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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.

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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.

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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

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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

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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.

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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.

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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

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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.