A parametrization of 8x8 magic squares of squares through octonionic multiplication

TL;DR

This paper explores a novel parametrization method for 8x8 magic squares of squares with orthogonal rows using octonionic multiplication, extending quaternionic techniques but highlighting limitations in further Cayley-Dickson extensions.

Contribution

It introduces a new octonionic approach to parametrizing 8x8 magic squares of squares, building on Euler's quaternionic methods and identifying the boundaries of Cayley-Dickson extensions.

Findings

Parametrization achieved via octonions for 8x8 magic squares of squares.

Extension to higher Cayley-Dickson algebras is not feasible.

Method parallels Euler's quaternionic construction for 4x4 matrices.

Abstract

In an analogous construction as by Euler for 4x4 matrices, a parametrization of 8x8 magic squares of squares with orthogonal rows is shown to be obtainable by extending the quaternionic method, as shown by Hurwitz, to octonions, but not possible to be carried even further in the Cayley-Dickson construction series.

Equations

$$pr+qs=0,\frac{a}{c}=\frac{-d(pq+rs)-b(ps+qr)}{b(pq+rs)+d(ps+qr)}.$$

$$\gamma\big{(}(a,b,c,d)\big{)}=(p,q,r,s)\quad\iff\quad q_{L}*(a+bi+cj+dk)*q_{R}=p+qi+rj+sk,$$

$$A\ast {\mathbf{e}}_i\ast P=(a+bi+cj+dk)\ast {\mathbf{e}}_i\ast (p+qi+rj+sk),$$

$$\begin{array}[]{c}{\displaystyle(a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2})(b_{1}^{2}+b_{2}^{2}+b_{3}^{2}+b_{4}^{2})=}\\ {\displaystyle(a_{1}b_{1}-a_{2}b_{2}-a_{3}b_{3}-a_{4}b_{4})^{2}+}{\displaystyle(a_{1}b_{2}+a_{2}b_{1}+a_{3}b_{4}-a_{4}b_{3})^{2}+}\\ {\displaystyle(a_{1}b_{3}-a_{2}b_{4}+a_{3}b_{1}+a_{4}b_{2})^{2}+}{\displaystyle(a_{1}b_{4}+a_{2}b_{3}-a_{3}b_{2}+a_{4}b_{1})^{2}}\end{array}$$

$$N(A{\mathbf{e}}_{i}P)=N(A)N({\mathbf{e}}_i)N(P)=N(A)N(P).$$

$$A=$$

$$P=$$

$$A,P\in$$

$$A=(-2,-3,7,-1,2,-11,2,-5),P=(-7,4,-4,-9,1,2,-5,3).$$

$$A=(8,-2,-4,8,-4,-1,-5,-4),P=\frac{1}{2}(5,7,-1,-3,-7,1,7,1).$$

Full text

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be symbolic octonions, (i.e., $a,b,\dots,h,\,p,q,\dots,w$ are variables, and $\boldsymbol{e}_{1}=1,\,\boldsymbol{e}_{i}$ are orthogonal basis elements of $\mathbb{O}$) and let $M\in\mathbb{R}[a,\dots,h,p,\dots,w]^{8\times 8}$ be given by the $i$-th row vector defined as the components of $A(\boldsymbol{e}_{i}P)$.

Then the matrix $M$ constitutes a symbolic semi-magic square of squares with entries in $\mathbb{Z}[a,\dots,h,p,\dots,w]$, with orthogonal rows and constant $N(A)N(P)$.

Consequently, supposing the matrix is evaluated at some $A_{0},P_{0}\in\mathbb{Q}^{8}$, such that all entries are distinct integers, this integer matrix constitutes an (integral) semi-magic square of squares with orthogonal rows and constant $N(A_{0})N(P_{0})$.

Proof  The assertions on $M$ follow from the fact that multiplication with unit octonions models $SO(8)$ (see [1]), hence right multiplication with $P$ and subsequent left multiplication with $A$ preserves the initial orthogonality, while multiplying all lengths equally. Also, all choices of orthogonal basis elements act as a permutation on each other, up to sign change, so that the coefficients of the matrix entries are all $\pm 1$. That all entries are different can be verified by calculation, e.g., using Sage [5]. Figure 2 also visualizes this in that no two $3\times 3$-subsquares have the same pattern combination of colors and white squares. The rest of the assertions then follows. $\Box$

– Having attained a parametrisation the next step is to find numbers such that the entries are indeed different and the constant is low. By plugging in concrete values for the $A,P$, e.g., this matrix with constant 9476 could be found:

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which only lacks the diagonal properties required for a fully magic matrix; this type is called semi-magic.

The diagonal properties in Euler’s matrix arose from choosing such a permutation of rows that in the diagonal all terms were present. Then calculating the sum of squares of the diagonal elements and subtracting the magic constant leads very naturally to the side conditions. – This does not work with the $8\times 8$ matrix, at least not in an obvious straightforward manner, so that for now we cannot ascertain whether or not a fully magic matrix exists. A small scale randomized computer search using parameters within the range $-32$ to $32$ did not produce any results.

It is not difficult to find at least parameters such that not only all entries but also all their squares (or absolute values) are distinct, one of low constant 43617 is given by

[TABLE]

When presenting the matrix in terms of the parameters, it is dependent on the specific implementation of the algebra. We chose the Cayley-Dickson extension of the quaternion algebra as given by the computer algebra system Sage [5]. (The according scripts are available from the author by request.)

The parameters for the 9476 matrix exhibit an interesting detail, in that one of them has half integers for coordinates:

[TABLE]

It is already the case in the quaternions that the appropriate integers to perform number theory contain half integers, this is even more true for the octonions (see a concise account in [1]). The half integers may be placed only in specific 4-subsets of the coordinates. It seemed appropriate to choose at least one factor of this form to achieve a low constant.

The next step by the Cayley-Dickson construction would be the 16-dimensional sedenions; however, the Hurwitz theorem states that there are no further normed real division algebras beyond dimension 8. In particular the multiplicativity of the norm does not hold any longer; consequently the analog matrix does not have orthogonal rows. Similar constructions might also be carried out in any other composition algebra, though a different definition of orthogonality would apply, i.e., instead of $MM^{\top}$ one would consider something like $MQM^{\top}$ for a matrix $Q$.