 English Japanese Home Site Map

# Terms

This site uses the following terms. Other sites, books, or papers may use different terms from these terms. These terms are part of the terms of magic objects.
(last updated on August 10, 2008)

## Types of magic objects

magic hypercube
a magic hypercube of dimension n and order m is defined as a n-dimensional hypercubical array of mn numbers in which the sum for every orthogonal and every n-agonal is the same value (called the (magic) constant).
Magic hypercubes of dimensions 2, 3, and 4 are called magic squares, magic cubes, and magic tesseracts, respectively.
semimagic hypercube
the same as magic hypercube except that the sum of each n-agonal does not have to equal to the magic constant. Every associated semimagic hypercube is a magic hypercube.
magic rectangle
a magic rectangle of dimension n and order (m1, m2, ..., mn) is defined as a n-dimensional rectangular array of m1m2...mn numbers in which for each direction, every orthogonal sums to the same value depending on the direction. There is no condition for r-agonals.
A magic rectangle in which every element of the order is the same is a semimagic hypercube.
Some construction methods of magic hypercube use magic rectangles.
magic pattern
A pattern of numbers in which the sum of every line of the pattern is the same value. Magic patterns is categolized into magic circles, magic stars, etc..
magic object
the general term for (semi)magic hypercube, magic rectangle, magic pattern, and so on.
Latin square
a (2-dimentional) Latin square is defined as an m by m array of m numerical or literal symbols in which each symbol appears exactly once in each row and each column of the array. It is not required that tha same condition apply to the diagonal. If they do, the square is called a diagonal Latin square. A Latin square can be regarded as a (non-normal) semimagic square if numerical symbols is used. A Latin hypercube of dimension 3 or higher is defined similarly.
Latin squares are frequently used for generating magic squares, and applied to other fields, for example, the design of experiments.

## Basic terms

normal (magic object)
a magic object which consists of consecutive integers start from 1.
According to circumstances, we use consecutive integers start from 0 for a normal magic object for mathematical convenience.
Aale de Winkel calls the former range the "regular number range" and the latter range the "analitic number range".
This site adopts the regular number range in principle.
A normal (semi)magic hypercube of dimension n and order m consists of consecutive integers from 1 to mn in the regular number range, and from 0 to mn-1 in the analitic number range.
non-normal (magic object)
a magic object which is not normal. In this site, we assume that every cell of a (non-normal) magic object is integer.
prime magic object
a magic object which consists of prime numbers. It is a kind of a non-normal magic object. The integer 1 is often used as an element of prime magic object though 1 is not a prime number.
constant magic object
a (trivial) magic object which consists of only a single number. It is a kind of a non-normal magic object. Of course, such a magic object is worthless.
row, column, pillar, file
directions of the first, the second, the third, and the fourth dimension, respectively, in a (semi)magic hypercube or a magic rectangle.
r-agonal (of a hypercube)
an r-dimensional space 'diagonal'. Specially, 2-agonal, 3-agonal, and 4-agonal are also called diagonal, triagonal, and quadragonal, respectively. 1-agonal (monoagonal) means orthogonal.
pan-r-agonal (of a hypercube)
a broken r-agonal parallel to some r-agonal of the hypercube. See the figure below.
Pan-2-agonal, pan-3-agonal, and pan-4-agonal are also called pandiagonal, pantriagonal, and panquadragonal, respectively. Pan-1-agonal (panmonoagonal) is a synonym of 1-agonal (monoagonal).
 X X X X X

(magic) constant
the sum which requires to be the same value in a (semi)magic hypercube or a magic pattern. Also called the magic sum.
The constant of a normal magic hypercube of dimension n and order m equals to m(mn+1)/2 in the regular number range.
In a magic rectangle, the constant is defined for each direction (the row constant, the column constant, etc.).
singly even number
an even integer which is not divisible by 4. A singlly-even magic hypercube is difficult to construct.
doubly even number
an integer which is divisible by 4.

## Conditions on (pan-)r-agonals

r-agonal magic hypercube (cube, tesseract)
a magic hypercube of dimension n such that every r-agonal of the hypercube sums to the magic constant.
2-agonal, 3-agonal, and 4-agonal are also called diagonal, triagonal, and quadragonal, respectively. The description 1-agonal (monoagonal) is usually left out.
We also use expressions like 2,3-agonal (that is, both 2-agonal and 3-agonal).
A hypercube of dimension n is a magic hypercube if and only if the hypercube is 1,n-agonal.
pan-r-agonal magic hypercube (square, cube, tesseract)
a magic hypercube of dimension n such that every pan-r-agonal of the hypercube sums to the magic constant.
pan-2-agonal, pan-3-agonal, and pan-4-agonal are also called pandiagonal, pantriagonal, and panquadragonal, respectively. The description pan-1-agonal (panmonoagonal) is usually left out.
We also use expressions like pan-2,3-agonal (that is, both pan-2-agonal and pan-3-agonal).
panmagic hypercube (square, cube, tesseract)
a pan-(1,)n-agonal magic hypercube, where n is the dimension of the hypercube.
Panmagic square, panmagic cube, and panmagic tesseract are identical to pandiagonal magic square, pantriagonal magic cube, and panquadragonal magic tesseract, respectively.
strictly magic hypercube (cube, tesseract)
a 1,2,3,...,n-agonal magic hypercube, where n is the dimension of the hypercube. The term strictly was suggested by Walter Trump.
Strictly magic cube is identical to diagonal magic cube. Every magic square is strictly magic.
No strictly magic cube of order 3 or 4 can exist and no strictly magic tesseract of order 3, 4, or 5 can exist, either.
Nasik magic hypercube (square, cube, tesseract)
a pan-1,2,3,...,n-agonal magic hypercube, where n is the dimension of the hypercube. Also called strictly panmagic hypercube.
Nasik magic square, Nasik magic cube, and Nasik magic tesseract are identical to pandiagonal magic square (or, panmagic square), pan-2,3-agonal magic cube, and pan-2,3,4-agonal magic tesseract, respectively.
Note that the condition of Nasik is stronger than that of panmagic and strictly magic.
pantriagonal diagonal magic hypercube (cube, tesseract)
a magic hypercube which is both pantriagonal and diagonal. Harvey D. Heitz calls this property PantriagDiag.
There exists a pantriagonal diagonal magic cube which is not Nasik.
pan and strictly magic hypercube (cube, tesseract)
a magic hypercube which is both panmagic and strictly magic, namely, a pan-n-agonal and 1,2,3,...,n-agonal n-dimensional magic hypercube.
Pan and strictly magic cube is identical to PantriagDiag magic cube.
pseudo-pan-r-agonal magic hypercube (cube, tesseract)
a magic hypercube of dimension n such that every pan-r-agonal on its oblique (n-1)-dimensional slices is magic.
Pseudo-pan-3-agonal is also called pseudo-pantriagonal.
pseudo-panmagic hypercube (cube, tesseract)
a pseudo-pan-n-agonal magic hypercube, where n is the dimension of the hypercube.
Every panmagic hypercube is pseudo-panmagic.
horizontal-diagonal magic hypercube (cube, tesseract)
a magic hypercube such that for some aspect of the hypercube, every square of horizontal slices of the hypercube is magic.
Every diagonal magic cube is horizontal-diagonal.
horizontal-pandiagonal magic hypercube (cube, tesseract)
a magic hypercube such that for some aspect of the hypercube, every square of horizontal slices of the hypercube is pandiagonal magic square.
Every pandiagonal magic cube is horizontal-pandiagonal.
s-magic cube
a magic cube such that each of six squares on the surface of the cube is magic. The prefix 's-' means 'surface'. This term is named by Walter Trump.
Every diagonal magic cube is s-magic.
simple magic hypercube (square, cube, tesseract)
a magic hypercube of dimension n which is not satisfy any condition of (pan-)r-agonal except n-agonal.
Simple magic squares are not pandiagonal, simple magic cubes are neither pantriagonal nor diagonal, and simple magic tesseracts are not panquadragonal, triagonal, or diagonal.
class of magic hypercube
classification of magic hypercubes of dimension n by the magicness of (pan-)r-agonals for each r such that 2 <= r <= n.
Magic squares are classified by the two classes,
simple (2-agonal) and panmagic (pan-2-agonal).
Magic cubes are classified by the six classes, simple (3-agonal), pantriagonal (pan-3-agonal), diagonal (2,3-agonal), pantriagonal diagonal (2-agonal and pan-3-agonal), pandiagonal (pan-2-agonal and 3-agonal), and Nasik (pan-2,3-agonal).
Magic tesseracts are classified by the 18 classes.

## Additional conditions

complement(ary) pair (of a magic object)
a pair of cells whose sum equals to 2*c, where c is the average of all the cells of the magic object. If a magic hypercube of dimension n and order m is normal in the regular number range, the sum of such a pair equals to mn+1.
One number of a complement pair is called the complement(ary) number of the other number of the pair.
complement(ary) pair line
the line that join the two numbers of a complement pair of a magic hypercube. The pattern of all complement pair lines of a magic hypercube is called the complement(ary) pair pattern of the hypercube. Magic hypercubes are classified by their complement pair patterns.
associated (associative) (magic hypercube (square, cube, tesseract), magic rectangle)
the property that every pair of cells diametrically equidistant from the center of the hypercube or the rectangle is a complement pair. If the order is odd, the double of the center cell need to equal to the sum of complement pairs. Also called a symmetrical magic hypercube.
axis/plane symmetrical (magic hypercube (square, cube, tesseract), magic rectangle)
the property that every pair of cells diametrically equidistant from some axis/plane of the hypercube or the rectangle is a complement pair.
complete magic hypercube (square, cube, tesseract)
a magic hypercube of dimension n and even order m that every pan-n-agonal contains m/2 complement pairs spaced m/2 apart. Complete is not defined for an odd order magic hypercube.
Every complete magic hypercube is panmagic.
rD-compact magic hypercube (cube, tesseract)
an n-dimensional magic hypercube with the feature that the 2r cells of every order-2 subhypercube of dimension r, where 2 <= r <= n, sum to the same value. The smaller r is, the stronger this condition is.
For example, in a 2D-compact magic hypercube, the four cells of every 2x2 subsquare sum to the same value, and, in a 3D-compact magic hypercube, the eight cells of every 2x2x2 subcube do so. Also called a 2r-uniform magic hypercube. 2D-compact is simply called compact for magic squares.
Every 2D-compact magic hypercube is panmagic. On the other hand, not every rD-compact magic hypercube is panmagic for r > 2.

## Terms for multimagic hypercubes

d-ply-r-agonal magic hypercube (square, cube, tesseract)
a magic hypercube in which every k-th powered hypercube is r-agonal, where k = 1,2,...,d.
2-ply-r-agonal, 3-ply-r-agonal, etc. are also called bi-r-agonal, tri-r-agonal, etc., respectively.
d-ply-magic hypercube (square, cube, tesseract)
d-ply-1,n-agonal magic hypercube, where n is the dimension of the hypercube. The number d is called the degree of the hypercube. Also called multimagic hypercube of degree d.
2-ply-magic, 3-ply-magic, etc. are also called bimagic, trimagic, etc., respectively.
In a d-ply-magic hypercube, every k-th powered hypercube is also a magic hypercube, where k = 1,2,...,d.
strictly d-ply-magic hypercube (square, cube, tesseract)
d-ply-1,2,...,n-agonal magic hypercube, where n is the dimension of the hypercube. Also called strictly multimagic hypercube of degree d.
stricty 2-ply-magic, strictly-3-ply-magic, etc. is also called strictly bimagic, strictly trimagic etc., recpectively.
In a strictly d-ply-magic hypercube, every k-th powered hypercube is also a strictly magic hypercube, where k = 1,2,...,d.

## Terms for inlaid magic hypercubes

inlaid magic hypercube (square, cube, tesseract)
a magic hypercube that contain another magic hypercube of lower order within it.
In an inlaid magic hypercube, we call the hypercube itself the main hypercube and a hypercube inlaid within it a sub hypercube.
concentric magic hypercube (square, cube, tesseract)
an inlaid magic hypercube that contain sub magic hypercubes of dimension n and orders m-2, m-4, ... concentrically, where n and m is the dimension and the order of the main hypercube, respectively. The least order of the sub magic hypercubes concentrically inlaid is 3 or 4.
bordered magic hypercube (square, cube, tesseract)
a concentric magic hypercube in which every concentric sub hypercube consists of consecutive integers. Also called consecutively concentric magic hypercube.
Every sub hypercube of a bordered magic hypercube can transform into a normal magic hypercube by subtracting some constant from all cells of the sub hypercube.
Note: The term bordered is often used as a synonym of concentric, but this site distingushes the two terms as above.
bordered diagonal magic cube
a bordered magic cube which is diagonal and every subcube whose order is higher than 4 is also diagonal. (It is impossible to make the subcube of order 3 or 4 diagonal.)
This magic cube was first constructed by the author in 2004.
The author proved that a bordered diagonal magic cube exists for any even order higher than 5.

## Other terms

XmlHypercube format
a description format by Aale de Winkel to descript magic hypercubes. In the old version, it was called HyperCube Language.
perfect magic hypercube (square, cube, tesseract)
People use this term in plural senses. This site does not use the term perfect except the case of quotation so as not to confuse the reader.

References
Harvey D. Heinz & John R. Hendricks, Magic Square Lexicon: Illustrated, self-published, 2000, ISBN 0-9687985-0-0.
Harvey D. Heinz's site http://www.magic-squares.net/
Aale de Winkel's site http://www.magichypercubes.com/Encyclopedia/index.html
Christian Boyer's site http://www.multimagie.com/indexengl.htm
Walter Trump's site http://www.trump.de/magic-squares/ Top Home Site Map

Mitsutoshi Nakamura (Feedback)    To send an email to me, please enable JavaScript on your browser.
[Legal indication] Prohibit the send of any spam or advertising.

This page was last updated on August 8, 2016.
"Magic Cubes and Tesseracts"   http://magcube.la.coocan.jp/magcube/en/
Copyright © 2004-2016, Mitsutoshi Nakamura. All rights reserved. Magic Cubes and Tesseracts 