R: Standard form for magic squares

as.standard {magic} | R Documentation

## 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 -> 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))

[Package magic version 1.5-8 Index]