What’s New in this Book

A necessary condition for a square to be magic in this book is that it must have equal quadrals everywhere, that is, it must have the corner numbers of every possible rectangle symmetric about the square’s center all sum equally, including diamond patterns within odd-size squares. That’s another whole degree of equality imposed beyond what heretofore passed for being called magic. So it is to be understood that if a square is designated magic in this book then all of its quadrals are equal.

That's just for squares. There are geometrical counterparts of the quadral and diamond in squares, called octals and octahedrons in cubes, and hexadectals and icosahedrons in hypercubes. So for any table to be even called magic in this book, all of the geometries associated with its dimension must be equal throughout the table. That is a constraint that nobody has imposed on their number tables heretofore.

The designation of perfect means that every possible linear sum of numbers equals the characteristic number for the given size magic square, specifically, all the rows, columns and diagonals, both main and wrap, major and minor.

Perfect magic squares are demonstrated for sizes 5x5, 7x7, 8x8, 9x9, 11x11, 12x12, 13x13, 15x15, 16x16, 17x17, 19x19, 20x20, 21x21, 23x23, 24x24, 25x25,  27x27, 28x28, 29x29 and 31x31. For the first time a prime-number-size perfect magic square as large as the 31x31 has ever been published.  Further, present-day investigations by others target the 8x8 magic square, and that magic square is not being presented in its perfect form as defined here. All other size tables  presented are grossly imperfect, either having unequal quadral summations or unequal summations for some or most of their minor and major wrap diagonals. No one as far as I can see, outside of the sizes 3, 4, 5 and 7, has gotten it perfect.

Squares that are near-perfect magic squares are presented for the first time for the 6x6, 10x10, 14x14, 18x18, 22x22, 26x26 and 30x30 tables. It takes a broad perspective to see that all even-size squares whose size is 2-times an odd number are at best near-perfect, and that it is a normal state of affairs. The proof of that is in my forthcoming book to be released in circa 2014 that deals with these technical matters .

For magic cubes to be perfect, it means that every planar sum of numbers equals the characteristic number for the given size cube. Besides having equal octals throughout the table, all its planar sums in all nine directions, including all the planar wrap diagonals, equal its characteristic number.

The existence of parallel planar equalities in all three axial and six diagonal directions for any given size cube is a major discovery. All cubes whose same-size square is perfect in 2-dimensions are found to be also perfect in 3-dimensions. In addition, these cubes also have linear sums (channels) in two horizontal directions and vertical sums (pillars) within the planar slices throughout the cube that are also equal.

These planar slices of equalities within the cube are found to have hyper-cubic counterparts in four and five dimensions.

The investigation of perfect magic cubes in all its generality has been done here for the first time from sizes 2 thru 11.

Here’s another topic that is brand new – within magic squares and magic cubes whose size is a multiple of 4, the maximally inscribed circle or sphere, respectively, also sums to the same characteristic number as do all the rows, columns and diagonals!

No one has ever attempted penetration into hyper-space with magic number tables before. Here, perfect magic quadracubes are shown for 4-dimensional squares for sizes 2 thru 7 and perfect magic quintacubes are shown for 5-dimensional hypercubes for sizes 2 thru 5. All have equal dimensional geometric summations throughout their number tables.

Moreover, all the embedded block-squares in both cubes and quadracubes have total sums equal to the table's characteristic number. This property is extraordinary in 4-dimensions because every embedded block-square anywhere within any quadracube, even crossing embedded cubic boundaries, sums to the characteristic number. It's called the continuous modularity property.

This property has never been seen before due to the fact that the perfect magic quadracube has never before been manifested. This property only exists at the 2-dimensional level  for magic squares whose size is a square of a prime number and only when that size square results from collapsing the same-size 5-dimensional quintacube to a 4-dimensional quadracube. That's why it has never been seen before in magic squares; its derivation takes collapsing a 5-dimensional table by taking depth-summations along one of its axes.

Another amazing property is that of magic squares of prime-number size: it is demonstrated that there are hidden complementary tiling patterns running throughout the entire number table that also sum to the table's characteristic number. These are demonstrated to possess more equalities than all the linear summation equalities combined by an order of magnitude. These tiling patterns have never been exhibited before because they have never even been recognized.

The most profound discovery in the book is the 100% correlation of the number of octals in characteristic spheres within perfect magic cubes which equal the cube's characteristic number with the number of elements in each row of the Periodic Table of Elements. And further, the simultaneously occurring tiling patterns hidden within perfect magic squares of prime-number size actually explain the fundamental geometry underlying the pairing of numbers in the Periodic Table table: {2,8,8,18,18,32,32}. They can account for actual physical patterns of electrons the outer electron shells in the sub-dimensional space of the atom.

A completely different genre of magic square, called the matchmaker magic square, or mm-square for short, is introduced for the very first time. The underlying pattern of mm-squares is depicted for the general case of a square of any size.  It’s extension to mm-cubes and mm-hypercubes shows that they continue to maintain their equality patterns in higher dimensions. But most exciting is that this completely different type of magic square is described by the same set of equations for every size of table and dimension as are the regular magic squares! That is formulated and proven in the Appendix.

A third configuration, that of magic diamonds, distinct from magic squares, is shown for the first time. They are always of odd-size and are the most complex of magic number tables to be manifested. There are two types: one which has all its row and column sums proportional to their length and another which has all its diagonal sums in each crisscross direction proportional to their length, each sum an exact multiple of their central pivot number.  It is the first time that these are shown anywhere.  Grid-diamonds are depicted for sizes 3 thru 29 and diagonal diamonds for sizes 7 thru 27.

What is really astounding is that all these different magic number tables of all sizes have identical formulations, even these diamond-shaped ones!

In summary, almost everything that is presented in this book is being shown to the world for the very first time. It is the end result of aiming for the BIG PICTURE by deciphering the inherent number patterns of intelligent design that are pervasive and common among the all magic number tables.

This book takes what was heretofore merely recreational math and turns it into the exploration of virtual space that may well have profound impact upon our interpretation of our place in the universe. With these discoveries, we are able to get a grasp of the actual inherent  properties of the 4th dimension.

With these discoveries, we as a species are now being challenged to recognize the reality of the 4^th dimension and perhaps eventually penetrate it, if not just conceptually, in actuality.  So strap yourself in and get ready for a real mind-trip into the exciting realms of discovery in hyperdimensional space!

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