BY: Dwane H. Campbell and Keith A. Campbell
Last updated November 20, 2013.
The primary focus of this site was originally the set of 8 by 8 by 8 Magic Cubes that are often called perfect magic cubes of order-8, nasik magic cubes of order-8, or order-8 pan-2,3-agonal magic cubes. A subset of these cubes that can be easily manipulated using binary math is described in detail. There are 3,295,497,267,707,904,000 (3.3 quintillion) visually different magic cubes described in this subset. All cubes in the set have over 10,000 different ways to sum to the magic constant.
In the last few years, I have extended the concept behind the creation of the magic cube to nasik magic figures of nearly all sizes and dimensions. The basic properties of these figures can be described by an examination of the set of 4 by 4 Pan-Magic Squares, the smallest recognized member of the set of n-dimensional nasik magic figures. The principles of the square creation can then be extended to make order-o^{p} Magic Squares, Magic Cubes, Magic Tesseracts and higher dimensional figures with similar magic properties.
There are even Pan-Anti-Magic Squares.
A unique feature of the subset of magic cubes described on this site is that they can be generated from the numbers in just one line of each of the dimensions of the figure. I call these lines Master Base Lines. For instance the three line sets 0 30 97 127 388 410 485 507, 0 166 265 431 83 245 346 508, and 0 329 150 479 40 353 190 503 are a row, column, and pillar of a cube that was published in 1979^{6}. From these three lines, the remainder of the order-8 cube can be easily constructed. Based on this feature this is a new class of cubes that I call a Magic Line Figure.
The concepts behind the figures are actually quite simple. An attempt will be made to describe the construction and characteristics of the figures in terms that are understandable to anyone with minimal mathematical background. There are interactive demonstrations that are designed as aids on some of the pages of this site and also downloadable cube and tesseract generators. It is suggested that one read the Magic Squares and Magic Cubes sections before attempting to use one of the generators as they give background on how and why they work. The Cube Generator Guide and the Tesseract Guide for each generator should also be perused, as use of some functions is not intuitively obvious.
Numerous sites provide background information about magic squares, cubes, tesseracts, etc. Although this site will use some of the definitions and conventions from these other sites, some concepts will be described differently. An attempt will be made to provide explanations for any terms used. Better descriptions will often be provided at the other sites. Some definitions of terms as I use them on these pages are shown in Definitions.
There are six major sections on this site, each with several subparts. The Magic Squares, Magic Cubes, and Magic Hypercubes sections progress from the simple order-4 pan-magic square to multidimensional hypercubes of order-2^{n}. The Other Order-8 Cubes discuss all possible order-8 nasik cubes. Miscellaneous information and the cube, tesseract, and hypercube generators are available from the Miscellaneous and Overview sections.
The Overview section and its subparts give some general information and background for the site. The History subpart describes my personal path toward the creation of this site. A Definitions subpart is included as my construction is new, requiring new terms and altered definitions of other terms.
There is a list of References and Web Sites. There is also some Contact information.
Various Downloads are available. The most useful and interesting for the casual reader are the cube and tesseract generators and the SquareLines, CubeLines, and TesseractLines spreadsheets. The former allow the user to easily make and manipulate their own unique order-8 cubes and order-16 tesseracts. The latter are compilations of figures of different orders with many feature variations.
The main page of the Magic Squares section is an introduction to the method of construction of the magic line figures created on this site. The discussion focuses on the order-4 pan-magic square but the concepts can be generalized to the larger figures. There are interactive parts to aid in the discussion.
The subsections of Magic Squares divide the pan-magic squares into categories that each require somewhat different construction methods.
The Order-2^{p} Squares subsection discusses order-8, -16, and -32 pan-magic line squares. There are many examples with different features shown. There are two subparts to this subsection, the Order-8 Generator and the Order-16 Generator. These generators allow the user to create magic squares with various feature combinations online.
The Order-3^{p} Squares subsection contains order-9 and order-27 pan-magic line squares. Their construction and features are discussed. The order-3 magic square is also shown to be generated from base lines.
The Prime Order Squares subsection describes the method of creating pan-magic line squares of prime order. It is an alternate approach to the method of using Graeco-Latin patterns. Pan-magic squares of various prime orders are illustrated.
Composite Squares are squares whose orders are composite numbers other than those included in the types above. These require a combination of techniques to create. Both even and odd composites are discussed.
The order-8 nasik magic cube was the original motivation for this site. Much of the material in the original site is still contained in some of the subparts of the Magic Cubes section. Most of the focus for this site was built around the desire to understand the properties of the set of magic line cubes that I first discovered in the 70's. All other parts of the site are offshoots of that effort.
Currently, the main page of Magic Cubes is a continuation of the discussion in Magic Squares It applies the concepts to three dimensions, specifically the order-8 cube. The concepts can be extended to cubes of other orders.
The Order-2^{p} Cubes subsection shows many order-8 and order-16 pan-magic line cubes. Most are nasik. Each has different sets of features, most of them unique. Order-32 nasik cubes are also described but not shown on the page. They must be downloaded. There are three subparts to this subsection. All focus on the order-8 nasik cubes.
The Prime Order Cubes subsection discusses the creation of magic line cubes of prime order. All the cubes discussed are made from base lines, from the simple order-3 cube to nasik cubes of order-11. All the cubes are shown and the order-3 cube can be manipulated interactively.
The Composite Cubes subsection discusses cubes of orders not discussed in the other sections. At this point it is mostly speculative although order-27 cubes are discussed and are available by download.
The Magic Hypercubes section is an obvious extension of the order-4 pan-magic squares and order-8 pan-2,3-agonal magic cubes to 4-, 5-, and multi-dimensional figures. The main page describes the difficulties inherent in manipulating these large figures. The Statistics page in the Miscellaneous section provides information about the size of these n-dimensional figures.
Order-16 and order-32 magic line tesseracts with various features are described in Order-2^{p} Tesseracts. They are not shown on that page due to their size. They are available by download.
The set of 16 by 16 by 16 by 16 nasik magic tesseracts or order-16 pan-2,3,4-agonal magic tesseracts can be made and manipulated using the tesseract generator. Creation and manipulation of this program are described in the Tesseract Guide subpart of the Order-2^{p} Tesseracts subsection. There are ~1.8E63 unique tesseracts in this set with over ten million ways to sum to the magic constant.
The Prime Order Tesseracts subsection describes magic line tesseracts of prime order. The subsection starts with an interactive order-3 tesseract that has no pan features and ends by describing tesseracts of order-17 that can be nasik. Only the order-3 tesseract is shown, the others can be downloaded.
The 5-D Hypercube subsection, of course, describes 5-D hypercubes. Order-32 and 37 nasik hypercubes are described. A rudimentary sum checker for each can be downloaded. There is also a 5-D hypercube generator that can make nasik order-32 5-D hypercubes or order-32 pan-2,3,4,5-agonal magic 5-D hypercubes. Building this hypercube is difficult even with the help of the description in the 5-D Hypercube Generator subsection.
Finally, a method for making n-dimensional nasik hypercubes is described in the n-Dimensions subsection and 6-D Master Base Lines subpart. It is possible to give the parameters for making such figures without testing for validity because when built according to the rules provided the resulting figure will always be a nasik magic figure.
The Other Nasik Cubes discusses all of the order-8 nasik magic cubes that can be made. Those that are described in the Magic Cube section constitute only a miniscule portion of those that can be made even though as mentioned above there are more than a quintillion of them. The section groups the order-8 nasik magic cubes into nine types and attempts a description of the types. However, in the final analysis, they are all fundamentally the same because they all have the same addition properties. The only difference is that these others cannot be made using master base lines. This section confirms my obsession with the order-8 cubes. I would be interested to know if anyone else has an interest in the composition of these other cubes.
The Miscellaneous section contains subsections that don't readily fit elsewhere.
The Statistics subsection focuses on figures of n-dimensions and order-n. It shows how various size parameters increase as one progresses from an order-4 square to an n-dimensional hypercube.
The Überprüfen subsection is my attempt to mathematically describe the process of creating magic figures of any size using base lines. Much of the material is covered in other sections but this subsection attempts to provide a more cohesive proof for the construction.
The Feature Builder is an interactive description of the method of building a magic square. The process is applicable to multiple dimensions.
The Pan-Anti-Magic Squares subsection doesn't appear to fit with the rest of the sites focus. My youngest son Neil generated the pan-anti-magic squares using ideas he gleaned from my construction. His idea was to make the anti-magic squares pan and to include features. Impressive!
The truncated icosahedron contains a symmetrical pattern of pentagons and hexagons on a spherical surface. This pattern allow the creation of a unique 3 dimensional magic figure at href="magicBuckyball">Magic Buckyball.
There are questions scattered in various places that are answered in the Answers subsection.