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Cubology - Biomatics.org

Cubology

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Cubology -The study of the cube

The subtle properties of the cube are central to information processing phenomenon in biological systems. Note that in this context that the term cube is used in it's general sense. A general cube of dimension n where n is greater than or equal to zero. Thus a single point, a line and a square represent n-cubes or hypercubes of dimension 0,1,2 respectively. The hypercube of dimension 4 has the special name tesseract].

The 3-cube, or cube, has many important qualities. To game players the number of sides is paramount as in the game die or dice.  The cube also has 8 corners, 12 edges, and a total of 64 possible transitions from one corner to another.  There are a total of 11 distinct nets for the cube (Turney 1984-85, Buekenhout and Parker 1998, Malkevitch). Considering travel along the edges of the cube results in the abstract algebraic concept of a group, which is a small algebraic system.  This is important because it introduces the primitive notions of "set" and "operation" into a community of carbon atoms. That is what a "group" in the abstract algebra sense is- a set together with an operation on the elements of that set.  The integers and addition is an example of a group.

The study of the cube is also as it turns out the study of the carbon atom. Thus the connection to biology and organic chemistry.

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Molecules and Rubiks's Cube- Molecules such as the Histone proteins and Rubik's cube can both be described in similar group theoretic terms.

 


Numerical vs. Analytic Solutions

In the best case scenario one can use calculus, trigonometry, and other math techniques to solve a problem, yielding a model that predicts behavior under any circumstances.  This is the analytic solution.
 
For some more complex situations, the math may become too complicated.  Numerical methods of solving the equations may offer an alternative.  The results are only an approximation, but it can be a satisfactory approximation under certain circumstances. 

The process may require thousands or millions of repetitive calculations, and the result is a long list of finite-precision numbers, rather than an equation. These are techniques that biological systems may utilize.



Fractal Groups



The Fibonacci sequence


 



Subgroups of the cube.  

CubeSubGroups.jpg

 


 The numbers five and eleven.

http://www.madewithouthands.net/3-Structure-of-Diamond/html/Cube-11-Nets.html 


Balanced Ternary

Balanced ternary is a non-standard positional numeral system (a balanced form), useful for comparison logic. It is a ternary system, but unlike the standard (unbalanced) ternary system, the digits have the values −1, 0, and 1. This combination is especially valuable for ordinal relationships between two values, where the three possible relationships are less-than, equals, and greater-than. Balanced ternary can represent all integers without resorting to a separate minus sign.

Balanced ternary is counted as follows. (In this example, the symbol 1 denotes the digit −1, but alternatively for easier parsing "−" may be used to denote −1 and "+" to denote +1.)

Balanced ternary
Decimal −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6
Balanced ternary 110 111 11 10 11 1 0 1 11 10 11 111 110

Unbalanced ternary can be converted to balanced ternary notation by adding 1111.. with carry, then subtracting 1111... without borrow (the string of 1s must be the same length, so if the result of the addition has more digits, subtract nothing from these extra digits). For example, 0213 + 1113 = 2023, 2023 − 1113 = 1113(bal) = 710.

Balanced ternary is easily represented as electronic signals, as potential can either be negative, neutral, or positive. Utilizing a third state encompasses more data per digit; linearly approximately 1.5849 (log23) bits per trit.

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