Semantics of the term 3D

[Jo-Jo]

Hello, I have been thinking about the term 3D. If you provide the length, width and height of an object, how would you tell a square or a circle apart? According to Wikipedia, a sphere has four points. Three values show the location of the center of the sphere and the fourth the radius. But how do you represent a sphere without a fixed point? Or is it's given location at a given time always required for calculations?

The example of an ant was given in another thread; well on a sphere an ant can go in infinite directions. It seems like 3D is a useful term, but that real space would be described by infinite points?

 

[HallsofIvy]

Hello, I have been thinking about the term 3D. If you provide the length, width and height of an object, how would you tell a square or a circle apart? According to Wikipedia, a sphere has four points. Three values show the location of the center of the sphere and the fourth the radius.

I doubt that Wikipedia actually says "four points". The four "values" you state are numbers, not points.

But how do you represent a sphere without a fixed point? Or is it's given location at a given time always required for calculations?

In that case, use a variable or expression for that "point"

The example of an ant was given in another thread; well on a sphere an ant can go in infinite directions. It seems like 3D is a useful term, but that real space would be described by infinite points?

"3D" is "three dimensions" NOT "three points". You are making the same mistake as above. Any single point in space requires 3 numbers to precisely describe it. In a "Cartesian coordinate system" those would be "x", "y", and "z", the distances of the point from an arbitrarily chosen "origin" point along three arbitrarily chosen orthogonal axes. In "spherical coordinates" the numbers would be "\(\displaystyle \rho\)", the distance from an arbitrarily chosen origin, "\(\displaystyle \theta\)", the angle the projection of the line from that point to the origin to an arbitrarily chosen plane through the origin makes with an arbitrarily chosen line in that origin, and "\(\displaystyle \phi\)", the angle the line from the origin to the point makes with the perpendicular to that plane, through the origin. In "cylindrical coordinates" the numbers are "r", the distance to the point from an arbitrarily chosen origin to the projection of the point in an arbitrarily chosen plane through that origin, "\(\displaystyle \theta\)" the angle the line from the origin to the projection, and "z" the distance to the point along a line perpendicular to that plane through the point.

Different ways to describe the precise position of the point, but always requiring three numbers- "three dimensions" or "3D".

I haven't seen the Wikipedia article you refer to, and you don't give a link, but I imagine that the point of the article was that the "space of all possible spheres" was "four dimensional" because we can precisely describe any specific sphere by telling where its center is using three numbers and then a single number to tell its radius.

In physics, we deal with "events", that happen at a specific point at a specific time. We require three numbers to specify the point and one number to specify the time. That is what Albert Einstein meant when he said that "we live in a four dimensional space-time continuum".

 

[Jo-Jo]

On the sphere page, points are mentioned a lot; "A sphere is uniquely determined by four points that are not coplanar. " https://en.wikipedia.org/wiki/Sphere#Equations_in_three-dimensional_space

If time and the radius are the fourth value (is that right?) why is the radius not "another dimension"?

This all started when I saw something about 4D video games, but I can't figure out what makes them 4D other then a gimmick.

 

[Jo-Jo]

I should have included the link for the "4D" game that claims to portray a 4th dimension of space:

 

[Dr.Peterson]

On the sphere page, points are mentioned a lot; "A sphere is uniquely determined by four points that are not coplanar. " https://en.wikipedia.org/wiki/Sphere#Equations_in_three-dimensional_space

If time and the radius are the fourth value (is that right?) why is the radius not "another dimension"?

This all started when I saw something about 4D video games, but I can't figure out what makes them 4D other then a gimmick.

How many points (4 for the sphere), or measurements (1, the radius, for the sphere), determine a figure is a different thing from the dimensionality of the space it is in. In the section you refer to, what makes the space (not the sphere) three-dimensional is that each point needs three coordinates (x, y, z) to identify it.

Did you try taking the link in the first paragraph of the article? https://en.wikipedia.org/wiki/Three-dimensional_space

Then you can try https://en.wikipedia.org/wiki/Four-dimensional_space

The idea of "walking through walls" using a fourth dimension (which would not be time, but an imagined space dimension) comes from the idea that if you lived in two dimensions (on the surface of a paper), you would be amazed to know that you could be lifted off the page and set down in a different place. Look up "flatland" for more about this idea. (I didn't look at the video.)

 

[Jo-Jo]

Thank you, I wanted to make sure I understood. I had read some of the 3D page but it did not help me differentiate points from dimensions. The 4D article is very interesting, I thought time was the only 4th dimension and that Einstein had proposed the concept. I now have a basic understanding that "four-dimensional space is simply a space with four spatial dimensions, that is a space that needs four parameters to specify a point in it". The video game is exploring the idea of a 4D space in a 2D interface.

Thank you both for your time, it helps me to discuss what I have read.

 

[Cubist]

This all started when I saw something about 4D video games, but I can't figure out what makes them 4D other then a gimmick.

I had a look at the game video. It seems very well done, and it seems a nice idea, and I love the transition animations when the character "sidesteps" in the fourth spatial dimension BUT the character's vision seems limited to a set 3d slice within the "4d game environment".

Therefore you could equally think of this game as having a transport button that takes you from your current (x,y,z) location within one 3d game map, and dumps you into the same (x,y,z) location within a different 3d map. Clever use of this transport button would let you avoid obstacles that exist in one map but not another.

An analogy that has one less dimension would be a flat character within a book containing a few pages of (2d) game maps. The book is closed, and the player is able to jump from his position on one page to a corresponding position on an adjacent page. But unlike a bookworm who is truly burrowing through 3d space as it eats from page to page, the player can't turn their view to look "out of the page".

 

[Steven G]

Young kids can do amazing things! When I played basketball as a teenager I put all my five fingers on the ball. How did I do this? Simply amazing.

You claim that the fourth point on a sphere is the radius of the sphere. Excuse me, but the radius is NOT a point. It is a length!

Please be more careful with what you say.

 

[Jo-Jo]

Young kids can do amazing things! When I played basketball as a teenager I put all my five fingers on the ball. How did I do this? Simply amazing.

I am not sure who are addressing, but with the capacity I have available to me now, it seems that you are either mocking someone who is taking their first steps towards higher learning, or those who are helping someone do so. It is not constructive in any way.

You claim that the fourth point on a sphere is the radius of the sphere. Excuse me, but the radius is NOT a point. It is a length!

You have not provided a quote, so I am not sure what part of this thread you are referring too. It does not seem like you read the entire thread. On my part however, my misunderstanding and misuse of the term point was the source of my confusion, as was explained by your colleagues. If I understood what I was getting wrong I would not be here seeking help. Would you criticize a toddler for misstepping?

Please be more careful with what you say.

This is very good advice, that perhaps you should consider yourself.

 

[Jo-Jo]

I found this video helpful for this topic:

This is where I have landed to learn physics basics, before I continue to more complex ideas:
www.khanacademy.org I like that it reintroduces me to actual math, while also providing visual and auditory content that reinforces it (at least on the oscillator section that I am watching now).

 

[Steven G]

I am not sure who are addressing, but with the capacity I have available to me now, it seems that you are either mocking someone who is taking their first steps towards higher learning, or those who are helping someone do so. It is not constructive in any way.



You have not provided a quote, so I am not sure what part of this thread you are referring too. It does not seem like you read the entire thread. On my part however, my misunderstanding and misuse of the term point was the source of my confusion, as was explained by your colleagues. If I understood what I was getting wrong I would not be here seeking help. Would you criticize a toddler for misstepping?



This is very good advice, that perhaps you should consider yourself.

I noted that you can put 5 fingers on a basketball showing that a sphere has at least 5 points.

Three values show the location of the center of the sphere and the fourth the radius. You really should know, and I suspect do know, that a radius is not a point.

 

[Jo-Jo]

I noted that you can put 5 fingers on a basketball showing that a sphere has at least 5 points.

Thank you for clarifying.

You really should know, and I suspect do know, that a radius is not a point.

I do know this now after coming here, and at some point I am sure I did, but really confused myself when reading the Wikipedia pages. I have since started uses sources that don't assume a person knows all the terms.

 

[Jo-Jo]

a sphere has at least 5 points

Could you please explain this to me? How would this be shown in a formula?

 

[Jo-Jo]

In "spherical coordinates" the numbers would be "ρ\displaystyle \rho", the distance from an arbitrarily chosen origin, "θ\displaystyle \theta", the angle the projection of the line from that point to the origin to an arbitrarily chosen plane through the origin makes with an arbitrarily chosen line in that origin, and "ϕ\displaystyle \phi", the angle the line from the origin to the point makes with the perpendicular to that plane, through the origin.

How would this be shown in a formula? Or a diagram? I do not entirely understand how to describe a sphere.

 

[Cubist]

There's more than one way to specify a sphere.

I think your original post was probably written after reading this quote from the sphere Wikipedia page :- "a sphere is uniquely determined by four points that are not coplanar." (EDIT: These points all touch the sphere's surface.) This is true, but it's not a common way to specify a sphere in mathematics. This specification requires 12 numbers (or variables) when you write the points down as coordinates (x1, y1, z1), (x2, y2, z2), (x3, y3, z3), (x4, y4, z4). However the four point method is very interesting to think about, it's worth reading the first 2 paragraphs in that link.

But it's far easier to specify a sphere by its centre and radius which only requires 4 numbers:- centre is at (x0, y0, z0) and radious "r". The equation that allows (x,y,z) to be any point on this sphere's surface is (x-x0)^2 + (y-y0)^2 + (z-z0)^2 = r^2.