To go directly to the Book List Click here. 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 n^{k} 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.
Here to the right is the perfect size-5 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 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 n^{2} 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 7^{2 }: ( 2.1 ) x_{ij} ≡ modulus [ (w_{ij }– 1) | 7 ] + 1 ( 2.2 ) y_{ij} ≡ integer [ (w_{ij} – 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 = 5^{3}. 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.)
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The sum of all the numbers in every
embedded · 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
·
Both the
rows and columns in each block-square sum · 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
· 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 n^{2} by n^{2} 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.
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.
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:
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 n^{2} 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. 3. The Totality Property: All the numbers in the square sum to n^{3}, 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.
#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 = 2x^{2} 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:
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 n^{k}. 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. #1 Number Magic 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
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. n, the book makes the undeniable case that space itself is quantized within each size confined space with a spatial weave of interwovenWho 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, 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 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 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,
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