What is a Tesseract anyway?
If you already have a good idea of what a tesseract
is, feel free to skip down to the next section.
What is a tesseract? That's hard to explain straight
away. However, I'll assume that the gentle reader is familiar with the
concept of real dimensions. Dimensions are used to describe the size of
real objects. A point is a dimensionless object; it has negligible length,
width, and height.
The next step up from a point is a line. If you
move that point along a straight path, you have a line, an shape with one
dimension, which we'll call length. Realistically, it could be width or
height or some diagonal dimension, but the idea is the same.
The next step is a two dimensional object. This could be any of a number
of things: triangles, circles, rhombuses, hexagons. But for the purposes
of this discussion, let's assume we want a square. We can make a square
by placing four lines end-to-end with right angle at each corner. We now
have a square, a 2 dimensional shape.
The next step after that is a cube--a 3 dimensional
object, which we can make by connecting the edges of 4 squares.
So far everything is grounded in reality. The next
step is where we stretch our neurons. I've shown you how to go from 1 to
2 and from 2 to 3 dimensions. Now imagine that we can make a four dimensional
shape. Just as a 3 dimensional cube is formed of 2 dimensional squares,
a 4 dimensional tesseract is formed of 3 dimensional cubes.
It is impossible for our minds, evolved on a 3 dimensional
world, to fully imagine what a 4 dimensional object looks like. However,
in the diagram at the right, we can see a partial representation of one.
If you stack several cubes together as shown, you have a structure similar
to the one shown. This is called an unfolded tesseract. It is called
such because it is analogous to unfolding a box. If you unfold a box, you
can lay it out flat. It essentially becomes a 2 dimensional object, an
assortment of cardboard squares. If you unfold a tesseract, you have a
3 dimensional object, an assortment of cubes.
The thing to remember (and that I didn't understand
when I first went to find out about tesseracts) is that this is only a
3 dimensional representation of the object. There are missing links we
have to think about. Note that when you unfold a box, only one of the panels
touches all the side panels that it did when it was folded. So it is with
the unfolded tesseract. There is only one cube that touches one other cube
on each of its faces. In a folded tesseract, all cubes touch all but one
of the other cubes through some 4th dimension.
The easiest way to picture this is to take the above
diagram and imagine the missing connections. The right surface of the bottom
block would connect to the bottom surface of the right block, for example.
All the outer surfaces of the bottom 5 blocks connect to the topmost block,
thus if you go down from the bottom, you end up on the top of the top block.
If you go left from the topmost block, you end up on the left end of the
leftmost block.
Navigating a
Tesseract
Even given the above, a tesseract can be a confusing
beast to understand and get around in. But before we even start, we need
to make a few assumptions about what our tesseract is going to be like.
This will take a few definitions, since we aren't used to dealing with
4 dimensional objects in our everyday life.
The first definition is cells. This is what
I'll call the 3 dimensional cubes that make up the 4 dimensional tesseract.
Cells of a tesseract can be like any real 3 dimensional space. For ease,
assume that it's the shape of a 3 dimensional cube. Cells can be number
of things; they can be like rooms or chambers in a dungeon, or they can
be solid blocks of material through which tunnels and rooms are bored.
In the below discussion, I use cell and cube interchangeably
Interfaces are the faces of the individual
cubes, and each such face is shared by two of the cubes. This interface
can be a wall with one or more portals in it, may require transit via some
magical means. Or, the boundary may not even be noticed by someone lost
in the tesseract--they may be walking down a hall and pass from one cube
to the next without noticing. In the below discussion, I use interface
and face interchangeably.
Once you've decided what your tesseract is like
inside, you next need a way to keep a way of where the players are. For
this, the image at the left is provided; I call this image an interface
map. It is an alternate way of representing a tesseract. Unlike the
"unfolded tesseract" above, it shows all of the interfaces intact. Instead
of being an "unfolded" tesseract, it is a "squashed" one.
The interface map shows 8 images. All images are
repeats of the same representation of a tesseract, but each one highlights
a different one of the 8 cells. Though the cells look like skewed cubes,
this is needed for the diagram to work. In four dimensional space, the
corners of the cells can be perfectly square.
Now, note each diamond (4 sided figures, "squashed
squares") in the figure. Each one of these represents one of the interfaces.
Each one is shared by exactly two of the shaded figures. This can tell
you what cubes are connected? For example, look at the topmost shaded diamond
in figure 1. See if you can find what other cell has that exact same diamond
shaded.
Find it yet? It's number 3. Once you get the hang
of it, you can use the interface map to tell which faces of what cubes
lead to what other cube. If you don't get it at first (I didn't) it might
be easier to make a chart to find your way around. A sample chart might
be like this. Note that I've used numbers instead of directions here, since
depending on how you orient gravity, directions might be totally subjective.
Cube faces separated by a slash denote cube number and face number respectively:
Cube face table: body if table is destination
cube and face when leaving the listed cube and face. The cube faces are
arranged around a cube like pips on a dice are arranged around a dice.
Face=>
Cube #: |
1
|
2
|
3
|
4
|
5
|
6
|
|
1
|
2-6
|
5-2
|
6-3
|
7-4
|
8-5
|
4-1
|
|
2
|
3-6
|
5-6
|
6-6
|
7-6
|
8-6
|
1-1
|
|
3
|
4-6
|
5-5
|
6-4
|
7-3
|
8-2
|
2-1
|
|
4
|
1-6
|
5-1
|
6-1
|
7-1
|
8-1
|
3-1
|
|
5
|
4-2
|
8-5
|
6-2
|
7-2
|
3-2
|
2-2
|
|
6
|
4-3
|
5-3
|
7-4
|
3-3
|
8-3
|
2-3
|
|
7
|
4-4
|
5-4
|
3-4
|
6-3
|
8-4
|
2-4
|
|
8
|
4-5
|
3-5
|
6-5
|
7-5
|
5-2
|
2-5
|
Example: A PC in cube 5 walks out a portal in face 3 of that cube. The
PC emerges from a portal is face 2 of cube 6 (listed as "6-2" in the chart.)
Note that a players who aren't aware that they have
crossed the boundary of a tesseract (or not aware that they are in a tesseract)
will have a heck of a time trying to map the thing until (and unless) they
figure out that they crossed a boundary (or that they are in a tesseract.)
For example, lets say you PCs are in cube 3 as described above. They exit
the cube via face 5. They emerge from face 2 of cube 8. Then they take
a left and exit via face 3 of cube 8. They then emerge in face 5 of cube
6. If they take another left and leave via face 4 of cube 6, entering face
3 of cube 3, the cube that they started in. A player mapping this might
assume that they drew the map wrong, or arrived in a fourth room identical
to cube 3. But in fact, this is the exact cube they started from.
Uses for a Tesseract
So now that we know what a tesseract is like, what
the hell use is one in a role playing game? There are a couple uses that
spring to mind:
-
As a trap or prison. One thing you'll notice about a tesseract right
away is that every face of every cell leads to another one of the constituent
cells. Therefore, you can't just burrow your way out of one (unless you
can burrow through a fourth dimension, which most PCs aren't capable of.)
-
As a dungeon, lair, dwelling or stronghold. Some crazed undead wizard
of demigod might find a tesseract an engaging project as well as a secure
stronghold.
-
As a demi-plane, layer, or campaign setting. If you're looking for
a truly bizarre place for your PCs to adventure--permanently or temporarily--try
a tesseract world. For some ideas how to do this, see the below section
on "Gravity."
Ways In and Out (and more on traps)
As noted above, since any way you travel inside
a cell of a tesseract, you can only get to other cells of the tesseract.
This little property has led to the concept of tesseracts being a trap
among RPGers who have taken the time to ponder tesseracts. However, the
converse of this property means that you cannot just wander into a tesseract
from the outside.
So to get in and out you need some other means.
This usually means some magical (or ultra tech) portal, gate, space folding,
or teleportation. Now if that means of transport is one way, the trap is
complete unless the PCs have access to some means of transport of their
own.
You may choose to secret the "exit" portal somewhere
within the tesseract, or make special procedures that are required to open
the portal. One way is to make a solution that requires that the PCs understand
the geometry of tesseracts. One way is to require that they visit some
or all the rooms in a certain order in order to open the portal. They may
also try to communicate with four dimensional creatures that can move along
a fourth dimension--for them moving out of the tesseract would be much
easier as they can move directly along a 4th diminution.
For more on portals, gates, and way to use them,
take a look at TSR's PLANESCAPE campaign setting.
Gravity and Cell Interiors
Once you've settled on making a tesseract as a setting
for your PCs to bang around in, there are more questions to consider. The
foremost is gravity. Once we introduce a fourth dimension, the "up" and
"down" of the real world become meaningless. We have to decide for ourselves
what the insides of the tesseract's cells will be like, with regard to
gravity and other factors. Some possibilities:
-
Arbitrary external reference: Choose a direction for gravity that
would make sense if the tesseract was a dimensional structure. For example,
take a look at the first figure above, the "unfolded
tesseract." As shown, assume that for each cube, face 6 is on the top,
face 1 is on the bottom. Cubes 1 to 4 are the central cells, top to bottom.
Cells 5-8 are the cells around cell 3. (This is the assumption that I used
when I made the cube face table.) Then "up" would
be towards face 6 of each cube. "Down" would be towards face 1 of each
cube.
This leads to some interesting situations when you
consider that you're really navigating a four dimensional space. For example,
let's say that your intrepid PCs are in cell 2. They walk out face 2 of
said cell. This leads to cell 5, face 6. This is the "ceiling" of said
cell; as soon as the first PC steps through the threshold of the interface,
the orientation of gravity shifts--and the character goes plummeting to
the bottom of cell 5! (If the party was running from some beasties that
you stocked the tesseract with in the first place, several PCs might take
the fall--ooh, this is beginning to sound like an entry in Grimtooth's
Traps...)
-
Random: Pick a face of each cube to be the floor--or roll a d6 for
each cell. Then the PCs will really have to be on their toes, as they never
know from on cell to the next what direction gravity will be towards. Of
course, clever PCs combined with obvious clues (such as opening a portal
and seeing something crawling across the opposite "wall", actually the
floor of the next cube) will give it away, but things like darkness spells
and magic portals that don't transmit anything short of a living being
across the threshold will perturb such efforts.
-
Towards cube faces: This is a little simpler than either of the
previous versions. Simply assume that gravity is towards each cube face.
Where two cube faces meet, presumably the direction of the gravity field
is split along a 45 degree line from the corner where the two faces meet.
A character walking towards a corner that is not careful will find more
and more of their weight being pulled "forward" instead of the current
"down. When more than half of their weight is being pulled "forward" instead
of "down", the would likely tumble over and fall to the ground of the face
they were moving towards.
If there are long term inhabitants of the tesseract,
they will have known about this phenomenon for a long time. Accordingly,
they might build steep ramps that allow a traveler to make a smooth transition
from one face to the next. You would go "up" one side of the ramp and "down"
the other. To an observer unfamiliar with tesseracts, it would look like
you walked up the ramp and up the wall.
Another thing to consider with this scheme is portals
through the interfaces. You can forget the idea that portals are just like
the door to your house or any dungeon room. These portals would invariably
be in the floor, i.e. "down". This creates some interesting situations
as well, because the guy on the other side of the interface from you thinks
that you are down from him, too.
At simplest, in a small tesseract, this could be
a trap door. You would have to be careful as you go through. You
couldn't just "hop" through, because as soon as you go through the portal,
the direction of gravity is reversed- you would almost immediately reverse
direction and fall back the way you came. You would bob back and forth
between the tow cells a few times, but would eventually come to rest, suspended
in virtual weightlessness at the interface between the two cells. If the
portal is small enough, you might be able to reach out and pull yourself
to the side that you were trying to get to anyway, but it would probably
have been simpler to crawl over the edge to the other side in the first
place.
Another variation of this would be instead of a
simple trapdoor for a portal, a long shaft. The simplest variation of this
would be to have a ladder (or rope) from one end of the shaft to the other.
Climb down one end of the shaft to the interface, turn your body around
so your head is facing the new "up" (this could be tricky), and climb up
to the destination cube.
Mean DMs can exclude the ladders and give it slick
walls and razor sharp protrusions (which nick characters as the fly by
- remember that they bob up and down and might make several passes through
the interface, thus hitting any protrusions several times.) Even if there
are no protrusions, remember that any thing that falls down the shaft will
eventually come to rest halfway down the shaft. A falling PC will smack
into anything resting there. Then, unless the PC has a means of ascent,
they will be stuck at the middle of the shaft until something else happens
along. If if nothing else happens along, the PC is likely to die of starvation
(or, the PC might be eaten by the last now-starving beastie that fell down
the shaft.)
One final suggestion for the shaft version is for
a magic item of sorts that might make getting across the shaft easy. If
identified by a PC, tell them that the item increases falling damage and
removes the upper limit on falling damage. The PCs might think it a cursed
item and might even toss it. Of course, what this item does is compensates
for energy lost to air friction. A character that falls into one end of
the above mentioned shaft/portal would fall all the way to the other end
("down" the middle and then "up" the other end.) A character that jumps
down a shaft with this item will fall down to the middle, where gravity
reverses, and by then is already moving quite quickly, and will make it
all the way to the top of the other end of the shaft, where they will stop
and start falling the other direction. Of course, since you are nearly
stopped at the end, it's pretty safe to reach an arm out and crawl out
on the other end. If they don't, they will just bob up and down forever.
A friendlier way to make the interface is illustrated at the right. The
portal might have a circular "lip" that when stood upon, gravity always
points towards the center of the lip. Thus someone who might want to get
to the other side of the portal would simply walk up to the "hole" and
keep walking around the edge of the lip to the next cell. The picture at
right shows a cross section of the portal and an intrepid adventurer walking
to the other side of the portal. The arrows show the direction of gravity
(i.e., "down".)
Of course, any of the portals may be guarded or
covered by fortresses or other structure to test your PCs mettle before
they pass on to the next layer.
-
Radially outwards, spherical floors: In my opinion one of the best
options for a tesseract if you want to make a large tesseract like a demiplane
or mini-world setting. Imagine that each cube is solid with a spherical
section removed from the middle. The spherical cavity would touch the walls
of the cube at 6 points--at the center of each face of the cube. This is
perfect spot for any of the portals described above in "Towards Cube Faces."
Inhabitants and visitors on the inside of the spherical
cavities would simply walk (or ride, but you get the idea) wherever they
need to in the cell. If the tesseract is large enough, the upward curving
would be gradual and you wouldn't have any problems with sudden gravity
shifts like you would anywhere else.
Presumably, if you want to have terrestrial flora
and fauna in the shell, you need to put some form of light/heat source,
or artificial sun, at the center of the spherical cavity. A bizarre addition
to this is to say that this source also generates the outward "antigravity"
force that makes everything inside seem to be pulled outwards towards the
"ground" of the spherical cavity. If this behaves like any known force,
the repulsion would get stronger as you approach the center, effectively
making it so you can never reach the artificial sun. Additionally, this
repulsive force would keep the artificial sun balanced in the center of
spherical cavity.
-
"Multiworld Tesseract in Disguise": So far, we have assumed that
you can travel from one cell to adjacent cells by traveling "outward" from
the cell via a portal. In reality, you would have to transit a fourth dimension,
and you might have to travel inward or a variety of directions if the portals
are situated in such a manner. Indeed, this allows you to come or go from
the tesseract outwardly, but by traveling in an inward direction, you can
travel to the other cells of the tesseract.
So far, so good. Now apply this to a whole campaign
world--that is a full size, everyday, run-of-the-mill planet like Earth
with the gravity towards the center. Assume that at six points on the planet's
surface, you have a portal--presumably one at each pole and four evenly
spaced along the equator. By entering one of the portals, you can enter
another campaign world. You effectively have 8 linked campaign worlds (effectively
in different planes.)
This could be a curious entity indeed. Let's say
that the portals are the lip-type described under the descriptions of "towards
cube face" configuration tesseracts. You walk into the portal and you end
up in one of the other worlds. Your players might assume initially you
are introducing them to a hollow would situation. But the conditions on
any of the linked campaign worlds would be at least superficially similar
to their home world. The sun would be in the sky, the planet would be spherical
so the land curves down not up (and thus there is sky, not ground, overhead.)
Considering how big worlds are, your players might take quite a while to
figure out that the portals don't just link two worlds, but eight (if you
let your PCs know there are 6 evenly spaced gates on the world, they might
think they all connect to the same world--but each portal leads to a different
world.
How Did the Tesseract Come About?
This is a question you might want to think about,
as it will likely affect what the interior of your tesseract is like. Here's
some food for thought:
-
Natural Phenomenon in a Normal Universe: Unlikely at best, since
all but the multiworld tesseract do not provide any means of access or
viewing by anyone in a normal 3 dimensional universe. All cubes are surrounded
by other cubes, so how would you ever touch or see a tesseract?
-
Alternate Universe with Different Physical Laws: Though this is
more likely than the above, it's still not too likely, but does bring up
some interesting possibilities. In a 4 dimensional universe, what's to
keep us 3D people from slipping along a fourth dimension and spontaneously
disappearing from the sight of our 3D buddies who cannot see or sense a
fourth dimension? A little mind-bending to contemplate.
-
Alien Habitat: This a variation of the alternate universe. Say that
some 4 dimensional beings need a 4 dimensional habitat to live in, but
they live in a special cranny of our own universe or one like it. Three
dimensional beings might co-exist with the 4D beings. But the 4D beings
would presumably be able to travel along a fourth dimension, and thus would
be able to hop from cell to cell without using the portals.
-
Construct: This seems to be the most likely result. Some crazed
wizard, god, pantheon, or magically talented race decided to construct
the tesseract for a variety of reasons: as a plaything, experiment, prison,
or refuge come immediately to mind. It might also have some 4D beings on
it; either created or imported by the creator of the tesseract, or somehow
evolved.
Taking This Further
This is just the tip of the iceberg--or just one
cube of the tesseract. Mathematicians who have no concept of what a role-playing
game even is sit around and think about this stuff all the time. The idea
of extra dimensions was first spawned by a book called Flatlander.
In addition to this classic, you can find many more modern, updated books
on the topic--go to a good library and look under the topics of tesseracts,
polygotes,
or n-dimensional geometry.
Note that we have stuck with the idea of tesseracts
in this articles. A tesseract is just a specific case of a polygote.
A tesseract is a cube expanded into a fourth dimension. A polygote
can be nearly any regular shape (look at your dice bag...) in any number
of dimensions (four, five, whatever you want or need). If you find that
your players have an easy time navigating a tesseract, you might want to
try something more complicated.
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