Magic tesseract
Encyclopedia
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 diagonal
s 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
The number n is called the order of the magic tesseract.
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.
.
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.
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Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a magic tesseract is the 4-dimensional counterpart of a magic square
Magic square
In recreational mathematics, a magic square of order n is an arrangement of n2 numbers, usually distinct integers, in a square, such that the n numbers in all rows, all columns, and both diagonals sum to the same constant. A normal magic square contains the integers from 1 to n2...
and magic cube
Magic cube
In mathematics, a magic cube is the 3-dimensional equivalent of a magic square, that is, a number of integers arranged in a n x n x n pattern such that the sum of the numbers on each row, each column, each pillar and the four main space diagonals is equal to a single number, the so-called magic...
, 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 diagonal
Space diagonal
In a rectangular box or a magic cube, the four space diagonals are the lines that go from a corner of the box or cube, through the center of the box or cube, to the opposite corner...
s is equal to a single number, the so-called magic constant
Magic constant
The magic constant or magic sum of a magic square is the sum of numbers in any row, column, and diagonal of the magic square. For example, the magic square shown below has a magic constant of 15....
of the tesseract
Tesseract
In geometry, the tesseract, also called an 8-cell or regular octachoron or cubic prism, is the four-dimensional analog of the cube. The tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of 6 square faces, the hypersurface of the tesseract consists of 8...
, denoted M4(n). It can be shown that if a magic tesseract consists of the numbers 1, 2, ..., n4, then it has magic constant
The number n is called the order of the magic tesseract.
Perfect magic tesseract
If, in addition, the numbers on every cross sectionCross section (geometry)
In geometry, a cross-section is the intersection of a figure in 2-dimensional space with a line, or of a body in 3-dimensional space with a plane, etc...
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 ClassesMagic cube classes
Every magic cube may be assigned to one of six magic cube classes, based on the cube characteristics.This new system is more precise in defining magic cubes...
.
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
British Columbia
British Columbia is the westernmost of Canada's provinces and is known for its natural beauty, as reflected in its Latin motto, Splendor sine occasu . Its name was chosen by Queen Victoria in 1858...
in 1999 with the help of Cliff Pickover at the IBM Thomas J. Watson Research Center in Yorktown Heights, New York
New York
New York is a state in the Northeastern region of the United States. It is the nation's third most populous state. New York is bordered by New Jersey and Pennsylvania to the south, and by Connecticut, Massachusetts and Vermont to the east...
after about ten hours of computing time on an IBM
IBM
International Business Machines Corporation or IBM is an American multinational technology and consulting corporation headquartered in Armonk, New York, United States. IBM manufactures and sells computer hardware and software, and it offers infrastructure, hosting and consulting services in areas...
IntelliStation computer system.
See also
- Magic hypercubeMagic hypercubeIn mathematics, a magic hypercube is the k-dimensional generalization of magic squares, magic cubes and magic tesseracts; that is, a number of integers arranged in an n × n × n × .....
- Magic hypercubes
- Nasik magic hypercube
- John R. Hendricks
External links
- John Hendricks Math
- University of Calgary, Strens Rec. Math
- [http://math.ku.sk/~trenkler/06-MagicTesseract.pdf Marian Trenkler: An algorithm for magic tesseract]
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