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Ostia Antica: the geometry of a mosaic involving a meander with a rhombus and tiling of the plane; Update #1

Authors:
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Ostia Antica:
The geometry of a mosaic involving a meander
with a rhombus and tiling of the plain. Update 1
Dimitrios S. Dendrinos, Ph.D., MArchUD, Dipl. ArchEng.
Emeritus Professor, School of Architecture and Urban Design,
University of Kansas, Lawrence, Kansas, USA.
In residence at Ormond Beach, Florida, USA.
Contact: cbf-jf@earthlink.net
July 14, 2016
Ancient Ostia’s site plan.
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Table of Contents
Introduction
Tiling of a plane and why the Ostia mosaic is of special interest
The geometry of the ideal meander-rhombus Ostia mosaic: its grid, modulus and code
Imperfections in the Ostia mosaic: ideal vs actual design
Conclusions
Bibliography
Acknowledgments
Ostia, Neptune baths, built by Hadrian circa 130 AD: mosaic of Neptune (the Roman equivalent
of the Greek god of the seas Poseidon) holding a trident, riding a 4-horse driven chariot,
accompanied by dolphins, tritons and nereids. Neptune sports a scarf, which we also encounter
in that exact configuration with Artemis Tavropolos’ renditions. The links to Greek late 4th
Century early 3rd Century BC Art are unmistakably strong.
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Introduction
Ostia is a well visited, richly written about, abundantly photographed ancient Roman site. Among
its many historical layers of construction activity (from late Republican Rome and Julius Caesar,
to early and mid-Imperial Rome, buildings by Augustus, Marcus Vipsenius Agrippa, Claudius,
Nero, Trajan, Commodus and Septimius Severus) many points of attraction are found [1] [2] [3]
[4] [8] [10].
Somewhere among all the glamor points of Ostia, lost between the Piazzale de Corporazione (and
of course the portico of the Ostia Theater) and the Horrea of Hortensius (in effect the warehouses
which stored the grain that fed Rome’s population during the Era of Imperium), lies a little
noticed photographed or written about outdoors mosaic, Figure 1. Largely lost among the more
glamorous and visually impressive imagery of the well-known Ostia mosaics, like for example the
pre-amble photo of this paper (from the Neptune baths section of the city) this un-assuming
mosaic is in fact rich in content.
The mosaic is referred to simply by a non-descript term, as “mosaic #432”, in G. Becatti’s work
[9], where two more photos of the mosaic are found (see Figures A.1 and A.2). It turns out that
Begatti’s short (about ten lines long) rudimentary description of this otherwise fascinating mosaic
is apparently the only reference in the literature on it. This single reference the author was able
to obtain due to Professor Jan Theo Bakker’s recommendation. Indicative of its neglect, Ostia’s
Piazzale Della Vittoria largely unnoticed mosaic is left to the ravages of time, slowly fading away.
Yet that mosaic tells a very interesting story. On it something of critical importance and value is
written and recorded. It reveals the level of mathematical sophistication mastered by its artist,
during the late Republican and early Imperial Rome, thus marking the time it was made. Further,
it also reveals that the mosaic’s artist was Greek and well versed into the mathematics of the
meander, a Greek eternity symbol of unparalleled sophistication in both its artistry and its
mathematics, and most of all in its extraordinarily impressive use in mosaics.
But the mosaic, as shown in the photos here, contains a number of impurities and imperfections,
the nature of which are also informative as to the construction and design conditions under which
the mosaic was made. It turns out that the theoretical design of this mosaic was significantly
dwarfed by the forces that ultimately contributed to its making. Its design impurities and
construction imperfections offer a window into the social, labor and artistic realities of the day
then. In that capacity, these impurities might even be more telling than the mathematical
achievements hidden in the intricate flowing and spinning of the mosaic’s meanders and their
imagery.
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Figure 1. Ostia Antica mosaic tessellation, involving a meander and a rhombus, in a pattern
which tiles the plane. Photo taken in June 2016; credit: Nancy Cowans. This is the photo that
triggered the author’s interest in this mosaic. Notice the two meanders at the left of the photo:
the one closer to the observer is spinning clockwise; the one next to it towards the upper left
corner of the photo is spinning counterclockwise. The reader should keep in mind this
difference, as it turns out that it is critical in discussing and analyzing the mosaic.
Tiling of a plane and why the Ostia mosaic is of special interest
Tessellation, or the tiling of a plane (in 2-d Euclidian geometry), i.e., the way by which a set of
polygons (from a single one to any number of polygons) perfectly cover an unbounded flat
surface is an interesting mathematical (geometric as well as algebraic) problem. Tessellation
implies a periodic pattern that conveys also from an Art and Architecture viewpoints considerable
aesthetic value, beyond pure mathematical interest.
In Archeology, the first case encountered involving a mosaic depicting tessellation of a (cylindrical
in fact) surface is the Sumerian Uruk rhombus, dated to the latter third of the 4th millennium BC.
It is shown in Figure 2. The stairway shown belongs to a Temple built during the late (fourth) city
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of Uruk’s period and belongs to the Eanna District, where the earliest cuneiform writings were
also found. A number of tiling patterns are shown in the above photo, but one is of special
interest here, the second from the right cylindrical surface. The other three are simple equilateral
triangle tessellations. The case of the above depicted rhombus is of interest because in the
scheme picked up by the Uruk cylinder, two equilateral triangles have merged to form a new
regular polygon, a rhombus, a special case of a polygon by itself in the mathematics of tessellation
of a plane.
The rhombus which tiles the cylindrical surface of Figure 2 contains within it a set of four
concentric and progressively smaller rhombi, each of a different color (white, dark purple, light
grey, dark grey). A closer look into this mosaic reveals that the pattern is quite elaborate, as it
consists of hexagonal pebbles. The hexagon is another single polygon which also can completely
tile a plane. These pebbles are set so that each rhombus of a specific color is formed by a double
strip of pebbles of that color. Double strips of pebbles or tesserae is a very common practice in
much later mosaics we encounter in Greek and Roman mosaics, and in fact it’s the case for the
mosaic to be analyzed here as well.
Notwithstanding the extraordinarily high artistic value of the above shown artifact, what is of key
import here is the fact that we have a case showing the simple geometry (and thus algebra) of a
single regular (convex) polygon tiling of a plane. The mathematics of single (referred to as
“convex”, meaning that any straight line joining any two of its points falls within it) polygon
tessellation in effect is called in abstract mathematics “regular tiling” and it is subject to the
“transitivity rule” [5]. Thus we have evidence that they were known quite early on in antiquity.
Not only was it known then that the rhombus (in fact the triangle, since a rhombus is two
isosceles triangles joined at their base) was capable of tiling a plane but also that a hexagon could
also do the job as well. It is noted that a hexagon is simply the special case of a triangle – in effect
a set of six identical equilateral triangles appropriately joined so that all share a vertex.
We now know, from analysis, that a triangle, a square and a hexagon are the only regular (convex)
polygons which can entirely tile a plane, in the case of a single regular polygon and regular tiling.
Of course there are also non-regular single polygons capable of tiling a plane in a non-regular
manner, but these irregular polygons and their tiling do not convey as much aesthetic or
architectonic quality, thus they are only of pure mathematical interest. It must be noted that the
field of tessellation is from a topology and set theory perspective a very advanced field, and the
mathematics of it rather demanding. The non-mathematically inclined reader can obtain a good
introductory review of the subject of tessellation in [6].
When we transit from a single regular polygon tiling a plane, to a combination of regular polygons
that tile a plane, of course things get quickly quite complicated. Searching for the geometry of
such a tiling in Art, we find it in the early and unique case involving a rhombus and a square in
the Roman mosaic shown in Figure 3. A rhombus and a square of equal sides always fully tile a
plane, no matter the angles of the rhombus. In the special case that the smaller angle of the
rhombus collapses to zero degrees, and thus the larger angle converges to 180 degrees, then the
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tiling becomes a simple tiling of the plane by a square. The case of Figure 1 demonstrates to us
that the 2-polygon tiling of the plane mathematical problem, had spilled over by that time from
the area of pure mathematics to the world of mosaic art.
Figure 2. Tiling of a cylindrical surface by a rhombus: Sumerian city of Uruk period IV (circa 3400
– 3100 BC). More details about the rhombus tiling and its elements, as well as the rest of the
tiling shown here, are offered in the text.
The rhombus-square combination tiling (as any regular tiling) is said to be “periodic”. This period
is easily detectable through the following easy exercise: by drawing any (random) straight line
but parallel to either the X or Y axis of the pattern’s Cartesian coordinate system (see Figure 4 for
the Cartesian coordinate system of the Ostia mosaic), and then by measuring the lengths of the
segments, where the line intersect any of the (regular) polygons involved in the tiling of the plane
(in the Ostia mosaic’s case the rhombus and the square), one sees that these lengths produce a
recurring (periodic) pattern, with a set period. That period is independent of the (parallel to the
coordinate system’s axis) line drawn, but a function of the number of polygons involved in the
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tiling of the plane. In the case of the rhombus/square combination, that period is four (meaning
that the fourth length count will equal the first count).
Figure 3. Ostia Antica: another Roman tesserae type mosaic, involving the tiling of a plane by a
rhombus and a square. In the text, this particular tiling and mosaic is referred to as “generic”.
The Ostia mosaic of Figure 1, and the subject of our analysis here, is a special case (in so far as
the tiling of the plane is concerned) of that shown in Figure 3. We shall refer to the pattern of
Figure 3 as a “generic” case of tiling a plane with a rhombus-square combination. However, the
Ostia mosaic of Figure 1 is far more complex than the “generic” case, since in the space of the
square shown in Figure 3, a complex meander is inserted. It is the presence of this meander, and
how its two crossing branches meander, that make this tiling of the plane pattern of such
significant interest as well as outstanding quality. We now turn to the mosaic in Figure 1.
In presenting the analysis of the mosaic, the distinction will be made between the “ideal” design
of the mosaic, and the “actual” design and execution of it. A huge design difference in fact will
be exposed between the two, this having to do with the direction of the flow in the meanders
contained in the mosaic. Whereas ideally, there should have been half of them flowing clockwise
and half counterclockwise, the actual floor mosaic contains only one flowing clockwise, and all
the rest spinning in a counterclockwise fashion. This source of imbalance creates numerous, and
not only design related, problems; but it’s the source of mostly construction related failures.
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Figure A. 1. The Ostia mosaic, referred to as mosaic #432 in Scavi di Ostia IV, by Giovanni Becatti
[9]. The photo shows that the length of the floor area accommodates eight meanders; whereas
the width of the floor’s area is taken up by five meanders. Two squares, referred to in this paper
as RS (or grids) of the pattern take up, in an isolated manner, the upper and lower right hand side
(in the photo) sections of the floor area. This angle reveals a host of construction related
imperfections. Location (4, 6), meaning the meander at position 4 from the lower left hand side
corner, and position 6 from the bottom as we move up in the photo, is taken by a clockwise
spinning meander (referred to as M+ in the text), as is the meander at location (3, 7). These two
meanders are the only ones of the M+ type in the entire floor. All the rest are M-.
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Figure A.2. Partial view of the Ostia mosaic, #432 in “Scavi di Ostia IV” by G. Becatti [9].
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Figure A.3. This is a photo of the Ostia mosaic, taken by Professor Thomas G. Hines, on May 26th
2001 and gracefully made available to the author. It offers a view of the mosaic along the width
of the area’s floor. The only two clockwise spinning meanders are the ones close to the middle
of the left hand side of the photo and the one diagonally next to it past the intervening
rhombus between them.
The geometry of the ideal meander-rhombus Ostia mosaic:
its grid, modulus and code
Considering that the meander is contained within a square, the abstract geometry of the square-
rhombus combination that tiles the plane of this ideal version of the mosaic is shown in Figure 4.
The diagram of Figure 4 depicts in effect the tiling of the plane consistent with the tiling type
shown in Figure 3, thus it is the ‘generic” representation of the Ostia mosaic under ideal design
conditions. Before we enter the analysis however, a note is in order.
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Map 1. Google Earth map of Ancient Ostia. The excavated area shown in this May 3, 2015 map
is approximately four fifths of a mile long (from bottom left to upper right) and about one third
of a mile wide. The location of the mosaic of Figure 1 is near the center of the photo (slightly to
the upper left) according to the person who took the photo of Figure 1 (Nancy Cowans).
It is noted that the author in drafting the first version of this paper had only seen just two photos
(Nancy Cowans’ photo of Figure 1, and Professor Thomas G. Hines’ photo of Figure A.3) of this
mosaic. It so happened (and, one might infer, not haphazardly) that both of these photos
contained in them the (only) two clockwise spinning meanders. They possibly attracted the
observer’s interest because they contained in them these two clockwise spinning meanders.
Moreover, in spite of an extensive search for literature or documentation on this mosaic, the
author was (till the completion of the first draft) unable to find any relevant material. The author
attempted to contact three entities (one of them being the site www.ostia-antica.org) plus two
researchers, whose work the author sited in the references. An exhaustive search on “images”
from both search engines (Google and Bing) had returned a single image of an equivalent (but in
no way identical) to this mosaic, see Figure B. The mosaic of Figure B is of interest because
although it isn’t an exact (albeit colorful) replica of the mosaic of Figure 1 (and Figures A.1, 2, 3),
it certainly belongs to the same “school” of mosaics. It must be of a much later time period, and
made by an artist who must have been a student of the Art school which created the Ostia mosaic
studied here. It will be further discussed later in this section.
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Further it should be noted that the author has not visited the Ostia site himself to acquire a critical
measure shown in Figure 4, namely the length x of the square’s side, and to verify the size of
angle , estimated to be 22.5, as it will be shown in the analysis which follows. These two
variables (x and ) are required to fully describe the mosaic. As a result, therefore, the analysis
is carried out on purely algebraic terms.
Figure B. An image of a tesserae mosaic in principle similar to the Ostia mosaic of this study, but
in color; it involves a much simpler mender design. The image is identified as a “Roman
Byzantine Church mosaic” in Bing’s “images of Roman mosaics” without any references to
either its exact location, time period or provenance. It is noted that all meanders in the pattern
flow clockwise.
After this brief introduction to the literature and image search, the analysis now turns to the
derivation of the governing grid pattern, the modulus of the mosaic, and its embedded code.
Again, it is emphasized that this analysis refers to the ideal design pattern, not the exact actual
one. The ideal design could have avoided many of the construction imperfections that we shall
identify later, and it would had been more aesthetically appealing.
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In Figure 4 the replicating square (ABCD) (to be referred as RS from now on) is shown. In effect,
it is this RS which basically tiles the plane. This is the grid pattern of the Ostia mosaic. Within this
grid, a replicating pattern (to be referred thereon as RP) exists which contains rhombi (actually,
one whole rhombus and eight sections of a rhombus, to be more exactly and fully describes in a
bit) and four meanders. RP is replicating in a second type of tiling of the plane. Thus we are
dealing with a tiling within a tiling, or a two-tier simultaneous tiling of a plane. This is a key
innovation of this particular mosaic, in the evolution of meander containing mosaics of the Greco-
Roman Art world.
Within the Cartesian coordinate system, with P as its origin, the grid’s X-axis and Y-axis identify
lines over which RP exhibits two symmetries. See also Figure 5, where the meanders are shown;
the meander flow shown in Figure 5 will be further elaborated and explore later in this section.
The section of RP to the left of the Y-axis is a mirror image of that at the right; and the upper part
of RP above the X-axis is the mirror image of (or is reflected in) the part below the horizontal axis.
In addition, there are two more symmetries (or transformations) in this RP: the four quadrants
of the RP, pairwise and facing each other, are symmetric to the origin P; that is, the upper right
and lower left quadrants, as well as the upper left and lower right are symmetric in reference to
the origin P. This fact presents the first case of a meander involved in the tiling of a plane, where
there are both clockwise and counter-clockwise meanders. Again, it is emphasized that this
represents the ideal design of the pattern.
Already mentioned is the set of rhombi (and their sections) involved in the RP of Figure 4. The
constituent parts of that pattern which involve a rhombus are as follows: a whole rhombus, at
center, four half rhombi (middle of the RS’s four sides), and eight quarters of a rhombus (two in
each quadrant of the RP in the RS). In total, the area of four rhombi is replicated by the RP of
RS. If by S we designate the area of a square in RP, and by R the area of a rhombus, then
(RP=4S+4R).
A unique feature of the Ostia’s Figure 1 (and Figure 3) rhombus-square combination of PR in RS
is this, see Figure 4: each of the diagonals in a square becomes the side of the next square in the
RP. This unique feature translates in the following statement (theorem): angle is exactly 22.5.
Proof of the above theorem is straightforward and simple: since the diagonal (in this case the X-
axis) bisects the rhombus’ smaller angle; and since the smaller angle in the rhombus of Figure 4
is necessarily 45, it follows that is half of 45, that is, 22.5.
The unique feature just mentioned of both Figures 1 and 3 (but especially of that in Figure 1),
namely that a square’s diagonal becomes the next square’s side, needs in situ verification of
course. However, inspection of all photos, in Figures 1, 3, A.1, A.2, and A.3 seem to fully confirm
this supposition.
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Figure 4. The grid, or the repeated square (in red) of the Ostia mosaic, RS, containing the
square (in green) within which the meander(s) unfold, and the resulting rhombus. Quantities x
and are the two variables completely defining the system.
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On that basis, one can derive the value of the length designated as L in Figure 4, as a function of
the (unknown to this author) value of the square’s side, length x. The length L is simply (from
trigonometry) the sum of the sin and cos times the length x. Since sin=.3826834…. and
cos=.9258795… the length L = 1.3x (approximately). Thus we have the full specifications of the
RS’s RP as a function of x. The side of RS is of course twice that of L (or about 2.6x). Finally, on
the algebra/geometry of this pattern, as the area of the square in the RP is simply x^2, one can
easily estimate that the area of a rhombus in this RP is about .7x^2. This implies that about 58.8%
of the (ABCD) is taken by the meander containing squares; whereas the remainder,
approximately 41.2% of the RS’s area is taken up my rhombus containing space.
We now turn our attention within the square S of the PR, specifically onto its two meanders: a
clockwise spinning meander (to be designated as M+) and counterclockwise spinning meander
(to be designated as M-). In Figure 5 the two opposite spinning meanders and their tentacles’
extensions are shown, as they form the next meander as well as they act as an envelope, by
forming the rhombus’ outer boundary.
Immediately apparent becomes the flowing pattern embedded into the two constituent and
crossing branches of each meander (the purple and brown branches of Figure 5.) It must be noted
that Figure 5 is schematically drawn to showcase the pattern of counter-flowing meanders and
their two moving and spinning branches. The ingenious manner in which the mosaic artist
intermingled the two branches allows for an uninterrupted flow from an M+ to an M- so that:
the purple branch of the meanders does not cross the boundaries of the left section (strip) of RS,
moving generally in a vertical manner and never crossing the Y-axis; whereas, the brown branch
of the meanders never crosses the X-axis, moving generally horizontally, and alternatively
forming M+ and M- type meanders. It must be noted that this is how ideally (according to this
author) the two counter-flowing meanders, M+ and M-, ought to have been drawn and made.
This is a magnificent interplay of flowing, spinning tentacles of an eternity symbol, unsurpassed
in Art by any meander seen to that date. The artist involved must have been a master of the kind.
Since only Greeks were specialists at that level of sophistication of meander design, it must be
concluded that a Greek mosaic artist must have been involved. More on this at the Concluding
remarks.
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Figure 5. The schematic unfolding of the two meander types, M+ flowing clockwise, and M-
flowing counter clockwise, in the ideal version of the Ostia mosaic of Figure 1 and their
horizontally and vertically resulting rhombus spaces. It also results in a motion as the eye
moves rapidly along either the X or Y axis.
Here however, and at the pinnacle of meander design, two critical design issues appear in the
manner this complex meander-rhombus tiling pattern fit the floor of the area covered, and these
two design issues are plainly shown in Figure A.1. The first design issue was the result of the
decision to only contain five (an odd number) and not six (as it ought) meanders. Thus, it does
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not fully contain the grid of the pattern all three times it was supposed to be repeated
horizontally in Figure A.1. Instead, a decision was made to add the column of counterclockwise
spinning meanders discussed in the legend of Figure A.1. It’s quite unclear as to why this decision
was made. It seems that the total width of the floor space covered could accommodate the
uninterrupted continuation of horizontally three repeated whole grids.
As a consequence of the fact that the ideal design for the mosaic was not followed, the peculiar
feature noted in Figure A.1 occurs, whereby only two of all meanders in the actual floor pattern
in the Ostia mosaic are of the M+ type, and all the rest are of the M- type. This incongruity in the
design pattern of the actual mosaic cascades into a number of other shortcomings to be more
fully addressed in the next section.
A closer look at the ideal design pattern of Figure 4 (and as noted in that Figure’s legend) is a
complex impression of another motion (besides that of the individually flowing and spinning M+
or M- type meander) created in the eye simultaneously by the two counter spinning meanders,
M+ and M- as the eye moves rapidly up or down the grid. It is possible that the artist, having
experimented with the pair in positions (4, 6) and (3, 7) may have realized that the complex
motion created was either too dizzying or confusing for the human eye, or not to the patron’s
liking. Thus (s)he may had abandoned the pattern at that stage. Although this might be simply a
speculative statement, and we may never actually know for sure the motivating factor behind
that decision, this could be considered a likely explanation.
Be that as it may, the manner in which the borders of the square containing each meander (of
both the M+ and M- type) were actually made differs from what it was presented as an ideal
design earlier (and shown in Figure 5). As made, the meanders are (unnecessarily) totally
enclosed (by a link shown in Figures 1 and A.1-3, and in a close up of Figure 7) to create a frame
within which to place the rhombus. This extra link presents, beyond a visual negative impression,
a design problem, as the continuous flow embedded in the spinning meanders is interrupted.
This problem is further discussed in the next section.
We now turn to the details of the meanders, both M+ and M- since we are still analyzing the ideal
case. In Figure 6 the detailed meanderings of the two branches forming the two types of menders
are shown for exposition purpose and for saving space only the M+ case is presented. The
design of M- is equivalent, only flowing the other direction. The main objective of Figure 6 is to
identify the detailed spacing among the meander’s branches, in the case of the ideal design, and
thus it constitutes a “construction detail document”. Moreover, its detail reveals the pattern’s
modulus.
At the outset it should be noted that the outline of the square which embeds each meander
(ABCD) of Figure 6 is the green lined square. The reader is cautioned not to confuse the square
(ABCD) of Figure 6 to the square (ABCD) of Figure 4 that is the pattern’s grid. This is the part which
is included within each quadrant of the RS in the mosaic of Figure 4. Specifically, this M+ meander
is found in the upper right and lower left of each RS.
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Figure 6. The detailed drawing of the meander (M+). The modulus of the pattern is y/2 or x/24.
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Ideally, the pattern must consistently exhibit distances between the two meander branches
equal to the branches’ width – in effect two strips of tesserae. If we designate by y the width of
each branch, then the total length of the square containing the meander (x) can be expressed as
a function of y, in this case x=12y. More precisely, since each y contains two lines of tesserae (see
the majority, although not all of meanders’ branch width as well as the background’s space taken
up by two lines of tesserae) one derives the modulus of the whole pattern. In effect the modulus
of the RP in this mosaic floor is M=x/24.
Looking at the construction detail (at the very lever of the individual tesserae square piece) one
could potentially count the total number of tesserae needed by color (light grey-blue and white)
to construct each square containing a meander. In total, 26^2=676 tesserae are needed per
square (containing the meander), out of which 398 are light grey/blue tesserae (59%) and 278
white (41%). Performing the analysis described by the author in [7], which however involved a
double mender, i.e., a set of two 2-branch meanders, one can obtain the periodic motion of
Ostia’s single 2-branch meander of Figure 1 (and Figure B), as a sequence of the number of
grey/blue and white tesserae encountered as one moves to either right or left scanning the
mosaic from side to side (within this square of Figure 6). This sequence, it was shown by the
author to depict the “code” of the mosaic’s meander, in the case of a meander with two crossing
branches. The two cases are equivalent, since in both cases the thickness of the meander’s
branches are equal to their spacing in the white background.
However, there’s a big difference between the codes of the two mosaics. Here, in the case of
Ostia’s mosaic the code is much simpler (since it involves a single meander) than the Kasta
Tumulus’ tomb mosaic complex code (which involved a double meander). Here the sequence
(number of light grey/blue {2x2} squares of tesserae encountered by a horizontal motion
scanning vertically), in effect the mender’s code, is simply: 12-2-11-4-8-6-9-6-9-4-11-2-12.
As in the case of Kasta, and qualitatively equivalent (but not quantitatively), there is the recurring
pattern {12-2-11-4}. {9} is the central count, and the two counts leading to the central count {9}
are the pairs: {8-6} to the left and {6-9} to the right. That difference (8 and 9) in these two pairs
is what gives the motion to the meander. This code is identical whether one scans the meander
horizontally or vertically. Since the meander types M+ and M- are for the purpose of deriving the
code exactly equivalent, the code is the same for both the ideal and the actual Ostia mosaic
design.
It was not so in the case of Kasta, since that case involved a double meander, and the recurring
pattern was not exactly a square. There, the code involved two sequences, one being (moving
horizontally) 3-13-8-12-8-13-3; while the other one (moving vertically) was: 2-12-7-11-7-12-2-15.
This completes the analysis of the ideal design pattern (and in some instances, as noted, of the
actual design as well) of this extraordinary mosaic from Ancient Ostia. It’s suggested that in all
likelihood what was depicted in Figures 4, 5, and 6 was the original design by the mosaic artist.
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The key clue is the presence of two M+ in the floor covering pattern, or put differently in the only
one complete ideal grid.
But as in all designs, there are costs associated with its benefits. The elaborate pattern and its
requirement for meticulous and extremely careful construction to attain the perfection sought
due to its complexity in design (and possibly due to lack of time and or highly skilled labor)
produced some impurities, to which we turn in the section that follows.
Figure C. A detail from the double-branch, counterclockwise spinning meander of Kasta
Tumulus’ tomb from the pebble mosaic floor Chamber. The double meander/waves pattern is
framing an iconographic representation depicting the abduction of Persephone by Hades, while
Hermes is watching. The mosaic was analyzed by the author in [7].
The key result of the possibly extraordinary demands for highly skilled labor, potentially absent
then and there, may have led the mosaic artist into the compromise we see in the photos of
Figures 1, 3, A.1, 2, 3. The fact that (s)he left two M+ type meanders (or one single and full grid
as per the ideal design) there, could have been a message to the future and the fellow mosaic
artists that he knew and thought about this ideal design pattern of Figures 4, 5, and 6 but reality
and its constraints ultimately prevailed over ideal design objectives.
Imperfections in the Ostia mosaic: ideal vs actual design
There are three types of imperfections detected on this mosaic from Figures 1, 3 and A.1, 2, and
3. Setting aside for the moment the underlying reasons of the failure to attain the ideal state of
Figures 4, 5, and 6, we focus on what was actually constructed. One could classify these impurities
as macro and micro imperfections. First, we shall address the micro types.
One impurity of the micro type is the apparent design imperfection involving an unnecessary link
in the meander, touched on in the previous section. A second imperfection is the inconsistent
spacing between stripes, as they seem to vary between two and three lines of tesserae at times
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in the unfolding of both the meanders and the rhombi. A third, and maybe not always so obvious,
imperfection is the failure to always attain the exact location at the meander’s corners within
each RS. A fourth failure is that almost no rhombus has exactly the same sizes with the other
rhombi, and often each of them has uneven sides, and ditto for the squares containing the
meanders. A fifth micro type failure is that at times the full swing of a meander’s two branches
is left incomplete – see for instance the meander located at position (6,7) of Figure A.1, i.e., at
the meander located six positions from the left bottom and seven positions up towards the top.
There are a few more minor (micro type) construction imperfections, but these five seem to be
the most noticeable ones, especially from Figure A.1. Most of them became apparent to the
author upon inspecting Figures A.1 and A.2 from the citation in [10].
However, the most critical shortcoming of the Ostia mosaic are the two macro type
imperfections, namely the decision by its artist to include only five (instead of six) repeated
meanders horizontally in Figure A.1, and the related design decision to interrupt the flow of the
pattern by the single meander column to the right of Figure A.1; and the decision to only contain
two M+ type meanders in the entire floor pattern.
It is not always possible to identify with certainty the errors due to the original workers while
setting up the grid and placing the pieces on the canvas (base); or to attribute these errors to
those who did the restoration (if any). Especially, since the author has no access to documents
identifying the various degrees of reconstruction efforts and what exactly these reconstruction
efforts really entailed
Since there are some construction (micro type) or design (macro type) imperfections of special
interest, a closer look will be taken at them. We’ll analyze these errors in turn, after a few general
remarks.
It is obvious that the mosaic was completed in a hurry, as some of these micro imperfections
clearly indicate. Most likely, it was not framed exactly according to the plan to fit its base. This
might had to do with the overall condition of the space (floor) the mosaic was to cover. In Figure
A.1 and A.3, the section of the floor covered by the meander-rhombus combination pattern is
shown to be interrupted by a strip comprising a column of crossed meanders (not very clear as
to why from the photos) and then the RP continues, shown in Figure A.3, as two isolated RPs top
right and top left.
Since the details of its reconstruction (and obviously some reconstruction was undertaken, as
seen in the lower right hand side part of the photo in Figure 1) are not available, one can’t be
absolutely certain that the last two of the impurities shown involving misalignments and lack of
consistently equal spacing are the initial workers’ fault or due to reconstruction workers’
renovation efforts. It is noted that the unnecessary link is not present in the much later “replica”,
done in color, of the Ostia mosaic, shown in Figure B.
22
The “unnecessary” link involving the margin of the meander, pointed out in the previous section,
needs special mention. If the link is there to complete the frame for the rhombus, it does so at
the expense of the continuity in the flow of the meander. The meander cost (to this author) is
not justified by the gain in the rhombus’ frame form. The very presence of this tradeoff might be
construed as a design shortcoming, too small however to override the significant benefit of this
very innovative mosaic pattern. For this shortcoming, one can’t blame either the original workers
or those who worked during the reconstruction of the mosaic.
Figure 7. The (unnecessary) link closing of the meander loop in order to frame the rhombus’
margin. This photo is a close up from Figure 1.
Figure 8. Misalignments present in the Ostia mosaic. Here we are observing two types of
misalignments: a corner misalignment, and uneven strips’ width. This photo is a close up of
Figure 1.
23
To close this discussion about the micro type impurities encountered in the mosaic’s making
during construction, a brief look into Figures 8 and 9 is taken. Both identify construction
imperfections. The one in Figure 8 identifies failed attempts to precisely derive joining (or rather
slight “touching” of the single M+ and almost all M- meanders) a touching at a single point as
shown (ideally) in Figure 4. In Figure 9, the case of a rhombus’ white tesserae made frame where
it should had ideally been “touching” that single point has slightly failed, making an awkward
junction. However, as already mentioned, these impurities in no significant degree subtract from
this extraordinarily complex and from a design viewpoint very sophisticated mosaic.
Figure 9. A clear misalignment of a rhombus corner and a meander’s corner. This photo is a
close up of Figure 1.
Before we conclude this analysis a few points will be made in reference to the rhombus-meander
combination of the colorful mosaic shown in Figure B. All meanders of the unfolding pattern there
(at least those shown in the photo) are of the M- (counterclockwise flowing) type. Moreover, the
spacing of the two constituent elements (rhombus-meander) seem to be only slightly better than
the majority of the spacing encountered in the Ostia mosaic analyzed here. These observations
seem to suggest that the mosaic in Figure B is of a much more recent time period.
However, the key difference (beyond the one involving the presence of only M- meanders) is that
the colorful mosaic of Figure C entails meanders that have less “spin” on them – only two levels
in the branching as opposed to the triple spinning of the meanders of the Ostia mosaic.
If one is to assume that some progress was made between the time periods of the Ostia mosaic
of Figure A.1 and that of Figure C, then this progress was painfully slow, and not always advancing
smoothly. Moreover, it also seems to suggest that although the mathematics of the tiling were
advanced, the art of implementing them in the design of mosaics at the time period from Ostia
to the time of Figure B mosaic was not progressing as fast. In effect, here we potentially identify
something like a 2-speed development pattern, whereby the artistry of mosaic making was
24
lagging in reference to the speed of movement in the development of the underlying
mathematics.
If one is to include a third speed in this 3-tier evolution, one might be forced to say that
development of skills in actual mosaic labor was even slower that the speed of the other two,
mosaic artistry and of course underlying mosaic mathematics’ sophistication.
Conclusions
The mosaic at Ostia, of Figure 1, tells us that the mathematical problem of how to tile a plane
using two polygons was a problem which had spilled over from the world of pure Mathematics
to the world of Art. Through the various experimentations we come across at Ostia, Figures 1
(including Figures A.1, A.2, and A.3), 3, and Figure 10 (at the end of the paper), we clearly
recognize the level of mathematical sophistication of the Era (most likely the 1st century BC).
It seems that during the period of Ostia’s early construction phase, there was plenty of trials
involving the meander design as an eternity symbol, under different configurations, and possibly
under different mosaic artists. Stylistically, one can distinguish early and late attempts. The
mosaics shown in Figures 3 and 10 seem to belong to an early phase of construction, definitely
earlier than the one of Figure 1 (and A.1, A.2, A.3), the subject of the analysis here. A much later
case is that depicted in Figure B, a colorful “replica” (or rather attempt to improve building on)
the mosaic of Figure 1, although it involved only a single type meander flow (M-) and a simpler
spin in the meanders’ double (crossing) tentacles.
All of these early attempts must be attributed to Greek mosaic artists, since the meander played
a significant role in 4th and 3rd century BC Greek mosaics throughout the Helladic space, but
especially in the mainland Greece region, including Macedonia. Quite likely, mosaic artists that
worked at Ostia at the early phases of construction (the first century BC) must have come from
or had been educated by artists belonging to the school that created the very similar mosaics at
Pella and Amphipolis, not to mention the even earlier ones from middle 4th century BC at
Olynthos, see Figure C.
Of course Greece by then was subjugated to Roman rule. The Greek artist of the mosaic had as a
patron a Roman at Ostia. It was the Roman’s desires (including the aesthetic, economic, and
cultural constraints of the Roman patron) that the Greek artist was supposed to serve. At the
end, the patron’s wishes (as they always do) overtake the artist and the architect’s wishes and
the artistic values are subjugated to the stark realities of the day. This is what the differences
between the ideal and actual configurations of the Ostia mosaic reveal, in a final analysis.
Only two colors seem to dominate almost the entirety of Ancient Ostia’s mosaics: grey (light or
dark) and white. This bi-color trait is not only characteristic of the mosaic under analysis here,
but it’s also the case with a great deal of mosaics in Ostia. This is shown by the photos of the
25
Neptune bath mosaic, and by all three mosaics of Figures 1, (A.1, 2. 3), and Figures 3, and 10.
Colorful mosaics are not common at Ostia, apparently. The cause for this must be sought by a
more comprehensive study and analysis of the area’s mosaics.
Figure C. Pebble type mosaic from the city of Olynthos.
But more specifically, a more detailed examination of the total space which houses the mosaic
examined here is of course needed. Such examination might answer not only the questions raised
by the mosaic’s impurities, but also place this mosaic in its broader perspective in reference to
the other nearby mosaics in Ostia Antica’s Piazzale Della Vittoria, and potentially all the rest. In
effect, this close examination might uncover the economics that at the end dwarfed the art and
mathematical sophistication of the mosaic.
Methodologically, beyond the social and cultural study of the mosaic and its environs, some
dynamical statistical analysis involving attributes of mosaics and time period involved in their
making might provide some useful clues as to the developmental speeds of the three factors
mentioned affecting mosaic quality (labor skills, artistic sophistication, mathematical
sophistication). Moreover, it would place this extraordinarily complex and sophisticated mosaic
(when examined from its ideal design perspective) in a temporally firm frame of reference.
26
Figure 10. Ostia Antica: more (unsuccessful in this case) experimentation with meander and
nonconvex shapes in an effort to tile the plane.
Some Notes
1. The author used in this work, and in drawing the graphs of Figures 4, 5 and 6 just pencil, a ruler
and a compass, as the artist of that time would had done. Parenthetically, the same conditions
the author followed in the drawing he prepared for the papers cited in [7]. By doing so, one is
able to obtain an understanding of the processes the artist/mathematician/builder went through
back then. It also enables one to depict the errors or imperfections embedded in the design as
well as in the making of the mosaic (mostly the construction) process.
2. This paper appears on the one-year anniversary of another paper by this author on the double
meander found as the frame of the Persephone abduction mosaic at Kasta Tumulus’ tomb, in
Amphipolis, Greece, [7] mentioned above.
3. It is noted that at times the author uses his two literary pseudonyms, George Watkins and J.
Peters.
27
Bibliography
[1] http://www.ancient.eu/Ostia/
[2] https://www.bluffton.edu/homepages/facstaff/sullivanm/italy/ostia/ostiaindex.html
[3] http://www.ostia-antica.org/
[4] http://www.historvius.com/ostia-antica-339/
[5] Branko Grunbaum, 1975, “Venn diagrams and independent families of sets”, Mathematics
Magazine, Vol 48, No. 1, pp: 12-23.
[6] https://en.wikipedia.org/wiki/Euclidean_tilings_by_convex_regular_polygons
[7] Dimitrios S. Dendrinos, 2015, “The mosaic of Kasta Hill’s tomb: the algebraic wonders of its
MAIANDROS , including an exact ratio embedded in it, and its magnificent cycles” available here:
https://www.academia.edu/16554729/The_Double_Meander_and_Waves_of_Kasta_Hills_Mos
aic_Frame
[8] http://www.ostia-antica.org/dict.htm
[9] Giovanni Becatti, 1961, “Mosaici e pavimenti marmorei” in Scavi di Ostia IV, Part 1, pp: 232-
233. Roma.
[10] http://archeoroma.beniculturali.it/siti-archeologici/ostia-antica
Acknowledgements
The author wishes to thank his Facebook friends Nancy Cowans and Guiseppe Giusti for
comments and suggestions. Nancy Cowans was the person who took the photo in Figure 1. That
photo prompted the author’s interest in this subject, and the writing of this paper. Besides these
two FB friends, the author has benefited by interacting with all his FB friends over the course of
the past two years. In the course of this study, the author was graciously offered the photo shown
in Figure A.3 from Dr. Thomas G. Hines, Professor of Theater at Whitman College, Walla Walla,
Washington. The author wishes to express his gratitude to Professor Hines.
Not less importantly, the author received additional information about the mosaic from
Professor Jan Theo Bakker of the University of Leiden, the Netherlands. Professor Bakker is an
expert on, with a rich record and long term involvement in, Ostia’s archeology. Leads and
references which Prof. Bakker provided to the author significantly improved the exposition of
this paper. The reference to G. Becatti’s photos made clear to this author that the M+ meanders
at position (4, 6) and (3, 7) in Figure A.1 were in fact the only such meanders in the Ostia mosaic’s
28
pattern under analysis here. The author is grateful to Professor Bakker for offering the Becatti
reference.
Finally, the author wishes to acknowledge his daughter Daphne-Iris’ contribution to the paper. It
must also be mentioned that the author feels very grateful to his family for putting up with his
late hours of work on these papers, his wife Catherine and his daughters Daphne-Iris and Alexia-
Artemis.
Legal Notice
The author, Dimitrios S. Dendrinos, retains all legal copyrights to the contents of
this paper. No part(s) of it, or the whole, can be reproduced by any means without
the explicit and written consent by the author.
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