(Re)visiting 4D space with Miegakure

Miegakure is possibly the best educational game of the century. For those who have been waiting for years, check out the byproduct: http://4dtoys.com/ available on iOS and Steam, I believe.

Right, so, 4D space. We're not talking about space-time in this conversation; every dimension considered here is a spatial dimension, i.e. dimensions of space: left, right, up, down, forward, backward, and 4D-forward or 4D-backward. There are major differences between the 4D spacetime construct and the actual 4 spatial dimensional world, all of which are not discussed in this blog entry.

Navigating the 3D world(s) using the 4th dimension as a corridor

We start with a highly relatable scenario.

Figure 1.

Depicted in the picture are:

• our protagonist in a red shirt (bottom left), 
• a giant wall acting as a barrier, and 
• a damsel in distress fighting a purse-snatching thief on the other side

We would like to cross this wall to aid the lady. But how? Miegakure suggests that we step into the 4th dimension, go to a parallel 3D world where the wall isn't there, cross the non-existent wall, then pop back into our original 3D space.

How does that work? The video does an excellent job explaining the mechanisms of the game; this blog post will attempt to do the same while giving the readers an intuition for how those mechanisms would behave outside the constraints of the game.

A highly useful technique

in struggling with 4D concepts is to relate the experience to that of a person living in a 2D world trying to understand 3D space. There are many ways to do this.

Consider a simple "barrier crossing" example in 2D space.

Figure 2.

In this world, there are two meaningful directions; we can name them according to our view: up/down and left/right.

Young blue circle can't seem to get to his beloved, no matter which combination of those directions he tries.

However, since we the viewers are 3D beings, we perceive something that blue circle does not: the third dimension. In our world, the problem looks more like this:

Figure 3.

So we can circumvent the wall entirely by moving "through" it; that is, we move around the wall in an alternate world in which the wall doesn't exist.

To achieve this effect, a couple steps need to happen, we need to:

Figure 4.

1. rotate our entire world (in this case 2D) perpendicular to the original world, a.k.a. The Turn. This puts us into a 2D plane we've never been in (the grey plane)
2. move along the 3rd dimension (which is now one of the two dimensions we exist in after The Turn)
3. rotate our entire world back to the original orientation. We are now in a world "parallel" to the original. As we move around in this parallel world, our "shadow" on the original world moves with us (through the wall, if we wish)
4. do another pair of rotation to move along the 3rd Dimension Corridor and arrive back on our original 2D plane

A couple things of interest:

Every plane is a separate 2D universe

In order to "pass through" the wall, blue circle has to traverse through a total of three foreign worlds (grey, pale blue, and another--different--grey worlds). These worlds are just as large as the original world: infinitely long and wide; they merely exist in a different orientation.

At any moment, blue dot can only occupy and observe one world; his eyes are not capable of 3D vision. All that blue dot is aware of are the things happening inside each world in which he is currently situated. This means blue dot has no idea what's going to happen every time he initiates The Turn. Depending on his location in the original world, he could be turning into any of the infinitely many "vertical" worlds.

Once in the vertical world (grey), blue dot can traverse along the z-axis to "elevate" himself away from his original world. The purpose of this movement is to find the blue plane, i.e. the world in which the barrier doesn't exist. Since blue dot is residing inside the vertical corridor, he cannot observe the parallel blue plane, so it's pretty much trial and error as far as finding a blue plane as desired is concerned.

The Corridor worlds have only ONE meaningful direction

Consider the grey plane in the picture, reposted here for convenience:

Figure 5.

After the first Turn, blue dot arrives in the grey vertical corridor world. This world has 2 dimensions (just like every world that blue dot can exist in), one of which is the same as in the original world: sideways parallel to the barrier.

As intuited from the figure, moving along this direction achieves nothing; only movement in the vertical (aka green) direction is meaningful for the purpose of scaling the barrier.

One step in the green direction gets us to world Slightly Blue. Maybe we'll enter this world right in the Oval office; maybe we'll enter this world inside a bubbling volcano. Who knows. Instead, if we take two steps, we end up in a different parallel world: Slightly Bluer, and our entry into this world opens up in the Andromeda galaxy.

There is continuity in the 3rd dimension, an example that extreme would not happen if we were to travel in 4D space in a manner similar to what blue dot employs for his endeavor. However, what the extreme example conveys is that we will fail to comprehend the things we see as we make these "turns". Just as a 2D person being capable of seeing 3D spheres only as expanding circles, we as 3D beings can only see 4D objects as shrinking and expanding 3D structures. So every time we enter a different "slice" of reality, we get a completely different and unpredictable 3D view. I find it intuitive to consider this different view a different 3D world.

How does Miegakure explain this same approach?

A side-scrolling 2D Mario is slightly different from our protagonist Blue Dot. Mario has "forward / backward" and "up / down" as his dimensions. To him, turning "left" and "right" might as well be witchcraft.

So the same procedure applies:

First we initiate The Turn to rotate into a vertical world

Then we move forward in this new, vertical, corridor world. Forward now has a new meaning.

We reverse the rotation to arrive in the parallel world (Note that as far as the damsel can tell, we've disappeared without leaving a trace.)

In this parallel world, there is no wall so we move our physical body beyond it. Correspondingly, our shadow in the original world also passes through the wall with us.

All that's left is to reverse the process and arrive back at the original world (be careful not to overstep or we'll end up in a different parallel world!)

Now let's try the 4D equivalent:

Initiate The Turn

Walk forward in this new, vertical, corridor world

Reverse the rotation to arrive in a parallel 3D world

Move our physical body in the direction towards the wall and cross it. Our 3D shadow in the original world also passes the wall as we move.

Then we do a pair of rotation and come back to our original world, arriving in a different location than when we left.

In this Miegakure world, we have to be mindful about what our movement means. Sometimes taking a step forward means "choosing a parallel world to drop into", while at other times moving around simply means "manipulating my spatial position in the original world".