
Xkcd Makes 4D Miegakure the Most-Sought Indie Game - m0th87
http://www.geekosystem.com/miegakure-4d-game-info-no-demo-download/
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MikeCapone
I saw the video and it just looked like turning "no clip" on for a piece of
the puzzle and moving it where it's supposed to be. I'm probably missing
something about how this is supposed to work..

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lmkg
So, what's going on, is that one of the dimensions in the game world is
getting switched out for a fourth dimension. The visual transformation that
you see isn't very clear about what's going on, and I guess it's basically not
possible to be clear about this in a rectilinear coordinate system.

I'll try to explain. At 0:12, the game world has 3 axis, aligned with the
sides of the blocks. By convention, let's say the vertical axis is the Z-axis,
that the one going up and to the right (on the screen) is the X-axis, and the
one going up and to the left is the Y axis. There's also a "hidden" axis,
which I'll call the W-axis, which can't be displayed (for obvious reasons).

At 0:13, the player switches out the Y-axis for the W-axis. We can no longer
see the extent of objects in the Y direction. However, we can now see the
extent of objects in the W-axis direction. For example, in the XY-plane, the
ground was all green, but in the XW-plane it looks stripped with green and
brown. This means there are a series of XY-planes sitting "next to" each other
in W-space, alternating in colors. The character was originally sitting on one
of the green ones. Now that the Y-axis was switched for the X-axis, we see
X-axis "slices" of these XY-planes, arranged along the W-axis.

Note that the big rock structure is located on the W-axis only in the XYZ 3-D
"slice" that the character originally occupied. The character then proceeds to
move the wooden ring along the W-axis to a different XZ-slice, where the rock
structure isn't in the way.

At 0:18, he switches axes back to the XYZ plane, although this isn't strictly
necessary. You'll notice he's now on an XYZ slice that, in W-space, has brown
ground. The rock structure isn't in the way because it's on a different
location on W-axis than everything on-screen (it's displayed as transparent,
for reference). He moves the structure along the X-axis, past the rock
obstruction. He then switches back to XWZ-space, moves the wooden thing back
along the W-axis again, and switches back to the original coordinate system.

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MikeCapone
Thanks for taking the time to explain. I think I get it better now, but it's
probably one of these things that I'll have to try for myself to really
understand.

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zackattack
The Cartesian plane is in ℝ^2. Any point can be expressed as a combination of
(x,y) where both x,y ∈ ℝ. That means that both x and y are members of the set
of real numbers.

A 3D graph is in ℝ^3. Any point can be expressed as (x,y,z), where x,y,z ∈ ℝ.

A 4D graph is in ℝ^4. Any point can be expressed as a combination of
(x,y,z,w), where x,y,z,w ∈ ℝ. Unfortunately, human vision is limited to three
dimensions, so only four different perspectives are available (4 choose 3):
(x,y,z), (x,y,w), (y,z,w), and (x,z,w). Moreover, the 3D must be collapsed
into a 2D for presentation on the screen.

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psygnisfive
For those interested, the traditional names for the directions in 4 dimensions
(equivalent to "up" and "down", etc.) are "ana" and "kata".

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joeld42
I watched people playing this for about 15 minutes at GDC. Didn't try it
myself and I didn't know at the time that it was 4D, thought it was just some
weird xform of the world.

Looked amazing but was completely unintuitive to me. I didn't get it at all
but if playing for a while helps me learn to visualize 4D space then I really
want to try it.

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jerf
I don't know if you'll find this helpful or not, but there is one deceptive
aspect in the video. Visually, you see a world composed of cubes, but
geometrically that is not what is actually happening. In fact the "cubes" are
actually points being represented as cubes on the screen. The game world is
actually 10x4x6x4 in a discrete grid of points. (The 6 for height is an
estimate.)

When the first dimensional swap is done, if the world was literally as it
appeared, the big ring would become a two-dimensional square ring as you
swapped out its depth for the 0 it has in the fourth dimension. Instead, since
it is really just a structure composed of 12 points represented by 12 cubes,
the points remain in your 3D space and it seems untouched. That is because it
was 2D in the first place, despite being drawn as 3D, and swapping around the
two dimensions it didn't exist in in the first place leaves it untouched.

So to some extent it is cheating a bit with the fourth dimension, in the sense
that you aren't actually dealing with a continuous world, you're dealing with
a very coarse-grained discrete one that is being represented as if it
continuous. (I am not objecting. There is no practical other choice. I am just
pointing out that if you are already confused, this unavoidable deception may
make it more difficult to understand what is going on.)

I will be intrigued to see how well the puzzle designers learn to cope with
4D. I was initially excited, but I'll want to see more than what they showed.
I want to rotate the fourth in on the other height axis. (Now, that's a
sentence I don't have the chance to write very often.) Keeping gravity
constant would require that the height dimension be held constant, and that
would probably be an acceptable compromise, vs. freely rotating in all
directions. But we've seen a mere minute of a game, so who knows. (The number
of people in other discussions I've seen about this video that implicitly
assume the interlocking ring thing is somehow the _only_ puzzle that could be
given is driving me nuts.)

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anonjon
So, the world is composed of hypercubes. Neat.

If you want to go the other direction dimensionally and technologically, you
can get 'Flatland, a romance of many dimensions' off project Gutenberg.
<http://www.gutenberg.org/etext/97>

It is a pretty amusing read.

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albertsun
I'll echo that. Flatland was a great read.

[http://www.amazon.com/Annotated-Flatland-Romance-Many-
Dimens...](http://www.amazon.com/Annotated-Flatland-Romance-Many-
Dimensions/dp/0465011233/)

The above is the version I read with a ton of great annotations and stories
about the author Edwin A. Abbott. I'd highly recommend it.

