# Some mathematical and geometrical interpretations of the Sator Square

Paul Dario TOASA CAIZA
Karlsruher Institut für Technologie. Versuchsanstalt für Stahl, Holz und
Steine
Otto-Ammann-Platz 1. 76131 Karlsruhe. Germany
<EMAIL_ADDRESS>
In 1738, the King of Naples and future King of Spain, Carlos III, commissioned
the Spanish military engineer Roque Joaquín de Alcubierre to begin the
excavations of the ruins of the ancient Roman city of Pompeii and its
surroundings, buried by the terrible explosion of Vesuvius in AD 79. Since
that time, archaeologists have brought to light wonderful treasures found in
the among ruins. Among them, the Sator Square is one of the most peculiar,
apparently simple but mysterious. Supernatural and medicinal powers have been
attributed to this object and its use was widespread during the Middle Age.
Studies to explain its origin and meaning have been varied. There are theories
that relate it to religion, the occult, medicine and music. However, no
explanation has been convincing beyond pseudo-scientific sensationalism. In
this study, the author intends to eliminate the mystical character of the
Sator Square and suggests considering it as a simple palindrome or a game of
words with certain symmetrical properties. However, these properties are not
exclusive to the Sator Suare but are present in various mathematical and
geometric objects.

##  1 Introduction

I was very intrigued by an article I read about the enigmatic Sator Square on
the BBC Spanish service. And the intrigue was because all the mentioned
explanations were spiritual and pseudo-scientific. Despite the fact that the
object presented certain symmetries, no geometric or mathematical study was
mentioned in this regard. So I searched for mathematical studies on this
object and only found countless sensational and absurd works. So I asked
myself: Is it possible to consider the Sator Square as a mathematical object
and study it?

ROTASOPERATENETAREPOSATOR Figure 1: Sator square

Among the many wonderful historical artifacts found in the ruins of Pompeii
caused by the eruption of Vesuvius in A.D. 79, Sator Square is one of the most
unique and unusual, see Figure 1. Particularly, during the Middle Age, the
square was used as talisman or amulet, against rabies and other diseases and
complaints, such as fever, intestinal gout and rose, jaundice and consumption,
cramps, headache and toothache, birth complications and worms. It was also
regarded as a remedy against thieves, witchcraft and fire, as a spell of
exorcism, blessing of bullets and hunting aid, even as a love magic [1].

An immense amount of works have been written in order to explain its origin,
meaning and, in the case of being a riddle, its solution. All these works
suggest a relationship with Christianity, Judaism, Satanism, philology,
numerology, esotericism, magic, mysticism and even medicine [2], [3], [4].
However, none of these works provides a complete and convincing explanation
about the origin and meaning of the square.

The magic of this square rest surely on the perfect symmetry of its component
letters which yield the same combinations in four different directions [4]. It
has 5 Latin words, which can be read from left to right and from top to
bottom, or from left to right and bottom to top and their “message” is the
same, see Figure 2

SATOR AREPO TENET OPERA ROTAS

Figure 2: Five words palindrome

In [4], the author analyses how difficult to build a rebus with twenty five
letters in five Latin words is. Moreover, it is suggested that the words were
written by Latin-speaking Jews.
Other similar squares have been written by considering other words and other
languages like Greek and Hebrew see [2].
Even in the pop culture the Sator Square has inspired the creation of Tenet,
which is a 2020 science fiction action thriller film directed and written by
Christopher Nolan. In this film the Protagonist master the art of ”time
inversion” as a way of countering the threat that is to come.
In [5] the origin of AREPO is discussed and it is proposed that the square has
Egyptian origin.
In [6] the work of Moeller [7] is reviewed. Moeller proposes that Sator is
Saturn and assures ”I now present these findings in the firm conviction that I
have come upon the correct solution“. This declaration received a lot of
critisism, saying that this work went in the wrong direction. Others called it
provocative but no persuasive.
In fact, about half the text deals with numerology and it also suggests that
probably the square has Christian origin, fact that Hemer supports in [8].
This is based on the fact that the phrase “Pater Noster” and additionally the
letters A and O, standing for Alpha and Omega, the first and the last were
obtained by rearrangin the twenty five letters of the square.
In [9], the work of Moeller is compared to a sandcastle.

Even in the classical music the influence of the square can be noted. The
Austrian composer Anton Friedrich Wilhelm von Webern1113.12.1883 – 15.09.1945
considered the square strucuture in its composition “Concerto for Nine
Instruments, Op. 24”222Konzert für neun Instrumente, written in 1934 [10].

An extensive study about several interpretations of the square, with a
abundant bibliography is available in [11].

##  2 Reading simmetry

The most discussed chracteristic of the Sator square is related with the
concept of palyndromes. The Sator square has five words, located in a
particular way, which makes possible to obtain several types of reading
simmetry.

###  2.1 Clasic palindrome

The word TENET is a palyndrome, and it is written in the middle column and row
of the square. If two perpendicular axis are drawn trought the center of the
square, the palindrome TENET coincides with these axes, see Figure 3.

ROTASOPERATENETAREPOSATOR$\mathbf{S_{y}}$$\mathbf{S_{x}}$ Figure 3: The
palindrome TENET of the Sator square

Moreover, by writing consecutively the five words, a longer palindrome is
obtained as well, see Figure 2.

###  2.2 Palindrome in both directions

The four words SATOR, AREPO, OPERA and ROTAS behave like a palindrome. They
can be observed and read in the traditional western way from left to right and
from top to bottom or from right to left and from bottom to top. Depending on
the reading direction these words are located in the ith-row and ith-column.
For example, OPERA is located in the 4th row and the 4th column from Figure
4(a) and in the 2nd row and 2nd column from Figure 4(b), see the corresponding
green arrows.

ROTASOPERATENETAREPOSATOR (a) Simmetry from left to right and from top to
bottom

ROTASOPERATENETAREPOSATOR (b) Simmetry from right to left and from bottom to
top

Figure 4: Reading simmetry of the Sator square in two ways

If both squares from Figure 4 are joined, the five words of the Sator square
are reduced to three SATOR, AREPO and TENET, see Figure 5. Moreover, it is
also possible to observe that the Sator square keeps its form after rotating
one time around the horizontal axis and one time around the vertical axis.
This kind of simmetry will be discussed in the next section.

ROTASOPERATENETAREPOSATOR Figure 5: Reading simmetry of the Sator square in
four ways

##  3 Geometric symmetry

The reading symmmetries observed in the previous section can be described
trought different types of geometrical symmetries. This description will allow
to find a general way to generate Sator Squares of higher dimensions.

###  3.1 Point symmetry

When every part of an object has a matching part, and both parts have the same
distance but in the opposite direction from the centre point, this is known as
point symmetry. It’s also known as “Order 2 rotational symmetry.”
Because the “Origin” is the primary point around which the shape is
symmetrical, point symmetry is also known as origin symmetry.
When a figure is rotated $180^{o}$, and it keeps its original shape, it is
said that it has point symmetry.
Thus, the palindrome TENET in the Sator Square has a point symmetry located at
the letter N, see Figure 3. In other words, the axes $S_{x}$ and $S_{y}$ have
a point symmetry with respect to the center of the square.

###  3.2 Reflective or mirror symmetry

Reflective symmetry, also called mirror symmetry, is a type of symmetry where
one half of the object reflects the other half of the object. In other words,
in this type of symmetry one part of the image or object represents the mirror
image of another part of the image. Reflexive Symmetry is described by a line
of symmetry.
In the Sator Square two lines of symmetry $S_{1}$ and $S_{2}$ are present, see
Figure 6.

ROTASOPERATENETAREPOSATOR$\mathbf{S_{1}}$$\mathbf{S_{2}}$ Figure 6: Lines of
simmetry of the Sator square

If a mirror is located along the lines of symmetry $S_{1}$ and $S_{2}$, the
letters in the Sator Square will be reflected as it is shown with dotted lines
in Figure 7

ROTASOPERATENETAREPOSATOR$\mathbf{S_{2}}$ (a) Reflexions relative to symmetry
line S2

ROTASOPERATENETAREPOSATOR$\mathbf{S1}$ (b) Reflexions relative to symmetry
line S1

Figure 7: Simmetry lines and their corresponding reflection in the Sator
Square

###  3.3 Rotational simmetry in 3D

Besides the 2D symmetries observed in the Sator Square, a symmetry in 3D is
presented as well. If the Sator Squere is rotated $\pm 90^{o}$, and afterwards
it is rotated $\pm 180^{o}$ by considering as rotation axis the symmetry line
$S_{x}$ or $S_{y}$, the Sator Square keeps its original form, see Figure 8.
Doing these operation in a reverse order, the original form is kept as well.

ROTASOPERATENETAREPOSATOR$\pm 90^{o}$$N$SATORAREPOTENETOPERAROTAS$\pm
180^{o}$$S_{x}$ROTASOPERATENETAREPOSATOR Figure 8: Rotational simmetry in 3D
of the Sator Square

##  4 Satorian functions and squares

Considering as reference the reading and geometrical simmetries observed in
the Sator square, it is possible to find a general discrete way to create
Sator squares of higher dimension.

However, before proposing a general definition and for didactic purposes, let
us consider the following function, which satisfies the point simmetry along
$S_{x}$ and $S_{y}$ corresponding to the center of the square and the lines of
symmetry $S_{1}$ and $S_{2}$.

$f(x,y)=\left\\{\begin{array}[]{rl}\sqrt{x^{2}+y^{2}}&\text{if }xy\geq 0,\\\
\\\ -\sqrt{x^{2}+y^{2}}&\text{else},\end{array}\right.$ (1)

where $x,y\in\mathbb{Z}$.

$\cdot$$\cdot$-2$\cdot$$\cdot$$\cdot$$\cdot$-1$\cdot$$\cdot$-2-1012$\cdot$$\cdot$1$\cdot$$\cdot$$\cdot$$\cdot$2$\cdot$$\cdot$
(a) $(x,y)$

$\sqrt{8}$$\sqrt{5}$2$-\sqrt{5}$$-\sqrt{8}$$\sqrt{5}$$\sqrt{2}$$1$$-\sqrt{2}$$-\sqrt{5}$$2$$1$$0$$1$$2$$-\sqrt{5}$$-\sqrt{2}$$1$$\sqrt{2}$$\sqrt{5}$$-\sqrt{8}$$-\sqrt{5}$$2$$\sqrt{5}$$\sqrt{8}$
(b) $f(x,y)$

Figure 9: Application of the Equation (1) for $|x|,|y|\leq 2$

Figure 9 shows the application of the Equation (1) in the set given by
$|x|,|y|\leq 2$. Consequently, a square of dimension $n=5$ similar to the
Sator Square is obtained. In fact, the only difference is that instead of
letters, numbers are used. It is worth mentioning that in order to keep the
expected simmetries, the dimension of the generated square has to be odd.
By assignng colors to the values of the square in Figure 9(b), it is possible
to observe the expected simmetries in a easier way, see Figure 10.

(a) $SS_{5}$ in gray scale

(b) $SS_{5}$ in color scale

Figure 10: Graphical representation of a Satorian square $SS_{5}$ in gray
scale and color

The Equation (1) is a particular case of a function, which will be called
Satorian function and it will be defined as follows.

######  Definition (Satorian function).

Let $(x,y)$ be a number in $\mathbb{Z}^{2}$, a function $f(x,y)$ is called
Satorian when $f(x,y)=f(y,x)$ and $f(x,y)=f(-y,-x)$.

The squares shown in Figure 10 are called Satorian squares and are defined as
follows.

######  Definition (Satorian square).

The square obtained by applying a Satorian function $f(x,y)$ on a a discrete,
centered and symmetric set of numbers of dimension $n\times n$, where $n$ is
odd, is called Satorian square of dimension $n$ and it is noted as $SS_{n}$.

It is not necessary to define a Satorian function as an explicyt equation, it
can be also given by a set of functions or by a set of discrete values. In
fact, the archeological Satorian square is given in this way.

###  4.1 Archeological Satorian square

Once the definitions of a Satorian function and square have been given, it is
possible to realize that the archeological Satorian square is no more than a
particular case of a $SS_{5}$, where the Satorian function is given by a
discrete set. Observe that the difference with the $SS_{5}$ given by Equation
(1) and shown in Figures 9(b) and 10(b), are the values written in blue in
Figure 11(a).

$\sqrt{8}$$\sqrt{5}$2$-\sqrt{5}$$-\sqrt{8}$$\sqrt{5}$$\sqrt{2}$$1$$\sqrt{8}$$-\sqrt{5}$$2$$1$$0$$1$$2$$-\sqrt{5}$$\sqrt{8}$$1$$\sqrt{2}$$\sqrt{5}$$-\sqrt{8}$$-\sqrt{5}$$2$$\sqrt{5}$$\sqrt{8}$
(a) $f(x,y)$

(b) Square in color

Figure 11: Archeological Satorian function and square

Generating Satorian squares by using numbers is much easier than using
palyndromes or even letters. As example, I tried to build a Sator Square of
dimension $n=7$ by using spanish words. To do this, it is necessary 4 words,
one palindrome of 7 letters and 3 words, which satisfy the corresponding
simmetries. Unfortunately, I found only three words333The author will
appreciate if an example with four words is given.. The palindrome ANILINA
(aniline), ECUADOR, CIANURO (cyanide) and the no meaning word UANIMUD, see
Figure 12.

RODAUCEORUNAICDUMINAUANILINAUANIMUDCIANUROECUADOR Figure 12: Satorian Square
of dimension $n=7$. It is built with three spanish words, ANILINA(aniline),
ECUADOR, CIANURO (cyanide) and one word without meaning UANIMUD.

##  5 Satorian matrices

By considering a Satorian Square as a matrix, it follows that it is a
bisymmetric matrix, see [12].

######  Definition (Bisymmetric matrix).

An $n\times n$ matrix $A=(a_{ij})\in\mathbb{R}$ is a bisymmetric matrix if
$a_{i,j}=a_{j,i}$ and $a_{i,j}=a_{n+1-j,n+1-i}$ for all $i,j=1,2,...,n$, that
is,

$\left(\begin{array}[]{ccccccc}a_{11}&a_{12}&a_{13}&\ldots&a_{1,n-2}&a_{1,n-1}&a_{1,n}\\\
a_{12}&a_{22}&a_{23}&\ldots&a_{2,n-2}&a_{2,n-1}&a_{1,n-1}\\\
a_{13}&a_{23}&a_{33}&\ldots&a_{3,n-2}&a_{2,n-2}&a_{1,n-2}\\\
\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\\
a_{1,n-2}&a_{2,n-2}&a_{3,n-2}&\ldots&a_{33}&a_{23}&a_{13}\\\
a_{1,n-1}&a_{2,n-1}&a_{2,n-2}&\ldots&a_{23}&a_{22}&a_{12}\\\
a_{1,n}&a_{1,n-1}&a_{1,n-2}&\ldots&a_{13}&a_{12}&a_{11}\\\ \end{array}\right)$
(2)

Bisymmetric matrices have several applications, such as numerical analysis,
graph theory or structural mechanics, vibration of beams and eigenavalue
problems see [13], [14], [15], [16].

######  Definition (Satorian matrix).

The matrix defined by the values of a Satorian function or a Satorian square
of dimension $n$ is called a Satorian matrix and it will be noted as $SM_{n}$.

A Satorian matrix $SM_{n}$ is a bisymmetric matrix.

##  6 Satorian surfaces and solids

If the values of a Satorian Matrix are considered as the values on the
$z$-axis, symmetric surfaces and solids can be obtained. Thus, the Satorian
matrix given by Equation (3) corresponds to the Satorian Square $SS5$ shown in
Figure 10.

$SM_{5}=\left(\begin{array}[]{ccccc}-\sqrt{8}&-\sqrt{5}&2&\sqrt{5}&\sqrt{8}\\\
-\sqrt{5}&-\sqrt{2}&1&\sqrt{2}&\sqrt{5}\\\ 2&1&0&1&2\\\
\sqrt{5}&\sqrt{2}&1&-\sqrt{2}&-\sqrt{5}\\\
\sqrt{8}&\sqrt{5}&2&-\sqrt{5}&-\sqrt{8}\\\ \end{array}\right)$ (3)

Observe that the matrix $SM_{5}$ depends on the function $f(x,y)$ given by
Equation (1) and shown in Figure 9(b). The Figure 13 shows the surface and
solid corresponding to the Matrix $SM_{5}$.

(a) Surface

(b) Solid

Figure 13: Satorian surface and solid corresponding to the Satorian matrix
$SM_{5}$

The matrix corresponding to the Archeological Satorian Square is given in
Equation (4), and the surface and solid corresponding to the Archeological
Satorian square are shown in Figure 14

$ASM_{5}=\left(\begin{array}[]{ccccc}-\sqrt{8}&-\sqrt{5}&2&\sqrt{5}&\sqrt{8}\\\
-\sqrt{5}&\sqrt{8}&1&\sqrt{2}&\sqrt{5}\\\ 2&1&0&1&2\\\
\sqrt{5}&\sqrt{2}&1&\sqrt{8}&-\sqrt{5}\\\
\sqrt{8}&\sqrt{5}&2&-\sqrt{5}&-\sqrt{8}\\\ \end{array}\right)$ (4)

(a) Surface

(b) Solid

Figure 14: Satorian surface and solid corresponding to the Archeological
Satorian matrix $ASM_{5}$

As it was mentioned before, the Satorian functions can be defined by a
discrete set of points or by a set of real functions. Thus, let us consider an
application of a Gaussian function given by

$f(x,y)=\left\\{\begin{array}[]{rl}|x+y|&\text{if }xy=0,\\\ \\\
\exp{(-x^{2}-y^{2})}&\text{if }xy>0,\\\ \\\ -\exp{(-x^{2}-y^{2})}&\text{if
}xy<0\end{array}\right.$ (5)

By aplying the Satorian function given in Equation (5) in a square of
dimension $n=5$ the corresponding matrix is obtained, see Figure 15.

$\cdot$$\cdot$-1$\cdot$$\cdot$$\cdot$$\cdot$-0.5$\cdot$$\cdot$-1-0.500.51$\cdot$$\cdot$0.5$\cdot$$\cdot$$\cdot$$\cdot$1$\cdot$$\cdot$$x$$y$
(a) $(x,y)$

0.1350.2871-0.287-0.1350.2870.6070.5-0.607-0.28710.500.51-0.287-0.6070.50.6070.287-0.135-0.28710.2870.135$x$$y$
(b) $f(x,y)$

Figure 15: Application of the Equation (5) on a discrete square

By assigning colours to the Satorian square shown in Figure 15(b) and by
plotting the values on the $z$ axis, the corresponding square and surface are
obtained.

(a) Square

(b) Surface

Figure 16: Satorian square and surface corresponding to the function given by
Equation (5)

The Figure 17(a) shows the solid correspondig to the function given by
Equation (5) and the Figure 17(b) shows the surface when the function $f(xy)$
is evaluated in $x,y\in\mathbb{R}$.

(a) Solid

(b) Surface in $\mathbb{R}$

Figure 17: Satorian solid and surface corresponding to the function given by
Equation (5)

##  7 Arabic architecture. The Alhambra

The kind of symmetry observed in the Sator Square has been also found in the
alicatados444Mosaic formed of polygonal, coloured glazed tiles. Made up into
geometric patterns. used for paving Spanish and Moorish patios but also for
wall surfaces. The Figure 18 show how a Satorian mosaic can be creted from
three Satorian squares $SS_{5}$. To do this, nine squares $SS_{5}$ are joined
to form a bigger square, which does not satisfy the simmetry $S_{2}$ and
afterwards, the design is turned $45^{o}$. However, by combining properly nine
squares $SS_{5}$, a Satorian square $SS_{15}$ can be obtained as well.
The real picture of the Satorian mosaic in the Alhambra is showed in Figure
19.

Figure 18: Creation of a Satorian mosaic Figure 19: Alicatado on one wall at
the Court of the Myrtles from the Comares Palace inside the Alhambra

A fascinating work about simmetry in the Alhambra and other fields can be
found in [17].

##  8 Conclusions

For centuries the Sator Square has been considered as a mysterious riddle.
Most of the studies about this archeological object have been based on
religions or pseudo-sicentific disciplines, such as Christianity, Judaism,
Satanism, Philology, esotericism, magic and mysticism. Nobody has been able to
propose a rational and convincing explanation about its meaning.

In this work, the author proposed to see this square as a simple symmetric
game of words. However, as every symmetric object, its structure also can be
related to mathematical objects. In fact, the structure of the Sator Square
can be described as a function, a matrix, a surface or a solid. By defining
the mathematical properties of a Sator Square, it is possible to create
Satorian structures of higher dimension.

Perhaps if the Sator square is analized or studied from an analytical
perspective, a reliable explanation about its origin and meaning can be
acquiered.

## References

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