Boxes, and the Art of Thinking Outside of Them

[Icarus_Fallen]

All,

I'd like to introduce you to Box Theory: a geometrically-based theory of sets. The beauty of this theory is two-fold. First, all denoted boxes within a boxset must comport to all of the known physical laws of the phenomenal world, as they're intended to correspond directly to containment units beyond of the abstract realm of mere mathematics. That is, for all intents and purposes, these are real “boxes” I’m talking about here. Second, but no less importantly, Box Theory sets have container/containment values that comport to ordinary numbers. This means that a denotation such as '{{},{},{},{},{},{},{},{},{},{}}' can be simplified thusly: '11=1o/10e'; with “11” denoting the total number of boxes; “1o” identifying the number of occupied boxes (in this case, 1); and “10e” denoting the number of empties (10). Note the correlation of the numeric value of the outermost container (11) to the sum of the occupied and empty boxes. I believe these aspects work in concert to render the idea more readily accessible to non-specialists.


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Questions?

[Icarus_Fallen]

Okay, let's open a boxset and see what, if anything, the number of 'immediately visible' members tells us. Consider 5=2o/3e. Since it may be helpful for some to visualize this in *set theoretical clothing, here's what the set in question looks like in brackets: {{{},{},{}}}. Now, under the relevant parameters, upon opening the outermost box, we'd find only 1 immediately visible resident, namely the other 'occupied' box. If we cracked open this secondary (or 'sub') container, we'd find that it contains 3 immediately visible members in its own rite: the empties. Adding up the occupied and empty sums, the total numeric value of the boxset is 5. But knowing the box-sum in advance would afford the ability to draw certain conclusions from the 'immediate visibility' (or lack thereof) of the members and sub-members. For example, from the fact that the encompassing box in 5=2o/3e has only 1 immediately visible resident, by simple subtraction, we could determine that the number of boxes contained by it (the subset) ...is 3.

Enough for now.

*NOTE: I'm disregarding the commonly-accepted postulates that only one 'null set' exists and that it's a subset of all sets, since according to my theory, "{}" simply denotes an empty box, ...and we all know there exists many more than one of those, right? But if anyone's interested in the evidence that supports my proposal WRT the multiplicity of empty sets, ...well, I suggest that you have a look around the dumpster area at your local grocery store.

[h0bby1]

did you ever looked on how octree works ?

Octree

From Wikipedia, the free encyclopedia


Left: Recursive subdivision of a cube into octants. Right: The corresponding octree.


An octree is a tree data structure in which each internal node has up to eight children. Octrees are most often used to partition a three dimensional space by recursively subdividing it into eight octants. Octrees are the three-dimensional analog of quadtrees. The name is formed from oct + tree, and normally written "octree", not "octtree".
Contents

• 1 Octrees for spatial representation
• 2 Common uses of octrees
• 3 Application to color quantization
• 4 See also
• 5 External links

Octrees for spatial representation

Each node in an octree subdivides the space it represents into eight octants. In a point region (PR) octree, the node stores an explicit 3-dimensional point, which is the "center" of the subdivision for that node; the point defines one of the corners for each of the eight children. In an MX octree, the subdivision point is implicitly the center of the space the node represents. The root node of a PR octree can represent infinite space; the root node of an MX octree must represent a finite bounded space so that the implicit centers are well-defined. Octrees are never considered kD-trees, as kD-trees split along a dimension and octrees split around a point. kD-trees are also always binary, which is not true of octrees.



it is like a common problem in computer simulation when working on hudge 3D world, to being able to quickly get the number of element you have to deal with in a particular space, specially for collision detection and also drawing, for that you can imemdiatly know what polygons to draw or test in the whole universe from a point and a range, without having to test all the world, and it divive the space into a recursive set of box, each parent box being bigger than the child, and you can quickly eliminate all the child of a box which are not visible, and like this you can quickly get only the box containing the visible polygon, witout having to parse the whole world because you can quickly discard large part of it by discarding a big parent box that is not visible, and you don't have to test all the box/polygon inside then goal is also to have box that have little empty space and are close to polygon boundaris, like this the box do not come selected in testing a thing in empty space


well seemed a bit linked to what you say hehe

[Icarus_Fallen]

h0bby1,

That's an interesting application, but cubes and boxes are geometrically and physically distinct. Internal/spatial division notwithstanding, dividing and "recursively subdividing" a "parent" box (ad infinitum) wouldn't render even a single "child" box, because the prospective parent's internal space is non-material. So, whereas the parent would be comprised of six material sides, the hypothetical octuplets of the first iteration could have only 3 material sides each. What's more, the material would have to be divvied up unevenly for any generation beyond the first iteration. The application only flies on the wings of material (or non-material) continuity across the solid regions or empty spaces divided. In short, it doesn't apply to boxes generally, nor does it have any relevance to my theory in particular.

[h0bby1]

yes was not sure it totatally relate , but it made me think of it lol

if this notation '{{{},{},{}}}' that made me think of this

[Icarus_Fallen]

h0bby1,

The denotation "{{{},{},{}}}" expresses a number of structural containment (not necessarily compositional) relationships. Physically speaking, material composition would have zero bearing on a given box's classification as a "box". Cardboard, wood, metal (ETC.), a box is a "box" solely by virtue of its geometrical structure and the resulting storage capacity of that form.

Now, while something similar can be said of “cubes” in general, the cube figures from the Wiki article exemplify a compositional principle (namely continuity), simply because of the logical need for material or non-material continuity across the board. In order for the Octree Application to work as a method of physical procreation, whatever the cubes are made of to begin with must be evenly distributable for each and every iteration to render 8 smaller versions of the preceding generations (bear in mind that size and position can be the only differentials). So, if a cube has some form of internal storage capacity (similar to that of a box), the material composition of that storage area mustn’t differ from the walls, the floor, or the ceiling, in order for the cube to be a viable candidate for ‘parenthood’.

[Icarus_Fallen]

h0bby1,

Funny. Somehow, I get the sense that what we're discussing here may have larger (if not universal) implications.

[h0bby1]

well the octree system is also used to work with color and is not necessary working in space with cubes like that, it is just of matter of recursivly delimiting subspace, the octree leaf or cubes are more to be seen as simple boundaries made with a range of inferior/superior values on each dimension to be able to define if something is 'inside' or 'outside' =) and it suppose that if something is inside of a subbox, it is also on inside the parent box, and that if something is out the parent box it also out of any of his child box

there are more complex system that do not work with cubes as bsptrees, or portal trees or various other systems

[Icarus_Fallen]

Okay, I'm bored again, sooo ...the following examples are intended to highlight an interesting correlation.

Consider 56: 30e/26o. If you opened this encompassing box, you'd see only 30 smaller boxes inside, because 25 would be hidden by virtue of sub-containment. What's more, of the 30 immediately visible boxes, only five would be empty, as the other 25 empties would be residents of 25 of the visible, occupied boxes.

Now consider the converse approach, in which the amount of contained empties is considerably fewer than their occupied counterparts. Upon opening 43: 2e/41o, you'd immediately see only two smaller boxes inside. One of these immediately visible boxes would be empty; the other would be occupied with boxes within boxes adding up to the remaining 40, the inner-most sub-contained box being the other empty.

The point here is that the respective amounts of empty boxes in the above boxsets correspond to the numbers of boxes that are "immediately visible" when the encompassing boxes are opened.

However, this interesting correlation is not necessarily limited to the outer-most box's containment-value (specifically where a singular, immediately-visible resident sub-contains the balance of the encompassing box's containment-value thereby shifting the correlation's applicability to itself), nor is it limited to the sum of the empties (it can also involve the occupied sum). Therefore, the manner of containment/sub-containment apparently determines whether the correlation involves the empty, occupied, or both sums simultaneously, and whether it applies to the outer-most container or to *one* immediately visible resident.

Example 1): {{{},{{}}}} expresses a total numerical value of five: two empty; three occupied -- simplified thusly: 5:2e3o}. However, due to the manner of containment/sub-containment, upon opening the outer-most container, only one of its residents would be immediately visible, so the correlation would apply to *its* containment-value instead of the all-encompassing container's. It's important to note here that another set could express an identical containment-value identical with the IV-E/O correlation applied to the outer-most container, but it would be doing so by virtue of the manner in which its residents are situated.

Example 2): {{{},{}}} expresses a total containment value of three: two empty; one occupied. In this case though, the IV-E/O correlation involves the occupied sum. Note that {{},{{}}} expresses an identical containment value, namely 4: 2e2o, but again, due solely to the manner of containment/sub-containment, the IV-E/O correlation involves the empty sum.

Lastly, there's another interesting feature to ponder. While an outer-most container can hold any number of occupied boxes (including zero), it can hold no less than one empty box. That is, at the very least, any occupied container has an empty sum of 1.