Magic Squares 

Here to the right is the perfect size-5 magic square.  It is composed of numbers from 1 thru 25 = 5^2.

·  It has all its rows, columns and diagonals summing to 65, its characteristic number.

·  Its central number equals 13 = 65/5 = 325/25 and is the dimensional average of the sum of all the numbers in the square.

·  Every symmetric pair of numbers, each number opposite and equidistant from the center, equals  26 = 2 x 13. This is a property that holds for all odd-size magic squares but not for all even-size magic squares.

·  Every set of 4 numbers, called quadrals, corresponding to a rectangle symmetrically located about the center, including diamonds, sums to 52 = 4/5 x 65. Although that is an obvious corollary to the previously stated property, the previous property does not hold for half the magic squares of even-size, whereas this latter one here applies to magic squares of all sizes and is therefore a general property for 2-dimensional squares.

Moreover, this 5x5 square is magic-perfect by reason of the fact that all the row, column and diagonal summations are equal to 65, and the sums of all rectangular-spaced numbers about the center, including diamonds, are equal to 52, a specific fraction, 4/5ths, of the characteristic number 65 for the size-5 square.

Now, observe that every "+" and "x" in the size-5 magic square also sums to the characteristic number 65. This is called a complementary pattern in that two distinct patterns occur simultaneously. Not only that but these patterns interlock into a tiling pattern for the entire table. It can be dragged across the table and wrapped onto the opposite edges and still the patterns sum equally to the characteristic number. Every number in the table can function as the center of either tile pattern in the tiling pattern.

There are more of these characteristic patterns than there are row, column and diagonal equality summations combined by an order of magnitude. Whereas there are only 20 linear summations here, there are 50 pattern summations equaling the size-5 square's characteristic number. The book depicts these unique hidden tile patterns for every perfect magic square from size 5 thru 31. These hidden characteristic patterns have never been detected nor seen before this.

There's more: observe that all the numbers in each of the four quadrant triangles in the size 5 square above sum to 195. Further, this is 3 times 65 exactly. See the size 7 square below. The corner triangles each sum to 700 = 4 x 175. All odd-size squares greater than 3 in my book have the property that all corner triangles of the same size as the square sum equally. And that sum is always a specific multiple of its characteristic number !! For instance, the corner triangles of the size 9 perfect square sum equally to 5 times its characteristic number. Do you see the pattern of the multiple?

And there's even still more.  If a table of differences between the numbers in each cell and it's adjacent cell is constructed in any one of the four directions: rows, columns and the two diagonal directions, employing wrapping with the extension to the other side of the table when encountering the edge cell, then each complementary tile applied to this difference-table will sum to 0 anywhere in the table, including edge-wrapping of the tile pattern !!!

This is labeled the zero differential property which holds for all perfect size squares. What this amazing property implies is that these perfect number squares are perfectly balanced, specifically, if a square table were loaded with weights equal to each cell's number, the table could be spun like a top without wobble. And further each tile pattern loaded with contiguous weights from anywhere in the table could also be spun without wobble.

Magic squares with this level of perfection have never been seen before.

All squares of odd-size and even-size divisible by 4 are perfect. Even-size squares not evenly divisible by 4 are only near-perfect (next).

Near-perfect Magic Squares

At right is a size-6 near-perfect magic square. It is the first time the size-6 magic square has been depicted with all its diagonals equal to its characteristic number. All squares whose size n is two times an an odd-number are only near-perfect.

Near-perfect squares are only a hair’s breadth away from being perfect – these squares have only two rows which do not sum to the characteristic number but differ from it by amounts ± n/2 when n = 6 and  ± (n/2)²  for n > 6.

Equality in these two unequal rows can be obtained by merely swapping one pair of numbers from a single column between the two unequal rows whose difference corrects the amount of row-inequality (purple-shaded cells). Making any one of these exchanges, however, then trades two row-equalities for four diagonal-inequalities in complimentary minor and major wrap-diagonal sums. It’s a choice of having all row-equalities or all diagonal-equalities, but not both.

There are more equalities in the near-perfect form so that’s the nearest to perfect status that they come. Outside of this, it possesses all the other equality relationships ascribed to perfect status except that near-perfect squares do not possess tiling patterns nor the zero differential property.

Matchmaker's Magic Squares

A brand new kind of magic square, called the matchmaker’s magic square (mm for short), is introduced for the very first time.

At right is an example of one of size-5. Here, the numbers range from 1 thru 9 with some numbers involved multiple times.

·  It has all its diagonals summing to 25, its characteristic number.

·  Every set of 5 numbers corresponding to a matching of one row to one column, with no row or column being duplicated nor omitted, sums to 25.

·  There is always only one row and one column which sum to 25.

Every sub-square in the bigger square has identical properties, most impressive in larger size squares like the 9x9 where all 362,880 possible matchings are equal and all 40,320 possible abbreviated squares of size 5 (smaller squares derived from the larger square, for example, from selecting any 5 rows and any 5 columns independently of each other)  have the same properties. This holds for any size abbreviated square.

The chapter on matchmaker's magic squares shows these kinds of magic squares for sizes 4, 5 and 9; a size-4 magic cube; and a size-5  magic hypercube of 4-dimensions. Because it is such a simple procedure, the chapter on mm-squares also shows how to construct this kind of square for any size and across as many dimensions as you like. It also describes what constitutes a matching in cubes and hypercubes.

When this kind of magic square is put in normalized form, it is shown to have many of the general properties of the regular magic square of the first kind eventhough it has a different range of numbers and a unique set of properties all its own!

 Something regular magic squares have in common with matchmaker magic squares: 

Moreover, these 5 distinct matchings cover the entire square without duplication of any cell. That is, together they form a continuous tiling pattern that exactly covers the entire square without overlap.

Magic Diamonds

Shown for the very first time are magic diamonds. They are always of odd size. They involve the numbers 1 thru N used once and only once. The pivot number is always central and is the average of all the numbers in the table. Their rows and columns or diagonal sums are multiples of the pivot number according to their respective lengths. The symmetric pairs always sum to 2 times the pivot number. Their characteristic number is always their size-multiple of the pivot number.

There are two distinct types of magic diamond: one where the rows and columns all sum to a multiple of the pivot number, called the grid-sum diamond or just magic grid diamond; and the other where it's the diagonals which sum to a multiple of the pivot number, called the crisscross-sum diamond or just magic crisscross diamond.

The magic grid diamond for size-13 is depicted here below:

The pivot number is 43, the 2-dimensional average. The diamond's characteristic number is 559 ( = 13 x 43). Each row and column excluding the extreme ends sums to a multiple of 43. Both major and minor diagonals of the largest embedded square sum to a multiple of 43. That multiple is the number of cells in the row, column or diagonal.

The colored sections sum to the numbers at left. Each sum is an exact multiple of 43 and these multiples sum to 85, the largest number in the table, N. 

Every symmetric pair sums to 86 and that equals 2 x 43 = 2/13 x 559.

In the book, every magic grid-diamond is shown for sizes 3 thru 29.

Here below is the size-15 magic crisscross diamond:

The pivot number is 57, the 2-dimensional average. The crisscross diamond's characteristic number is 855 ( = 15 x 57). Both the central row and central column sum to the characteristic number 855 .  All the major and minor diagonals sum to either 456 or 399, a multiple of 57.  That multiple is the number of cells in the corresponding diagonal.

Every symmetric pair sums to 114. So every quadral sums to 228 ( =  2 x 114 = 4/15 x 855). The sum of all the quadrals (7+6+5+4+3+2+1) x 228 = 6384 is an exact multiple of the largest number in the table less 1, namely 112  So adding in the pivot number for completeness, all the numbers in the diamond sum to 6441, an exact multiple of 57, namely 113 , the largest number in the table, N.

In the book, magic crisscross diamonds are shown for all sizes 7 thru 27. Magic crisscross diamonds do not exist for sizes 3 and 5 .

While these equality sums have been obtained for magic diamonds, all of this is the same as for the odd-size magic square of the same size! Their generalized formulations are identical! These are general equal-summation properties that hold in common for every pair of both types of  magic diamond and both types of magic square of the same size.

If you play the California Super Lottery, here is an application of a grid-diamond that may have some real use to you.

As you know, playing the California Super Lotto requires you to mark 5 numbers ranging from 1 to 47 and one number not necessarily different from the first 5 between 1 and 27 as the mega number.

I have taken the size 47 geonometric grid-diamond with equal summing equal-length rows and columns and converted it into a modulus 47 diamond table; that is, the numbers in the size 47 diamond range from 1 to 47 where each number is replicated either 22 or 23 times.

This diamond table can be used as a dart-chart to generate your own sets of lottery numbers that you can use consistently over and over again. If used in the manner described on the first page of my book California Super Lotto Dart-chart, your odds will be increased 100 times over just accepting a different random number generated for you by the lottery, if requested, each time you play. The book contains five duplicate 15x17 dart-charts that you can remove and put on an 18x24 inch cork bulletin board and make a Lotto dart-chart board.

Click the Buy button to read the instructions and see the dart-chart
You can purchase an 8½x11 hard-copy or download the book and print-out  the sectional charts on your printer.

Triangles

At right you see a perfect size-9  magic triangle. All the same-length diagonal and horizontal summations are equal throughout the table. The  corner size-2 triangles also sum equally. They are shown in the Geonometrics book ranging sequentially from sizes 2 thru 18.

Just as for magic squares, there are both perfect and near-perfect magic triangles. These perfect and near-perfect magic triangles have never been seen before.

Absolutely Perfect Magic Cubes

Cubic number tables are easily viewed as a sequence of block-squares stacked like cards in a deck of cards. Each block-square is a left-to-right vertical planar slice through the cube.  View the top square as the furtherest from the front of the cube and the bottom square as the front of the cube facing you.

Below at right is the absolutely perfect size-5 cube of numbers 1 thru 125 = 5³. Observe that for the 5x3D cube:

·        All the planar rows and planar columns add up to the same number 1575.

·        All the planar levels from top to bottom add up to the same number 1575 (sum of identically located rows among all block-squares).

·        All the six planar main diagonals each add up to 1575 (the sum of identically located diagonals among all block-squares).

·        All of the planar-wrap diagonals in all six directions (two crossing each of the three axes) sum to 1575 – making the cube absolutely perfect.
(All not shown here.)

·        The sum of all the numbers in every 5x5 block-square is 1575.

·        Every octal (pair of identically located quadrals from oppositely positioned block-squares) adds up to 504.

·        Each of the only two possible octahedrons sum to 378 = 3/4 x 504. (crimson and plum colored cells)

·        Both the rows and columns in each block-square equal 315.

·        The centers of all the block squares sum to 315.

·        Each quadral in the central square #3 sums to 252.

·        Every pair of numbers symmetric with the central pivot sums to 126.

·        The central pivot number is 63,  the dimensional average of all the numbers in the cube.

All these relationships are correlated with a list of formulas for the size table at the end of each table’s section.

Chapter III on cubes shows perfect cubes for sizes 2 thru 12. All cubes shown, except the size-10, are absolutely perfect! It is outlined there just how the size-10 magic cube can be manifested absolutely perfect by going into the 4th dimension and collapsing that magic table along its B-axis by taking depth-sums.

Perfect Magic Hypercubes

The picture at right shows what a 4-dimensional table looks like after it is contorted into a torus. The spirals depict  the major diagonals in the table. (See: http://markorodin.com for the current 4-dimensional mathematical configuration of the universe.)

The picture below right depicts what a 4-dimensional box looks like in 3-dimensions so you can relate the individual embedded cubes in the hypercube with interlaced cubical blocks of numbers in the hypercube's table, below left. 

Recall from the preceding section that cubic tables are rectangular, so these embedded cubes overlap in the 4-dimensional table just as they do in the 4-dimensional view of cubes at right. The reason there are 10 cubes in the table and only 4 in the picture below is that there are multiple embedded cubic tables passing through each 4-dimensional cubic representation, just like multiple smaller boxes contained in a bigger shipping box. The center of the table corresponds to the center of the torus and is the common point among all 4 dimensions.

The book has two chapters on hypercubes, Part IV and V. Perfect hypercubes are shown from size 2 thru 7 in 4-dimensions and from size 2 thru 5 in 5-dimensions in very easily observed number arrays of 2-dimensions.

One of the fundamental discoveries revealed in the book is that of the continuous modularity property of 4-dimensional tables: Given a 4-dimensional magic table of size n (n² by n² table) it is shown that every contiguous block-square of size n by n embedded anywhere in the table sums to the same characteristic number of the table that all the rows, columns and diagonals do. There are some real surprises discovered there that are not revealed here in this synopsis!

The book The New Geonometry - The Natural Geometry of Natural Numbers describes all these geometries depicted above and depicts all the "geonomic" tables that can be printed legibly.

Geonometry is the discovery and exploration of the natural geometry that lies hidden in the natural number series 1, 2, 3 ... . When this continuous stream of numbers is placed in a confined array with geometric boundaries such as a square, diamond, triangle, cube  or hypercube of 4 or more dimensions, amazing equal-summation patterns emerge that are so integrated into each other that it appears as a woven fabric mimicking the boundaries of its confining geometry. This is no illusion because this fabric extends beyond the table like a draping table-cloth and can be pulled across the number table either horizontally, vertically or diagonally in either direction and all the while the tile patterns' equalities still persist eventhough the numbers within them change!

These complementary tiling patterns found for prime-number size squares are extended to include all perfect squares. It is demonstrated how these "geonomic" squares can be woven into equal-summing quilt patterns.

Each page in the book is double-side printed in black and white on 8½ by 11 inch pages. It comes in a glossy colored soft-cover perfect-bound publication of 276 pages.

To see a 60-page overview of the book's contents click here.  You will see the enormous amount and depth of material presented in this book. This is the most extensive work on so-called "magic" number tables in the world.

Book B&W interior $19.95 / Color download $14.95. To the buy the book click the button here.  

 Treatment of the Subject

Even though the subject is somewhat technical, it is presented in a systematic manner so that most intelligent readers can readily understand it without becoming stymied by the lack of a math background. Although it is number-based, it is the patterns of summation equalities that are important, not the numbers per se. A liberal use of consistent coloring (or in the case of black & white, different shades of gray) is employed to identify equality patterns throughout the presentation thereby freeing the reader from performing the many additions that would otherwise be involved to observe the equal summations. Only a few spot-checks along the way will suffice to understand and confirm what’s being presented.

And the methods employed to expose these amazing tables are not described in my book. These methods of manifestation are my intellectual property and I'm not revealing them without appropriate compensation. Sales of my book do not compensate me for the years of development of this subject to just lay it all out there for the unscrupulous or for the academicians to steal. I'm not accusing all academicians of being unscrupulous but it only takes a few and these assholes do exist in academia.

Nonetheless, everything is spelled out; so for the perennial insipid doubters, it’s all there for a thorough re-examination. And the number of equality summation patterns is maximized including characteristic circles in squares, characteristic spheres in cubes and characteristic toruses in 4-dimensions (doughnuts). For the given number of dimensions,  there can be no more basic summation equalities than those demonstrated for each and every magic number table.

These tables were printed directly from Microsoft Excel program immediately after the summations were verified so any error is highly improbable. I'll stake my reputation on it because I'm the one who did the calculations and verification of every table in each book and personally transferred the tables either directly to Microsoft Word or indirectly through Adobe Photoshop when they were too large for Word to handle. I made no shortcuts nor assumptions; everything was verified.

Every size table that can be printed on one or two 8½x11 pages is depicted in readable detail. In fact, the only exception is the size-5 square in 5-dimensions; if it was to be printed at all, had to be printed in a fashion that its 5 embedded quadracubes were spread over five consecutive pages and that is as far as it is possible to go and still maintain legibility and coherence.

I have relegated the general algebraic formulas to the end of each table's section in a summary list where you can see the algebraic pattern among all the different equalities that were enumerated, in just one table. The more formal formulas were relegated to the Appendix.

Reading the Appendix is not necessary to grasp the big picture, but for those who have had at least an introduction to high school algebra, the Appendix is very enlightening.

In the Appendix, it is proven that these number arrangements are pervasive throughout any number of dimensions and are the properties of space and not the properties of the numbers per se'. The Appendix  serves well as a teaching guide for demonstrating how patterns are formulated and then unified into generalized formulations.

The Geonomics book contains a lot of new information and many neat ideas never discussed anywhere before. Don't let anyone claim otherwise.

Making the Case for Intelligent Design Patterns

This book demonstrates that there exists an underlying pre-existing numerical pattern of space that is not only consistent and pervasive but, since the measures form a progressive pattern themselves, are of definite intelligent design.

To make the case for an intelligent pattern underlying multidimensional space, there is no need to address the details of all the methods that were used to uncover these number patterns; only the existence of intelligent patterns need be demonstrated – and that’s what the primary book is all about. It is 276 pages of solid discovery, including the informative Preface, Foreword and Glossary.

Let me remark: I didn’t invent these numerical tables – I discovered them just as the first prospectors discovered oil.  These discoveries of pre-existing conceptual patterns underlying square, rectangular, diamond-shaped and triangular number arrays of the natural number series is way more than my mere creative handiwork; it is something much more fundamental on a cosmological scale. It says something profound about regular and hyper-dimensional space.

These equality properties hold for all sizes of number-tables from 1 to infinity and have existed ever since the beginning of time, if in fact there was such a beginning. They were only discovered recently between January 2005 thru April 2010 through the use of my PC computer employing Microsoft’s Excel program; they were not invented nor constructed from imposed patterns like all the limited published work on magic squares so far. The computer functioned as my microscope, my eyeglasses, my telescope and, as you will see in the Postscript, also my transporter. This book lets you see what I saw in the grandest fashion with very little effort.

It is demonstrated by showing numerous specific examples in a continuous series that multi-dimensional space can be measured and mapped by four distinctly different measures and yet still independently exhibit identical dimensional geometric equality patterns among all the dimensions. Specifically, the multiple cross-directional summations within a consecutive number series, arranged in a square, rectangular, diamond or triangular table spanning multiple dimensions, all adding up to the same number is clearly not just happenstance, nor is it a predictable or  contrived occurrence. And all the embedded block-squares in higher dimension tables summing to the same number in addition to the summations just cited is not something that can be concocted. There are consistent equality patterns so complex that it is beyond human intelligence to fabricate them. They are an intellectual design property of space as well as major fundamental discoveries.

The number patterns manifested herein stand on their own without having to resort to unrelated non-numerical patterns such as chess-board moves of the knight or sacred science for their manifestation. Moreover, the patterns that emerge are not strictly numerical either; they are shown to be essentially the property of space itself.

These natural-number summation-equalities that arise again and again in larger and larger tables and in higher and higher dimensions from increasingly multidimensional perspectives can only be attributed to the pre-existing intelligent design of multidimensional space.

Note: This book is not intended to be a proof of creationism trumping evolution. It is merely a numerical exhibition that there is definitely an intelligent numerical order underlying multi-dimensional space. Amazingly, these intelligent patterns actually have correlations with the physical universe.

The book makes the case for intelligent design in two physical instances, one in astronomy and one in atomic physics: in the spacing of planetary orbits in our solar system and in the number of elements in each period of the Periodic Table of Elements, neither of which has been recognized heretofore.  See the 4th book below.

If there is to ever be any basis for a Mastermind’s shaping of the universe, this book furnishes that basis by demonstrating the pre-existence of pervasive numerical and geometrical patterns which are purely an intellectual product that no one can deny. And it's all there for you to see for yourself.

If you think you know all about this subject, you're in for a real disillusionment. You can no longer deny the existence of these patterns of superior intelligent design because my book shows that they are pervasive on every dimensional level using 4 distinctly different measures. And despite the uniqueness of their patterns among these 4 different measures, they all have the same mathematical formulation for their characteristic number and inherent dimensional geometries. The content of my book will profoundly change your world view just as it has changed mine, this I promise.

A note to academicians (and laymen alike)

Here's proof that these magic number tables cannot be generated
by formulating and solving a set of simultaneous linear equations

1.   Even if all the equal-summation constraints for magic number tables of 2-dimensions were to be formulated into a set of simultaneous linear equations whose variables were restricted to be only the numbers in the table, there would not be a sufficient number of independent equations to solve them simultaneously.

(For the layman: To do this mathematically, each equation must be independent from any other and the number of equations must equal the number of variables exactly.)

2.   Furthermore, even if that were possible, there is nothing to prevent all the variables from equaling the dimensional average for the number table – there are no linear expressions to eliminate this possibility without introducing yet even more variables into the equations.

(For the layman: When inequalities are introduced into the formulation for, say, forcing half of the variables to be larger than the known dimensional average, they must be turned into equations by introducing yet another variable, called the slack variable for taking up the slack when all the other variables sum to a strict inequality.)

3.   And still, if such a solution were to be obtained, there is no guarantee that the resulting values for the variables would be all positive unique integers, viz. some or all values for the variables could be duplicated, be fractions or even be negative.

(For the layman: Mathematicians have no computationally tractable way of solving a complex set of simultaneous linear equations like these in all positive integers, even using the powerful linear-programming optimization algorithm which would eliminate negative values for all the variables.)

4.  And if even all of the above constraints were possible, imposing the constraints for simultaneous complimentary tiling patterns on squares of prime-number sizes would require knowing apriorily just what forms these patterns would take. So other than those patterns exposed in this book, how would mathematicians know just what form these patterns would take for larger prime-number sizes ?

(For the layman: One can't prove that he/she can make these patterns happen without knowing apriorily what form they will take in order to formulate them in advance.)

5.  There is a branch of mathematics called integer linear programming that employs an algorithm developed by Ivanescu & Rudenu from Romania back in the '60's that was published by Springer Verlag in Germany. It solves linear programming problems in non-negative integers. However, that algorithm is only useful when there are integral coefficients of the variables that differ from 1 and from each other. In linear formulations of the equal-summations patterns in magic squares, all the variables in every equation have the coefficient 1. And with no objective function that values any one variable over any other, that algorithm is useless for arranging numbers in complex equal-summation patterns.

(For the layman:  Using the Integer linear programming optimization algorithm of Ivanescu & Rudenu becomes equivalent to trying all possible combinations and that becomes intractable in short order. For instance, the number of combinations to examine for finding all the patterns of the  perfect  square of size 11 is almost 40 million: 39,916,800.)

In plain and simple terms, even the best of the best mathematicians using high-powered computers, cannot manifest by computational means alone these pervasive and ubiquitous numerical patterns for magic squares of any substantial size by imposing constraints upon the natural number series. They would only end up trying all possible combinations and examining them for unknown patterns. So it is quite evident that these equal-summation patterns are beyond human intelligence to fabricate apriorily. The peculiar patterns can only be at first discovered for prime-size number tables and be subsequently extended to larger-size composite-tables and then to ones of higher-dimensions.  QED.