| void-cluster | | points distributed fractally, so small bunches with small gaps make up larger bunches with larger gaps. Like clouds where the water droplets are about the same size but the distance between them is not uniform but clustered. There must be areas without points. |
| void-tree |   | points connected without cycles or loops, so a dendritic pattern of zero or negligeable width. |
| void-sponge | | points connected in loops but not enclosing any gaps in an airtight way. Water can pass through.
No part is dense and solid. |
| void-shell |   | small planes branching off larger planes recursively, no loops, this form can hold water like a bucket. The left image shows 8 shells back to back points in the 8 octant directions.
A shell encases a tree, in the right picture this shell encases an order 1 tree, i.e. a tree with only one sub-segment for each segment, angled so it is in a spiral shape. |
| void-foam | | membranes that surround pockets of air. A few large bubbles pressed against many smaller bubbles recursively. It is airtight but can still flex. |
| void-solid | | a void-solid is a sea of border points, and classifies all space filling curves and fractal tilings. The left picture is a 3d Moore curve.
void-solids have volume but are not solid, so not defined as rigid, a good natural analogy is liquid or water. |
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| cluster-tree |   | a tree of objects connected at points or edges (kissing).
In the right picture the earth is the base object, then about 5 cactus buds come from this, each one has several smaller child buds and so on. So it is a cluster-tree up to about 3 levels. |
| cluster-sponge | | a cluster sponge is a cluster of kissing objects that include loops, e.g. A touches B and C, B touches C.
The pebbles show a few large pebbles and many more tiny pebbles and finally sand and dust below. |
| cluster-shell | | This is a watertight casing made out of a fractal distribution of objects connected along edges.
The tortoise shell is made of a few thick hexagonal shapes that touch together, connected to many smaller shapes that fit together. |
| cluster-foam | | This is a connected set of sealed gaps (bubbles in the pic) and solids (pebbles in the pic). You will find some dense volumes and some void volumes. Smaller volumes in higher frequency as usual. |
| cluster-solid | | A cluster-solid is a cluster of objects that are packed so tightly that there is no air between them, a good example is this seabed where the larger pebbles are really embedded amongst the smaller pebbles, sand and dust. |
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| tree-sponge | | This is a tree which has its branches touching each other (kissing) but not welded together, there are no loops in the solid.
The examples are these plant roots and a type of coral. |
| tree-shell | | A tree shell can be thought of as a tree which is connected along edges to form a water-holding shell.
The left image is made of boxes on the faces of larger boxes and forms an impermeable shape, without loops or tunnels.
This fan coral is a good example. Even though there are visible gaps between the tree branches you can see that the fractal nature of this coral is to fill the gaps with smaller branches. This is because the coral wants to capture as much surface as it can, so it is creating a shell out of a tree structure. |
| tree-foam | | Tree-foam is a tree that connect with itself to form pockets of air. Hard to find examples, but the best might be to look at seaweed, it grows dendritically but needs pockets of air to float, seen as the yellow sacks in this image. |
| tree-solid | | A tree-solid is a tree that is so fat that it is in contact with itself everywhere (apart from the outer border).
Alternatively it is a solid with shell-like cracks in it. Such fractal fault lines are the normal form that cracks take.
So this image of a cracked rock is a tree-solid. If the cracks had joined up (a cluster-solid) then this rock would have fallen apart. |
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| sponge-shell | | This is a shell that is touching itself at points so as to cause loops, alternatively you can think of it as encasing (i.e. complement of) a tree-sponge (a tree that touches itself). A good example is lungs, the gap (the air in the bronchi) is in the shape of a tree which gets arbitrarily dense, so the solid, the actual lung itself is a sponge-shell. |
| sponge-foam | | A sponge foam is a solid with a fractal size distribution of bubbles in it, where the bubbles touch each other.
So just like this image, there are connected bubbles, but also areas of dense solid between them. Unlike void-foam this has a mass and can be heavy. But like other foams it is airtight and non-porous. |
| sponge-solid | | This is a solid with a connecting network of hairline cracks (or thin tunnels) in it.
This picture is of woodworm, the tunnels connect together and are distributed and clustered randomly. There must exist volumes that are dense too. |
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| shell-foam | | Shell foam is a shell of dense material that connects to itself along edges to hold air gaps or bubbles. It encases a cluster-tree.
The right image is of tree caverns in China. If we assume the rooms inside are connected by small doorways and they are singly connected (no loops) and contain a few large rooms and many small rooms, then the rooms form a cluster-tree so the mountain itself is a shell-foam. |
| shell-solid | | A solid with dendritic cracks in it, like this glass. |
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| foam-solid | | This is a solid with a cluster of isolated points inside it. The complement of a void-cluster, which often looks like a cloud, this image is a solid rock with a cloud of points inside it which aren't part of the rock. So the rock is a foam-solid. |
