as.standard: Standard form for magic squares

In RobinHankin/magic: Create and Investigate Magic Squares

Standard form for magic squares

Description

Transforms a magic square or magic hypercube into Frenicle's standard form

Usage

as.standard(a, toroidal = FALSE, one_minus=FALSE)
is.standard(a, toroidal = FALSE, one_minus=FALSE)

Arguments

a	Magic square or hypercube (array) to be tested or transformed
toroidal	Boolean, with default FALSE meaning to use Frenicle's method, and TRUE meaning to use additional transformations appropriate to toroidal connectivity
one_minus	Boolean, with TRUE meaning to use the transformation x\longrightarrow n^2+1-x if appropriate, and default FALSE meaning not to use this

Details

For a square, as.standard() transforms a magic square into Frenicle's standard form. The four numbers at each of the four corners are determined. First, the square is rotated so the smallest of the four is at the upper left. Then, element [1,2] is compared with element[2,1] and, if it is larger, the transpose is taken.

Thus all eight rotated and transposed versions of a magic square have the same standard form.

The square returned by magic() is in standard form.

For hypercubes, the algorithm is generalized. First, the hypercube is reflected so that a[1,1,...,1,1] is the smallest of the 2^d corner elements (eg a[1,n,1,...,1,1]).

Next, aperm() is called so that

a[1,1,...,1,2] < a[1,1,...,2,1] < ... < a[2,1,...,1,1].

Note that the inequalities are strict as hypercubes are assumed to be normal. As of version 1.3-1, as.standard() will accept arrays of any dimension (ie arrays a with minmax(dim(a))==FALSE will be handled sensibly).

An array with any dimension of extent zero is in standard form by definition; dimensions of length one are dropped.

If argument toroidal is TRUE, then the array a is translated using ashift() so that a[1,1,...,1] == min(a). Such translations preserve the properties of semimagicness and pandiagonalness (but not magicness or associativity).

It is easier (for me at least) to visualise this by considering two-dimensional arrays, tiling the plane with copies of a.

Next, the array is shifted so that a[2,1,1,...,1] < a[dim(a)[1],1,1,...,1] and a[1,2,1,..,1] < a[1,dim(a)[2],1,...,1] and so on.

Then aperm() is called as per the non-toroidal case above.

is.standard() returns TRUE if the magic square or hypercube is in standard form. is.standard() and as.standard() check for neither magicness nor normality (use is.magic and is.normal for this).

Note

There does not appear to be a way to make the third letter of “Frenicle” have an acute accent, as it should do.

Author(s)

Robin K. S. Hankin

See Also

magic, eq

Examples

is.standard(magic.2np1(4))
as.standard(magic.4n(3))

as.standard(magichypercube.4n(1,5))

##non-square arrays:
as.standard(magic(7)[1:3,])


## Toroidal transforms preserve pandiagonalness:
is.pandiagonal(as.standard(hudson(11)))


## but not magicness:
is.magic(as.standard(magic(10),TRUE))

RobinHankin/magic documentation built on Jan. 17, 2024, 8:36 p.m.