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11-2017 By Douglas A. Engel, Littleton, Co

(symmetry, math, games, magic squares, fractals…)

Jain magic square space number analysis. Ref. link: https://en.wikipedia.org/wiki/Most-perfect_magic_square .

The Parshvanath Jain temple in Khajuraho India has a most perfect magic square, meaning a magic square where the greatest number of possible magic sums appear. It was produced in about the year 1000. It is a 4x4 square with numbers 1 thru 16 placed in its 16 interior squares. The numbers are arranged so that all columns, main diagonals and 2x2 corner subsquares all add up to the magic constant 34. Many other symmetrical patterns also add up to 34. It is also known as a diabolical square since it has so many ways to derive the magic constant.

Since it is a 4x4 square and 2^4=16, it can be seen as four two dimensional power patterns, 2^0, 2^1, 2^2, 2^3 as shown above by the black and white 4x4 cells. Think of each power pattern as having zeroes in the white squares and the power number in the black squares. Thus the 2^3 pattern has 0’s and 8’s. The numbers of the magic square are produced by adding the power numbers in each position together. Then the lower right Jain square numbered cell would equal 0+0+2+1=3. Space numbers subtract 4 from the magic constant changing it from 34 to 30(numbers 0 thru 15) but it remains just as magic. Looking at the four power patterns you see immediately why it is so magical. For instance you can see that all the columns and rows have 2 black squares and two white squares and each 2x2 corner subsquare has two black squares and two white squares and similarly for the two main diagonals, and so on for the other magic patterns of the Jain magic square. In addition we can shuffle the power patterns(such as let the 2^0 and 2^2 power patterns exchange positions) however we like and the square stays most perfect, only the numbers change positions. We can also reverse the black and white of any power pattern and the magic is retained. Rotation of individual power patterns is not allowed as the two pairs of the power patterns are already ninety degree rotations of each other. Thus with shuffling and reversing we can have 24x16=384 different number arrangements of the Jain magic square. Of course many of these will be rotations or reflections of others others, but it shows the ease with which space numbers can be used to make more magic squares. The 4x4 magic square shown in Albrecht Durer’s 1514 engraving ‘Melancholia’ can be analysed the same way and produces a very similar space number with 384 number arrangements. Other magic squares could be produced using space numbers. For instance a 12x12 magic square would need to use power patterns in base 2 and base 3.

The Durer magic square as a space number

Here is an illustration of Albrecht Durer’s magic square from his 1514 engraving Melancholia. It is not quite as magic as the Jain magic square but has several interesting features detailed here https://en.wikipedia.org/wiki/Magic_square#Albrecht_D.C3.BCrer.27s_magic_square

The four power patterns can be exchanged, and black and white reversed for a total of 4!x(2^4)=192 new magic squares. There may be some duplicates since reversing all four is the same as a 180 degree rotation.

The figure below shows the Mars magic square as a base 5 space number. Since 5 is prime this is the only way it can be shown as a space number. It shows how much simpler an odd order magic square can be as opposed to an even order magic square.

Below we show the Sol magic square as a space number. Since it’s prime factors are 2 and 3 it is necessary to use both binary and trinary base numbers to break it into a space number. From this you see it has some complexity since it contains both even order and odd order properties. All even order magic squares are more complicated than odd order magic squares.

Next we show the Venus 7x7 magic square below. It can only be shown as a base 7 space number since 7 is prime. You can see it is very simple with a diagonal symmetry of numbers similar, to the order 5 Mars magic square.

The 8x8 magic square below is presented as a binary space number with 6 power patterns. You can see that the columns and rows all have 4 black cells and 4 white cells. For each column and row the black cells must intersect equal numbers of black and white cells and this is also true for the white cells. The same is true for the two main diagonals. This is the equal fractional intersection rule for a space number to uniquely number all the cells and be magic when combined. By considering the power pattern symmetry operations of rotations, shuffles(exchanges), reverses of black and white we get 92,160 magic squares from this space number.

The same 8x8 magic square above can be analyzed with a base 8 space number. Shown below you can see it is really four 4x4 magic squares. On some permutations it probably produces many semi magic squares while the binary 8x8 space number above produces only fully magic squares. The base 8 patterns each combine 3 of the binary power patterns into one power pattern. Thus less freedom exists to move the power patterns around by shuffling and color permutation. This is partly made up for since more permutations are allowed with the base 8 pattern.

We can also use a base 4 space number to analyze the Barink 8x8 magic square. This results in the figure below showing three 4x4 power patterns that add together to equal the 8x8 magic square. Using a different but compatible base produces a different set of magic squares where the space number symmetry operations of power shuffling, rotation, reflection and permutation are performed. It is interesting that the 4x4 pattern produces the greatest number of new magic squares.

This is but a tiny fraction of the total number of possible 8x8 magic squares which is a large number. Using space numbers this huge number can be reduced by about 10^5 since each space number representation could be taken to represent all the magic squares that can be generated from it.

The Jain magic square can also be presented as a base 4 space number. The figure below shows this along with a color permutation of the Jain space number and a vertical reflection of the Jain space number. The color permutation results in an Albrecht Durer type magic square while the reflection produces a Jain magic square. Using base 4 instead of base 2 here seems to limit the power of the space number system. However it is possible to do other kinds of permutations that always produce Jain magic squares. It does show that using a different base space number can produce unexpected results.

Introduction to the concept of Space Numbers

Space numbers are a democracy of numbers where all numbers and dimensions have equal importance. While appearing complex Space Numbers are wonders of simplicity and symmetry.

Historical: The idea for space numbers came to me one morning around 1968, waking up at a Mining camp in the Colorado mountains, altitude over 12000 feet. I was learning to work with an IBM 360. It filled a whole room. Now you have many times the thinking capacity in a smart phone.

Looking at the walls there appeared different checkerboard patterns and the idea that they could combine to create numbers when overlapped. Several years later this was published as “Number Patterns in More than One Dimension” in the Journal or Recreational Mathematics, edited by my friend Harry L. Nelson, working at Lawrence Livermore labs in California. The paper did not excite any interest.

Space Numbers

I have continued to work with this idea, now calling them space numbers Sn, and Sn() which is the set of all possible space numbers. I have devised some entertaining space number computer games. These games use rules of symmetry to recombine the space number patterns in many interesting ways to create different symmetrical number lists.

A Simple Description of Space Numbers and their symmetries

A space number consists of a set of power patterns each occupying an identical cellular geometry with one base position number, b^p, in each corresponding cell of each separate cellular geometry. The power patterns are added together to make a combined identical shaped cellular geometry. This is done by adding the number in identical cells of each power pattern. The cells of the combined pattern can be a linear list, like a line of square cells, or a square grid of cells or a cubic grid of cells, or a four or larger dimension grid of cells, or any symmetrical cell geometry such as the surface of a dodecahedron with each face divided into a pentagonal grid of cells. The power patterns might be rotated, reflected, permuted, and shuffled(where b^1 becomes b^3 and b^3 becomes b^1 for instance before being re-combined. These operations are only permitted, assuming we always want a complete numbering of the cells, depending on the design of the power patterns and the symmetry of the cellular geometry.

According to this description a group of cells could be randomly scattered but would still have translational(shuffle, swap, or exchange) symmetry since it could be superimposed over itself, and permutation symmetry if some path or linear set of paths connect the cells in a way that respects their power pattern symmetry. All sorts of other detailed dimensional operations within a given power pattern. are possible depending on design.

Mathematical Description

Given positive integer base b, a power pattern is a list of n numbers 0 thru n-1 filling the n cells(1 number per cell) of a symmetric geometric pattern, where you take all the ones positions in the base b number list as the b^0 power pattern, all the b^1 positions as the next power pattern up to b^(k-1), as the final power pattern. For a single base number b and a simple geometry, line, square, cube, etc., b^k=n is the total number of cells in the pattern. The power patterns consist of positive base integers and zeroes so that the integers and zeroes can be given colors where the zeroes are usually white.

A 4x4 square grid of cells contains 16 cells. Using a binary base the numbers 0 thru 15 can be gotten using four base 2 power patterns, 2^3, 2^2, 2^1, 2^0 of 4x4 cells in each pattern. The 2^3 pattern will have eight 8’s and eight 0’s in its 16 cells, while the 2^2 pattern will have eight 4’s and eight 0’s, and the 2^1 pattern will have eight 2’s and eight 0’s and the 2^0 pattern will have eight 1’s and eight 0’s. If the power patterns have symmetry then rotation, reflection, reverse of 0’s and 1’s(2 color permutation) are all possible. In addition exchange(shuffle, or swap) of powers is always possible since this is translational symmetry. For instance the 2^0 power pattern could exchange values with the 2^3 power pattern so that 1’s and 8’s in the two patterns would change places. If the power pattern numbers do not have rotation or reflection symmetry then the only allowed symmetrical operations are translation, and usually permutation of colors.

The base b could consist of a combined set of bases. For instance with a 6x6 square of 36 cells the numbers 0 thru 35 can be gotten with four power patterns by using two base 2 power patterns and two base 3 power patterns. This is done by multiplying rightmost base position number with leftmost base power numbers as required. This definition leaves open all sorts of other ways to form the number list and thus the base positions, such as complex numbers, negative numbers functions and so forth.

Linear Example

The simplest Sn examples are linear power patterns or just a list of numbers in a line. The simplest linear example employs the binary base using zeroes and ones. Write down eight binary numbers in a column. This is an element of Sn(2x2x2) shown here with decimal equivalents. A second column is shown with the decimal values of each binary pattern. Adding these decimal values for each triplet produces the column of decimal numbers. The last three columns show the power patterns separated.

Binary Dec. equiv. Power patterns 2^2 2^1 2^0

000 =0 000 =0 0 0 0

001 =1 001 =1 0 0 1

010 =2 020 =2 0 1 0

011 =3 021 =3 0 1 1

100 =4 400 =4 1 0 0

101 =5 401 =5 1 0 1

110 =6 420 =6 1 1 0

111 =7 421 =7 1 1 1

Notice that the binary listing(and 3 column decimal listing) forms three symmetrical columns of zeros and ones. The three columns show what the black and white colored cells look like where the 1’s are generally a black or colored square and the 0’s are white squares. Each of theses columns is referred to as a power pattern, Pp. The solution to the power pattern is the decimal sum of the three columns.

The left column is the 2^2 power pattern, the middle column is the 2^1 power pattern and right column is the 2^0 power pattern. Now try flipping the left column over, top and bottom. This results in a list of the numbers 0 thru 7 as follows:

4, 5, 6, 7, 0, 1 2, 3.

In fact you can rotate any of the columns and will always get the complete list of integers, just in a different or permuted order. You can also exchange columns(also called shuffle or swap). Exchange the left and right columns to get this solution. 0, 4, 2, 6, 1, 5, 3, 7. Exchange is the most powerful operation possible because it always works to produce a full list. Rotation may not work when the geometry of the numbers matches another power pattern after the rotation, causing duplicate numbers to appear. For binary patterns a simple symmetry operation is reversal of zero’s and non zero’s in a pattern thus the 1’s become zero’s and the zero’s become 1’s. Another symmetry operation is circular permutation. Another symmetry operation is mirror reflection.

Operations:

Rotation

You can rotate a power pattern about an axis of rotational symmetry if the separate power patterns have a geometric symmetry and a group symmetry of intersection. The group symmetry/intersection rule means that each pair of power patterns intersect each other by the same fractional amount’s after the symmetry operation as before the symmetry operation for a given color(base number) of their cells. For instance a linear list of 16 binary numbered cells can can have any of its power patterns flipped end for end so that the 0’s will intersect eight 0’s and eight ones of each of the other power patterns and that is true for all the other intersecting pairs of power patterns.

Exchange

You can exchange(also called shuffle or swap) any two power patterns(example a^0 with b^1 become a^1, b^0) Exchange is the least restrictive and therefore the most powerful symmetry.

Permutation and Reverse(reverse applies only to binary base and means zeros and non zeros exchange places(white=0 and black=1 usually are the binary colors used)

You can permute colors(numbers) in any one n dimensional power pattern if this meets the group intersection rule above. Other kinds of group permutations may be possible depending on symmetry of the power pattern(s) and symmetry of the overall geometry of the cells. Pairs of columns or rows could be exchanged if the number of pairs is even.

Reflection

Mirror reflection is also possible if this meets the group intersection rule stated above.

Dimensional symmetry

Normally a power pattern is a product of two or more positive prime numbers equal to the number of cells. The dimensions of the cellular geometry are each represented by one or more of the prime numbers. An example is a 3x3x3 cube must use a trinary base or base 3 numbering consisting of 3^2, 3^1 and 3^0 Pp’s. The cube could also be a base 9 and a base 3 consisting of 9^0(3) and 3^0. This results in a nine color pattern and a three color pattern. The nine colors are the number symbols (0,1,2,3,4,5,6,7,8) times 3 giving 0,3,6,12,15,18,21,24, while the three colors are symbols 0,1,2. Thus the largest number is 24+2=26=27-1 as required. If the power patterns are shuffled you would have 3^0(9) and 9^0. This produces (0,1,2) times 9 giving 0,9,18, and 0,1,2,3,4,5,6,7,8. Thus the largest possible number is 18+8=26=27-1 as required. A 4x9 rectangle could have two binary base power patterns and two trinary base power patterns with multipliers of the left base positions as needed to form a complete numbering. Alternatively a 4x9 rectangle could have a base 4 and a base 9 power pattern. This is illustrated by using this game: http://www.puzzleatomic.com/GAMES/esnbase2-3Advanced/esnbase2-3Advanced.html.

Symmetrical S (also called Balanced)

All symmetry operations of the cellular geometry are allowed

Semi symmetrical Ss

Simi symmetrical or Semi balanced meaning that only some symmetry operations are allowed.

We can use a shorthand for denoting a Sn with certain properties:

S=symmetrical or balanced(all symmetry operations(rotation, reflection, shuffling, color permutation) are allowed)

Ss= Semi symmetrical or semi balanced(has at lease one symmetry but not all main symmetry operations)

Dsn=degenerate(semi space number)the numbers in the list can change or have duplicates after an exchange operation thus the number listing and its base or geometry are incompatible)

E = exchange(patterns can be swapped or shuffled, always produces a unique renumbering)

R = rotation(about all symmetry axes). Rs is simi Rotatable

Rs= semi rotatable(rotatable about some but not all symmetry axes)

P = permutation(colors or numbers can be permuted, each color takes the place of another color in a specified order) Ps is semi permutable meaning only some of the Pp can be color permuted or there is a restriction on how a power pattern can permutted.

M= mirror reflection allowed across all bilaterally symmetrical axes. Ms is semi reflectable.

Rules for producing balanced or semi balanced space numbers.

Overall symmetry of geometry: Thus a square or rectangle has this symmetry since it can be rotated, and mirrored. Many other shapes also have this symmetry such as a polygon, cube, 4D cube and so forth.

Intersection rule: For binary power patterns any pair of power patterns must overlap to produce an equal number of the four paired numbers 00,01,10,11. Example call the four 4x4 square power patterns A, B, C and D. Each of these is divided into a group of 8 white and 8 colored cells The white cells are zeros and the colored cells are ones (with place value of the 1’s depending on the power of the power pattern,2^0,2^1,2^3, or 2^4). Thus A /\ B must produce the following pairs of overlapped cells(where /\ means intersection or exact overlap), four each of 00, 01, 10 and 11.

Then A/\B/\C must produce two groups of 000, 001, 010, 011, 100, 101, 110, 111 to satisfy the pair rule for A^B, A^C and B^C. Then A/\B/\C/\D has to produce the full 0000 thru 1111 sequential binary numbering of all 16 cells. Any exchanging or shuffling of this intersection such as D/\A/\C/\B does the same but moves the numbers around in the solution pattern. If the patterns all have the correct intersections for the operations rotation, mirror, permutation the result is a balanced set.

As mentioned for Three and higher dimensional Pp the extra axes of symmetry cause some rotations to produce duplicate numbers so some figures may require a larger base number for a balanced solution. For instance for the 2x2x2 cube a balanced solution was found using one base 4 Pp and one base 2 Pp since 4x2 = 8 = 2x2x2. A fully balanced base 2 solution has not been found to date and is thought not to exist.

Enlarging

The 4x4 patterns can be enlarged to produce square symmetrical patterns for any 2^x where x is odd. For instance the six patterns 2^0 thru 2^5 result in six 8x8 power patterns using 2x8 bands and 4x4 checker board areas for the two added power patterns.

Mixed base Sn

You can mix different number bases to produce a balanced Sn(). The caveat is that the prime factors of the product determine the minimum sizes of the bases that must be used. A balanced set of power patterns for a 2x3x5 array of cells can be designed. This requires a binary, 2^0, 0,1, a trinary, 3^0, 0,1,2, and a pentary, 5^0, 0,1,2,3,4, power pattern with multipliers for the higher place number patterns to the left. The multipliers produce numerical alignment of the bases that allows this mixing of bases. For instance if you arrange the different power patterns as 2^0,3^0,5^0 you put zeros and decimal place numbers (2^0)x3x5, (3^0)x5, 5^0 into the cells of the three power patterns. With a different order of the power patterns you multiply with the base numbers to the right such as (5^0)x2x3, (2^0)x3, 3^0 with decimals. Thus you multiply the patterns to the left by any base numbers to the right. For two dimensional patterns this has been found to work for any integer product if it is possible to divide them geometrically to have the correct intersecting properties. It is usually not difficult to do and each solution leads to an infinity of larger solutions. A balanced game with these three base patterns has been implemented and is available for free play on the puzzleatomic.com website: http://www.puzzleatomic.com/GAMES/esnbase2-3-5hex/esnbase2-3-5hex.html .

Zero sum power patterns.

The sum of all the numbers of a zero sum power patterns always equals zero. One way to do this is to subtract 1 from the highest power pattern and then make it negative. The 4x4, 2^3 binary Pp gives (8-1)(-1)= -7’s in the cells of the 2^3 Pp. Alternatively you could have 7’s in the 2^3 Pp and make the numbers in the other three patterns negative. This results in two similar sets of numbers in the solution pattern where one set is positive and one negative, ie. 0,0,-1,1,-2,2,-3,3,-4,4,-5,5,-6,6,-7,7. A zero sum game like this has been implemented and is available for free play on the puzzleatomic.com website: http://www.puzzleatomic.com/GAMES/esn4x4-zerosum/esn4x4-zerosum.html .

Fractional Pp

You can use negative powers for the power patterns. For instance 2^0, 2^-1, 2^-2, 2^-3 produce fractions 1/8, 2/8, 4/8, 8/8. The total sum of the 16 cells is 15, and the highest sum of 4 cells is 15/8.

Irrational Pp

Irrational power patterns can be used in four 4x4 Pp such as rt2^0, rt2^1, rt2^2, rt2^3 giving decimals 1, 1.414, 2, 2.818. This gives a highest sum of 4 cells 7.232 and a total sum for the solution pattern of 56.856.

Imaginary Pp

Imaginary numbers can be used in four 4x4 Pp, 2i^0, 2i^1, 2i^2, 2i^3 producing 1, 2i, -4, -8i. For the solution pattern this results in a highest sum of 4 cells of -5 -6i.

Product combining operations for Pp

Product power patterns multiply the pattern numbers to produce a solution pattern by changing all the zeros to ones in the power patterns before multiplying. If this is not done all products will be zero. A binary power pattern product has all powers of 2, 2^6 thru 2^0 for k cells. There will be duplicates since 2^2 x 2^1 = 2^3 and so forth. For combined bases the solution pattern would have a products of prime powers in the cells with duplicates. If the duplicates are shown as a single color the solution pattern would have a symmetrical kind of pattern with different sizes of colored areas if the Pp have a symmetrical design. The highest possible power would have the greatest number of duplicates. This is somewhat similar to the idea of squaring to determine the probability of an occurrence. Of course there is no probability to determine but the math has the same kind of exponential similarity so the increase of in size of factor numbers = much bigger increase of their product.

Division combining operations for Pp

Division power patterns can be of two kinds. As for product power patterns all zeros must be changed to ones in the power patterns before dividing. Then divide either from highest to lowest or lowest to highest Pp. For a 4x4 binary Sn you would have 2^3 / 2^2 / 2^1 / 2^0 or 2^0 / 2^1 / 2^2 / 2^3. There will be a similar number of kinds of duplicates as for product Pp.

Power patterns and digital color graphics

A digital color graphic consists of mixtures of three colors in pixel groups. These could be compared to a base system of three additive color patterns known as RGB or red green blue. The usual method is to use two hexadecimal numbers for each color. Thus each color could be seen as a combination of two base 16 power patterns(degenerate) giving a total of 255 colors for each of the three additive color patterns. Exchange is only possible for each pair of base 16 numbers. For instance we could exchange the two red base 16 numbers. This would cause some of the colors to change appreciable other very little to not at all depending on the two number. If both numbers are zeros no change. If the left number is a zero and the right number a 15 then the color would go from a very dim red to a fairly bright red and similarly for the green and blue number pairs. Another way to see the two hexadecimals as power patterns is to use base 2, binary as eight binary power patterns. This would allow more color variation equivalent to eight factorial possible shufflings instead of two. However even with just the division into 3 sets of additive power patterns much manipulation of the graphic should be possible. simply produce the same graphic with drastic color changes. If the graphic is rectangular rotation and reflection would be possible as well causing much mixing of the graphic and perhaps making it unrecognizable.

Permutation of the different Pp in various ways would produce a large number of different graphics. For instance a 100 by 100 pixel Pp would have up to 10,000 different permutation possibilities along x multiplied by y giving 10^12 total for the three RGB Pp. This might be a method of encoding or encrypting a message or graphic. Decoding just reverses the encoding instructions. In reality it might be necessary to add some kind mixing to the individual Pp so that individual Pp could not be easily analyzed for holistic appearance. Encoding or mixing graphics as Pp in these different ways could also produce graphics with a very nice artistic appearance and give the brain a kind of engaging and entertaining entanglement.

A graphic could be tested for the number of duplicate numbers/total no. of cells, its non repetitive fraction.

Fractal Pp

Fractal Pp have a fractal geometry. This geometry generally has rotational and/or reflection symmetry operations. Simple power pattern systems are fractal like. They can be enlarged by enlarging the existing patterns and making additional power patterns designed by enlarging internal pieces of existing patterns. The path taken by following the sequential numbering is a fractal where different magnifications have a similar path when taken to infinity. As the system grows the number of possible power pattern combining operations increases exponentially faster creating a greater infinity of possible fractals.

Space Numbers and Mathematics

What are Space Numbers for? Often new mathematical abstractions have no practical use. The games are useful as mental exercise. Magic squares are a natural subject for space numbers. Space numbers might improve the ability to recognize symmetry, and number base systems by viewing the patterns as colored graphics and mentally working out the the combination as numbers and colored patterns. Investigating number properties by using space numbers and computers as a tool should not be discounted.

Space number patterns are quite similar to waves. Natural waves are generally continuous, such as sound or water waves. Digital waves are often used to simulate a continuous wave, for instance a digital wave generator. A balanced Sn() has the sequential number exclusion principle where no two cells can have the same number for all the allowed operations. Similar to analog waves analyzed as a group of additive sin waves, you can analyze integer number patterns by separating them into Pp. Degenerate systems of Pp can be used both for encoding and artistic renditions of computer graphic RGB patterns.

Our place value number systems uses the idea of the combinatorics of linear ordered pairs, triplets, … n places. Going to the left increases value exponentially for each movement. Going to the right decreases value exponentially for each movement. All ordered combinations of n places using base b(b symbols) as left-right place ordered produces b^(n-1) possibilities(always including zero as a symbol). Zero is the most frequent symbol since its purpose is to hold a place but not increase or decrease that place. We do not write zeroes that are leftmost or rightmost past the units place(a decimal point would be necessary making the numbers real instead of integers!) but they are mentally and mathematically always there and so as far a space numbers are concerned there are always a ghostly infinity of them.

Rising-falling number symmetry

Symmetrical space numbers have symmetry in their geometric power patterns. How can we see this symmetry when there are all sizes of numbers mixed together in the combined pattern? One technique is an idea borrowed from a mathematical property of some intersecting circles. It boils down to a way of finding the average twist of a linear set of circles coded as a sequential group of integers(in some given order) where every pair of circles is linked. This twist property is an invariant. The method for doing this can be found here http://www.puzzleatomic.com/ALL%20CIRCLE%20LINKS.htm in some expositions and a circle twist calculator. Power patterns are not linked circles. Since only a sequential set of ordered integers are needed to find the twist number we can use the same twist calculation on the combined Sn columns and rows of numbers if it is an orthogonal pattern. The calculation works by testing all possible ordered pairs of numbers. If the number to the right of a number is larger +1 is added, less -1 is added to the twist number. Some additional manipulation of the integer code is done but this is gist of the twist calculation. The twist numbers of columns and rows show where the columns and rows have symmetrical behavior if they are the same and in other ways. So instead of twist the same kind of calculation can show numerical symmetry between different looking rows and columns of numbers. Here is and 8x8 binary space number game that calculates linear twist, LTT, and total matrix twist, MTT, of rows and columns of for the mix and solution combined Sn. http://www.puzzleatomic.com/GAMES/esn8x8v1/esn8x8v1.html . In some instances all rows and columns add to zero twists. For other solutions there are some different numbers listed but always lots of repeats of numbers. It shows the left to right rising symmetry for positive twist numbers and left to right falling symmetry for negative twist numbers. These numbers are like vectors pointing in the direction of increasing/decreasing values. If the number of increases matches the number of decreases is zero the vector or twist number is zero.

You could also calculate vector rise along diagonal lines such as xy, -xy, etc. Using analysis in this manner enables such calculations to be applied to analogue waves as well(by using equal gaps) but the picture of the analogue waves pretty much tells the same story. The Sn numbers are all different but the vector numbers shows the symmetry of how they are different by showing the symmetrical and non symmetrical patterns. Even though the sizes of the numbers are all quite different, they exhibit a kind of scaleless symmetry.