Magic tesseract

In mathematics, a magic tesseract is the 4-dimensional counterpart of a magic square and magic cube, that is, a number of integers arranged in an n × n × n × n pattern such that the sum of the numbers on each pillar (along any axis) as well as the main space diagonals is equal to a single number, the so-called magic constant of the tesseract, denoted M₄(n). It can be shown that if a magic tesseract consists of the numbers 1, 2, ..., n⁴, then it has magic constant (sequence A021003 in OEIS)

$M_4(n) = \frac{n(n^4+1)}{2}.\,$

The number n is called the order of the magic tesseract.

Perfect magic tesseract

If, in addition, the numbers on every cross section diagonal also sum up to the tesseract's magic constant, the tesseract is called a perfect magic tesseract; otherwise, it is called a semiperfect magic tesseract.

Alternative Definition

The above assumes that one of the older definitions for perfect magic cubes is used. See Magic Cube Classes. The Universal Classification System for Hypercubes (John R. Hendricks) requires that for any dimension hypercube, all possible lines sum correctly for the hypercube to be considered perfect magic. Because of the confusion with the term perfect, nasik is now the preferred term for any magic hypercube where all possible lines sum to S. Nasik was defined in this manner by C. Planck in 1905. A nasik magic tesseract has 40 lines of m numbers passing through each of the m⁴ cells.

The smallest possible nasik magic tesseract is of order 16; its magic constant is 524296. The first one was discovered by retired meteorologist John R. Hendricks from British Columbia in 1999 with the help of Cliff Pickover at the IBM Thomas J. Watson Research Center in Yorktown Heights, New York after about ten hours of computing time on an IBM IntelliStation computer system.

See also

• Magic hypercube
• Magic hypercubes
• Nasik magic hypercube
• John R. Hendricks

References

• Andrews, W.S., Magic Squares and Cubes, Dover, Publ., 1960, this is a facsimile of an Open Court 1917 edition. Two essays on 'octahedrons' (pages 351 - 375 written by Kingsley and Planck.

• Hendricks, John R., Magic Squares to Tesseract by Computer, Self-published, 1998, 0-9684700-0-9

• Hendricks, John R., All Third-Order Magic Tesseracts, Self-published, 1999, 0-9684700-2-5

• Hendricks, John R., Perfect n-Dimensional Magic Hypercubes of Order 2ⁿ, Self-published, 1999, 0-9684700-4-1.

• Heinz, H.D., & Hendricks, J.R., Magic Square Lexicon: Illustrated, HDH, 2000, 0-9687985-0-0

• Unfortunately, all of Hendricks books (except the Lexicon) are now out-of-print. Some are available for download in PDF from his web site. All are available at the Strens Recreational Mathematics Collection (Univ. of C.)

External links

• John Hendricks Math
• [http://www.ucalgary.ca/lib-old/sfgate/strens/index.html University of Calgary, Strens Rec. Math
• Marian Trenkler: An algorithm for magic tesseract

Collections]

• 11 pages about magic tesseracts