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Mathemagics The New Wow Math

(The Tapping, Measuring and Mapping of the  Spatial Fabric of Space itself)

There is a new elementary math that has arisen in the last 5 years. It is essentially simple arithmetic on 2 or more dimensions. The math takes place in square and rectangular tables of numbers ‒ "square" for even dimensions and "rectangular" for odd dimensions.

The numbers in the tables range sequentially from 1 thru nk where k is the dimension of the table. Mathemagics is the discovery of numerical patterns in these tables which are all related to a number that is characteristic to the size n of the table, called its characteristic number. This number is fundamental in that it is related to the table's size and not the distribution of numbers within it.

The math is distinctly different than matrix algebra. In fact, none of matrix algebra applies to Mathemagics. Besides plain arithmetic and a few algebraic expressions, there are just two arithmetic functions that are involved. These are the modulus and integer functions of algebra. They have been reformulated to replace 0's with numbers. They are readily accessible in Microsoft's Excel program. In fact, Mathemagics requires Excel to operate -- it's laptop math entirely.

 1 2 3 4 5 5x5 65 65 65 65 65 1 20 8 21 14 2 65 2 11 4 17 10 23 65 3 7 25 13 1 19 65 4 3 16 9 22 15 65 5 24 12 5 18 6 65 65 65 65 65 65 65 65 65 65 65

Here to the right is the perfect size-5 square.  It is composed of numbers from 1 thru 25 = 52.

·  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 squares.

· Every set of 4 numbers, called a quadral, that corresponds to a rectangle or rhombus symmetrically located about the center, sum 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 squares of even-size, whereas this latter one here applies to squares of all sizes and is therefore a general property for 2-dimensional squares. Squares with this property are called pangenic.

Moreover, this 5x5 square is 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.

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

There are more of these characteristic tile 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 prime-number size square from size 5 thru 31. These hidden characteristic tiling patterns have never been detected nor seen before this.

Shown in the book is a whole new world of mathematics discovered after any perfect number square has been reduced to its modulus and integer loom-tables All the prime-number size tables yield so many equal-summing string series running in four cross-directions that contain only the numbers from 1 thru n exactly once, that the table takes on the role of a 4-way loom-table. As such, it provides the means for weaving many more perfect squares of the same size. And these all have the same complementary tile patterns which contain all the numbers from 1 thru n exactly once too, regardless of where they are centered in their table!

And there's even more: All of the tile centers sum to the characteristic number too. And if these complementary tiles are transposed, not only do their centers sum to the characteristic number, but their centers placed uniformly anywhere across the table do too, for an additional n2 equal characteristic summations.

Here is an example of the size-7 square's complementary loom tables X(n) and Y(n) derived from the original mathemagical square W(n) containing the numbers 1 thru  72 :

( 2.1 )      xij ≡ modulus [ (wij – 1) | 7 ] + 1

( 2.2 )      yij  ≡   integer [ (wij – 1) / 7 ] + 1

X is a table of the square's modulus values and Y is a table of the square's integer values as redefined here by (2.1) and (2.2) respectively.

Observe that every row, column, diagonal and tile contains the numbers 1 thru 7, exactly, independently and simultaneously. What holds for X also holds simultaneously  for Y

These loom-tables can regenerate the original table in a simple operation of addition, subtraction and scalar multiplication. The most amazing thing occurred when it was discovered when interchanging the roles of X and Y in the generation formula ‒ it produced another perfect square of the same size with all the same equal-summing patterns as the original. It was designated as the square's dual.

Besides having all the same properties, the only other thing that they had in common were the locations of the 7 numbers derived where the loom-tables were equal! These numbers are the bright blue boxes for w in the loom-table number distribution diagram above. And these numbers all sum to the characteristic number for the size-7 square.

These same square loom-tables come into play when generating cubes and quadra-cubes of Class-1 tables. There's more surprises when loom-tables are recovered from  tiling patterns in higher-dimensioned tables.

Systematically rearranging the numbers in loom-tables can yield nearly 600 more perfect squares of the same size with the same properties.

Absolutely Perfect Geonomic 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.

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. That's now a total of 9 planar slices all adding to the same number simultaneously!  (All not shown here.)

·      The sum of all the numbers in every embedded
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 sum
to
315.

·        Both  main and all wrap diagonals in both directions in each block-square sum to  315.

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

·       The same tile patterns in the size-5 square above sum to 315 continuously in every embedded block square of the cube, making it "ultra- perfect".

·        Each quadral in the central block-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.

All prime-number size cubes are shown to have complementary tiling patterns that are each unique to the square's size! And furthermore, the center numbers of the individual tiles in each block-square embedded in the cube taken together form a loom-table that will generate yet even more perfect cubes!

You have to see it to believe it. It can't get any better than this! These cubes are designated as absolutely ultra-perfect.

Perfect Geonomic Hypercubes

The picture at left 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 is the common point among all 4 dimensions.

Further, within the 3-dimensional cubic blocks are characteristic hemispheres, 1/2 from the primal table and the other 1/2 from its dual table, both comprising characteristic hyper-spheres in 4-dimensional space.

One of the fundamental discoveries revealed in the book is that of the continuous modularity property of 4-dimensional tables: Given a 4-dimensional table of size n (an n2 by n2 square table) it is shown that every contiguous block-square of size nxn embedded anywhere in the table sums to the same characteristic number of the table as do all the cubical rows, columns and diagonals.

There are some more amazing properties discovered there which uncover what the 4th dimension really looks like and the implication this might have on the whereabouts of the antimatter in 3-dimensional universes!

The tables in Mathemagics can be segregated into 6 distinct classes based on their size. Each class has its own set of equal-summing patterns. The numbers in blue are the sizes of squares not shown in these books for loss of legibility.

Here is a sampling of the patterns found for other than Class-1 squares.

Class-2 size-10:

The tile pattern consists of a double ring-diamond pattern of 38 connected cells that together sum continuously to 2-times the size-10 characteristic number when the green highlighted cells that are shared between the two integrated rings are counted twice. The ring-diamond pattern integrates to cover the square completely with overlap of 2 cells each at the top or bottom, or 1 cell each at the left or right depending whether it is propagated vertically or horizontally , respectively.

Class-3 size-15:

Here is the size-15 square. Its characteristic number is 1695.

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 15x15 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1 20 108 221 64 77 170 33 146 214 2 95 183 71 139 152 1695 2 111 204 42 85 198 36 129 192 10 123 186 54 117 160 48 1695 3 207 100 13 176 144 57 175 88 26 219 132 25 163 101 69 1695 4 153 41 134 222 15 78 191 59 147 165 3 116 209 72 90 1695 5 49 137 180 18 106 199 62 105 168 31 124 212 30 93 181 1695 6 70 83 171 39 127 220 8 96 189 52 145 158 21 114 202 1695 7 86 179 67 135 173 11 104 217 60 98 161 29 142 210 23 1695 8 182 150 38 151 119 32 225 113 1 194 107 75 188 76 44 1695 9 203 16 84 197 65 128 166 9 122 215 53 91 159 47 140 1695 10 24 112 205 68 81 174 37 130 218 6 99 187 55 143 156 1695 11 45 133 196 14 102 195 58 121 164 27 120 208 46 89 177 1695 12 136 154 17 110 223 61 79 167 35 148 211 4 92 185 73 1695 13 157 125 63 201 94 7 200 138 51 169 82 50 213 126 19 1695 14 178 66 109 172 40 103 216 34 97 190 28 141 184 22 115 1695 15 74 87 155 43 131 224 12 80 193 56 149 162 5 118 206 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695

In keeping with 60 cells per diamond in a size-15 square, each solid diamond tile sums to 4 times the characteristic number .

Tile pattern sums are always equal to or a multiple of the square's characteristic number.

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1 20 108 221 64 77 170 33 146 214 2 95 183 71 139 152 1695 2 111 204 42 85 198 36 129 192 10 123 186 54 117 160 48 1695 3 207 100 13 176 144 57 175 88 26 219 132 25 163 101 69 1695 4 153 41 134 222 15 78 191 59 147 165 3 116 209 72 90 1695 5 49 137 180 18 106 199 62 105 168 31 124 212 30 93 181 1695 6 70 83 171 39 127 220 8 96 189 52 145 158 21 114 202 1695 7 86 179 67 135 173 11 104 217 60 98 161 29 142 210 23 1695 8 182 150 38 151 119 32 225 113 1 194 107 75 188 76 44 1695 9 203 16 84 197 65 128 166 9 122 215 53 91 159 47 140 1695 10 24 112 205 68 81 174 37 130 218 6 99 187 55 143 156 1695 11 45 133 196 14 102 195 58 121 164 27 120 208 46 89 177 1695 12 136 154 17 110 223 61 79 167 35 148 211 4 92 185 73 1695 13 157 125 63 201 94 7 200 138 51 169 82 50 213 126 19 1695 14 178 66 109 172 40 103 216 34 97 190 28 141 184 22 115 1695 15 74 87 155 43 131 224 12 80 193 56 149 162 5 118 206 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695 1695

Class-4 Size-24:

Here is the size-24 square. It has a 24-cell interlocking ring-diamond patterns which sum exactly to the square's characteristic number 6924. It's complementary tile pattern is the 24-cell size-12 X-pattern, indistinguishable from the ring-pattern when integrated into a complete tiling pattern.

And that's still not all. Here is the same square that also possesses a quilt pattern of size-3 diamonds shown here. These sum continuously only vertically along their aligned centers. They sum exactly to the square's characteristic number.

There are yet other patterns on this same size-24 square that sum equally and continuously, such as:

• size-11 and size-12 solid diamonds integrated into a continuous quilt pattern which also covers the square exactly.

• a size-6 solid diamond pattern consisting of 2x2 block squares that sums continuously to 6 times the square's characteristic number

• a size-3 diamond composed of 3x3 block-squares that sums to 9 times the characteristic number.

• a pyramidal diamond pattern of size-5 that sums continuously to 40390 surrounded by a size-6 ring-diamond that sums to the square's characteristic number, both patterns do so equally and continuously and completely cover the square with edge- wrapping imposed.

Here is what a size-5 pyramidal diamond sum looks like:

The numbers in the cells are the multiples of the cell's number counted in a multi-layered lamination forming a pyramid of level 5.

Class-5 Size-35:

Here is the only printable size of Class-5 square. It possesses a continuous tiling pattern of a 5x7 staircase structure which sums to the square's characteristic number. It's complementary pattern is horizontal flip of the staircase pattern. In both patterns, the tile centers all sum to the characteristic number too.

When the staircase tiles are transposed, keeping the numbers in the same location in the table,  all the tiles will still sum to characteristic number. And that's for every location within the tiles too, not just their centers. Those are referred to as anchor-dot patterns because their centers are the anchors for dot patterns identically spaced relative to the centers in each tile. There are usually n2 such different  patterns in any one square.

Class-6 Size 25

The size-25 square is continuously 5x5 modular, so patterns can be made from 5x5 block squares. One tiling pattern spaces the blocks uniformly vertically and the complementary pattern spaces them uniformly horizontally. Each block sums to the square's characteristic number. All the block centers sum to the characteristic number too in each version. In fact, because the vertically spaced pattern is the transpose of the horizontally spaced pattern, the square has an anchor-dot pattern running through all its block-squares of either orientation.

The Matchmaker’s magic square.

Here is an example of a size-9 numeric table of what is called the Matchmaker’s magic square.

A matching in a matchmaker’s square is a selection of a different column for every distinct row and summing the numbers in the cells at their intersections.

The Five Basic Properties of the Matchmaker’s Magic Square

1.    The Dimensional Average Property: All the numbers average to its size n, here 9.

2.    The Matchings Property
: All matchings sum to the same number, n2; here 81.

3.    The Totality Property: All the numbers in the square sum to n3, here 729.

4.    The Subset Property: Any smaller square, called an abbreviated square, taken from anywhere in the table has all possible matchings summing to the same number. This number is different for different abbreviated squares. Moreover, these squares need not be composed of numbers from adjacent rows and columns -- any number of m independent rows and m independent columns may be selected at random and used to make an abbreviated m-by-m square from the cells common among their intersections -- of course keeping their places in the abbreviated square relative to the bigger square. And every matching within the abbreviated square will sum to the same number! This is just an amazing property that derives from both the numbers and the space in which they are strewn, that is, their underlying organized pattern.

5.    The Invariance-under-permutation Property: The rows and columns may be swapped one pair at a time, that is, rearranged into whatever order you like, by interchanging the position of any two rows or any two columns, any number of times in series, like spinning a Rubric’s Cube, and the square will still have the same properties #1 thru #4 ! Formally, the matchmaker’s square is said to be invariant under column and row permutation.

When I showed this to my sponsoring professor at Stanford, the late George B. Dantzig , he exclaimed " I'll be damned!". Why? Because this example was a counter example to his optimal matching algorithm. This had never been seen before in academia; he was in contemplative shock! He asked "How did you come up with that?" The books show how to readily create a matchmaker's magic square of any size.

Applications to Date

#1   The new math found application in explaining in mathematical terms just how the electrons are distributed in atoms in shell-pairs and their fill sequence in going from atom to atom in series by the addition of one more electron. It also described their organized orbital paths which Science has not yet been able to do experimentally. That's explained in detail in Part VI  of Laptop Mathemagics and Program 5 of Geonometry - Equal-sum Patterns in the Natural Number Series.

Synopsis of #1

This book develops for the first time ever a plausible basis for the pattern underlying the number of elements in sequential rows of the Periodic Table of Elements, namely the series {2,8,8,18,18,32,32}. Atomic physicists and chemists have yet to offer a fundamental explanation for this particular numeric pattern.

This feat is accomplished by showing that magic squares and magic cubes together hold the answer. The number and location of octals incident to the characteristic spheres in a series of perfect cubes are shown to offer a 100% correlation with the number and orbital paths of electrons in the layered electron-shells of the noble elements. Then, continuous complementary characteristic tiling patterns hidden within perfect squares of associated sizes are shown to provide the missing links needed to explain and account for the number-pairing of electron-counts in shell-pairs. And further, it shows that these dual loom tables balance-out the parity of coupled electron shells. This is one of the times that such a profound connection between the natural geometry inherent in the natural number series and the real physical world has  been made.

At left is one of the size-17 dual loom-tables that underlies the two spherical surface tiling patterns at left and right. The table is duplicated six times to demonstrate the patterns which contain the numbers 1 through 17 exactly once; namely each row, column, diagonal, both main and wrap in both directions, and each characteristic tile pattern throughout the table. That is a complex intricate detailed pattern that is so beyond the mind of Man to concoct; nor is it one that can be derived through classical mathematics! That is a pre-existing mathemagical  pattern.  The simultaneously complementary tile patterns are unique for every size of square.

These fundamental, tightly inter-connected patterns that exist among the series of consecutive natural numbers are shown to be of superior intelligent design, i.e. to be beyond human intelligence to fabricate. It then follows that the basic fabric of space, being of an exceptionally complex weave, must itself be the scaffolding upon which the atoms are built. Consequently the structure of the atoms can only be the result of a superior intelligent design.

Now let me make it clear here: I'm not attempting to provide grist to the creationists who use intelligent design as their camouflage for countering Darwin's theory of evolution. I'm just saying that there are fundamental numerical spatial patterns among the series of natural numbers which can account for the sequence {2,8,8,18,18,32,32} on the right-side of the Periodic Table above that exhibit an undeniable intelligence in their own configuration. They follow the formula y = 2x2 for x = 1,2,3,4. That is, there is an underlying numerically complex structure of the  atoms  that  we  no  less can immediately recognize mathematically as being far from random or coincidental.

This is a major discovery in mathematics, never seen before, that is presented for the very first time in four books: #1, #2, #3 and #4 below. It has direct application to the quantization of space and is a mathematical demonstration of the natural pervasive quantized vibrations found in String Theory to explain the hidden fabric of space. However String Theory is still stuck in the muck of 1D. Geonometry goes well beyond that!

Professor of Engineering, Dr. Phil Keston, of Santa Clara University commented that he knew of no mathematical description yet of the electron distribution of the noble elements.

#2     Properties of the 4th dimension were applied to describing the relationship among quarks in 1/4D sub-dimensional space and to demonstrate just why the neutron and proton are the only stable pair of the 6 basic quarks. That's detailed in Part VII of Laptop Mathemagics and Program 6 of Geonometry.

#3     Mathemagics was extended to describing hexagonal tables of sequential numbers starting with 1 thru 3(n-1)xn that summed equally along the sides of nested hexagonal frames. This was able to describe mathematically how snowflakes formed such perfectly uniform tri-axial patterns. That's explained in Part XI of Laptop Mathemagics and Program 10 of Geonometry.

#4     This new math is just now being released and is not yet taught in schools or colleges. It promises to offer some  capability to the emerging technologies. One of these is levitation and space propulsion. You just have to see it applied mathematically to grasp its possibility and potential capability. That's all described in  Part VII in Laptop Mathemagics and Program 9 of Geonometry.

This is just a tip of the iceberg, a mere sampling of the amazing equal-summing patterns that exist in the fabric of space. Now it should be quite evident that this new math is truly wow math!

There are characteristic circles, spheres and hyperspheres that also sum to the characteristic number of the square, cube and hyper-cube, respectively.

What was equal-summing 4-corner patterns in squares, called quadrals, becomes 8-cornered boxahedron  patterns in cubes, called octals, and 16-corner boxahedron patterns in quadracubes, called hexadectals.

The books Laptop Mathemagic and Geonometry uncover the source of these amazing patterns and shows how they originate in squares of a composite size derived from two smaller squares. Here is a list of continuous patterns:

1. Ring-diamonds that integrate like a flat stack of chain-link fences;

2. Solid diamonds that integrate like a quilt pattern;

3. Triangular and diamond patterns that integrate together like a woven tapestry;

4. 3D pyramidal square and diamond patterns on multiple laminations of identical tables;

5. Unique patterns for each square of a prime-number size;

6. 8 simultaneously occurring staircase structures for Class-5 squares of the order ab where the size of square n = axb and a & b are prime numbers such that a>b>3;

7. These patterns when integrated to cover the square and the center of the square is also the center of the central tile, then all the tiles' centers together sum to the square's characteristic number;

8. The centers of the transpose of each tiling pattern, integrated across the square, not only sum to the characteristic number too but do so continuously  ̶   here the center of the central tile need not be located over the square's center. The centers are now described as  an anchor-dot pattern because their overall distribution serves as a general equal-summing dot pattern anywhere across the square.

Each style of pattern sums individually to the square's characteristic number.  Further, these same equal-summing patterns exist continuously within the embedded squares of larger tables of higher dimensions.

Now here is an amazing discovery: Class-5 squares actually have 12 different patterns in the same square, each of which consist of n numbers but  which are different from each other, and all of these sum to the square's characteristic number simultaneously! This is beyond the mind of man to accomplish; it could only have been discovered by stumbling upon it. This result demonstrates that the fabric of space is an amazing weave of wonder! Mathemagics uncovers these amazing patterns in all number series running from 1 up thru nk. These patterns will be important in the future for the engineering fields of levitational and propulsion technology, synthetic chemistry, micro-chip technology, synthetic biology and  spatial telecommunications.

The primary books #2 and #3 below  describes all these geometries listed above and show you how to systematically generate all of them on your own.

How to get these books

To purchase any of the the Mathemagic books listed below from publisher CreateSpace, click on one of the preferred book title's below. If you wish to order different titles at the same time, click the "Return link" after updating your CreateSpace shopping cart to return to the top of this page here. Then click on the jump-link there to get to this book list. You can make another selection here and your shopping cart will be automatically updated when you are again transferred back to CreateSpace for a secure transaction.

Before you decide on which book(s) to purchase, click on the book's price and you will go to a preview page for that book on CreateSpace.

Continue this process until you are ready to complete your shopping and checkout via CreateSpace. All the books in your shopping cart will be delivered together. And unless you select expedited shipping, your order will be delivered to you by FedEx at minimum cost.

Note - you can't get any of these books on Amazon. This is the sole link to and from the publisher, Create Space, for these books. Use code EP2BTYAY  for a 10% discount on CreateSpace purchases.

To order, click on the title of any book of interest. To select multiple books in a single order return to this page after the last order before checking out your shopping cart on CreateSpace and click on the next desired title; it will automatically direct you to the CreateSpace page for the title where it can be added to your CreateSpace shopping cart.

360 pages, black & white print, a complete source of all sizes of mathemagic tables from size 3 up thru size  31 for squares, size 2 thru 15 for cubes, size 2 thru 7 for quadracubes and size 2 thru 5 for 5-dimensional quintacubes, plus magic triangles, diamonds, and hexagons -- \$31.95

#2 Geonometry - Equal-sum Patterns in the Natural Number Series 500 pages, printed in color, presents an integrated series of 10 amazing lectures for teaching Mathemagics -- \$90.00. A set of Power Point slides on 10 video CDs for screen projection of the color-patterns in the number tables covered in the book is available. Teachers should contact me directly as CreateSpace doesn't yet produce slide-oriented CDs. Click here to read the Table of Contents

#3 Laptop Mathemagics  400 pages, printed in color, contains everything presently known about this topic -- \$74.95   The book has been converted to a PowerPoint presentation with full script for passively learning/teaching Geonometry. It is available on a CD. Teachers should contact me directly as CreateSpace doesn't yet produce slide-oriented CDs.   .

Here's Mathematical Proof.  42 pages in color print (Based on Parts VI and VII of Laptop Mathemagics)  -- \$14.95

30 pages in color print  (Based on Part XI of Laptop Mathemagics)  --  \$15.95

Each page in all but one the books is printed in color on 8½ by 11 inch pages in large 12pt. lettering. All my books come in a glossy colored soft-cover, perfect-bound publication.

Treatment of the Subject

It is demonstrated throughout the chapters by showing numerous specific examples in a continuous series that multi-dimensional space can be tapped, measured and mapped by two 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 or rectangular 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 throughout in addition to all 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 (See the patterns above where they are depicted for the size-17 square). They are major fundamental discoveries.

These equality properties were only discovered recently between January 2005 thru September 2015 through the use of my PC computer employing Microsoft’s Excel program. The tables themselves would enhance the teaching of Excel because of the amazing pervasiveness of the equal-summing patterns. Plus, the amazing properties are all visible geometrically.

Through the use of two nxn square loom tables that are dual to each other, both derived from the original prime-number size  square of size n, the book makes the undeniable case that space itself is quantized within each size confined space with a spatial weave of interwoven strings from 4 different linear directions and within 2 simultaneously-occurring tiling patterns where  each contains the numbers 1 thru n exactly once, all simultaneously. These loom tables regenerate the original square as well as generate the original square's dual counterpart, and visa-versa. And it is this double-duality that actually accounts for the pairing of the numbers in the rows of the Periodic Table of Elements. The book explains all this in detail. This is an actual manifestation from the bottom-up of multi-dimensional String Theory. You have to see it for yourself to comprehend it! And it exists nowhere else.

Now, having observed the properties of multidimensional space as discovered in Mathemagics, I have presented in Part X of Laptop Mathemagics and Part II of Program 9 in Geonometry a radically new view of multi-dimensional space like it has never been seen before, but is strongly supported by the relationships among same-size characteristic numbers of various dimensions and some obvious physical properties that are side-stepped in academia. You just have to read it to appreciate it because the math eliminates any denial.

Who would benefit from knowing Mathemagics

Math education today at the K12 level lies in yesteryear: teachers are still writing with chalk on blackboards and talking while they’re writing, then quickly testing on what they presented to see that the students got it. Further, highschools are only authorized to teach what has been proscribed by the Superintendent of Education and his/her staff in their district. Nothing new is permitted because it may not have any use in preparing students for achievement testing or college entrance examinations. Teaching today is primarily targeted to getting students ready for what they will be tested upon, on and on, grade after grade. Teachers are rewarded upon the students' achievement on standardized tests. There’s no more room for a new math that nobody tests against.

Today is another turning point of how math will be taught in the near future. Changes have been seen throughout the last half century: using the adding machine, to the digital calculator, to the desktop computer and now to the portable laptop computer. Today many States’ Learning Centers are undergoing evaluations of how to best educate students via digital learning, that is, through the use of the laptop computer. Just do an internet search on “digital learning” and see for yourself.

It will be seen that this math does have practical applications in real space. This new math will clearly give attendees an advantage over their future classmates who will remain ignorant of the subject as this math is not taught in schools. It has yet to suffer a single criticism that is based on technical merits.

Parents today have their day completely filled with getting kids to and from school, doctors' appointments, sports practice, band practice, extracurricular art classes, preparing meals, on and on, even when school is not in session. Parents rarely have time to teach their children anything beyond discipline and primarily don’t even have time for each other. So inviting working parents to comprehend anything new, especially in math, is impossible because they have more pressing and disruptive things they must do to keep it all going.

So when does something so new like this new math of Mathemagics ever get reviewed, digested, and applied? The only place that I see currently is retirement or disability. So if you know of anyone who is in either of these categories and has or had an interest in numbers, Sudoku or math, you might consider giving them one of my books as a present. It would give them something positive and exciting to which to devote their idle time, especially if they are familiar with using Microsoft’s Excel software. My books Laptop Mathemagics and Geonometry - Equal-sum Patterns in the Natural Number Series show how to create everything discussed in it systematically from scratch in Excel. It sure would be more fulfilling, enlightening and even more promising than playing cards, games, bocce-ball, golf, watching sports on TV, visiting neighbors, gossiping at the local pub, etc. etc..

Even retired grandparents may introduce their grandchildren to Mathemagics. What my paternal grandmother taught me in math and art later paid off handsomely in my educational and productive years: I received, as the  school's only recipient, the Carnegie Tamoshanter Scholarship in gradeschool  to study art for 4 years and, as the  company's only recipient, the US Steel Fellowship to attend Stanford to study the mathematics of Management Science for 2 years.

My success came from knowing things about subjects beyond what my peers were taught by spending a year and a half out of highschool at the Carnegie Library in Pittsburgh PA researching math. By working at US Steel for 10 years using advanced algorithmic methods, I became an expert in numerous computational methods, one of which was linear programming. It hasn’t stopped even in my retirement, viz. Mathemagics! When I was at Stanford, I observed that just about everybody in my group sessions knew things about math that I was never taught because they had their training as exceptional protégés in a foreign country. Of course, so did I know things they didn't: the French students always came to me for answers in linear programming which I learned at US Steel. So right here's an opportunity to give your child or grandchild something new and exciting that very few will ever get a chance to experience and may just come into play in their vocation.

Let me explain something that is very fundamental in space exploration. Today the US has moved into a new era of rocket technology: the Orion rocket. The rocket is designed to take astronauts to asteroids and even farther to Mars. Guess what the round trip to Mars will take using gravity to make it all possible? 14 months! Absurd!! Only the emotionally introverted autistic beings would submit to being confined to a space capsule for 14 months? That's where Geonometry comes in. Both books, Laptop Math and Geometry, show how the vibrational fabric of space exposed in Mathemagics could be harnessed for levitation and space propulsion. You could be a part of this impending development. I myself am turning to experimentation in this regard with researchers experimenting with planar phaser vibrations arranged in loom-table patterns. Get involved now; it's coming and it will change the course of mankind.

About me, the  explorer of virtual-space and author of these books

I hold Mathematics Degrees from Penn State University (BS 1964) and Stanford University (MS 1971) with a good foundation in the sciences. Just because I don't hold a Ph.D. degree, doesn't imply that I don't know what I am talking about. Further, I don't stutter or use the word "perhaps" over and over again like most intimidated academicians do. I am not intimidated by academicians with Ph.D.s. I am affirmative throughout these books because I verified every property that I state as fact. The tables and formulas speak for themselves. My derivations are available to any reviewer who wants to be convinced that what I claim is true actually is true. I can pull up all the summations in Excel in an instant.

My Masters Thesis at Stanford unified a whole field of divergent mathematics in linear programming and covered over 160 pages of formulas and tables. Prior to that, many practitioners in the field were putting their own spin on the topic. That practice ended after it was given a common methodology with the publication of my Masters thesis as a book by US Steel Corp. entitled "Decomposition of Large Scale Systems for All Known Structures" (Library of Congress ref no. A417086, reg. 3/15/73). That book was presented at the Eighth International Conference on Mathematical Programming. I've done the same thing here in this book with magic number tables.

Any previously published work on so-called “magic squares“ is so myopic and the field of numerical spatial geometry so virgin that its presentation had to be broadened to establishing first-principles first. It takes many consecutive examples to see the broader pattern for the big picture to emerge. Specifically, one has to look beyond the smaller size squares to surmount the anomalies that occur there for embryonic size tables in 2-dimensions to see the bigger patterns and to get beyond those dimensions into 3-dimensional and hyper-dimensional space to see the really profound amazing relational patterns. So the smallest first-round introduction to this new subject is not  a life-long series of published papers in some abstruse math journal, but an entire self-published book. This book took retirement and 12 years part-time to write and get it all correct, cross-referenced and theoretically coherent.

But instead of trying to appeal to academicians, I kept the presentation pictorial and in summary form for the reader with an above-average IQ. I also refrained from using formal terminology that was not really informative and high-sounding although I had to introduce numerous new terminologies for the many new concepts that have never been seen before.

The book provides an excellent introduction to practical algebra for K12 students because all the many pictured patterns are formulated there. Further, the relationship among the characteristic numbers of the same dimension and different dimensions lead to some amazing formulas that have been given special names in classical Algebra. Here is where students can experience what these algebraic forms really relate to geometrically.

There are even more fundamental discoveries where the math is no longer rudimentary. The book Laptop Mathemagics shows all the mathematics underlying all these amazing magic number tables for the very first time ever. It's classical in its originality but not yet recognized as classical mathematics that should be taught in schools. It's all new, complete in theory, totally integrated within itself, and simplified in every operation. However, it is encountering resistance with educators who generally don't think outside their comfort zone, even when they claim otherwise. They need to protect their asses by not undertaking anything radical in their own fields of study for sake of discrimination by their own peers.

The really major consequence of this new math is the recognition of the structure of the multi-dimensional universe and the sub-dimensional spaces within them. Don't believe it? Then get the Laptop Mathemagic book or the book Geonometry - Equal-sum patterns in the Natural Number Series.

Einstein was only partially correct in his claim of slower time passage at the light speed limit. In fact, Geonometry claims that the light speed limit is pegged to the distance from the nearest star and demonstrates this mathematically using orbital diameters of planets in our own solar system.

Unlike tenured academicians, I was not paid for my research by any institution or grant money and consequently I am not under any obligation to give my results and discoveries away without reasonable compensation. I will not succumb to the pressure to go public with my achievements by formally publishing my results in mathematical journals. Further, it takes decades for academicians to get around to examining something so new and profound as Geonometry until it comes back to haunt them. So the way you get access to my work is to purchase my books, CDs or  Power-point Lectures and refrain from duplicating them or presenting the math therein as your own. I have no problem with your teaching this math on your own, earning an independent income by doing it and in that context I will assist you in any way I can.

My books are not available outside the United States of America, Canada or the UK for purposes of copyright protections.

Contact

If you wish to contact me, use this email address:  Mr.Math@Live.com

Every time I come across a numeric square on a website that someone touts as being perfect, I check it with my EXCEL functions for the size square by plopping the numbers as presented into my pre-formulated squares of that size. With the exception of the size-8 square and on only one occasion, the size-5 square, I have never yet found these numeric tables to satisfy all the linear and geometric equal-summing properties for that size that appear in my books here. And for the one incidence of the size-5 square being perfect, it did not possess any tiling patterns.

The perfect size 9x9 square which appears in my books on squares is made public here for the very first time. No one has ever produced a perfect size-9 square with all its rows, columns, main and wrap diagonals and all embedded 3x3 block-squares (continuous 3x modularity) each summing to the square's characteristic number, plus having equal quadrals symmetric with the square's center summing to 4/9ths the characteristic number everywhere throughout the square.

All perfect and near-perfect magic squares of the same size with all these properties are derivative works of tables that first appeared in the original book Number Magic and were copyrighted there  back in 2007.

The tables and pictures on this page and in these books have been formally copyrighted between 2007 and 2015 and may not be reproduced on other websites, in wikipedia, in presentations, or other publications without written permission by me or my authorized legal agent. These copyrights will be strictly enforced.

Let me make it perfectly clear to academicians right here: Although it be true that one cannot copyright a mathematical formula, these tables of numbers are not formulas, but patterns. These patterns and all their derivatives are copyright-protected by the copyrights already issued. Even though all the mathematical properties of these patterns still hold when the columns or rows are permuted and when the number tables are rotated 90º, 180º or 270º, or all the numbers are subtracted from the largest number plus 1, all of these are recognized as derivative patterns of the original by the registered copyrights. Be so advised. To publish a mathemagic table, you must generate your own original version. There are 596 possible versions for each size of square table, so have at it. It took me 12 years.

Intellectual property is the easiest thing to steal because nobody witnesses the immediate theft; it's done by stealth. This material is protected by 11 separate US-issued copyrights with 1 more pending and may not be reproduced in whole or in part. Taking intellectual property without adequate compensation to the author cannot be construed as being "borrowed".

If any unregistered copies of these books, CDs or portions thereof, hardcopy or otherwise, are ever discovered, both the institution and the teacher or Department Head associated with it will be sued for copyright infringement in Federal or International Court.

Because of the timely filing of these copyrights within the 60-day time limit of each books' publication date, any losing defendant will be required by law to pay for all of the author's legal expenses and their assets of equal value confiscated.

 Reward: Anybody who is first in reporting an infringement is entitled to an immediate reward of \$100 upon verification of the infringement and identification of the infringer, plus a 5% share of the judgment award or settlement. Please contact the author if you have such information -- Your identity will be held with the strictest of confidence. You notify me of the infringement and I'll take it from there.