Which notation do you prefer?

[pcwiz]

Okay then how would you explain the Cartesian Coordinate system?

 

[blade740]

The cartesian coordinate system is the most commonly used today. You measure distance away from an origin in each dimension. It can be 1-dimensional, 2-dimensional, 3-dimensional, 19872-dimensional. If you're using (x, y) you're using the cartesian system. If you're using (x, y, z) you're using it. It's just an extra dimension.

 

[pcwiz]

The cartesian coordinate system is the most commonly used today. You measure distance away from an origin in each dimension. It can be 1-dimensional, 2-dimensional, 3-dimensional, 19872-dimensional. If you're using (x, y) you're using the cartesian system. If you're using (x, y, z) you're using it. It's just an extra dimension.

So how would you explain it to a person who has no idea what the Cartesian Coordinate system is and doesn't know what it's for?

 

[brunson]

So how would you explain it to a person who has no idea what the Cartesian Coordinate system is and doesn't know what it's for?

Well, first I'd ask if they've ever graphed anything or seen graph paper, but since there are people on the list that my not have done that I'll take this approach:

Look at a wall in the room your in. Pick one that, while you're facing it, meets the floor and another wall in the lower left hand corner, I'm going to refer to this corner as the "origin". Now, spit on the wall. Where did the spit land?

There are a few simple ways you can describe where your spit landed. Let's say you drew a line straight down from the loogie to the floor and another one horizontally to both walls. Now if you were to take a tape measure and measure the distance along the floor from the origin to the vertical line you drew, then a second measurement from the origin straight up to the horizontal line, you could say, 5 feet horizontally and 4 feet vertically from the origin. The uniquely identifies where the spit is on the wall.

Hey, guess what? You've just used two dimensional Cartesian coordinate to describe where you hocked your phlegm. If you were to lay out a grid on that wall drawing a vertical line every foot from the origin and a horizontal line every foot from the floor, that would be a two dimensional coordinate system.

Rene Descarte, the French philosopher and mathematician, was the first to formalize this system of describing location. The corner formed by the wall and the other wall to the left, he designated the y axis and the corner formed with the wall and the floor he called the x axis. So, if you spit again you only have to say, "It landed at (4,6)" to describe the location 4 feet from the left wall and 6 feet from the floor.

This can be extended to 3 dimensions by adding an additional axis along the floor and the wall to the left, and we'll call that... the z axis! So, standing in the middle of room you can describe the location of your head as 4 feet from the wall on the left, 5 feet off the floor and 5 feet from the front wall (measured along the z axis) or (4, 5, 5) in three dimensional space. Notice that our concept of two or three dimensions is a direct derivative of the Cartesian coordinate system.

So, an excellent model of this is the core of your 3x3x3. Take your cube apart right now because you know it's been too long since you cleaned it. Look at the core, it has six arms extending from the center. Think of the center of the cube as the origin, the horizontal arms pointing left and right are the X axis, the vertical arms as the Y and the ones pointing towards and away from you are the Z axis. An x rotation of the cube turns it around the x axis, similarly with y and z rotations, they turn the cube around the y and z axes.

Now the only question is which way to turn it. That's where the left hand rule come in. In a standard cartesian system with your origin in the lower left hand corner of your room the origin it 0,0,0 and x increases positively towards the right and negatively towards the left, y grows up and z grows as it comes towards you.

So, if you take your left hand as described in my previous post and point your thumb in the direction that x gets larger, i.e. to the right, then your fingers curve down and away from you. Turning the top face of the cube away from you and down is a positive x rotation, turning it the other way is a negative x rotation, or an x' turn. The same with y, pointing your thumb up your fingers turn away and to the right, so a cube rotation in that direction is y and the opposite is y'. Z rotations work the same way, just point your thumb at your face and a clockwise rotation, just like your fingers curl, is z and counter clockwise is z'.

So there you have your first two weeks of Algebra 2 summed up in a single posting. You can paypal me a dollar for the education. ;-)

Seriously, though, I hope that was clear and helpful.

 

[fanwuq]

Pcwiz,

I thought your were sarcastic! I mean, everyone should know what the Cartesian coordinates are. What are in? 3rd grade? That is why x,y,z is so much easier to understand.

(r) (u) (f) are ambiguous. I thought they are optional or fingertricks for double layer turns.

I do like 001 for R, but it is too long to right. I say the current system has it's faults but is understandable in general. We do need a new system when the 6x6+ Vcubes come out however.

 

[Feanaro]

I have never even seen the second kind of notation. Do the ()'s make a r different than a (r)?

 

[MistArts]

I have never even seen the second kind of notation. Do the ()'s make a r different than a (r)?

Yes... r is a double layer turn and (r) is an x rotation

 

[Feanaro]

Wow, that is a lot easier. I also never connected the x,y,z notations to the cartesian plane.

3x3x3 PB: 18.22 sec
3x3x3 OH: 1:17
3x3x3 BLND: 2:55
2x2x2 PB: 7.35
4x4x4 PB: 2:30

 

[pcwiz]

So how would you explain it to a person who has no idea what the Cartesian Coordinate system is and doesn't know what it's for?

Well, first I'd ask if they've ever graphed anything or seen graph paper, but since there are people on the list that my not have done that I'll take this approach:

Look at a wall in the room your in. Pick one that, while you're facing it, meets the floor and another wall in the lower left hand corner, I'm going to refer to this corner as the "origin". Now, spit on the wall. Where did the spit land?

There are a few simple ways you can describe where your spit landed. Let's say you drew a line straight down from the loogie to the floor and another one horizontally to both walls. Now if you were to take a tape measure and measure the distance along the floor from the origin to the vertical line you drew, then a second measurement from the origin straight up to the horizontal line, you could say, 5 feet horizontally and 4 feet vertically from the origin. The uniquely identifies where the spit is on the wall.

Hey, guess what? You've just used two dimensional Cartesian coordinate to describe where you hocked your phlegm. If you were to lay out a grid on that wall drawing a vertical line every foot from the origin and a horizontal line every foot from the floor, that would be a two dimensional coordinate system.

Rene Descarte, the French philosopher and mathematician, was the first to formalize this system of describing location. The corner formed by the wall and the other wall to the left, he designated the y axis and the corner formed with the wall and the floor he called the x axis. So, if you spit again you only have to say, "It landed at (4,6)" to describe the location 4 feet from the left wall and 6 feet from the floor.

This can be extended to 3 dimensions by adding an additional axis along the floor and the wall to the left, and we'll call that... the z axis! So, standing in the middle of room you can describe the location of your head as 4 feet from the wall on the left, 5 feet off the floor and 5 feet from the front wall (measured along the z axis) or (4, 5, 5) in three dimensional space. Notice that our concept of two or three dimensions is a direct derivative of the Cartesian coordinate system.

So, an excellent model of this is the core of your 3x3x3. Take your cube apart right now because you know it's been too long since you cleaned it. Look at the core, it has six arms extending from the center. Think of the center of the cube as the origin, the horizontal arms pointing left and right are the X axis, the vertical arms as the Y and the ones pointing towards and away from you are the Z axis. An x rotation of the cube turns it around the x axis, similarly with y and z rotations, they turn the cube around the y and z axes.

Now the only question is which way to turn it. That's where the left hand rule come in. In a standard cartesian system with your origin in the lower left hand corner of your room the origin it 0,0,0 and x increases positively towards the right and negatively towards the left, y grows up and z grows as it comes towards you.

So, if you take your left hand as described in my previous post and point your thumb in the direction that x gets larger, i.e. to the right, then your fingers curve down and away from you. Turning the top face of the cube away from you and down is a positive x rotation, turning it the other way is a negative x rotation, or an x' turn. The same with y, pointing your thumb up your fingers turn away and to the right, so a cube rotation in that direction is y and the opposite is y'. Z rotations work the same way, just point your thumb at your face and a clockwise rotation, just like your fingers curl, is z and counter clockwise is z'.

So there you have your first two weeks of Algebra 2 summed up in a single posting. You can paypal me a dollar for the education. ;-)

Seriously, though, I hope that was clear and helpful.

Great Description! I can easily understand it and I think a beginner would too!

 

[pcwiz]

Pcwiz,

I thought your were sarcastic! I mean, everyone should know what the Cartesian coordinates are. What are in? 3rd grade? That is why x,y,z is so much easier to understand.

Well I'm just saying because I didn't learn it in 3rd grade (at least not the z axis) and nobody here mentioned it except Brunson. I never knew the simple ordered pairs was the Cartesian coordinate system. I just thought it was the coordinate system.

By the way I'm not sarcastic a lot, and if I tried, it would be the worst sarcasm you've ever seen.

 

[pcharles93]

Wait, why is the "Cartesian" necessary? If it's the only coordinate system out there, why label it with an unnecessary adjective?

 

[pcwiz]

Maybe the person who invented the Cartesian Coordinate system has the last name of Cartesian, and the want to be credited for their invention?

This is just a logical guess.

 

[blade740]

It's not the only coordinate system out there. It's just the most commonly used. The spherical system comes to mind. (r, θ, φ)

 

[brunson]

It was named after Rene Descarte and there are other coordinate systems.