Hypernomials and Array Sizes


As I looked at arrays beyond dimensional arrays, I've uncovered an interesting way to extend the non-negative polynomials into uncharted territory. This page will look at these "hyper polynomials" - or "hypernomials" for short. Hypernomials relate to arrays in two ways, they can act as sizes to the arrays and as positions of entries within the array. Hypernomials can expand to include not only tetrational arrays, but pentational, hexational, and beyond.

Non-Negative Polynomials

First lets start with typical non-negative polynomials, I've excluded the negative ones since they don't contribute to the arrays. Non-negative meaning that there are no minus signs in the polynomial anywhere. The smallest ones are the natural numbers: 0, 1, 2, 3, 4, 5, ..., n,... - where each of these can represent the size of an array, for example a size 7 array has 7 entries. They can also denote positions within the array - {0,1,2,3,4,...}. Position 0 is the first entry, also called the base. Position 1 is the second entry, called the prime. Position n is the 'n+1'th entry which passes up n entries. Interesting note, position n passes up n entries - this continues with all hypernomials where position k passes up a size k portion of the array. The natural numbers represent sizes of linear arrays.

Next in line is the polynomial X - which represents the size of a linear array of any size. Position X passes the entire first row of the array and is therefore the first entry of the second row. Continuing, we have X+1, X+2, X+3, ..., X+n, ...., 2X, 2X+1, 2X+2, ..., 2X+n, .., 3X,.., 4X, ..., nX, ..... - these represents the positions on the first plane, where position mX+n passes up m rows and n additional entries and is therefore the 'n+1'th entry on the 'm+1'th row - an array of size mX+n has m rows and n additional entries on the next row.

Next polynomial is X2 which is the size of a planar array, and the position of the first entry on the 2nd plane. Continuing we have X2+1,..,X2+n, .., X2+X, .., X2+mX+n,.., 2X2, .., 3X2, ...., nX2, ... - these represents positions throughout the next few planes - where position mX2+nX+p passes up m planes, n rows on the next plane, and p additional entries on the next row.

Next polynomials would be X3, X3+1,..,X3+X,..,X3+X2,..,2X3,....,X4,......,X5,.....,X6,......,Xn,....... - This is as far as polynomials go. These represents positions within a multidimensional array as well as the sizes. The array to use to solve the number Gongulus (which is a 10^100 array of 10's) is of size X100 (which represents a 100 dimensional array) - the last entry of the gongulus array is at position 9X99+9X98+9X97+.....+9X3+9X2+9X+9. In summary: integers represent the sizes of linear arrays, while polynomials represent the sizes of dimensional arrays.


Super Polynomials

Super polynomials allow X to be taken to any polynomial power, not just integer powers. The first one is XX which represents an entire Hilbert space (infinite dimensional space), in the same way X represents an entire row. An array of size XX can have any number of dimensions. Position XX represents the first entry in the second Hilbert space. Next would be XX+1, XX+2, .., XX+n, .., XX+X, .., XX+X2,..., XX+Xn,....,2XX,....,3XX,....,nXX,..nXX+p (p any polynomial), ..., XX+1. These super polynomials represent the positions on the next few Hilbert spaces, where position XX+1 passes up a row of Hilbert spaces. Position nXX+p passes up n Hilbert spaces and a p-sized array segment on the next Hilbert space.

Next we have XX+1+1, .., XX+1+p,.., XX+1+XX,..,XX+1+nXX+p, .., 2XX+1. These represent the second row of Hilbert spaces. Continuing, we have 3XX+1,...,4XX+1,...mXX+1+nXX+p,..... - these represent more rows of Hilbert spaces where mXX+1+nXX+p passes up m rows of Hilbert spaces, n more Hilbert spaces in the next row of Hilbert spaces, and a p-sized segment in the next Hilbert space, where p is a polynomial.

Next is XX+2 which represents a plane of Hilbert spaces. Lets jump the gun a little bit and look at aXX+5+bXX+4+cXX+3+dXX+2+eXX+1+fXX+p which is equal to qXX+p, where p and q are polynomials. This example represents an array consisting of a 5-spaces of H-spaces (Hilbert spaces), b 4-spaces of H-spaces, c realms of H-spaces, d planes of H-spaces, e rows of H-spaces, f more H-spaces, and finally a p-sized array in the last H-space. This could also be described as a q-sized array of Hilbert spaces plus a p-sized array of entries.

Next up is X2X which is a Hilbert space of Hilbert spaces - imagine the terror of what the Exploding Array Function would do to this sized array!

Next is pX2X+qXX+r, where p, q, and r are polynomials - this represents a p-sized array of H-spaces of H-spaces plus a q-sized array of H-spaces plus an r-sized array of entries.

Next up we have the sequence: X3X,......,X4X,......,X5X,......,XnX,....... - where each XnX can be multiplied by a polynomial and added together with others. An array of size XnX is a Hilbert space of Hilbert spaces of Hilbert spaces of......of Hilbert spaces - n times.

Next we have XX2 this represents a 2-superdimensional array, where a superdimension is a dimension of dimensions. Hilbert space is only 1 superdimensional.

Let p now represent any hypernomial from the polynomials 0,1,...X,...Xn,...up through the super polynomials of this sort - XnX,.. - then we could go through the next few 2-superdimensional spaces (or 2-superspaces) using the following hypernomials: XX2+p, 2XX2+p, 3XX2+p,...,nXX2+p,... Next up is a row of 2-superspaces: XX2+1 = X*XX2. We can keep on going through cases like this: XX2+nX+m+p = X(nX+m)*XX2+p which represents a size XnX+m array of 2-superspaces plus a p sized array in the next 2-superspace. In general we could include pXX2+q, pX2X2+qXX2+r, pX3X2+qX2X2+rXX2+s, ..., etc. Where p, q, r, and s are hypernomials up through the XnX cases. Notice that X2X2 is a 2-superspace of 2-superspaces, and X3X2 is a 2-superspace of 2-superspaces of 2-superspaces, etc.

Now lets look at a 3-superdimensional space - or 3-superspace array, it is represented by the super polynomial - XX3 - we can also let p and q represent hypernomials up through the 2-superspace cases and continue with pXX3+q which is a p-size array of 3-superspaces plus a q sized segment in the next 3-superspace. Lets continue: pX4X3+qX3X3+rX2X3+sXX3+t - where p,q,r,s,and t can go through the 2-superspace sizes - a X4X3 is a 3-superspace of 3-superspaces of 3-superspaces of 3-superspaces.

Notice the trend that we used up to this point, we can continue with the same trend to look at the 4-superdimensional arrays - XX4, 5-superdimensional arrays - XX5,..., up through the n-superdimensional arrays - XXn. Where the p in pXmXn can be hypernomials all the way through the (n-1)-superdimensional cases and where m and n are integers. All of this represents the super polynomials which represents the superdimensional arrays - and these are just the beginnings of tetrational arrays.


Trinomials, Quadranomials, and the Rest of Tetration

For trinomials, the power we take X to can now be a super polynomial - our first trinomial is XXX - This is X tetrated to 3, it can also be referred to as 1-trimensional - where a trimension is a dimension of dimension of dimensions. 1-trimensional space can be called a 1-trispace for short. As we increase we will encounter the following: XXX+p (p goes up to the super polynomials),..,nXXX (this is many 1-trispaces),..,XXX+q (q is a polynomial, this is a Xq-space of 1-trispaces),..,XnXX (this is a 1-trispace of 1-trispaces of 1-trispaces of...of 1-trispaces - n times),..,XXX+n,..,XXnX. We then continue to 2-trimensional space - XXX2, then higher trimensional spaces - XXX3,XXX4,..,XXXn.

For quadranomials, we now go to the 1st quadramension which is the 4th tetration - XXXX. Continuing onwards - XXXX+p (p can be a trinomial), nXXXX - many 1-quadramensional spaces, XXXX+q (q is a super polynomial), XnXXX (a 1-quadspace of 1-quadspace of....of 1-quadspace - n times), XXXX+p (p is a polynomial), XXnXX, XXXX+n, XXXnX, XXXXn - n-quadspace (or n quadramensions).

We continue with the tetration spaces: XXXXX, XXXXXX, XXXXXXX, n --> XXXX:::XX, n --> XXXX:::XX+p, n --> mXXXX:::XX, n --> XXXX:::XX+p, n --> XmXXX:::XX, n --> XXXX:::XX+p, n --> XXmXX:::XX, n --> XXXX:::XX+p, n --> XXXmX:::XX,...,n --> XXXX:::XX+p, n --> XXXX:::mXX, n --> XXXX:::XX+m, n --> XXXX:::XmX, n --> XXXX:::XXm, n+1 --> XXXX:::XXX, ..... - notice the "n -->" and the "n+1 -->" - this represents the number of X's in the powertower. Also notice how the +p and the m climbs up the powertower as we get larger - lets call this the "climbing method". This sums up tetrational spaces as well as tetrational hypernomals.


Pentational Hypernomials and Arrays

Here we start with X --> XXXX:::XX, there are now X X's in the tower - this is an X^^X array - we can also continue this using the climbing method until we get X+1 --> XXXX:::XX,X - where the comma separates the first set of X's from the second set in the tower. Lets continue to a very distant hypernomial - 2X --> XXXX:::XX,XXX:::XX - this is X^^2X. We continue using the climbing method to get to the next tetration level 2X+1, and again to 2X+2, and again to 2X+3, etc,...., up to 3X,...4X,...5X,...nX,...and X2. X2 --> XXXX:::XX - this is starting to look scary!

Continuing onwards: X3 --> XXXX:::XX,..,X4 --> XXXX:::XX,..,XX --> XXXX:::XX,...,XXX --> XXXX:::XX,.......X --> XXXX:::XX --> XXXX:::XX.

Below is the size of the array needed to solve the kungulus - it is a X^^^100 array where X is evaluated at 10.

100

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X --> XXXX::X --> XXXXXXX::X --> XXXXXXX::X --> XXXXXXX::X --> XXXXXXX::X --> ::::::::::: --> :::::::::::: -> :::::::::::: -> :::::::::::: -> :::::::::::: ::::::: --> XXXXXXXXXXXX::X

The climbing method works here too, by climbing on the rightmost power tower, then the second right tower, and so forth. These are the pentational hypernomials and array sizes.


Hexational Hypernomials and Arrays and Beyond

Here is an example of a hexational hypernomial which represents the size of some hexational array - this represents the size of a {X,n,6} array to be exact - notice how the size of an array can be an array!

X

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X --> XXXX::X --> XXXXXXX::X --> :::::::::::: -> :::::::::::: ::::::: --> XXXXXXXXXXXX::X

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X --> XXXX::X --> XXXXXXX::X --> XXXXXXX::X --> ::::::::::: --> :::::::::::: -> :::::::::::: ::::::: --> XXXXXXXXXXXX::X

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X --> XXXX::X --> XXXXXXX::X --> XXXXXXX::X --> ::::::::::: --> :::::::::::: -> :::::::::::: ::::::: --> XXXXXXXXXXXX::X

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X --> XXXX::X --> XXXXXXX::X --> XXXXXXX::X --> ::::::::::: --> :::::::::::: -> :::::::::::: ::::::: --> XXXXXXXXXXXX::X

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::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: n layers :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

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X --> XXXX::X --> XXXXXXX::X --> XXXXXXX::X --> XXXXXXX::X --> XXXXXXX::X --> ::::::::::: --> :::::::::::: -> :::::::::::: -> :::::::::::: -> :::::::::::: ::::::: --> XXXXXXXXXXXX::X

On the above hypernomial, we could use the climbing method to increase the size, by climbing the lower right tower first and moving left to the next tower, until we reach the end and then add a new tower - which adds "1" to the layer above it, then we do the climbing method on that layer, and continue upwards until we reach the top and finally to the point of adding a new layer. We could keep going like this to represent larger hypernomials such as the ones in the {X,n,7}, {X,n,8}, or {X,n,m} ranges and then even futher into {X,a,b,c} ranges - or even worse the A ranges where A is a linear or a multidimensional array - as a matter of fact it keeps on going where the new arrays generated can represent the size of a much larger array - it never ends - hahahahaha!!


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