The mechanics of how the 4th Dimension work IN RELATION to the 3rd Dimension

Let's begin with orthogonal geometry, since that's what we're talking about here:

(n = 0)
In zero dimensions you have a point. Nothing more, nothing less. It has no area, no volume—it exists only as a dot of infinite smallness.

(n = 1, corners = 2, edges = 1)
If we leave that point and create a second point, we've got one dimension. For purposes of example, we'll call the length of the line "one", so the two points have coordinates 0 and 1, with only one possible line connecting the two of them.
Points:
0
1

Edges:
0 - 1


(n = 2, corners = 4, edges = 4, planes = 1)
Making a parallel copy of that line distance one away from the original line at a right angle to the original, you get a square. Four points connected by four edges and those four lines define a plane, face or square.
Points (Corners):
(0,0)
(0,1)
(1,0)
(1,1)

Edges:
(0,0) - (0,1)
(0,1) - (1,1)
(1,0) - (1,1)
(0,0) - (1,0)

Planes (Squares):
(0,0) (0,1) (1,0) (1,1)


(n = 3, corners = 8, edges = 12, planes = 6, cubes = 1)
Making a parallel copy of that square distance one away from the orignal square at a right angle to the original, you get a cube. 8 points connected by 12 edges

Points (Corners):
(0,0,0)
(0,0,1)
(0,1,0)
(0,1,1)
(1,0,0)
(1,0,1)
(1,1,0)
(1,1,1)

Edges:
(0,0,0) - (0,0,1)
(0,0,0) - (0,1,0)
(0,0,0) - (1,0,0)
(0,1,1) - (0,0,1)
(0,1,1) - (0,1,0)
(0,1,1) - (1,1,1)
(1,1,0) - (1,1,1)
(1,1,0) - (1,0,0)
(1,1,0) - (0,1,0)
(1,0,1) - (1,1,1)
(1,0,1) - (1,0,0)
(1,0,1) - (0,0,1)

Planes (Squares):
(0,0,0) (0,0,1) (0,1,0) (0,1,1)
(0,0,0) (1,0,0) (1,1,0) (0,1,0)
(0,0,0) (1,0,0) (1,0,1) (0,0,1)
(0,1,0) (1,1,0) (1,1,1) (0,1,1)
(0,0,1) (1,0,1) (1,1,1) (0,1,1)
(1,0,0) (1,0,1) (1,1,0) (1,1,1)

Cubes:
(0,0,0) (0,0,1) (0,1,0) (0,1,1) (1,0,0) (1,0,1) (1,1,0) (1,1,1)


With me so far?

Let's take it to the 4th dimension!

(n = 4, corners = 16, edges = 32, planes = 24, cubes = 8, hypercubes = 1)
If we copy our cube and move it perpendicular in the 4th dimension to the existing cube, we get…

Points (Corners):
(0,0,0,0)
(0,0,0,1)
(0,0,1,0)
(0,0,1,1)
(0,1,0,0)
(0,1,0,1)
(0,1,1,0)
(0,1,1,1)
(1,0,0,0)
(1,0,0,1)
(1,0,1,0)
(1,0,1,1)
(1,1,0,0)
(1,1,0,1)
(1,1,1,0)
(1,1,1,1)

Edges:
(0,0,0,0) - (0,0,0,1)
(0,0,0,0) - (0,0,1,0)
(0,0,0,0) - (0,1,0,0)
(0,0,0,0) - (1,0,0,0)
(0,0,0,1) - (0,0,1,1)
(0,0,0,1) - (0,1,0,1)
(0,0,0,1) - (1,0,0,1)
(0,0,1,0) - (0,0,1,1)
(0,0,1,0) - (0,1,1,0)
(0,0,1,0) - (1,0,1,0)
(0,0,1,1) - (0,1,1,1)
(0,0,1,1) - (1,0,1,1)
(0,1,0,0) - (0,1,0,1)
(0,1,0,0) - (0,1,1,0)
(0,1,0,0) - (1,1,0,0)
(0,1,0,1) - (0,1,1,1)
(0,1,0,1) - (1,1,0,1)
(0,1,1,0) - (0,1,1,1)
(0,1,1,0) - (1,1,1,0)
(0,1,1,1) - (1,1,1,1)
(1,0,0,0) - (1,0,0,1)
(1,0,0,0) - (1,0,1,0)
(1,0,0,0) - (1,1,0,0)
(1,0,0,1) - (1,0,1,1)
(1,0,0,1) - (1,1,0,1)
(1,0,1,0) - (1,0,1,1)
(1,0,1,0) - (1,1,1,0)
(1,0,1,1) - (1,1,1,0)
(1,0,1,1) - (1,1,1,1)
(1,1,0,0) - (1,1,0,1)
(1,1,0,0) - (1,1,1,0)
(1,1,0,1) - (1,1,1,1)

(I'll leave the planes & cubes for you to work out, but it pretty much follows from the above...)

reply to post by JoshNorton


that was pretty labor intensive! nice job.


I was not long ago sorta obsessed about the cube and cubic sections. the 6 inward-facing tetrahedrons with edge length (sqrt3) is a real trip.

Originally posted by tgidkp
reply to post by JoshNorton

 

that was pretty labor intensive! nice job.

I feel bad for not having finished it, but I was getting hungry, and it's not the easiest thing to be able to reproduce from memory—it's been years since I've gone through that exercise…

Back when I was younger and first got into such things I used to play 4D tic-tac-toe on a 5x5x5x5 tesseract. Figuring out the diagonals was a bitch!

reply to post by JoshNorton


Stop making us regular Joes feel stupid.

Originally posted by JoshNorton
Let's begin with orthogonal geometry, since that's what we're talking about here:

(n = 0)

(n = 1, corners = 2, edges = 1)


With me so far?
NOT REALLY

The only way to get a corner, let alone edges is
2B square

The beauty is that you as a child and you as and old person exist at once. You are connected like a snake through time. You are also connected to every possible outcome of you. The thoughts, fears, intuition or inner voice is the many fragments of you.

For example, say that you got a call to go for a ride in your buddies new car, you contemplate, dont really want to but you do. later on you guys crash. After the accident you yell yourself over and over, "man, if I only stayed home and played some vid games or something none of this would of happen". This message resounds in you, you in the past, and you in the future, always you. You get a call to go for a ride in your buddies new car, you contemplate, get a bad vibe, and tell him that you are going to stay in and play a some vid games. You go to bed late that night.

Get in the habit of analyzing your day and think how you could of done it better, think in the past of things you wished you done differently. You will help yourself in the past, future, present and parallel states get in the habit of helping yourself with the gift of time. Its like having many yous scouting the future for you and reporting back to you the right choice you should take for you have lived and are living many outcomes right now This is just a slice of the snake of time you chose to see. Chose which one you prefer.

Next time you hear that voice in your head know that it is you.