The ideas behind Brightland:

1.  The Dimensional Analogy:

    It all really started with Edwin Abbott Abbott's Flatland.  The main relationship between Flatland and Brightland was in its presentation of a dimensional analogy. Flatland takes place within a planar universe.  The narrator, A Square, is guided by a 3-dimensional sphere to expand his horizons and try to understand the three dimensional universe.  To help him, the sphere presents a dimensional analogy.  By pointing out the differences between 0-, 1- and 2-dimensional universes, the sphere invites A-square to extend the patterns and conjecture about the third dimension.
    Brightland attempts to continue this analogy.  In Brightland, the analogy between 2-dimensional world of Flatland and the 3-dimensional world Monochromia is used to promote an understanding of the true 4-dimensional universe in which Brightland takes place.

2.  Embedded Shapes:

    One of the ideas that personally fascinated Jonathan was the idea that a lower-dimensional object could not exist unless it could be embedded in a higher-dimensional space.  An example of this is the surface of a sphere, which is itself two dimensional but cannot exist in any less than 3-dimensional space.
    Brightland, from the point of view of the player, is similar. The world of brightland appears three-dimensional at all times, because the player is three-dimensional.  But the space of Flatland acts in ways that a Euclidean three-space cannot.  So we learn about 4-dimensional space indirectly by seeing what 3-dimensional formations are possible through it.

3.  Marathon 2:

    Why Marathon 2?  Why not a more realistically rendered and more modern game engine, such as the one used by Unreal?
    Three reasons, two simple, one more complex.  The first simple reason is that Marathon 2's sophisticated terminals made telling a story easy.  The second is that Jon was already familiar with the editor for Marathon 2.  He has two other ongoing Marathon 2 projects: The Mission Unstoppable project and the Spontaneous Human Combustion project.  Both of these are somewhat larger than Brightland and use more facets of the game engine, but do not use the four-dimensional capabilties of the game.
    The more complex reason is that modern first-person games think about the world differently from second-generation games like Marathon 2.
    The first generation of games were based on square grids of tiles.  Wolfenstein 3D was a prime example.  The game stored the universe as a series of cubic objects which were either filled or empty.  Empty cubes represented the space inside rooms, while filled cubes made up the walls.
    Marathon was the first Mac game not the be square-based.  In addition, it was the first Mac game where altitude became an issue. But fundamentally, Marathon is still a two dimensional game.
    How is it possible to make a two dimensional game that looks so three dimensional when played?
    The answer can be found by looking at how the map editor utility works.

This is a map of a level from one of Jon's older projects, called "What's you can't see can hurt you."  As you can see from this picture, the world is defined as a series of connected convex polygons with up to 8 sides.  These polygons are called "sectors". Each sector has a floor height and a ceiling height.  And that's how it appears three dimensional.  You see height, but the computer is thinking about polygons.
    To have two spaces directly above one another, you have to make the polygons overlap.  How does Marathon 2 know which polygon you're standing in?  Well, to enter a polygon, you must either teleport or walk into it.  Teleporting requires that the level builder give the ID of the polygon to teleport into, so that's a definite case.  When you walk into a polygon, you must do so by crossing one of its sides.  Marathon 2 keeps track of which sides are shared by which polygons and calculates your current position based on which side you crossed and where you crossed from.
    Notice that altitude plays no role in this calculation.  That means that you can have two overlapping polygons with the same floor and ceiling heights, and Marathon will still regard them as completely separate spaces.  This is not Euclidean behavior.
    It is this feature of being able to share space between polygons that led to the decision to use Marathon 2 for Brightland.  Because altitude plays no role in the calculation of which polygon you're in, any value that could be assigned continuously could be used as a dimension, and Marathon 2 would ignore it just as much as it ignores altitude.  Luminosity (how much light is in a room) was chosen because the player would be able to detect it easily.

The best demonstration of how Marathon 2 lets the same space be part of two different sectors is the multi-player scenario 5-d space. A picture of the map is to the left.  Three parallel corridors running north-south appear to intersect three parallel corridors running east-west.
 
 

    They do share the same space, but since none of their sides have been adjoined together, it is impossible to get from one set of corridors to another.  You have to take one of the many twisted side-passages.  This makes for incredible disorienation at first.
    Although this configuration is impossible to build in 3-space, it is easy to build in Brightland.  Just take the north-south corridors and move them slightly brighter than the east-west corridors, and uddenly they don't intersect at all. Marathon simulates this behavior nicely.

Professor Banchoff's Class:

    We have studied several topics which are relevant to Brightland in class this year.  Coordinates are relevant to Brightland because all of the objects within the levels have to be identified by their X and Y coordinates plus the ID of the sector they are in (which provides their brightness and altitude).
    Slicing is key to understanding Brightland because Monochromia is a slice of Brightland such that the lighting levels are constant across the slice.
    There is a hypercube in one of the later levels.
    The Marathon engine uses perspective projection to change the 3-d space it creates for the player into a 2-d projection onto the computer screen.

The Fifth Dimension:

    In the level "With Thoughts Toward Higher Things", we discover a hallway that is, in fact, embedded in five-dimensional space.  This was the most difficult part of the story to make understandable because we don't have any direct access to the fifth coordinate (while we could tell how high something was and how bright and how far north and how far west.  The player has to infer the existence of a fifth dimension from a discontinuity in the brightness of the corridor.
    The reasoning which leads to this conclusion is explained in the walkthrough.

Social Commentary:

    Since there was social commentary in Flatland, we've decided to add a little social commentary of our own.
    There are four classes in Brightlandian society.  Blue is the ruling class, then green and purple, and orange is the lowest class. The official justification is that blue is an indication of intelligence.  The law against heresy is not an injunction against beliefs that are against the official doctrine, but against beliefs which (they believe) are manifestly false and could lead people do the wrong thing because of the false information. But the reality is that nobody has access to the truth.  The blues themselves only use this pretense to add legitimacy to their rule.
    Brightland is indeed a very dark, almost distopian presentation of what interdimensional contact might be like.  The four-dimensional creatures generally use three dimensional creatures for their own purposes.  And every dimension shows a contempt for the one below it.  This contempt is undeserved.  In our ending we point out that the Monochromians can make circuit boards which are three dimensional.  These are far superior to four-dimensional technology because they take up no four-dimensional space, so they can be used anywhere.  The moral of that part of the story is that we have just as much to learn from lower dimensions as from higher.