**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 that 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, and the
path behind you forms a line. In a hypermaze you move a line through it, and
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
dimensional environment, and 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,
in which 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 top of 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, then 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 "r".

**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 that
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, in
which 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, in which the
actual solution surface of course has no holes. Note the green figure-8 shaped
circuit near the far right edge. That 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, it requires
moving the line 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, because a
standard 3D non-hypermaze consists of a single connected tree, while a 3D
hypermaze has two separate trees attached to boundary surfaces. In the
hypermaze graphics pictured, you can treat the wall 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, and 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, and 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 also 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, then 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, because 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 versions 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 the boundaries.
Creating hypermazes is also similar to how one carves the tunnels in a 3D
non-hypermaze, with the only difference being that 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 versions 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 method of solving a standard non-hypermaze by
filling in all the dead ends. Although more technically, 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. the decision tree is defined by a
simple graph. This is true regardless of the type of Maze, and 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, and
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,
in which 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, in which the line consists of people tied along a
rope at different points, and they all need to cooperate and move together to
get through!