Xi-Module

A Xi-Module, often just called a "Module" by Xi-Theorists, is a fundamental building block to a truly massive "Archversal"-sized hypercomputer architecture system known as The Xicohtlxul, sometimes simply referred to as The Xul.
A Xi-Module is contained within the boundary of an icosahepteract (27D hypercube) in it's true form, though is often displayed as a 3D Menger Sponge-like structure for ease of understanding. It is composed of many many many many smaller components called Xi-Cells, which are the smallest units of construction in The Xul.
== Basic Structure ==
To understand a Module think of it like a giant borg-like-cubical "space-station" in 27-dimensions. It remains suspended at very specific coordinates within a UniXule, it's position stored as a sequence of 27-trit integers, the length of the sequence denoting the size of that particular UniXule. It is composed of smaller units called XiCells, where it is gauranteed that the number of XiCell Units to an edge is always a power of 3. This means by definition every Module is actually a Rubrik's-Stage of some sort, to be explained shortly.
A XiCell is a much smaller unit of construction. To get a picture how it builds up to a Module, think of a 3x3x3 arrangement of cubes. This is how it appears in 3dimensions, but in 27 dimensions its actually a 3x3x3x...x3 (27 3s) arrangement of icosi-hepteracts. Such an arrangement is called a Rubrik's-Stage. A Rubrik's-Stage is simply a 3x3x3x...x3 (27 3s) arrangement of XiCells or is composed of elements of one lower stage. A XiCell is definitionally a Stage 0 Rubrik's-Stage. The Rank of a Rubrik's-Stage is 1 more than the elements it is composed of.
== Behavior and Navigation ==
A Rubrik's-Stage often does not actually contain 3^27 elements, but slightly less, with at most 3^27-1 elements. The reason for this is it creates gaps to allow elements to reshuffle themselves. An example of a reshuffling sequence in 3 dimensions of a 3^3-20 case going back and forth between two configurations is shown in Fig. 1:
This process is iterated, leading to gaps of higher and higher Stage orders. The entire structure is constantly reshuffling itself, meaning, it's very difficult to know where you will end up at any time. Special Gateways can found be at the centers of each facet of the Module. If you can manage to get in the precise XiCell that is situated at these 54 locations it is possible to be launched through a special invisible high velocity "tube" that will launch you to the next neighboring Module in one of 54 orthogonal directions. Think like forward,backwards,left,right,up,down, ... and extend that to 27D. Just escaping a Module is a major feat, and requires careful study of it's patterns of shuffling, which are usually determined by simple algorithms, though some have behavior far more chaotic. It is not entirely known what the purpose of the shuffling is, but it's been known to happen that Rubrik's Stages can be traded between Modules at times. If you are lucky enough you can use this as a form of transit. It's believed that all this shuffling is actually a form of defragmenting that sometimes happens in real time while calculations are still being performing. There are various engineering problems The Xul would present to its designers, but the most important one is that given it's unbelievably vast recursive size, no system of coordinates could actually properly be stored within the units themselves. As such the literal physical location of things sometimes would have to serve as their only distinguishing mark.
== Typical Sizes ==
With those preliminary concepts in mind, a Module can have any number of Stages, but most commonly they are found to be at Stage 27. That is, the edge length of a typical Xi-Module is 327 (7,625,597,484,987) Xi-Cell Edges, and contains (327)27 = 3272 (6.628186x10347) Xi-Cells total. Given that the edge length of a Xi-Cell is approximately 7.1669 km, we can work out that a typical Stage 27 Module has an edge length of approximately 54,651,894,615,153 km, which works out to 5.776 light-years to an edge! This means it would be about the size of a stellar nebula, such as the Crab Nebula, for point of comparison. This is far larger than any star, and even most planetary orbits ...
... and these are only the smallest ones known. They are also the most common, being the basal element of about 88.892% (approximately 8 of every 9) known UniXules.
The next most common are Stage 54 Modules, accounting for approximately 2.5% of all UniXules. Stage 54 Modules have an edge length of 3^54 (5.814973x10^25) Xi-Cells, and contain exactly (3^27)^54 (4.393285x10^695) Xi-Cells. This might not seem much worse, but these are in fact much much larger than Stage 27. They have an edge length of approximately 44 trillion light-years (4.404545x10^13), making it 440 times larger than the Observable universe, and placing it on the low end of the Multiversal scale, and it gets even bigger than that ...
The third most common Modules are those of Stage 81, accounting for 2 out of every 243 UniXules roughly. They have 3^81 (4.434264x10^38) Xi-Cell edges to a side, and (3^27)^81 (2.911951x10^1043 or about a googolchime) Xi-Cells total. This works out to a side of approximately 336 septillion light-years (3.358728x10^26), which already exceeds the maximal extent of Xenoverses and Superverses, lying just below the hyperversal scale.
Modules continue on to higher and higher Stages, with the smallest case surpassing the size of hyperverses being the Stage 108 Modules which have an edge length of 2.5 duodecillion light-years (2.561231x10^39), that alone being 51 billion times larger than a typical hyperverse.
Beyond these common types exist a complex ever evolving ecosystem of different sizes. Some other known sizes include, Stage 30, 33, 34, 36, 66, 72, 111, 118, 144, 216, 222, 243, 333, 666, and the largest observed case of 729. In the largest case a Xi-Module would have an edge length of 3729 Xi-Cell Edges, and contain 3273 = 319,683 (approx. 1010,000 or googoltoll) Xi-Cells ... Yikes!
Below you can find a table of the most common known Module Types and their statistics:
Common Module Sizes
Stage	Edge Length	Edge Length
(Xi-Cell Edges)	Total Elements
(Xi-Cells) 	Percentage of UniXules
Composed of Stage
27 	5.7758 light-years
(Nebula-sized)	327 (7,625,597,484,987)	(327-1)27 (6.628186x10^347)	88.892%
8:9
30 	 	 	 	
33 	 	 	 	
34 	 	 	 	
36 	 	 	 	
48 	60 billion light-years
(60,416,908,030)
(60% Observable Universe)	 	 	1:7,772,324
54 	44.439 trillion light-years
(4.4439x10^13)
(Small Multiverse)	 	 	0.253245%
72 	 	 	 	0.004516%
1:22,143
81 	335.8 septillion (3.358x10^26)
(Larger than largest Xenoverses)	3^81 (4.434265x10^38) 	(3^27-1)^81 (2.912x10^1043)	6.838%
1:15
96 	 	 	 	SUPER RARE
1:209,852,764
108 	 	 	 	0.0007214%
1:138,601
144 	 	 	 	SUPER RARE
1:101,040,220
162 	 	 	 	0.019480%
(1:5133)
216 	 	 	 	1:287,859
243 	(6.603941x10^103)
(Too Large)	3^243 (8.718964x10^115)	(3^27-1)^243 (10^3130) 	0.526%
288 	 	 	 	SUPER RARE
1:2,728,085,932
324 	 	 	 	1:1,801,813
432 	 	 	 	SUPER RARE
1:1,313,522,861
486 	 	 	 	0.149%%
1:66,729
648 	 	 	 	1:3,742,167
666 	 	 	 	
729 	 	 	 	0.040459%
(1:2472)
There isn't an adequate theory about what purpose the different sizes serve. Some Xi-Theorists believe there is some erratic behavior that happens in The Xul that doesn't necessarily serve any particular purpose but is merely unintended consequences of it's evolutionary like programming. The Stage 34 Modules are believed to be a glitch that was never corrected because it does little to effect the computers operation, thus there is no evolutionary incentive to correct it. 34 is currently the only known case of a Module Stage which is not a multiple of 3.
Category:1st-class
Category:Xicohtlxul mythos
Category:Navigable
Category:Saibian's Pages