area.

> 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?

Yeah. I think maybe with our 3D minds we could travel through the fourth 
dimension, yet not be able to witness it directly as we do the 3rd. More 
speculation, I know. It might have been inspired by something...

Sam

Post a reply to this message

Invisible wrote:
> 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...

Oh I see, well even I like Math there is a point where get really 
complex, so unless you devote your life to it you won't get an A from a 
teacher I think or you are some kind of genius. I have been the victim 
of those intellectually wain teachers that won't let you get more grade 
than they think you deserve even when I have proven them wrong according 
to a particular Math concept that they taught us to begin with, so I 
wouldn't worry much about a grade, you know what you know and life will 
grade you more fairly when the grade don't depend on biased humans.

Post a reply to this message

stbenge wrote:
> It's good to know that some people are taking the idea of a fourth 
> spatial dimension seriously. A lot of people argue that time is the 
> "true" 4th dimension, but that idea just doesn't sit well with me, even 
> though I don't have a lot of knowledge in this area.

I agree that it seems sort of like cheating to call time the fourth 
dimension, but there are some extremely good reasons for doing so. 
Foremost among these is that, due to special relativity, time and space 
can't be cleanly pulled apart like they can under a more classical model 
of physics.

In the "common sense" viewpoint we have a three-dimensional universe 
which is changing over time, so that at each time the entire 3d universe 
has some "state".  If you wanted to look at this is a four-dimensional 
space, you'd just stack these 3d universe-states together along a single 
other dimension representing time.  This gives you a 4d space, but under 
this view it's perfectly possible to the time dimension as somehow 
"fake".  The key point here, however, is that the common sense view is 
possible because these 3d "slices" of things happening at the same time 
are always parallel (and thus never intersect).

Under special relativity, unfortunately, this model is no longer the 
most natural.  The reason is that there is no longer a well-defined 
meaning to "things happening at the same time" as there was before. 
Instead, this meaning is dependent on your velocity (or rather the 
relative velocities of objects to you).

If we go back to the 4d model of the universe we'd built up for the 
common sense viewpoint, the interpretation now is that the *angle* of 
the 3d slice corresponding to everything "happening at the same time" 
changes with your velocity.  Thus it's not possible to pull the space 
apart into a bunch of well-defined 3d universe-states anymore, but 
instead you need to keep time in there are a proper fourth dimension.

It is worth nothing, however, that the time dimension is *not* identical 
to the space dimensions (one would hope not!) and distances are measured 
differently in time than in space.  Nevertheless this doesn't prevent it 
from being a proper dimension, it just means that the space has a 
different geometry than you'd get from treating time just like another 
spatial dimension.

Post a reply to this message

Invisible wrote:
> 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...

Since clearly the electromagnetic forces would remain three dimensional? 
  I'm not sure it's possible to talk about what it would actually mean 
to superimpose a three and four spatial-dimensional laws of physics in 
this way in a coherent manner, and at the very least doing so would 
require a significant amount of analysis to make sure you've correctly 
coupled the two.

Post a reply to this message

stbenge wrote:
> It's good to know that some people are taking the idea of a fourth 
> spatial dimension seriously. A lot of people argue that time is the 
> "true" 4th dimension, but that idea just doesn't sit well with me, even 
> though I don't have a lot of knowledge in this area.

I have found that there are 2 theories (Wikipedia), spacial 4D and the 
one assuming the 4th dimension is time, I think both concepts are not 
compatible, are two different assumtions.

> Yeah. I think maybe with our 3D minds we could travel through the fourth 
> dimension, yet not be able to witness it directly as we do the 3rd. More 
> speculation, I know. It might have been inspired by something...

Or maybe you leave your 3D body and become a 4D spirit that then will 
incarnate a 4D being and so on...

Post a reply to this message

Saul Luizaga wrote:
> Darren New wrote:
>> I have. I liked Cube better. I didn't know there were more.
> 
> I liked Cube²:Hypercube better because of the wonderful weirdness 
of 
> waking up inside a 3d Cube which was part of a 4D cube,

OK, maybe i'm remembering the wrong movie. The second Cube movie I saw wa
s 
shown from the POV of the operators of the first Cube.

Maybe we're talking about entirely different series of movies. :-)

-- 
   Darren New, San Diego CA, USA (PST)
   I ordered stamps from Zazzle that read "Place Stamp Here".

Post a reply to this message

Saul Luizaga wrote:
> stbenge wrote:
> Or maybe you leave your 3D body and become a 4D spirit that then will 
> incarnate a 4D being and so on...

But if so, how could you possible remember and/or integrate the 
experience with a 3D mind? Maybe you could remember that *something* 
happened, with a gained benefit of a new spiritual perspective...

Post a reply to this message

Darren New wrote:
> Saul Luizaga wrote:
>> Darren New wrote:
>>> I have. I liked Cube better. I didn't know there were more.
>>
>> I liked Cube²:Hypercube better because of the wonderful weirdness of 
>> waking up inside a 3d Cube which was part of a 4D cube,
> 
> OK, maybe i'm remembering the wrong movie. The second Cube movie I saw 
> was shown from the POV of the operators of the first Cube.
> 
> Maybe we're talking about entirely different series of movies. :-)
> 
No no, the second movie you saw is 'Cube Zero: the beginning' or 
something like that, which is the fourth and latest of the series, to my 
knowledge.

Post a reply to this message

stbenge wrote:
> Saul Luizaga wrote:
>> stbenge wrote:
>> Or maybe you leave your 3D body and become a 4D spirit that then will 
>> incarnate a 4D being and so on...
> 
> But if so, how could you possible remember and/or integrate the 
> experience with a 3D mind? Maybe you could remember that *something* 
> happened, with a gained benefit of a new spiritual perspective...

Is a possibility, but again, who remembers being a 2D person before this 
life? maybe is part of the "instinct" of each person, but who knows...

Post a reply to this message

Saul Luizaga wrote:
> stbenge wrote:
>> Saul Luizaga wrote:
>>> stbenge wrote:
>>> Or maybe you leave your 3D body and become a 4D spirit that then will 
>>> incarnate a 4D being and so on...
>>
>> But if so, how could you possible remember and/or integrate the 
>> experience with a 3D mind? Maybe you could remember that *something* 
>> happened, with a gained benefit of a new spiritual perspective...
> 
> Is a possibility, but again, who remembers being a 2D person before this 
> life? maybe is part of the "instinct" of each person, but who knows...

You seem to be alluding to a sort of an evolution of consciousness. 
Which I won't refute, considering how this life seems to be evolving our 
consciousnesses.

I can see how having a working knowledge of the second dimension could 
help our understanding of this one, just as this would have an effect on 
the next :-)

Post a reply to this message

stbenge wrote:
  > You seem to be alluding to a sort of an evolution of consciousness.
> Which I won't refute, considering how this life seems to be evolving our 
> consciousnesses.

is just one of many possibilities you can speculate, since nobody 
actually knows for sure, what we were or what we will be before/after 
this life, all you can do is just choose to believe in something...

> I can see how having a working knowledge of the second dimension could 
> help our understanding of this one, just as this would have an effect on 
> the next :-)

yeah, would be logical, fair and a nice guiding way :-)

Post a reply to this message

Saul Luizaga wrote:
> stbenge wrote:
>  > You seem to be alluding to a sort of an evolution of consciousness.
>> Which I won't refute, considering how this life seems to be evolving 
>> our consciousnesses.
> 
> is just one of many possibilities you can speculate, since nobody 
> actually knows for sure, what we were or what we will be before/after 
> this life, all you can do is just choose to believe in something...

Which I do.

That's not _all_ we can do ;) Dreams, for instance, can facilitate new 
learning experiences. But that's not the end of the possibilities. Some 
are more morally accepted than others, though.

>> I can see how having a working knowledge of the second dimension could 
>> help our understanding of this one, just as this would have an effect 
>> on the next :-)
> 
> yeah, would be logical, fair and a nice guiding way :-)

Which, based on my experiences of this life, I can only guess that we'll 
have as much fun and as much good company as this one :)

Post a reply to this message

stbenge wrote:
> Which I do.

Me too.

> That's not _all_ we can do ;) Dreams, for instance, can facilitate new 
> learning experiences. But that's not the end of the possibilities. Some 
> are more morally accepted than others, though.

I agree.

> Which, based on my experiences of this life, I can only guess that we'll 
> have as much fun and as much good company as this one :)

I really hope so :)

Post a reply to this message

Saul Luizaga <sau### [at] netscapenet> wrote:
> I find it to be the most frustrating and fascinating thing ever. I did a
> search for its objects:

Yes, me too. Fascinating because I have a kind of feeling that it's sort of
real; frustrating, because I can't imagine it.
Over the past months I've read some stuff about it. But sometimes only part of
the books, because I often didn't succeed in understanding the examples (I have
a very bad understanding of all mathemetical related things), let alone
imagining it.
There are many interesting resources on the topic. I assume you already know the
Pov-Ray generated films:
http://www.dimensions-math.org/Dim_E.htm
I found them very intriguing, though too difficult to understand sometimes.

Apart from the classic novel Flatland there seem to have been written quite a
lot of books and essays on the topic between 1880 and about 1925, maybe because
the topic was inspiring to those day's broader interest in mysticism and
theosophy, e.g. Ouspensky's Tertium Organum and some essays by Rudolf Steiner.

I thought the books by "amateur-mathematician" Charles Howard Hinton most
interesting, because I got the impression that he really was capable of imagine
a little of the 4th dimension, at least in so far as it can be imagined.
Some of his books can be downloaded for free, e.g.:
http://www.archive.org/details/fourthdimension00hintarch

He developed a set of coloured cubes (that could be bought along with his books
at that time) which were supposed to facilitate the imagination of the hypercube
in his book "The fourth dimension":
http://www.greylodge.org/occultreview/glor_011/hintoncubes.pdf

I haven't tried yet to make these models, because I was too lazy and thought
maybe it wouldn't help and I would possibly be more frustrated than before. When
reading through part of "The fourth dimension" I noticed however, that there
would be no chance to understand it properly without models. I also thought of
recreating them in POV too, but it seems to be an awful lot of work.

However Hinton's approach to visualizing the 4th dimension is supposed to have
had a strong influence on Alicia Boole Stott, too, who herself had no formal
education in mathematics either but managed to get a honorary doctorate in 1914
because of her work on four dimensional polytopes. She made lots of beautiful
cardboard models and one gets the impression that she too was capable of
"imagining" at least some sort of 4th dimension.

Regards,

Simone



> 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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Interesting... yes I downloaded the movie but haven't seen it yet, yes I 
have trouble also imagining 4D object, I think reading the 4D Cartesian 
coordinate system and basic elements like a "line" and a "plane" could 
help imagining more complex forms, thanks for the tips, I'll check them 
out later.

Cheers.

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stbenge wrote:

> It's good to know that some people are taking the idea of a fourth 
> spatial dimension seriously. A lot of people argue that time is the 
> "true" 4th dimension, but that idea just doesn't sit well with me, even 
> though I don't have a lot of knowledge in this area.

An extra spatial dimension gives you 4D Euclidian space.

Time as an extra dimension gives you a 4D non-Euclidian space, which 
turns out to be useful in Einstien's various reletivity theories.

These are two seperate, unrelated geometries.

4D Euclidian space is interesting because it's a straight extension of 
3D Euclidian geometry.

And then, as I mentioned, there are other non-Euclidian geometries. For 
example, elliptic geometry. In 2D elliptic geometry, a "straight line" 
is in fact a line drawn on the surface of a sphere. Since the surface of 
a sphere is curved, all these "straight lines" curve around it and 
eventually close into circles. (E.g., the Earth's equator is such a 
circle.) Thus, in elliptic geometry, *all* lines intersect each other, 
and there is no such thing as "parallel" lines.

Somewhat weirder is hyperbolic geometry, where multiple "straight lines" 
through a single point do not intersect each other [except at that 
point]. You really need to play with this:

http://cs.unm.edu/~joel/NonEuclid/NonEuclid.html

I lost hours in it...

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stbenge schrieb:

> It's good to know that some people are taking the idea of a fourth 
> spatial dimension seriously. A lot of people argue that time is the 
> "true" 4th dimension, but that idea just doesn't sit well with me, even 
> though I don't have a lot of knowledge in this area.

Just one stupid question: What is there to be taken serious about - I 
mean, if not considering time to be the 4th dimension (which it actually 
is)?

The main point is that /if/ such a 4th ("space-like") dimension /does/ 
exist, then our 3 dimensions /already are/ embedded in that 4D world - 
the 4th dimension will not just suddenly appear out of thin air to 
entertain us.

There is no room for "serious" speculation about "what would it be like 
if there /was/ a 4th dimension" - either a 4th (space-like) dimension 
/does/ exist, in which case the natural laws in this 4D world /must/ 
result in the effects we observe in "our" 3D "slice" - or such a 4th 
dimension does /not/ exist, in which case there's no room for 
speculation left.

In any case we cannot in any way "leave 3D and enter 4D" - all we can do 
is act within our 3D world, and (possibly) find that the resulting 
effects can only (or best) be explained by assuming that our 3D world is 
just a "slice" embedded in a 4D world.

Scientists are, by the way, seriously discussing models according to 
which the universe does indeed have more than three space-like 
dimensions (quite a lot actually), but they appear to agree that in this 
case those extra dimensions must be quite small (think of the surface of 
a thin wire, with the length corresponding to our known three space-like 
dimensions, and the circumfence corresponding to the extra dimensions).

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Kevin Wampler schrieb:

> It is worth nothing, however, that the time dimension is *not* identical 
> to the space dimensions (one would hope not!) and distances are measured 
> differently in time than in space.

Well, AFAIK there's actually no fundamental reason to apply different 
"measuring tapes" to time and space: The constant vacuum speed of light 
can serve as a ruler for both, with the distance of a light second 
equating a second.

It just happens that it's still more practical to use meters for 
space-like dimensions and seconds for time-like dimensions.

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Saul Luizaga schrieb:

>>> Or maybe you leave your 3D body and become a 4D spirit that then will 
>>> incarnate a 4D being and so on...
>>
>> But if so, how could you possible remember and/or integrate the 
>> experience with a 3D mind? Maybe you could remember that *something* 
>> happened, with a gained benefit of a new spiritual perspective...
> 
> Is a possibility, but again, who remembers being a 2D person before this 
> life? maybe is part of the "instinct" of each person, but who knows...

You can't be (and never could have been in any prior life) a "2D 
person": Evidently, there /are/ 3 dimensions (or more), so each "2D" 
being must also be embedded in 3D space, and therefore /be/ a 3D being.

Similarly with 4 (space-like) dimensions: Either we /are/ 4D beings, or 
there is no such thing as a 4th dimension.

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>> It is worth nothing, however, that the time dimension is *not* 
>> identical to the space dimensions (one would hope not!) and distances 
>> are measured differently in time than in space.
> 
> Well, AFAIK there's actually no fundamental reason to apply different 
> "measuring tapes" to time and space: The constant vacuum speed of light 
> can serve as a ruler for both, with the distance of a light second 
> equating a second.
> 
> It just happens that it's still more practical to use meters for 
> space-like dimensions and seconds for time-like dimensions.

When you start measuring "events" which happen at different points in 
space and different points in time, it becomes worth using a "spacetime" 
measurement which simultaneously encodes both.

(Not that I know much about such things...)

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It seems to me people are confusing mathematical formalisms with reality.

4D Euclidian geometry is an interesting mathematical system. 4D 
time-dimension geometry is an interesting mathematical system. 2D Euclid 
is an interesting mathematical system. (Euclid was reputedly so 
fascinated by it that he was speared to death by some random Roman.)

Whether these systems have any baring on the Real World which we inhabit 
is an entirely orthogonal question. ;-)

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clipka wrote:
> Saul Luizaga schrieb:
> 
>>>> Or maybe you leave your 3D body and become a 4D spirit that then 
>>>> will incarnate a 4D being and so on...
>>>
>>> But if so, how could you possible remember and/or integrate the 
>>> experience with a 3D mind? Maybe you could remember that *something* 
>>> happened, with a gained benefit of a new spiritual perspective...
>>
>> Is a possibility, but again, who remembers being a 2D person before 
>> this life? maybe is part of the "instinct" of each person, but who 
>> knows...
> 
> You can't be (and never could have been in any prior life) a "2D 
> person": Evidently, there /are/ 3 dimensions (or more), so each "2D" 
> being must also be embedded in 3D space, and therefore /be/ a 3D being.

You're missing the point: by 2D person we mean you live in a 2D world as 
we live in a 3D world, without being able to experience any other 
dimension. Of course 2D, 3D, 4D, etc are included in a nD universe but 
we're not gonna call everything a nD being or object, we have to call 
the object dimensional feature by how many dimensions that being/object 
is able to experience not by how many dimension that being/object is 
included in.

> Similarly with 4 (space-like) dimensions: Either we /are/ 4D beings, or 
> there is no such thing as a 4th dimension.

read above.

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ANd nobody knows if we were or not 2D beings.

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Invisible wrote:
> These are two seperate, unrelated geometries.
> 
> 4D Euclidian space is interesting because it's a straight extension of 
> 3D Euclidian geometry.

I should read more about this, sounds interesting.

> Somewhat weirder is hyperbolic geometry, where multiple "straight lines" 
> through a single point do not intersect each other [except at that 
> point]. You really need to play with this:
> 
> http://cs.unm.edu/~joel/NonEuclid/NonEuclid.html

I'll check it out.

I've been trying to figure out how to calculate hyperbolic geometry, so 
I may create new "circle limit" Poincare disk tesselations. I figure it 
must be possible to create such a thing with POV-Ray functions, but so 
far all my attempts have been total failures :(

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>> These are two seperate, unrelated geometries.
>>
>> 4D Euclidian space is interesting because it's a straight extension of 
>> 3D Euclidian geometry.
> 
> I should read more about this, sounds interesting.

3D Euclidian space = the place where we (apparently) live.

4D Euclidian space = the place where objects such as the hypercube live.

4D non-Euclidian space = any space with 4 dimensions that *isn't* 4D 
Euclidian space. This includes a system where time is the 4th dimension, 
but also many other kinds of space as well.

>> Somewhat weirder is hyperbolic geometry, where multiple "straight 
>> lines" through a single point do not intersect each other [except at 
>> that point]. You really need to play with this:
>>
>> http://cs.unm.edu/~joel/NonEuclid/NonEuclid.html
> 
> I'll check it out.
> 
> I've been trying to figure out how to calculate hyperbolic geometry, so 
> I may create new "circle limit" Poincare disk tesselations. I figure it 
> must be possible to create such a thing with POV-Ray functions, but so 
> far all my attempts have been total failures :(

I believe you can use ordinary 2D coordinates for hyperbolic space, you 
just need to remap them when plotting them in normal 2D space. 
(Unfortunately, I can't find any formulas for doing this.)

Note that there is more than one way to project hyperbolic space into 
Euclidian space; you probably want the Poincaré disk model if you're 
trying to copy Escher's Circle Limit.

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clipka wrote:
> Well, AFAIK there's actually no fundamental reason to apply different 
> "measuring tapes" to time and space: 

It's not measured *quite* the same way... The distance between two events is 
sqrt(x*x+y*y+z*z-t*t) Note the - sign.

-- 
   Darren New, San Diego CA, USA (PST)
   I ordered stamps from Zazzle that read "Place Stamp Here".

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Darren New wrote:
> clipka wrote:
>> Well, AFAIK there's actually no fundamental reason to apply different 
>> "measuring tapes" to time and space: 
> 
> It's not measured *quite* the same way... The distance between two 
> events is sqrt(x*x+y*y+z*z-t*t) Note the - sign.

Hence "non-Euclidian space".

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clipka wrote:
> Kevin Wampler schrieb:
> 
>> It is worth nothing, however, that the time dimension is *not* 
>> identical to the space dimensions (one would hope not!) and distances 
>> are measured differently in time than in space.
> 
> Well, AFAIK there's actually no fundamental reason to apply different 
> "measuring tapes" to time and space: The constant vacuum speed of light 
> can serve as a ruler for both, with the distance of a light second 
> equating a second.
> 
> It just happens that it's still more practical to use meters for 
> space-like dimensions and seconds for time-like dimensions.

I think that this becomes problematic for measuring between points which 
aren't connectible by a lightlike geodesic, but maybe there's some 
clever way around that (although I don't see how).

In any case, I was referring to the fact that under special relativity 
spacetime has a metric that differs from that of a 4D Euclidian space 
along the dimension corresponding to time.  For instance, the length of 
a vector can be negative, which is impossible under a Euclidean metric.

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Invisible wrote:
> Darren New wrote:
>> clipka wrote:
>>> Well, AFAIK there's actually no fundamental reason to apply different 
>>> "measuring tapes" to time and space: 
>>
>> It's not measured *quite* the same way... The distance between two 
>> events is sqrt(x*x+y*y+z*z-t*t) Note the - sign.
> 
> Hence "non-Euclidian space".

Yes. I wasn't sure if that counted as a euclidian space or not. :-)

-- 
   Darren New, San Diego CA, USA (PST)
   I ordered stamps from Zazzle that read "Place Stamp Here".

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Darren New wrote:
> 
> Yes. I wasn't sure if that counted as a euclidian space or not. :-)
> 

It doesn't, but it does share the property that the metric tensor is 
constant over the entire space.

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Darren New schrieb:
> clipka wrote:
>> Well, AFAIK there's actually no fundamental reason to apply different 
>> "measuring tapes" to time and space: 
> 
> It's not measured *quite* the same way... The distance between two 
> events is sqrt(x*x+y*y+z*z-t*t) Note the - sign.

It /is/ measured in the same way - it's just that the Pythagorean 
theorem doesn't hold in our peculiar (3+1)D universe...

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Kevin Wampler schrieb:

> I think that this becomes problematic for measuring between points which 
> aren't connectible by a lightlike geodesic, but maybe there's some 
> clever way around that (although I don't see how).

Well, if two points cannot be reached from one another - is there /any/ 
way to assign a distance to these points at all?

So I think this is a non-issue.

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clipka wrote:
> Kevin Wampler schrieb:
> 
>> I think that this becomes problematic for measuring between points 
>> which aren't connectible by a lightlike geodesic, but maybe there's 
>> some clever way around that (although I don't see how).
> 
> Well, if two points cannot be reached from one another - is there /any/ 
> way to assign a distance to these points at all?
> 
> So I think this is a non-issue.

In this context a lightlike geodesic refers to a path in spacetime which 
  light would follow, and I meant to imply that defining the distance 
between points which could only be connected by going *slower* than 
light would also be hard to define uniformly.

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clipka wrote:
> Darren New schrieb:
>> clipka wrote:
>>> Well, AFAIK there's actually no fundamental reason to apply different 
>>> "measuring tapes" to time and space: 
>>
>> It's not measured *quite* the same way... The distance between two 
>> events is sqrt(x*x+y*y+z*z-t*t) Note the - sign.
> 
> It /is/ measured in the same way - it's just that the Pythagorean 
> theorem doesn't hold in our peculiar (3+1)D universe...

I fail to see how stating that the Pythagorean theorem doesn't hold 
isn't just another way of saying that distances are measured differently 
in our spacetime?

Is you point that in space time there *is* a well-defined notion of 
distance which unifies both the spatial and temporal aspects, and thus 
we don't really need to use one set of units for space and another set 
for time?  Your comment about light-seconds would make more sense in 
this context.  If this is the case perhaps I've been misunderstanding 
your point form the beginning, since I'd surely agree with this.

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Kevin Wampler wrote:
> clipka wrote:
>> Kevin Wampler schrieb:
>>
>>> I think that this becomes problematic for measuring between points 
>>> which aren't connectible by a lightlike geodesic, but maybe there's 
>>> some clever way around that (although I don't see how).
>>
>> Well, if two points cannot be reached from one another - is there 
>> /any/ way to assign a distance to these points at all?
>>
>> So I think this is a non-issue.
> 
> In this context a lightlike geodesic refers to a path in spacetime which 
>  light would follow, and I meant to imply that defining the distance 
> between points which could only be connected by going *slower* than 
> light would also be hard to define uniformly.

In light of one of your other comments I think I now understand what 
point you're making: we can treat space-time distances in a common set 
of units by treating distances as times via the speed of light.  In 
which case I'd agree, and it fact the necessity for this clearly falls 
out of being able to have a single number represent the distance at all.

I still don't see how it's relevant for my initial comment that the 
space and time coordinates are treated differently though, since they 
are most definitely factor into the distance function in different ways. 
  Stated another way, swapping the time axis with a space axis is *not* 
in the symmetries of Minkowski spacetime, but swapping any of the space 
axes *is*, and this there's something "different" about the time axis. 
Otherwise why bother saying (3+1)D instead of just 4D?

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Kevin Wampler wrote:
> Darren New wrote:
>>
>> Yes. I wasn't sure if that counted as a euclidian space or not. :-)
>>
> 
> It doesn't, but it does share the property that the metric tensor is 
> constant over the entire space.



-- 
http://blog.orphi.me.uk/
http://www.zazzle.com/MathematicalOrchid*

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Orchid XP v8 wrote:
> Kevin Wampler wrote:
>> Darren New wrote:
>>>
>>> Yes. I wasn't sure if that counted as a euclidian space or not. :-)
>>>
>>
>> It doesn't, but it does share the property that the metric tensor is 
>> constant over the entire space.
> 
>  my head asplode


It was just a fancy way of saying that there's a sense in which the 
spacetime isn't "curved"; which I probably should have just said in the 
first place since the way I phrased it should really be made more 
precise to be accurate anyway.

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On 20-10-2009 12:41, 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
> 

Book reference: 'Doorways in the sand' Roger Zelazny

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