**Definition:** A hypermaze is a Maze in a higher dimension. A hypermaze
is more than just what is normally considered to be a "3D Maze". A standard 3D
Maze has passages like any other, just they tunnel through a 3D solid
instead of move across a 2D surface. A hypermaze however is a true multidimensional
Maze, and increases the dimension of the solving object and the passages
themselves. In a normal Maze you move a point through it, where the path behind
you forms a line. In a hypermaze you move a line through it, where your path
forms a surface!

**Dimensions: **In general, for any Maze, be it a standard Maze or a
hypermaze, the solution is represented by an n-dimensional figure. That
n-dimensional figure is formed by moving an (n-1)-dimensional object along the
solution path. The environment forming the Maze itself is a (n+1) or higher
dimensional object. For example, a standard non-hypermaze consists of moving a
0D object (a point), along a 1D solution path (a line), through a 2D or higher
dimensional environment (a plane, solid, etc). In a hypermaze you move a 1D
object (a line), along a 2D solution path (a plane), through a 3D or higher
environment (a solid, 4D solid, etc).

**Hyperhypermazes:** Hypermazes can be more than just moving a line. A
hyperhypermaze increases the dimension of the solving object and the solution
path again. In a hyperhypermaze, you move a 2D plane through a 4D or higher
dimension environment, where your path forms a 3D solid. A hyperhypermaze can be
considered a hypermaze of the 2nd order. A hypermaze of the 3rd order is even
more extreme, involving moving a 3D universe or hyperplane through a 5D
environment or beyond, forming a 4D solution path! A normal hypermaze is a hypermaze of
the 1st order, while a standard Maze involving moving just a point is a
non-hypermaze, or a hypermaze of the 0th order.

**Physical model:** In solving a non-hypermaze, you work with what amounts
to a point, such as a marble, a cursor, or yourself. In solving a hypermaze, you work with
a line, such as a long section of string. You grab and drag different parts of
the string, moving it up and down and back and around past the obstacles in the
hypermaze. Note the solving line in a hypermaze needs to act like an infinitely
long
line, or rather you shouldn't be able to retract the ends of the line to inside
the hypermaze. A mere line segment would be equivalent to a non-hypermaze, because you
can just crumple the section of string into a ball, and then move the point
through like in a non-hypermaze. As you move the line into the entrance face of
the 3D hypermaze object, through
the hypermaze across to the exit face, the ends of the line should stick out
both adjacent faces. This can be modeled by an infinite line without endpoints,
or a finite line segment where the ends are tied to points outside the
hypermaze. A example physical model of a hypermaze is a 3D
construct in the middle of a room. A piece of string has ends attached to
opposite walls of the room. You then have to move the middle part of that string
into one side of the 3D solid and out the other.

**Entrances:** The entrance and exit to a non-hypermaze is a point. The
entrance and exit to a hypermaze is of course a line. Multiple entrances on the
face of a 3D non-hypermaze consist of a set of points, like a piece of Swiss
cheese. Multiple entrances on the face of a hypermaze consist of crisscrossing
lines, like the canals on the surface of Mars. Note the outer surface or any
cross section of a hypermaze can form a non-hypermaze. In a 2D non-hypermaze, a
solution path starting on one edge and ending on another splits the walls into
two halves. In a 3D hypermaze, a solution surface starting on one face and
ending on another again splits the solid into two halves. In the physical model
above, the hypermaze could rest on a table, however the top half needs to be
suspended from the ceiling, since the solution technically cuts it into two
pieces with space between them.

**Obstacles:** A model of a hypermaze can have relatively sparse
obstructions and still be challenging. This sparseness can allow a physical
model to actually work where one can reach inside the solid at points to
manipulate the string. A non-hypermaze (especially in 3D) needs to be relatively
solid, i.e. basically needs to be formed of tunnels, because there are so many
places a point can fit. A "passage" for a hypermaze line to follow needs to at
least be a plane area, like a low ceiling warehouse or the area behind a
bookcase. Note a "dead end" in a hypermaze can be a mere column in the middle of
a room, and doesn't need to have walls all around, where 1D "tentacles" or lines
are enough to obstruct and form dead ends for the solving string. Picture a
parking garage with a concrete column down the middle, and you're walking a dog,
where the leash connecting you and the dog is part of the hypermaze solving
line. If the dog goes around one size of the column and you go around the other
side, then the line gets caught on the column, and column stops you from getting
across the room. If however the column is like a lamppost instead of connected
to both floor and ceiling, then you can throw the leash over the pole and keep
going. For another example, picture an upright letter "F" attached to the floor,
between a floor and ceiling, while the solution string approaches it from the
right. If the string goes between the two horizontal tines of the "F", or
between the bottom tine and the floor, it will get caught on the stem of the
"F", and that will act like a dead end for that section of the string (the
proper path goes over the top of the "F").

**Obstacle formula:** More generally, for any Maze or hypermaze, the
dimension of the obstacles needs to be at least (dimension of environment) -
(dimension of solving object) - 1. For example, for a standard non-hypermaze in
2D, you need at least 1D lines to obstruct (i.e. form tunnels and dead ends for)
a 0D solving point in the 2D environment. For a non-hypermaze in 3D, you need 2D
surfaces to obstruct (i.e. form tunnels and dead ends for) a 0D solving point in
the 3D environment. For a hypermaze in 3D (our example here) you need at least
1D lines to obstruct (i.e. form places to get caught on) a 1D solving line in
the 3D environment. For a hyperhypermaze in 6D, you need 3D solids to obstruct a
2D solving surface in the 6-dimensional environment. And so on.

**Alternate model: **In addition to an infinite line, another model that
works for a hypermaze line is an elastic circle or ring. The entrance and
exit are ring shaped holes in the sides of the hypermaze object, with at least one wall in the middle of
the ring preventing you from collapsing the ring to a point. You then move the
ring down the "rail", stretching and compressing its circumference as
appropriate. In this model, the hypermaze is divided into two pieces by the
solution - one piece for the outer Maze and one piece for the part inside the
ring, where the inner piece needs to be suspended in air, e.g. held at the right
point with wire or whatever in a physical model. Note in all general hypermazes,
the solving line is completely elastic, so can be stretched and bent around any
obstacle to form an arbitrarily curved solution surface. Of course there are
many possible variant rules one can add to a hypermaze for extra challenge, such
as one where the solving string can't be bent beyond some radius.

**Example models:** In the hypermazes pictured here, there's a ceiling floating above a foundation.
From the ceiling "stalactites" dangle down, and from the floor "stalagmites"
grow up, and they branch and twist around in a complex fashion like a briar
patch. The hypermaze consists of two separate solids, just as how in a 2D
non-hypermaze, with the entrance on one side and the exit on the other, the
solution path divides it into two separate shapes. Your goal is to move string
between the floor and ceiling, in one side and out the other, like a piece of
dental floss. Note for hypermazes of this model, it doesn't matter which side
you enter. You can enter from the north face and exit on the south or vice
versa, you can enter from the west face and exit on the east or vice versa, or
you can make the string enter at a corner.

**Small example:** Above is a perspective rendering of a simple 9x9x9
cube hypermaze. Note the Maze is reasonably sparse, where there are places
you can see through and out the other side. If you were to start by moving the
string in from the leftward (dark red) face, note there are 5 possible ways of
entering. The row across the very bottom is one, and there are four possible
combinations in the upper half. If you were to start by moving the string in
from the rightward (pink) face, there are 2 possible ways of entering. The row
across the very top is one, and the path from upper left to lower right is the
other.

**Solution coloring:** Above is another
rendering of the same 9x9x9 cube hypermaze, viewed from the opposite corner.
Here everything connected to the base is colored blue, while everything
connected to the ceiling is red. This is somewhat of a spoiler or a solution to
the hypermaze, because to solve it you just need to have all parts of the string
between a red and blue section at all times. (A similar method works to
solve 2D non-hypermazes, if you color the walls on one half a different color
from those in the other half.) Note the red pole surrounded by blue near the
bottom of the left hand face. Part way through solving, you'll need to loop the
string out and around that red pole, like throwing a rope over the top of a
lamppost.

**Large example:** Above is a orthographic
rendering of a very complex hypermaze (73x73x73 cubes). Note there's literally
billions of ways of arranging the string across and moving it into one of the hypermaze's faces. Because
this is a "perfect" hypermaze, only one solution and hence initial
starting configuration actually works! And that's just to get
started, much less to actually navigate through the hypermaze.

**Solution half: **Above is a picture of
just the bottom half piece of the same 73x73x73 hypermaze, with the top half
removed. This exposes the solution similar to how different coloring does, where
the solution surface always follows between the area removed and the area still
present.

**Solution surface: **Above is the unique
solution to the 73x73x73 hypermaze, showing the complex solution surface, or the
path taken by the string. The edge colored in red is the edge of the surface,
i.e. the entrance line that the string must enter (no other combination works)
if you enter via this face, or the exit line that the string leaves from if you
exit via this face. The edge colored in blue is the same entrance/exit line
along the adjacent face. Note there are some holes in the surface, whose
circular edges are colored green. These are places where the solution surface
goes back to and briefly touches a boundary face (like the red pole in the earlier
example). This is just a rendering effect to show the solution
surface clipped to the boundary of the hypermaze itself, where the actual
solution surface of course has no holes. Note the green figure-8 shaped circuit near the far right edge. This
shows an interesting point where two such holes touch. A hypermaze solution
surface will never actually intersect itself, but it can briefly touch itself at
various points, where some areas may need to be passed through by the solving
line more than once. The analogy here is having your thumb and forefinger form a
circle, almost touching, where to move the solving line over a solution surface
formed by your fingers, requires moving it through the narrow space between the fingers
twice.

**Hypermaze duals:** The passages in a 3D non-hypermaze are similar to the walls
in a
3D hypermaze. Basically, a 3D non-hypermaze and a 3D hypermaze are duals of each
other, where turning all barriers into open space and vice-versa turns one into the
other! Ok, there's a slight difference, where a standard 3D non-hypermaze consists of
a single connected tree, where a 3D hypermaze has two separate trees attached to boundary surfaces.
In the hypermaze graphics pictured, you can treat the blocks as passages, and
attempt to
find a path from the top face to the bottom face. The dual of a Maze is
basically the dual of its graph, with the new passages being the old walls and
vice versa. A good example of that is this 2D
non-hypermaze. Normally black
is wall and white is passage, where your goal is to navigate from the top to bottom. However
you can also try to solve the dual, and treat white as wall and black as
passage, where your goal is to go from the left edge to the right edge.

**Solving the dual:** For any Maze or hypermaze, if it has a solution, then its dual has no
solution. In the example 2D Maze, if there's a white path connecting top to
bottom, then there can't be a black path connecting side to side, and vice
versa. Similarly for the 3D hypermaze, if you can follow a solution surface
through the solid, which splits the environment into two pieces, that means the
dual 3D non-hypermaze is unsolvable, since there's no path from the top wall to
the bottom wall. The reverse is of course true too, where if there's a path
connecting the top wall to the bottom wall, then the hypermaze is unsolvable, as
there's no way to run a solving line through the solid without getting caught on
the column connecting the two halves. In a 2D Maze, it's always the case that
either the Maze is solvable or its dual is solvable. However in a 3D
environment, it's possible that neither Maze is solvable! Consider a hypermaze
where the top and bottom halves are separate, but linked like a chain. In such a
case the 3D non-hypermaze is unsolvable, because there's no way to get from the
top to the disconnected bottom. However the hypermaze is also unsolvable,
because there's no way to run a length of string between the two halves, because
they're inseparably linked.

**Dual formula:** The general rule is for a hypermaze of order p in a n-dimensional
environment, its dual is a hypermaze of order (n-p-2). For example, in a 2D
environment, the dual of a non-hypermaze (order 0) is another non-hypermaze (2-0-2
= 0), as seen in the 2D example above. In a 3D environment, the dual of a
standard hypermaze (order 1) is a non-hypermaze (3-1-2 = 0) and vice versa, as
seen in the 3D example above. The dual of the tunnels forming a 4D non-hypermaze
is a hyperhypermaze (4-0-2 = 2nd order hypermaze), which you should be able to
run a solving plane through forming a solution solid.

**Creation algorithm: **The ability to create
3D hypermazes is in the latest version of the Maze program
Daedalus. All hypermazes on this page were created and rendered by the
program. The algorithm for creating a hypermaze turns out to be pretty simple. Start with the top and bottom floors, and then grow a bunch
of random wall branches from them like a tree, where each appended branch
connects to an existing object at exactly one end. This is exactly the same as
how one adds walls to a 2D non-hypermaze when creating a Maze by adding walls,
which also simultaneously grows two trees of walls inward from boundaries.
Creating hypermazes is
also similar to how one carves the tunnels in a 3D non-hypermaze, with the only
difference being the tree structure in a hypermaze forms walls instead of the
passages, and in a hypermaze you grow two trees simultaneously. These hypermazes are
"perfect", in that there's exactly one solution surface, just as how a perfect
non-hypermaze has exactly one solution path and exactly one path from any point
in the Maze to any other point.

**Solving algorithm: **The ability to
solve hypermazes is also in the latest version of
Daedalus. To solve a hypermaze means finding its solution surface. The
solution surface can be generated by focusing on the different solids that
compose the hypermaze, similar to coloring them differently as described earlier. For any two adjacent bits of obstacle with unbroken space
between them, if the two bits are part of the same solid, then connect them.
This fills in all the false areas, leaving just the cracks between separate
solids. For a perfect hypermaze, the space left will be exactly the solution
surface. This solving method works similar to the way of solving a standard
non-hypermaze by filling in all the dead ends. Although in actuality, solving hypermazes is a 3D version of the "blind alley sealer" solving algorithm, which
also works by adding connections between contiguous areas of the Maze.

**State: **Hypermazes are special because they
go beyond having an easily definable state. Most Mazes (and puzzles in general) tend to have a finite number of easily
enumerable states, and a small number of actions you can take at any one state,
i.e. it's defined by a simple graph. This is true regardless of the type of Maze,
where it covers almost all logic Mazes or Mazes with special rules, where I call this concept
Maze equivalence. For example in a 10x10 orthogonal passage
2D Maze, there
are 100 possible locations where the solving point can be, and in any cell you
have up to four choices you can take to visit neighboring cells. However in a hypermaze
represented by a 10x10x10 arrangement of blocks, there are virtually an infinite
number of combinations of how the solving line can be spread out through it, and
ways in which you can manipulate the string at any one configuration, just as
there's virtually an infinite number of ways you can run a length of string
through a room. It's easy to count and enumerate all the states of a
non-hypermaze. Although the states of a hypermaze are still ultimately
finite, an analogy is that a non-hypermaze is "countable" like the set of
integers in a range, while a hypermaze is "uncountable" like the set of real
numbers in a range.

**Interface:** Because hypermazes are fundamentally different from other
Mazes and state based puzzles, there's a lot of potential to make interesting
new computer or physical puzzles involving hypermazes. The most challenging
aspect of hypermazes is easily interfacing with them. A computer interface to
navigate a hypermaze is challenging to make, since you need to be able to look
at all parts of the line, and the nearby parts of the hypermaze in the 3D
environment, and be able to select and drag different parts of the line, where hopefully the line acts like a real string where points near the grab
point get partially dragged too. A physical model of a hypermaze is also
challenging to make, because the model is delicate yet needs to be rigid, and it
tends to defy gravity.

**Philosophy: **A hypermaze is interesting because your focus is not on a
single point, or on your individual self, but is rather across multiple points,
where you need to deal with every point along a line at once. Similarly in life
one can think for just themselves, or can work for the environment and serve the
world as a whole. :-) An interesting team building exercise would be a large
physical hypermaze, where the line consists of people tied along a rope at
different points, where they all need to cooperate and move together to get
through!