A Fundamental Function Generates a Matrix for Operations
A fundamental function (novel as far as I know) generating a square matrix. The second performs the redistribution of values, drawing diagonals (division) within the previously declared matrix.
Is it just me? ..or does the second function resemble the engine of a train?
The first and shorter of the two functions pictured, establishes a square matrix n^2 (n*n) then substracts the remaining decimal value.
Though it is displayed otherwise, logically the matrix is a square comprised of n^2 smaller squares (4x4 or 4*4).
The second function includes, the squaring of the first then divides the areas of matrices produced by the matrix generating function.
1234321
A vertical two-dimensional (x,y) slice of a three-dimensional (x,y,z) pyramid that shares a base of seven with the square matrix it fits. A triangle fitting the larger seven squared (7^2) matrix of seven by as many squares.
The attachment on the front of the train can be changed and added to. I thank Lex Fridman for the thought inspiring post which started the tumble up the hill, walking down provides the clarity of retrospective.
The resulting distribution of values among the (n * 2) - 1 digits, is the result of squaring F, then divided by n^2.
F(n) = (n * 10^n) - (n)
--- ---
9 9
A function which fills each successive order of magnitude with n, for n number of times. The function completes by cleaning-up, subtracting the remainder in decimal from the result.
The following performs the matrix division operations upon the fillset of data points generated in the prior sequence of the function.
F(n) = [(n * 10^n) - (n)]^2
--- ---. ---
9 9 n^2
..concise and complete.
Now to see if the previously untested, proves true...
..the first time.
Now in one statement.
Thank you to Lex Fridman, the poster of the numbered series and operations that prompted the development of a function. Establishing a (square matrix)^2 for division over n^2.
https://www.facebook.com/101437778296134/posts/143461604093751/
An unrelated application where the mathematics are similar. What I see above, a method for enumerating increments that describe a common scale emanating from a point, within a spectrum.
DECIPHERING COLOUR-WHEEL BASED NUMBERING SYSTEMS AND MATHEMATICAL OPERATIONS
https://steemit.com/either/@deanpiecka/deciphering-colour-wheel-based-numbering-systems-and-mathematical-operations
A statistical perspective, when the number of values in a dataset equals the mode (most frequent) and the mean, the maximum value of the base is present in each order of magnitude.
Square matrices, of 11 - 22 to 111111111 - 999999999.
One square matrix is declared for each order of magnitude (exponent increment).
Cut one square matrix into equal rectangular halves, each is further divided diagonally such that four triangles containing one square-angled vertex remain. The four triangles can constitute two rectangles, one diamond or the square matrix of origin.
Division of squared values subtracts an area (two triangles) from a matrix to yield the even and linear change of values across the orders of magnitude (columns) for a given matrix.
In terms of the area declared within a square matrix of cols and rows, 12345678987654321 should occupy the same area as 11223344556677889.
22^3 / 2^3 = 1331
333^3 / 3^3 = 1367631
4444^3 / 4^3 = 1371330631
...
Multi-dimensionals
22^2 / 2^2 = 121
333^3 / 3^3 = 1367631
4444^4 / 4^4 = 1523548331041
The second and third dimensionals function, the fourth may be yielding an error due to the limitations of calculating successive orders of magnitude. An appropriate calculator may yield a different answer upon double-checking.
The current quantum computer architecture in D-WAVE systems, implemented a five-dimensions to each interconnected vertex design.
I can imagine computer scientists attempting certain multi-dimensional computations beyond the third, to realize that the inclusion of more dimensions negates an increasing amount of information when resolved in a reference frame of less dimensions, commonly two.
A three-dimensional pyramid can be viewed in two dimensions as a series of squares increasing in area, upon a matrix of squares used to display the given horizontal "slice".
A vertical two dimensional slice of a three-dimensional pyramid is represented. The functions are listing linear division into halves of squared matrices. This can also be understood as a subtraction from the two dimensional area represented by the linear and diagonal divisions declared.