"We own the mathematics; the mathematics doesn't own us."
Things and relationships between them
Our whole geometry depends on just 2 basic ideas:
Things and the relationships between them.
We will pick a few things to include in our
conceptual universe; we certainly are not going
to try to invent a geometry which can describe
every that might exist. We also pick and choose
the relationships we care about and ignore all
the rest.
To make some sense out of that, we will define some
jargon — special words that we use to mean
"things that we care about" and "relationships that
matter to us".
A NODE is a discrete thing
a Euclidean point, a desk, a tree in the forest
— we get to choose which things
something real or something conceptual
a countable thing: nodes map to positive integers
An EDGE is a relationship between 2 of the things
in our geometry
"edge" is a synonym for "relationship"
ANY 2 nodes can make an edge
(depending on whether we care about
the relationship between them)
— we get to choose which relationships
we care about
or: edges are pairs of nodes: {A,B}
and: every edge has EXACTLY 2 nodes
Our edges are not directional; relationships go both ways.
The edge from A to B is the same as the edge from B to A.
(Other geometries do use directional edges, but not ours.)
Paths from here to there
Next we'll define a few ideas that will help us
describe the bigger picture which these nodes
and edges make.
A PATH is a sequence of nodes where
each successive pair is an edge
or: where every successive pair has
a relationship
(A,B,C,D) if and only if the edges
{A,B}, {B,C}, and {C,D} all exist
so: a path has at least 2 nodes
(because that's how we defined an edge)
Two nodes may be CONNECTED
there is a path from one to the other
or: there exists a path which includes both nodes
The DISTANCE along a path is the count of the edges
or: the number of pairs of nodes
or: the number of nodes after the first node
or: the "degree of separation" along that one path
This means ordinary ideas of "distance" are unimportant.
And that means most ideas of "shape" are ignored.
A CIRCUIT is any path which begins and ends at the same node
a name for one kind of path (like a collie among dogs)
has at least 3 nodes (and so 3 edges)
can be described as begining with any
of the nodes: (A,B,C,D,A) or (C,D,A,B,C)
are made of the same nodes and edges
and so they are the same circuit
A SHAPE is a "shortest" circuit in a particular direction
a name for an important type of circuit
(like a border collie among collies)
or: a circuit with no available shortcut
so: ANY triangular, 3-node circuit is a
shape because there is no shorter path
so: any square, 4-node circuit will be
a shape unless there is also a
diagonal edge making a shortcut —
(A,B,C,D,A) is a shape unless {B,D} or
{A,C} exist to make a shortcut
so: any circuit which includes the same
node more than once (after the starting
node) will not be a shape because you
could leave out all the nodes between:
(A,B,C,D,E,C,A) implies the shorter path
(A,B,C,A)
Note that an edge could be shared by
multiple shapes, say a square on one
side and a triangle on the other.
These shapes share 2 nodes but a shape
(like any circuit) must have at least
3 nodes.
A path exists by virtue of the existence of the
edges, so paths don't need to be separately defined.
Paths exist because the edges which make the paths
are defined.
Since edges are not directional, neither are paths.
The path can be specified in either direction:
(A,B,C,D,E) or (E,D,C,B,A); the same edges are
implied either way. In the example, {A,B}, {B,C},
{C,D}, and {D,E}. Paths are ordered in the sense
of adjacency but not in the sense of direction.
A shape exists by virtue of the existence of a
circuit. Remember that our idea of a shape is
just a path; it isn't a new kind of thing.
The shape defined for a Euclidean solid isn't
the same. Our shape only includes the edges. We
haven't defined any meaning for the interior
of a polygon, so in our geometry there isn't
any such thing. Besides, we have no concept of
a "plane" and no definition of "straight".
1D and 2D structures
A STRUCTURE is a set of nodes and edges
where every pair of nodes are connected
where every node is part of AT LEAST 2 edges
or: each node has at least 2 relationships
or: no dead-end paths which can't be extended
1D means that every node is part of AT MOST 2 edges
(for fully connected structures, EXACTLY 2 edges)
or: when building a path, no choices about
the next step
2D means that every edge is part of EXACTLY 2 shapes
or: each is part of exactly 2 "shortest" circuits
This is not the same as Euclidean dimensionality!
Note there can be structures that are between 1D and 2D in the
sense that SOME nodes have 3 relationships (so they are more
complex than 1D) and some edges are part of only one shape (so
they are not fully 2D). Joined polygons are an example.
More complex structures
3D means some edges are part of 3 or more shapes
our 3D is a vague, "everything else" category
3D allows more complexity than 2D
we do not differentiate 4D or higher
A CELL is a subset of a structure
which is itself a structure
which is 2D
which includes all the edges
for the selected nodes
which may be the entire structure or just a part
(there is nothing "improper" about an improper
subset)
or: a cell is a "smallest" 2D substructure
(somewhat analogous to a circuit being a
shortest path)
an alternative name might be "cage"
because we only see nodes and edges
We define structures and then define cells as parts
of structures. We could have defined cells first
and then defined structures as combinations of cells.
Cells are 2D; combinations of cells can be 3D (more
than 2 shapes per edge).
In Euclidean space a cell defines a volume; each
cell divides the space inside the cell from the
outside. A polyhedron in Euclidean space would be
a cell, for example. We are more general; we can
have cells even without any definition of volume.
Our geometry could equally well apply to cells of
secret agents who have limited awareness of each
other.