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The Mathematics and Astronomy in Tutankhamun’s

Funerary Mask.

Dr. Dimitrios S. Dendrinos

Emeritus Professor of Architecture and Urban Planning, University of Kansas, Lawrence, Kansas, USA

In residence at Ormond Beach, Florida, USA

Contact: cbf-jf@earthlink.net

January 7th, 2016; 1st update 1/19/16

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ABSTRACT

It is established by this paper that the mask of Tutankhamun, see [1] for a broad description of it, is

governed by certain mathematical principles; it was made on a modulus; and that it contains in its form

specific astronomical measures, as known to the Ancient Egyptians at the time it was made. The paper

presents the mathematical (both algebraic and topological) foundations of the mask, but it is not written

for mathematicians. It’s addressed to an audience with only an elementary algebraic background. Given

the time the mask was made, the mathematical sophistication imprinted on it is indeed very impressive.

The astronomical foundations of the mask are revealed, based on the then lunar and seasonal calendar

the Egyptians used, to which Tutankhamun’s age (about 18 years, or close to 54 Egyptian Seasons) was

embedded. Further, the magnitude of the modulus used by the artist to create the mask is estimated in

this paper. Its length measurement is found to be approximately 1.786 (or about 1.8) centimeters. The

parabola used by the artist (evident from the back side of the mask) is also extensively discussed, as the

key to decoding the mask’s mathematical base.

The religious feature of the mask, expressed by frequently encountered patterns involving the number

“4” on the mask’s many decorative motives, is further analyzed and speculated upon as to its source.

PART I. Introduction.

Pharaoh Tutankhamun (born in around 1341 BC) died (unexpectedly as a result of an accident, or as a

result of an illness – the matter is still under debate) at the age of 18, circa 1323 BC. Thus it is reasonable

to assume that the mask was made sometime following his death (although this contention is also

disputed). These historical arguments and their eventual resolution, if ever achieved (since in History

and Archeology it is customary to encounter arguments, almost about everything, and rarely is there a

definitive resolution of them) most likely would not greatly impact the findings presented here.

What is of significance nonetheless, is the consideration of the knowledge the Egyptians possessed at

the later part of the New Kingdom, during the 18th Dynasty, the period that is when this mask was made.

Thebes was restored as the Capital of Egypt under Tutankhamun’s reign. It was a turbulent set of years

following the death of Tutankhamun’s father Akhenaten, and the transition back to polytheism after a

brief switch to monotheism (and god Aten) under Akhenaten. It was the tail end of the so-called

“Amarna” period in Egypt.

Analyzing the art and mathematics suggested to govern the mask, as well as any astronomical features

to be detected on it, must be drawn from and be based on the stock of knowledge the Egyptians

possessed at that time, as well as the artistic vein prevailing back then. The precise time of the mask’s

making, the latter part of the 14th Century BC, is close to the approximate time basic components of

algebra and astronomy were possessed by the Egyptians. We must then identify the level of these stocks

of knowledge at that period.

The Rhind Papyrus (a circa 1650 BC document) shows that the Ancient Egyptians could carry out

multiplications and divisions in a systematic fashion, as well as they could master fractions. Moreover,

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the Berlin Papyrus (a circa 1300 BC document) establishes that they could solve simple algebraic

quadratic equations [2]. Thus the mathematics of the mask can’t be far more sophisticated than that.

The reader is directed to [6] for a review of the state of the art in Egyptian mathematics and science at

the time of the 18th Dynasty (among other periods as well).

On the Astronomy front, the Egyptians of that era were into the recordings of the Moon’s cycles, in

addition to the setting of their Temples to align with major Astronomical events (like the Solstices and

the Equinoxes, as well as the rising of major Stars, especially Sirius (Sothis), and Constellations, in

particular that of Orion) [3]). It is thus known that rough estimates involving the Lunar calendar was

what the Egyptians at the time the mask was made were able to master [4]. Within that calendar, of

particular import is that the Egyptians back then considered three Seasons per year, four months per

season, and 30 days per lunar month. If there’s any astronomy, it’s those rough estimates then which

we should expect to record on the mask. As it turns out, some of that information is imprinted on it.

Let’s take a brief look at the specifics of the mask. Its overall dimensions are given in [1] Part 2, and they

are as follows: the overall height of the mask is 54 cm; the frontal width is 39.3 cm, and the depth (side

width) is 49 cm. It must be noted however, that the reader should not confuse the number 54 (in

centimeters) with the number 53 (as in the number of stripes encountered on the mask) we shall

encounter later. To the Egyptians back then, 54 centimeters meant nothing, as their units of

measurement were not based on the modern metric system.

Most likely, the artist who designed and (possibly) made the mask (it will be referred to here as if the

artist was a male, although there is no historical evidence that this is so – it could be a female as well)

didn’t have much choice regarding these general dimensions (except possibly at the margins). It seems

logical to assume that the broad dimensions of the sarcophagus, the tomb and the cost associated with

the making of the mask (and all the rest in Tutankhamun’s tomb) were political decisions taken by

higher ups in the Pharaonic bureaucracy at the time. So the problems facing the artist(s) most probably

were of a purely design nature, given some overall constraints he operated under and handed to him.

So, we are forced to ask the question, what was the key objective the artist had, what exactly was the

central design problem he dealt with, and how did he solve the design problem he confronted. By

looking at the mask today, the components of it, and the solution the artist provided we can attempt to

reconstruct the problem, thus decode the mask’s meaning and message. The most striking feature of

the mask is of course the nemes’ stripes. Why the nemes has the features it has is of no concern here.

The concern is the study of the stripes, their form, numbers and positions on the mask’s nemes, as well

the position of the nemes itself within the overall shape chosen for the mask.

That seems to have been the set of design problems facing the artist, and by answering these design

problems, we will obtain insights regarding the roots of the solution the artist derived. As it turns out,

the artist was trying to solve a very sophisticated topological problem, and by doing so successfully, he

offered to us a marvel of an artifact. Few artifacts evoke such an emotional and aesthetic appeal as does

the Tutankhamun mask, and it will be shown here why this is so. A key feature in this answer of course is

the astonishing mathematical sophistication one can discern on its decorative patterns and its overall

form and construction.

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Figure 1. The left hand side view of Tutankhamun’s mask. Source: [1].

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Figure 2. The back side of Tutankhamun’s mark.

It must always be kept in mind, this artifact was made at the Late Bronze Age, when the Minoan

Civilization was at its declining stage (at its post Palatial period), and the Mycenaean Civilization was at

its ascent. It is the period following the Amarna Correspondence under Akhenaten and Egypt’s

neighboring states to the East. This is the historical context within which the mask was made.

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PART II. Description of the mask.

A number of things become apparent from a close-up look at the back side of the mask. The mask is split

in two sections, the upper (head) piece (section A) consisting exclusively of part of the nemes and has

the approximate shape of a parabola; and the lower part consisting of the back side of the collar that

covers the shoulders and back and has the approximate shape of a rectangle, as well as the ponytail

(section B). The back side of the collar (on both the right and left) contains four full columns of

hieroglyphic inscriptions, and sections of a fifth far narrower column (most likely put there as a filler of

spare space, leftover by the design’s not being perfectly rectangular in shape at the back). For the

meaning of these hieroglyphs, one can read [1], Part 2 II. We shall come across this number “4” often in

the artwork of this mask. It should be noted here that decoding the back will take the analyst a long way

towards decoding the whole mask as it is in effect the key to the mask’s design.

The two back sections (A and B) are approximately equal. As the nemes moves back over the top of the

shoulder it curves down. At the top of the shoulder, Figure 1 side view makes that clear, each section of

the mask has a length (height) of about 27 cm. This documents the 2:1 ratio, as one of the underlying

principles used in the making of the mask, an element of symmetry found throughout the mask. It is a

simple principle, but a basic one in its making. This ratio must have been instrumental in determining

the total back as well as frontal length of the mask. As the nemes moves at the back side of the mask, it

slightly extends downwards, so that the point halfway at the back the nemes (at the point where the

ponytail starts) attains a height of about 28 cm, whereas the exposed collar height drops to about 26

cm. It will be seen later, that this difference is about equal to the modular length. It is also noted that

the ponytail, where all stripes (blue and gold) converge, of course does not start at the focus of the

parabola, which falls quite a bit higher (as we shall see in the last subsection of Part III); however, it is

located at a point on the parabola’s symmetry axis directly connected to the parabola’s focus. Since the

basic mathematics of a parabola involve simple quadratic equations, it must be assumed that the artist

was familiar with elementary parabolic functions. These will be extensively presented in Part III.

At the front side, the mask consists of the following three elements: (i) the nemes which has flaps

covering the head from the back of the ears and extends beyond the shoulders to the chest; (ii) the false

beard; and (iii) the collar (which is now covering the frontal part of the shoulders and the chest). The

collar is divided into twelve bands, in the form of recessed arc-shaped relatively wide stripes. Each band

is outlined on both sides by rim forming narrow gold rings, thirteen in all. The first and last gold rings are

much wider than the rest of the rings. The rims hold the much wider decorative bands in place. Of these

twelve bands, bands #4, #8 and #12 carry lapis lazuli. Here again, we detect the number “4” and its

multiples as being specifically emphasized by the insertion of lapis lazuli at the front of the mask’s collar.

It is noted that the overall shape of the collar isn’t of any special geometric kind (semi-circular for

example, elliptical, parabolic or any other recognizable shape). Quite likely, the shape of the arcs was

chosen to fill the space derived by the requirement we detected at the side and back sections of the

mask (them being of equal length), plus the additional requirement that the bottom of the nemes’ flaps

do not cross over the last four bands of the collar. The number “4” again pops up in the mask’s

description.

Now we reach the core of the analysis. The mask’s dominant features are the blue and gold stripes at

both front and back sides in the nemes part of the mask. The dark blue stripes consist of blue glass

pegged inside recesses created by the mask’s gold frame. The width of each recess equals the width of

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the golden rim between every pair of recesses, at each point along the length of the stripes, in a mastery

of art construction. In the design of these stripes, their size and their shape as they move along the

front, sides and back of the mask, the mathematics and astronomy the artist wanted to install are

embedded.

Of course, much symbolism is attached to every element of the mask, and a good exposition of a fair

amount of that symbolism is found in [1] among many other places. Part of that symbolism is found in

the insignia at the front top part of the mask, designating the Pharaoh’s reign of both Upper (through

the vulture) and Lower (through the cobra) Egypt and their relative size to the total mask’s height. The

length of the cobra for example is 0.1765 the total length of the mask, not an important or central

design feature. However, its total length obeys the modulus employed, as it will be seen later in the text

(Part III on the modulus of the mask). This and all other associated measures and symbolisms will not

preoccupy the analysis here, as the main focus is the search for the fundamental design and

mathematical principles with which this artifact was made. These principles are mainly found in the gold

and blue stripes of the mask where the mask modulus is also embedded.

We now turn into the search for the mathematics and astronomy printed on this artifact, one of the

most recognizable artifacts for both its beauty and sophistication, a combination (which garnished by

the mystery surrounding Tutankhamun in specific, and Ancient Egypt in general), have render this

artifact as one of the most striking and extraordinary artifacts of all Ages.

PART III. The mathematics, module and astronomy of the mask.

The front side of the mask.

In front, the number of blue stripes in section B (the lower part of the nemes with the flaps covering the

shoulders and chest) is fourteen (28 on both the right and left, running more or less horizontally and as

if in a continuum), see Figure 3. This section also contains fourteen gold strips on each side (another 28,

although the lowest gold strip securing the first bottom blue stripe is not counted, as being too thin). Up

till this level, the stripes move along two planes at the same frontal level (the more or less flat plane of

the two flaps at the frontal side of the nemes). Above this section, the lower part of section A

commences (with a blue stripe). There are six blue strips running more or less horizontally in a

seemingly smooth continuum, on both the left and right sides (for a subtotal of twelve gold and twelve

blue stripes, a total of 24 stripes). They also constitute the lower six blue and six gold stripes of the back

side of the nemes to the right, and an equivalent number of stripes to the left (a total of 24 stripes).

However, starting with blue stripe #21 (from the bottom) at the front, and stripe #7 at the head section

of the nemes at the back, the blue (and gold) stripes take a turn towards the front to cover the

mummy’s forehead. This bending of the stripes takes them to a third surface, the side of the head. The

bending lasts for three blue and four gold stripes on each side of the front of the mask (for a subtotal of

fourteen stripes in front of the mask, and of course as they continue at the back, the constitute fourteen

additional stripes in section A). The rest four blue and three gold (on each side) plus the blue stripe at

dead center (a subtotal of fifteen stripes) move smoothly and continuously from the front onto the

spheroid surface of the head piece at the back.

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As a result, if one were to count the total number of blue stripes seen on the frontal part of the mask,

on each side from the blue central stripe (located between the vulture and the cobra), it would be 27.

There are fourteen on the lower part, section B, left and right (28 subtotal]; and there are thirteen (for a

subtotal of 26) on the upper part of section A (out of which six are straight, three “bend” and four curve

smoothly). There’s also the very central blue stripe to be added. Correspondingly, for the gold stripes, it

would be a total of 27 as well. Number 27 includes the fourteen on the lower part, section B, at both

right and left hand sides; and the thirteen on the upper part, section A, where again, six are straight on a

plain, four “bend”, and three are moving smoothly on a curved surface.

Figure 3. The right side of the mark, where the number of blue and gold stripes on the nemes are clearly

discernible. At section B (the lower part of the nemes’ flaps) contain fourteen blue stripes; whereas the

upper part of the nemes, section A, contains thirteen blue stripes on each side (right and left). The outer

bending starts at blue stripe #7 (from bottom up, in section A), and #21 overall (from the bottom of the

flap).

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If one were to count the total number (blue plus gold) of stripes in the frontal part of the mask, from the

above provided counts, one derives a total of 109 stripes. However, if one wishes to avoid double

counting the stripes (on the right and left hand side of the nemes’ flaps) on the largely aesthetic basis

that they seem to constitute single stripes only interrupted by the presence of the mask’s face, but

include the forward bending stripes as separate stripes, one would come up with a total of 69 frontal

stripes.

This total of 69 stripes is obtained as follows. There are fourteen blue plus fourteen gold stripes (from

section B); it is noted that the thickness of these 28 stripes (to the right, plus 28 to the left side of the

mask) is about half that of the stripes in section A. We shall come back to this observation.

In addition there are six blue and six gold (total of twelve) straight stripes from section A; this total of 40

stripes appears on both the right and left hand side of the nemes, and will be counted once. In addition,

there are seven blue stripes with forward motion plus eight gold stripes with forward motion on each

side of the nemes (right and left) which account for an additional 28 stripes, plus one additional blue

stripe at the very center of the nemes. Thus the total number (blue and gold) of identifiable stripes

(avoiding double counting) on the front of the nemes is 69 (35 blue and 34 gold stripes in all).

The back side of the mask.

Now, it is far easier to count the stripes at the back parabolic side. It is noted that the first (bottom) six

blue and six gold stripes continue into the frontal area of the mask by a simple (almost 90-degree)

topological “fold”. The following three blue and four gold stripes undergo a double topological

transformation, as they initially curve around (fold) into the frontal part of the mask, then topologically

“bend” forward onto another plane, thus splitting and generating double the apparent number of

stripes; that is, they create an additional fourteen (2x{3+4}=14) stripes.

We shall see the implication of this set of topological transformations in a bit. The total number of blue

stripes on the back parabola of section A is 27, and the total number of gold stripes is 26. Put together,

there are 53 stripes in section A of the mask’s back side. Out of these 53, the top four blue and three

gold do not undergo any topological transformation, as they smoothly move along the top of the head’s

spherical surface.

Discussion and analysis.

Noted is the fact that the lower thin gold margins supporting the blue stripes at the back (section A) are

not counted. Along similar lines, it must be noted that the thin gold frame at the lower end of the flaps

holding in place the first blue glass stripe does not count as a stripe also. Since it is not clear that these

thin gold stripes were not intended to be so thin by design, so as to go almost unnoticeable, it will be

assumed that were on purpose made thin. They were the result of necessary welding and riveting during

the process of constructing the mask.

However, and what is far more important (and as noted earlier) is the differential in the thickness of the

frontal stripes. It is underlined that the stripes (blue and gold) in section B are noticeably thinner than

those found in section A. In fact, they are about half in thickness. Thus the question is, why is it so? Was

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there a reason the artist had in mind when designing them with that differential thickness so evident? It

turns out that there was, and it is related to the mask’s modulus.

There is a modulus used in the making of the mask, which attains its minimum (half in fraction) manifest

size in the thickness of these narrow stripes, and full size in the width of the wider stripes. This modulus

has not only a lot to do with the differential in thickness of these stripes, the stipes of section A versus

the stripes of section B. But it is also linked, through a marvel of mathematical and artistic skill, to the

topological folding and bending of the stripes of section A in the nemes of the mask.

Before we turn to the dimensions of this modulus, we observe the dominance of certain frequencies and

dimensions. So far, by looking at the most evident of stripes, and their differences in thickness, we can

arrive at some preliminary conclusions.

It is quite obvious that numbers, bunched in ranges as {3, 4, 6}, {13, 14}, {26, 27}, {53, 69, 109}, are those

which are keenly associated, with varying frequency (repetition) of stripes on the mask’s nemes.

Numbers {4, 8, 12} are associated with the mask’s collar, and the frequency (and location) of lapis lazuli

applications. Moreover, were the total number of bands in the collar to be included (by counting the

thinner inner and outer bands in it), then to total count of bands would reach fourteen as well. By far,

however, the most frequently encountered number in all decorative elements of the mask is number

“4”.

The frequent encounter of the number “4” must point to the four seasons the Egyptians back then

divided the year, see for example [4]. Actually they considered only three seasons, plus some extra days

constituting the very short in duration 4th season. Primarily however, the source for number “4” must be

the four months they included in each season. Other sources for this number could include four major

gods (and in [1] some suggestions are made, although Ra, Amun, Osiris, and Isis seemingly represent

what Egyptologists’ consensus is on this matter, see [5]). The number may also be drawn as well from

the four directions – North, South, East and West. Obviously, there could be many other interpretations

for this number “4”.

On the mask however, these are not the only places where certain numbers (number 14 in particular)

figure prominently. A close look at the back side of the mask shows that all stripes (blue and gold)

converge to a point where the ponytail begins. This ponytail however doesn’t start at the focus of the

parabola. The mathematics of this parabola, and the relative location of the point where all stripes

converge in reference to the focus will be addressed later. However, for the moment suffices to point

out that the number of rings around the ponytail are: 19 blue glass rings, and 20 gold rings. In so far as

the front of the mask is concerned, in an equivalent feature, the false beard contains 14 diagonally

running (in both directions) strips of gold. Ten were the number of days in an Egyptian week back then,

as already noted. Thus, reference to number 20 could be the two out of three weeks in an Egyptian

month, or the 3:2 principle. On the other hand, references to number “14” could be the approximation

to half a lunar month, and the 2:1 principle already discussed.

We are now close to deriving the modulus of the mask and of its nemes, but before we do so, another

transformation will be evoked, a transformation which will try to combine the frontal stripes of sections

A and B into area equivalents and link it to the major feature of the mask, and Tutankhamun directly. By

doing so, we take a big step in decoding the mask and deriving its underlying code.

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If we were to discount the number of (gold and blue) stripes found in section B, at the front of the mask

(the bottom part of the nemes) by their relative (approximately one half) thickness, then the total area

equivalent of the frontal stripes would drop from 69 down to 53. We note the total number of the back

(section A) stripes (blue and gold) is 53 as well (as derived earlier). Now we can perceive in full the

ingenious topological transformations we noted earlier: they guaranteed that the folding and bending of

the stripes, would exactly counter the discounting effect from the thinness in the frontal lower fourteen

sets of blue and gold stripes, by generating enough “new” strings (in thickness and number) that would

maintain the total number at 53 in terms of “equivalent area”.

However, what is more of import is the very nature of this number “53”. It is suggested here that this

number was chosen specifically to identify in Seasons Tutankhamun’s age. Tutankhamun, historical

sources seem to indicate, dies at the age of eighteen. Now it is unlikely that he died at his eighteenth

birthday. It is suggested thus here that he died probably in the last Egyptian Season of his eighteenth

year, and that this exactly for what the 53 stripes stand. Their gold and deep blue colors seem to identify

the Sun (day) and the Moon (night), according to this author.

Thus we conclude that the stripes were a key component in the making of the mask, and that the artist

who designed the mask wanted to keep this count in all three spatial dimensions constant (at 53). This is

an amazing and ingenious mathematical achievement: the “folding” and “bending” of a certain number

of stripes (nine blue and ten gold) enabled the artist to maintain the desired number 53 of area-

equivalent stripes in 3-d.

The mask’s modulus.

As the back view of the mask indicates, starting with the top blue stripe, the 53 stripes smoothly move in

succession, covering the mask’s nemes as they unfold from a vertical (the central gold stripe) to

horizontal (the stripes at the bottom of the nemes). The thickness of the top blue stripe identifies the

dimension of the modulus. As the stripes, on the outline of the parabola at the back, slide to an almost

horizontal position (the bottom two blue stripes), they seem to maintain (more or less) the module’s

dimension in their thickness. The stripes seem actually to ever so slightly have their thickness modified,

as if by a rule. However, since extremely accurate measurements need to be taken in situ to exactly

measure and verify this “decreasing in thickness” rule, if it really exists, something this author has no

way of obtaining at this stage, it will be assumed that a statistical average of a stripe’s thickness can be

used, and this is satisfactory enough for the purposes of this study. Slight variations in the stripes’

thickness could be also topical and the result of slight imperfections during the making of the mask. In

such a case, a statistical average can be used, as the modulus size estimated here must be viewed as

such a statistical average. Thus, all statements below need to be approached with caution, in view of

comments also made in the last subsection of this Part III.

It is concluded that the width of section A’s (blue or gold, since they seem to be the same) stripes at the

very top of the mask is the width of the modulus, and the width of the section A (blue or gold) strip at

the very bottom of section A is the height of the mask’s modulus (if actually different). From a visual

inspection of all photographs the author examined, these widths seem to be about the same. A

preliminary measure of this modulus length, obtained from the count given here (height 54 cm, width

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39.3 cm) seems to suggest that the size of the modulus was approximately 1.786 cm or about 1.8

centimeters.

In [6], the unit measure of length are offered applicable at the time of the 18th Dynasty. The length’s unit

dimension was the “finger”, or “djeba”, with dimension in the metric system approximately equal to

1.875cm. The proximity of this unit estimate to the length size of the modulus is noted.

Then the height (54 cm) of the mask is an exact multiple of this length: 54:1.8=30. This is precisely the

number of days back then the Egyptians considered the month to last.

The width (39.3 cm) of the mask is an exact multiple of this length as well: 39.3:1.786=22.

The depth (49 cm) of the mask is an approximate multiple of this length: 49:1.8=27.2

The ponytail’s length (equal to the distance between its uppermost point where the top gold ring ends,

and the very top of the mask) is exactly sixteen times the modular length.

The length of the mask’s face (supposed to depict the Tutankhamun’ exposed face) is exactly thirteen

times the module’s length.

The length of the false beard is also exactly thirteen times the modular length.

The width of the face proper is exactly twelve times the module’s length.

The length of the neck is six times the module’s length.

The total thickness of the front part of the collar is fifteen times the modular length.

All these ratios seem to suggest that there was a deeper meaning to this modulus’ length. Since this

study identified the number of stripes (53) as directly linked to Tutankhamun’s age (in Egyptian

Seasons), and in turn these necessitated such a modulus, given the overall dimensions of the mask, it is

concluded that this modulus is directly linked to Tutankhamun. In effect it is his signature on the mask,

or his architectonic fingerprint. It is thus expected to be found in all elements of this mask (as it was

shown above).

The nemes of the mask’s back side and its parabola.

Next, we seek to examine the mathematics of the parabola, the geometric shape seen at the back of the

mask, as an outline of the back’s side plan. We shall approach the analysis of this parabola by employing

mathematics at the level that the artist of the mask would command. In specific, we shall examine the

relationships between the parabola specifications, the modulus of the mask, and the number of stripes.

We shall understand then how he chose the particular proportions of the mask, and linked them to the

modulus and the 53 stripes. In effect, we shall seek the original mathematical problem the artist was set

up to solve.

It was pointed out that the Egyptians of that era were able to solve simple quadratic equations. They

also knew how to work with fractions. But at the back of the mask, we come across a parabola. The leap

from simple quadratic equations and fractions (the stock of mathematical knowledge of the latter part

of the 14th Century BC was not ready for the mathematics of a parabola. However, the artist of the

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Tutankhamun’s mask did produce and work with such a parabola. It is a significant achievement all by

itself. Combining this mathematical achievement with the aesthetic beauty of the stripes and the rest of

the mask’s components elevates this artifact to a unique standing in the pantheon of archeological

marvels. We thus turn to the study of this parabola, as it holds the key to decoding the mask’s

mathematics and astronomy.

As it is evident, the artist’s main concern was the division of the parabola’s perimeter, as shown in

Figure 3 (the nemes at the back side of the mask), into 53 equal in length arcs. The stripes, both blue and

gold, if extended they would meet at point where the second gold ring from the top of the ponytail is.

Let’s call this point Y(0). As noted earlier, this isn’t the focus of the parabola, which for notational

purposes will be called Y(1). Let’s also call the tip of the parabolic back side of the mask, in effect the

vertex, Y(2).

For the detailed description of a parabola (in effect the graph of a second degree equation) see Figure 4.

The pink lines indicate the basic property of a parabola, namely that the distance of any point on the

parabola, from its focus (which lies on the symmetry axis of the parabola), and a line called “directrix” is

equal. Let’s imagine the back side of the mask transposed, or inverted, so that it looks like the parabola

in Figure 4.

In the case of the parabola depicted by the back side of the mask’s nemes, the algebraic problem

becomes to locate these two elements of the parabola, namely the focus and the vertex, knowing a

point which lies on the parabola. That point, located at the lower edge of the nemes, is {(X=19.65cm),

(Y=v+27cm)}, where v is the size of the vertex.

Figure 4. A parabola and its main elements; the inverted nemes of the mask’s back side.

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Straight forward simple algebra then produces the following results, as the parabola of the mask has to

be thought of as inverted, with its highest point being at the vertex. The simple quadratic equation the

artist had to deal with is the following:

Y = aX**2 + bX + c where parameters (a, b, c) are to be estimated. (1)

We broadly know the shape of the mask, i.e., the shape of the parabola, since we know the number of

arcs from the widest point (a line defined as X*=19.65cm) to the top of the mask. The 26.5 modular

lengths (the module having been estimated at 1.786cm at the very top, but slightly declining in size

moving down – up in the above scheme – an issue we already discussed in the prior subsection) cover

the distance from the vertex to the point on the parabola set at distance of 47.33cm found from the

condition (19.65x26.5=47.33cm). Since the artist did not command either knowledge of the Pythagorean

Theorem, or calculus, he employed the approximations offered by the number of stripes needed to

derive the shape of the parabola.

Below, a rough estimate will be made mostly for illustration purposes, to derive the approximate

specifications of this parabola (focus and vertex) utilizing simple algebraic equations. We shall assume

that the average stripe at the back of the mask was 1.7cm (given our discussion earlier). Thus the 26.5

modular lengths correspond to about 45cm, which is an approximation of the parabola’s arc length. The

straight line distance between the vertex [point Y(2)] will be approximately ¾ of that, or about 33.5cm (a

result confirmed by the Pythagorean Theorem, by considering the hypotenuse of a right triangle with

sides 19.65cm and 27cm).

This approximation allows us to easily estimate the vertex and the focus, by simple application of the

Pythagorean Theorem: the straight distance from the point we just found, let’s refer to it as point R {at

coordinates (19.65, v+27)} and the unknown (but on the Y-axis) point F (the focus, with coordinates 0,

and 2v) generate two right angles with the following properties:

(19.65**2 + (27-v)**2 = (27 + v)**2 (2)

The left hand side of the equation above (2) is the result of the Pythagorean Theorem, and the whole

equation is the squared manifestation of the parabolic property just cited (that any point on the

parabola is equidistant from the focus and the directrix). Solving the above, one obtains the

approximate location of the focus in the parabola of the mask’s back side, at a point close to 3.6cm from

the top on the symmetry axis. This result would also place the directrix at around 3.6cm above the top

of the parabolic back side of the mask.

It is of interest that these lengths are about twice the modular length of the mask (1.8cm). This is then

where the point where all the stripes at the back side converge is located, two modular lengths below

the imaginary line linking the bottom edges of the nemes at the back, the points identifying the

maximum width attained in the mask. And this modular length is the length of the parabola’s focus.

Going back then to equation (1) stated above, we can now compute easily all three parameters.

Parameter c is simply the vertex (c=3.6) and twice the modulus in size. Parameter b is zero, as it can be

easily checked by elementary calculus. The slope of the parabola is given by the expression:

dY/dX = 2aX + b (3)

16

At point X=0, the slope is zero, and the above expression in (3) results in b=0. Finally, the coefficient (a)

is computed at the point of latus rectum, where X=Y=2c, and the expression becomes:

2c = a(2c)**2 + c (4)

Which results in a=1/(4c). Since c was shown to be twice the modulus, it turns out that the parameter

(a) is about equal to one eighth of the modulus (around .022cm). And this completes the mathematics of

this extraordinary artifact. From the above analysis it becomes now clear what the problem facing the

mask’s designer was: the estimation of the modulus, given the number of stripes he wanted to fit (53

being the Tutankhamun’s age in Egyptian Seasons), and the overall approximate size of the mask.

CONCLUSIONS

It is an extraordinary artifact, this mask of Tutankhamun’s sarcophagus. It is an artifact of extraordinary

beauty, not because of its gold and lapis lazuli and all the other precious and semi-precious stones it’s

decorated with and graciously carries, but because of the mathematics and astronomy and architectonic

sophistication it contains. The stripes that adorned the nemes of the mask are not only striking because

of their glittering gold and royal blue they are made, but mostly because of their elegant topology and

algebra. Working with a parabola during the late part of the Bronze Age isn’t a small feat. The solution

that the artist came up with in accommodating the 53 stripes he wanted to identify the person under

this mask, his fingerprint, is remarkable.

The combination of Astronomy, Mathematics, Architecture, Symbolism and Art the Tutankhamun mask

contains is unique for any artifact, of any Age. The emotional response it evokes even today 33 centuries

later, worldwide, attests to the extraordinary skills of the artist, designer, and creator of this unique

masterpiece.

REFERENCES

[1]. Joan Huber, “The Funerary Mask of King Tutankhamun” in:

https://www.academia.edu/7327556/The_Funerary_Mask_of_King_Tutankhamun

[2]. http://storyofmathematics.com/egyptian.html

[3]. http://starteachastronomy.com/egyptian.html

[4]. http://www.historyembalmed.org/ancient-egyptians/ancient-egyptian-calendar.htm

[5]. http://www.ancientegyptonline.co.uk/thegods.html

[6] Marshall Clagett, 1999, Ancient Egyptian Science, A Source Book, in Volume Three, Ancient Egyptian

Mathematics, American Philosophical Society, Philadelphia, available also here:

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https://books.google.com/books?id=8c10QYoGa4UC&printsec=frontcover&source=gbs_ge_summary_r

&cad=0#v=onepage&q&f=false

The author, Dimitrios S. Dendrinos retains all legal rights to this paper. No part of this paper can be

reproduced in any form without the author’s explicit and written consent.