Would it be easier to cut an apple in a 4 dimensional universe.

[Smilin]

So I'm a bit *relaxed* and cutting up an apple one day thinking about the universe...

I set the apple on the cutting board, slice it in half.
I then take advantage of the fact that I have multiple dimensions to work with. I place half an apple on the plane of the cutting board and then I make a cut through a plane perpendicular to the first. easy.
I do this with the remaining half. I now have four wedge shaped pieces. These, however are more difficult to slice. I can't cut perpendicular any longer. I have to either set the curved edge of the slice on the cutting board or the pointed edge of the wedge. Either way I no longer have a right angle I can use to press straight down to make the next cut.

Question: If I lived in and could see and act in a universe where I had 4 dimensions instead of three would I have an extra dimension to take advantage of? Could I somehow set this wedge of apple down in a manner that would again allow me to make a perpendicular cut?

I've always tried to visualize an additional dimension and I guess this little incident was small and simple enough that for the first time I almost 'got it'...almost. I realize that there is already a fourth dimension and more that we live in, we just can't interract with it other than continuously slipping forward in time. I'm kinda wondering more along the lines of 4 dimensions being viewable and say the 5th being time.

Does this make any sense or was I just too relaxed?

 

[Grasshopper27]

You have too much time on your hands...

It is an apple, you're just supposed to eat it...



Grasshopper

 

[dejitaru]

It would be like cutting a pizza. A red, juicy pizza.
You would be able to see and interact with all three dimensions simultaneously. You could core the apple without damaging the skin.
Or you could get an apple-shaped cutter to conquer any and all slices in one blow.

 

[Smilin]

Originally posted by: dejitaru

You could core the apple without damaging the skin.

I hadn't thought of that.

 

[everman]

Problem is, it's rather difficult to think of coreing and apple without damaging the skin. I've read some explanations of how dimensions beyond 3 can be thought of, too bad I don't remember which book that was and it was a few years ago.

 

[Cogman]

according to Einstine, (ok, so I cant spell) There are 4 dimetions

Length, Width, Depth, Time (might also say Motion)

Of course, I guess if time did not exist it would be pretty hard to cut an apple, as you would not be moving

 

[Shalmanese]

Yes, but you would need a 4 dimensional apple, If we think of the 4th dimension as something akin to time except it acts a normal dimension, the first cut would be to cut the past-apple away from the future-apple. Then next cuts would be the same as in 3 dimensions.

 

[Woodchuck2000]

Discounting time for a moment, (Good way to start a post eh?)
The shape of any object can be described in terms of three mutually orthogonal axes (i.e. XYZ for most people.)
Say we were to try and introduce a 4th spatial dimension, how would that fit in?

The problem with the question is that many of your definitions rely on a spacially 3D world - things like perpendicular, curved, pointed, and right-angle would mean completely different things in a hypothetical 4D world.

 

[Mark R]

What if the 4th dimension 'wrapped around' like the dimension of longitude does on earth? What happens if its range was so small that it was smaller than the diameter of a proton?

Would it make the apple any easier to cut? Probably not.

 

[DarkLance]

ok, my first post, just thought I'd add my cents. I like to think of the 4thD as a state, not a spatial fixture. You could not "cut" the fourth dimension with your knife. Now I say state, because you exist in a portion of the 4th dimension, you are now. So while you can say you are 5'10", we cannot not measure the 4thD like this yet.
But back to the apple, I believe you could, say, remove the core without damaging the skin. If you move (backwards) through the 4thD and cut the apple and remove the core, would your current apple have a core? Would it be cut? I always wondered if things actully would change in the current if you changed the past. I know movies say it would, but I don't put must trust in them? I think it would just cause a rift or something and screw the present. Because... the present has already happened, in truth the future may have already happened. Ack, now my head hurts...

 

[Geniere]

Imagine a cube made out of 12 toothpicks. This cube is suspended above a sheet of white paper. A light above the cube creates a shadow on the paper. This shadow is a 2 dimensional representation of the toothpick cube. Imagine what 4 dimensional object would cast a shadow that we see as a 3 dimensional cube. In the 4th dimension each point on the cube would be a plane, each line a plane, and each plane a cube. Of course in the 4th dimension, the light source would completly surround the object.

Regards

 

[Athelwulf]

Lets go the other way. You have a 2d sqare and you live in 2d space, and you want to cut out another square in the middle of this square. However, you can't access the middle of the squre without cutting the sides. But a person in 3d space could come down and cut out the middle square without affecting the sides. So if you can apply this (unlikely) to the 3d and 4th dimension then you could cut the core without breaking the skin.

 

[Shalmanese]

I tend to think the easiest way of visualisaing 4D is to use time as the 4th dimension and then envision us as some sort of god-like creature that can move through time at will.

This, a hyper cube is simple a cube that extends from 5s to 10s. A hypersphere is a sphere that appears as a point at 5s, grows to a full sized sphere at 7.5s and shrinks back to a point at 10s.

Your hypothetical hyper-apple would start off as a particularly funnily knobbled apple at 5s, trandform into several different funnily knobbled apples and finally disappear at 10s. To "cut" the apple, you would first seperate it in the 4th dimension so you would have a funny knobbly apple that goes from 5s to 7.5s and another funny knobbly apple that goes from 8s to 10.5s. Then you can treat each object as a seperate 3d object.

 

[Daovonnaex]

The ability to manipulate the fourth dimension probably wouldn't help you that much in cutting an apple. It would open up other possibilities, though. For example, if you could manipulate the fourth dimension, you could correct any mistake you make.

 

[Smilin]

Originally posted by: Shalmanese
I tend to think the easiest way of visualisaing 4D is to use time as the 4th dimension and then envision us as some sort of god-like creature that can move through time at will.

This, a hyper cube is simple a cube that extends from 5s to 10s. A hypersphere is a sphere that appears as a point at 5s, grows to a full sized sphere at 7.5s and shrinks back to a point at 10s.

Your hypothetical hyper-apple would start off as a particularly funnily knobbled apple at 5s, trandform into several different funnily knobbled apples and finally disappear at 10s. To "cut" the apple, you would first seperate it in the 4th dimension so you would have a funny knobbly apple that goes from 5s to 7.5s and another funny knobbly apple that goes from 8s to 10.5s. Then you can treat each object as a seperate 3d object.

That's it I think. I could cut the apple into 1/8th sized slices *before* I cut it into 1/2 or 1/4 sized slices. That way without the previous cuts it would be easy to lay flat on the cutting board.

Another question now... do you spose the universe is finite in size but has no end? (similar to the surface of a sphere?) What composes dimensions 5 and beyond?

 

[dejitaru]

The fouth dimension is not time. Time is the first (or zero).
Have one or two-dimensional universes no duration?

 

[Goosemaster]

Originally posted by: dejitaru
The fouth dimension is not time. Time is the first (or zero).
Have one or two-dimensional universes no duration?

So what then, partel, is the fourth?

 

[FuriousBroccoli]

Originally posted by: dejitaru
The fouth dimension is not time. Time is the first (or zero).
Have one or two-dimensional universes no duration?

Time is what all relativists call the fourth dimension. In general use when we speak of dimensions, we're just talking about spatial ones. Time is of course, not a spatial dimension.

 

[KevinF]

I highly recommend you read the book Flatland. It's an old book that is equal parts geometry and social satire of Victorian England. It's public domain and you can read it here.

The book helps you visualize the fourth dimension by telling the tale of a two dimensional being who is taken into the third dimension. It's not for everyone, let me tell you the way that helped me visualize the fourth dimension. I'm sure you know what Pascal's triangle is. Pascal's tetrahedron is the three dimensional version. Interestingly, the three sides are made up of pascal's triangle. Pascal's pentatope (a pentatope is the fourth dimensional version of the triangle) is the four dimensional version, and interestingly, the "sides" are all made up of pascal's tetrahedron. It's fun to play around with if you're ever that bored. Search on google for more information about that.

If you're interested in the mathematics of the fourth dimension, try mathworld.

 

[silverpig]

There are actually 2 "fourth dimensions"

The most popularly recognized one is time. It is a fourth dimension in which we exist.

There is however, a theoretical 4th spatial dimension. This is not time. You can sort of get an idea of how this dimension works if you consider the following:

Imagine trying to make a 2D map of a 3D terrain such as a mountain. This is easily done with a topographical map. I'm sure you've all seen one. A topographical map has 3 coordinates; an x coordinate (distance from the left side of the map let's say), a y coordinate (distance from the bottom of the map), and a z coordinate. Let's identify this z coordinate as a colour. Blue areas signify low points in the topology, and red areas signify high points, with a continual spectrum in between. You could look at this 2D map and picture a 3D surface by using the colour to represent the height of the mountain at any (x,y) coordinate you choose. Mathematically speaking, the height z, is a function called h of x and y.

z = h(x,y)

Now let's try to extrapolate to 4 dimensions by taking an intermediary step.

Imagine a room with a single heat source. Now let's look at this room with an infrared camera (heat vision). Hot areas are red, and cool areas are blue. Each point in 3 space will have a corresponding temperature value represented by a certain colour. Mathematically speaking, the temperature w is a function called W of x, y, and z.

w = W(x,y,z)

This equation looks very similar to our topographical map equation above. Let's change the meaning of w to be the "height" in the fourth spatial dimension along the w axis of our curve. This defines the surface (actually volume) of the curve as we look at it along the 4th dimensional axis.


Let me continue...

Let's go back to our topology map for a second. Let's draw lines on it where the colour is the same (these lines are called level curves). Many topological maps are drawn with these lines in them to show places where the elevation reaches 500 ft, 600 ft etc etc... If you were to look at the level curve corresponding to an elevation of 0 ft, you would see that it is a rather wide, closed shape. As you increase the z value, you will see the level curves get smaller and smaller until they reach a single point, which will be the peak of the mountain (let's assume this is a relatively simple topography).

Now, let's revisit our temperature room. Let's start at a certain point somewhere low on the w axis and look at the level surface. We may find it to be a large sphere centred about our heat source. As we move upwards along the w axis, we will see that sphere shrink down to a single point coinciding with our heat source.

Kinda hard to grasp, I know, but there you have it.

 

[kngai]

actually the temperature example said before can also have one more variable W=w(x,y,z,t)... just a mathetical model to define a state (in this case, the temperature at a pt in space at time t).. does not tell much about the 4th dimension argument...

This is actually a good topic.. make me to think for a while
i am kind of thinking 4th dimension exists only the first 3 are fullfilled... like.. you cannot draw a line without first putting a dot on a paper.. then after you have the line.. it opens up a "opportunity" for you to access the 3rd dirmension. just like you can't cut the core of an apple unless you have access to the 3 previous dimensions (which is to cut the apple open in the first place)...

damn.. back to study for my finals in 1 days...

 

[dejitaru]

So what then, partel, is the fourth?

spatial

 

[eakers]

i would think about the fourth dimension as time and you can see the fourth dimension as the change in the apple as time progresses.

my set theory prof brought in some blocks and had them in a bag was like "we are going to see a block in the fourth dimension" i dont really know what i expected but i was so disppointed when he showed four different blocks with respect to time.

anyways, thats my take on it.

 

[KevinF]

If you start with a dot, then drag it on a 2d plane you have a line. Take that line and drag it parallel to itself along a 2d plane and you a square. Take that square, drag it on a 2d plane, and you have a cube (this is what you are doing whenever you draw a cube by drawing two overlapping squares and then connect the corresponding points. If you take that cube and drag it on a plane, you create a a two dimensional representation of a Hypercube-4 or tesseract. You can take that hypercube-4 and drag to make a hypercube 5, 6, 7, 8, and on and on. It makes a really beautiful picture, especially once colorized. Come to think of it, I should load photoshop and make some really cool backgrounds. Maybe later.

You can see a picture of a tesseract here. There is both a 2d and 3d picture. The 3d picture is constantly shifting, in a way, it's simultaneously in all the different states pictured at once (the cube is pictured rotating in the fourth dimension).

Also, this is a great java applet. It lets you rotate a tesseract on all the different axises/planes. If you look around a bit, you should be able to find a program that'll let you manipulate and picture all the different polytopes. The 600-cell is a lot of fun...

Once again, I suggest everyone read Flatland. It's about 150 pages, public domain, and a very interesting read.

 

[serialb]

That Flatland link from U of Minnesota is dead...

I was reading Stephen Hawking's "Universe in a Nutshell". Near the end of the book it mentioned something about 11th-dimension superstring.