what is this simplex?

It is a classification table in the shape of a pyramid, or a triangle or just a line (a list) depending on how detailed you wish to be. 
It classifies the shapes that you find in nature. Natural processes are recursive/self referencing in general I think, which means that they can be described by power laws and visually are fractal-like in their shape and motion.
We can categorise these recursive shapes that are found in nature based on how they connect to themselves. So here goes, I'll take you through it...

Where do we start? well with nothing..... empty space.. void. You might say that most of the universe is made of this, so it is a good place to start. This is the 0-simplex, void.
Opposite to this is solid, a dense volume of matter. Out of void and solid we can make all sorts of different shapes. They fit into these seven core forms:
This is the 1-simplex, we have categorised the world into voids, clusters, trees, sponges, shells, foams and solids. 
That isn't to say that everything in the world is precisely one or the other, but it is a description of the raw elements. You could say that an alien has a tree-like head with sponge-like ears next to a cluster of shell-like objects.
A notable thing about this list is that if you flip it upside down then swap the solid with void then it is the same classification as the original list. The mathematical way to say this is that the complement of one of these forms is the vertically opposite form. 

If you take into consideration the border points of a shape then there are exceptions to the complement rule above which require the list to be broadened into a table. Every element in this table is the intersection of two of the seven forms. So you write Void-tree or Cluster-sponge etc. Here it is for 2d objects:
and for 3d objects (the 1-simplex is down the long diagonal):
Generally name1-name2 where name1 is the form if borders are not counted as connections and name2 is the form if borders are counted as connections. For example, a cluster tree is a cluster if you ignore the contacts, but a tree if you consider the contacts as connecting the shape together.

If you rotate the table a little then the seven forms can be seen easily down the right hand side. Here is a table of natural examples:
This is the 2-simplex.

At this point you may be thinking that nothing in the real world has zero volume like void-foam, nothing is completely solid, nothing is really a fractal tree of infinite depth. Yes that is true, but also true that there is nothing in nature that is exactly a cube or a sphere or a plane. Our descriptions always have an accuracy level to them, some objects fit more closely than others.

So far so good, but we're missing out an important part of the real world aren't we... time. What about moving objects?
The motion of an object can trace out all sorts of different shapes through time which you would imagine would explode this table of shapes into a vast, maybe infinite table of all the possible interactions that could happen to each shape over its lifetime.
Well, not to be overwhelmed, if we take it slowly and imagine for instance a tree blowing in the wind. If the wind is light then the tree remains a tree in its form... so there is no need for a new element in the table for this.
However, if the wind is strong enough that the big branches bang against each other (and the little branches and little twigs too) then at certain times the tree is becoming a sponge tree. It cannot go in the sponge-tree class because sometimes it is just a tree. Nor can you describe it as simply flipping between a tree and a sponge-tree because the contact times are fractally distributed in time and space; the little twigs will bang together much more frequently than the big branches. This moving tree has to go in its own class.
Taking the more alien example of a tree that as it moves its branches connect and weld together, then pull apart again. This is a case of a fractal motion between a tree and a sponge, another example of a new class? Well, in order to get from the tree to the sponge there has to be a point when a branch is just touching, so the motion must also include the tree-sponge, so it is still part of this new class. 
Let's try a new and different example of a changing object, what about the lifecycle of mushrooms which we can model as not existing, then growing from a point, with many living to a young age and a few living to an old age and growing large before dieing: http://vimeo.com/19769856. The mushrooms are a cluster of different sizes, but there is also a void form before a mushroom has started growing, and between not existing and existing the mushrooms must begin as single points, giving a void-cluster form. So again we see that these forms seem to form natural groups, what are the constraints on these groups?

Well there are two constraints, firstly as in the previous examples natural structures cannot jump from one class to another without taking a path in-between. So every group that we classify must be a connected set in the diagram.
Secondly, the horizontal variations (which fuse and unfuse objects) are independent of the vertical variations (which press objects together or lift them apart) which means that you can't pick and mix the variation in these directions, you simply get the bounding box of variations. If we only consider symmetric variation then we can define the set of groups to be every square set of classes that is possible on the 2-simplex.
Every 2x2 square gives us a 6x6x6 triangle table, every 3x3 square gives us a 5x5x5 triangle table and so on. These layers are referred to as the uniform layer, 2-form, 3-form etc and build into a triangular pyramid table which is the 3-simplex:

The 3-simplex is a table of 84 classes of object, characterised by its four opposing corners. Void represents emptiness and lack of connections, Solid is fullness with maximum contact between all points, Void-solid is a full set of border points, neither solid or gas; with volume but not rigidity. 7-form void-solid is an object that changes in time and space into any other object, you might say it has maximum 'density of movement' at all scales, so is in some sense the high energy or hot corner.

In fact this is not the end of the story. The 3-simplex only defines symmetric variation, to include all the groups we define one per rectangular set of classes on the 2-simplex. We can refer to this as the 4-simplex as it contains four degrees of freedom and can be written as in 2-3 tree-foam, 1-7 void-solid etc. However it is perhaps more accurate to think of this full classification as being defined by four magnitudes, of which the magnitude which provides variation has two components. In total there are 210 classes of dynamic fractal in the 4-simplex.

Update
It appears that we can logically extend the 2-simplex into a square table. The lower triangle of the table represents in-phase transitions, shown as the critical points between the diagonal fractals. The upper triangle represents anti-phase transitions where connections form fully over half the fractal. A real transition between the diagonal fractals would be some combination of these two extremes. 
We see three new fractals, the Tree cluster, a cluster of trees, the Sponge tree, a tree of sponges and a Sponge cluster, a cluster of sponges (not that this allows each object to contain itself within one of the holes. The Solid column and void row are quite unusual as they repeat the diagonal set. This is because fractals are recursively defined so a cluster is a Solid cluster but also a Cluster cluster. 
The most unusual entry is the Solid void, it is drawn here as an infinite filled half-plane, but could be drawn as a solid blob in a void background or a void blob on a solid background, these are all the same fractal class as we can define each fractal class as being invariant (not changing class) to any conformal transform. It is not necessary that a conformal transform produce exactly the same shape, just for it to remain in the same class.
In 3d we will get a similar addition, this time the addition of a Tree cluster, Sponge cluster, Shell cluster and Foam cluster; a Sponge tree, Shell tree and Foam tree; a Shell sponge and Foam sponge and a Foam shell. 

This changes the geometry of the classification, for the better... it isn't a simplex table, but a choice (based on your precision) between a classification list (the 1-simplex), a classification table (above) or the dynamic fractal table which is defined by each rectangular range on the above table.

Use of the word Fractal
I'm using the word fractal loosely in this site, what I am actually describing are symmetric objects which include scale symmetry. The Void-x objects are fractals, and others can have a fractal border, but are not strictly fractal sets. They would be better described as recursive sets.

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