Stonehenge

This work is based on the precise measurements given by Alexander Thom in his paper "Stonehenge" published by the Journal for the History of Astronomy (1974). Thom has also measured other megalithic sites in England and has found a unit of length called "rod"  equal to 2.5 megalithic yards (my) and approximately 6.803 ft.

According to Thom. the main circle of Stonehenge, the sarsen ring, consists of 30 large, upright stones. The inner faces of these stones is flat and polished and their width near the ground is 1 rod. The spaces between them are 1/2 rod, so the inner circcumference is 45 rods (30 x 1.5). The outer faces are rough and most of them rugged, but the mean thickness is 0.48 rods and the circumference 48 rods. These stones were capped by a complete ring of lintels that were cut to the curve of the circle and were all at the same level. Inside this sarsen circle there are three other rings of stones, the Bluestones and the Trilithons. On the outside, there are the Z, the Y and the Aubrey holes. Beyond the Aubrey holes is a ditch that surrounds the monument.

Thom writes that the Z and Y holes are not perfect circles but spirals with radii about 9 to 9.5 rods and 12.5 to 13 rods, respectively. The Aubrey holes have a radius of 141.80 ft and a circumference of 891.0 ft, almost precisely 131 rods. He adds that "if we assume that the intension was to make the circumference exactly 131 rods then we obtain a value for the rod of 6.802 ft which can be compared with the value found in Carnac of 6.803 ft at Le Menec and 6.808 at Kermario". On this circle there are "two so-called stations each of which consisted of a stone in the middle of a mound, the whole being surrounded by a ditch. The rectangle is completed by two station stones; both are still to be seen, one upright and one almost prostrate. There are idications in the underlying chalk that two other stones existed between the Aubrey circle and the bank".

It is obvious that the precise value of the "rod" is not well-known but it is approximately 6.803 ft. If the Aubrey holes have a mean radius of 141.80 ft, the circumference is 890.9557 ft (not 891.0). This means that if it was equal to 131 rods, the rod is equal to 6.8012 ft (not 6.802).

The sarsen ring
If the circumference of this circle is equal to 45 rods, the rod is 8 degrees, the spaces 4 degrees and the radius 45/2π rods. And if the value of the rod - according to Thom - is about 6.803 ft (2.07355 m), then 1 degree is about 0.2591943 m. But (π/2)-1 MC =
0.259173129 m! Thus,


1 rod = 4(π-2) MC = 2.073385 m
                               = 6.80244 ft


We also observe that the arc between the centers of two stones is 1.5 rods, or 6(π-2) ΜC = 3.110077 m
(φ^4 ΜC = 3.11216 m).

I have already mentioned that:

1. The height and the inside width of the Gate of the Lions in Mycenae is 6(π-2) MC (192 d or 3.11 m).
2. The second stone of the second row in the entrance of the Treasury of Atreus (south wall) is 4(π-4) MC (128 d = 1 rod). Also, the height of the first three rows on the same wall is 1 rod.
3. The length of this entrance (south wall) is 10(π-2) MC (319.646 d or 2.5 rods).
4. The width of the four doors in the palace of Tiryns is 4(π-2) MC (1 rod).
5. The diameter of the altar in front of this palace is 4(π-2) MC (1 rod).

The geometry of Stonehenge

(Using a ruler and a pair of compasses only).



Suppose that we draw a circle of radius 1. We inscribe this circle in the square ABCD and we bring the diagonals and the perpendicular lines in the middle. Using the four corners we write quarter circles of radius 2. Thus, we get the rectangle abcd and the points m, s, t and f. This is the basic geometry.

The rectangle abcd is about the same as the one in Stonehenge (formed by the "stations" in the Aubrey holes) and the stylobates of the Parthenon. We observe that the Y holes are inscribed in the quarter circles and the Z holes in the square formed by m, t, and their perpendicular lines on ab. Thus, if the radius of the Aubrey holes KB is 141.80 ft = 20.8437 rods, the radius Km of the Y holes is 12.21 rods and the radius of the Z holes is 8.63 rods. (The difference that exists is small).

PARTHENON

Measurements: 11-15 October 2002


The architecture of the Parthenon

The "Old Parthenon" on the Acropolis of Athens, made of poros stone, had been destroyed by the Persians in 480 BC. Thirty three years later, in 447 BC, Pericles ordered the construction of the new Parthenon, a Doric order, peripteral temple made of white Pentelic marble. The architects were Iktinos and Kallikrates and the sculptor, who supervised the work and the decoration, was Pheidias. Pheidias himself made the chryselephantine (gold and ivory) statue of Athena Parthenos that was standing on a pedestal inside the temple. Her head reached the roof of the sekos (cella) which was 12.45 m high.

In this photo, taken on October 12, 2002, we see that the "new" Parthenon was built almost exactly on the pedestal of the "Old Parthenon". The difference is about 1 m to the north. (This is the south side - looking east).

                                                                                  



The pedestal consists of a small base and three steps. The third step, where the outer columns of the peristyle stand, is called stylobates. There are 8 fluted columns in the narrow sides and 17 in the long sides. Thus, the total number of columns around the temple is 46, or 2(6+17) = 2 x 23. (*The number 23 is the arithmetic value of the Greek words "Η ΘΕΑ" (the goddess) if we add the numbers that correspond to each letter - e.g. Η=8, Θ=9, Ε=5, Α=1). In general, if the number of the columns on the narrow sides of an ancient Greek temple is α, then the number of the columns on the long sides is 2α+1 (twice the first number plus one).

Each column consists of 10 spondyloi (round pieces of marble put one on top of the other) and a capital (11 pieces). The total height of the columns above the stylobates is 23 MC (10.4433 m - 23 = Η ΘΕΑ). Above the capitals are the epistylia (= on the columns) that connect the columns. Above these long stones are the metopes serarated by the triglyphs. There are 92 metopes around the Parthenon, 14 in the narrow sides and 32 along the long sides. Now, the arithmetic value of the name ΑΘΗΝΑ (Athena) is 69 (Α=1 + Θ=9 + Η=8 + Ν=50 + Α=1).  Thus, the words Η ΘΕΑ ΑΘΗΝΑ (the goddess Athena) are equal to 23 + 69 = 92. Also, 69 is 3 times 23 and 92 is 4 times 23.

The height of the epistylia and the band with the metopes and triglyphs is 6 MC, and the height of the eaves and the pediment is 11 MC. Therfore, the total height of the Parthenon from stylobates is exactly 40 MC (23 + 6 + 11 = 40 MC = 18.162 m).

The entrances were on the east (main) and the west sides. After the first outer columns of the peristyle, there are two more steps and six smaller columns on the top of them and in front of the sekos (cella). The first part of the sekos on the east side - where the statue of Athena was standing - is called pronaos or prodomos and the second part on the west side opisthodomos (= back room). The total length of the sekos inside the walls is 44.166 m (29.7974 for pronaos + 13,2145 for opisthodomos + 1.154 for the wall between them). The width is 42 MC (19.065 m).

The dimensions of the sekos on its "stylobates" (including the walls and the 6 columns in the front and in the back) are 59.087 x 21.715 (m). The slabs of the frieze around the walls of the sekos were about 160 m in length and 1.05 m high. They were carved in situ and depicted the Panathenaic procesion.

The dimensions of the Parthenon

The width of the small base of the pedestal around the first step is 0.103 m and its height 0.30 m. The width of each of the next two steps is 0.70 m and their height 0.512 m. The height of the stylobates is 0.552 m.

The length of the base and steps on the four sides of the Parthenon is not exactly the same because the stylobates is not a perfect rectangular. The north side is 69.617 m, the south side is 69.5615 m, the east side is 30.9066 m and the west side is 30.963 m. The average is about 69.59 m for the long sides and 30.935 m for the short sides. In order to find the dimensions of the other steps and the base, we must add 1.40 m for each step and 0.206 m for the base.

The center of each column - with the exception of the four in the corners - has been put exactly on the joints of two adjacent blocks of the stylobates, so most of my measurements are between these joints (or the centers of the columns). For the four corner columns, I measured from the corners to the center of the next column. Because of the restoration work at that time, part of the north side was covered and I was not able to measure there. However, I took one measurement of the whole side.

For comparison, John Pennethorn (1878) writes that the dimensions of the Parthenon are 228.141 ft (69.537 m) and 101.336 ft (30.8872 m). According to Anastasios Orlandos (1949), the mean length is 69.556 m and the mean width 30.9205 m.

The mean distance between the centers of the columns - except for those in the corners - is 3π MC (4.28 m). In the corners, the distance is 10π/3 MC (4.755 m). Thus, the length of the east side is 65π/3 or 68.068 MC (30.9066 m).

In ancient times, the Parthenon was called "ekatompedos neos" (100-foot temple) because the narrow sides on the stylobates were 100 ft. The long sides were 225 ft, so the ratio is 9:4. In 1984, I made the observation that if the mean circumference of the Earth is 40,030,375 m (360 degrees), then 1'' is equal to 30.8876 m. This was published in my first book "Omphalos" (Jan. 1986, p. 278). However, at that time I had not measured the Parthenon yet and I used the width we find in most books (about 30.88 m). But after my measurements in 2002, I found that this number was wrong and that the mean width is about 30.935 m. So, if we use the equatorial circumference of the Earth (40,075,161 m), 1'' is equal to 30.92 m. Is this a ...coincidence?

According to these numbers, the true dimensions in megalithic cubits are about 153,262 x 68.13 (MC). But the measurements were made along the curves of the stylobates - not in a straight line - so they are a little longer. The difference, of course, is negligible.

In the short sides, the difference of the curve  of the stylobates from the straight line between the corners AC is about 6.64 cm. In the long sides AB, the difference is 12.28 cm.

If the curves of the stylobates are arcs of circles, the radii KA are 68000/2π for the long sides and 400π^2 for the short. This means that the circumference of the first circle is 68000. But the number 68 is the width of the short sides.

 The geometry of the stylobates

We draw a circle of radius 84 MC. The number 84 is the arithmetic value for ΘΕΑ ΑΘΗΝΑ (goddess Athena). The diameter is 168 MC or approximately 17π^2 (167.8) and the circumference is 17π^3, or 527 MC.

First we inscribe this circle in a square of sides 168 and we bring the diagonals and the perpendicular lines in the middle. Each diagonal is about 238, or 14 x 17, so AK = 7 x 17 = 119. But ΠΑΡΘΕΝΩΝ / Η ΘΕΑ ΑΘΗΝΑ (Parthenon/the goddess Athena) is 1095 / 92 = 11.9.

If we use the corners A, B, C, and D and write circles with radii 168 and 84, we get the points a, b, c and d. The dimensions of the rectangle abcd is 153 x 68 MC and the ratio 9 : 4.

 

 

 Restoration of a broken capital on the floor of the sekos (October 2002).

   






























Metopes and triglyphs.



Gla

Measurements: June 2002
Photos: June 7, 2002

*See "The Homeric Hyle" for new color diagrams and corrections.


The hill

The Gla hill is situated about 2.0 km SE of the modern village Kastro at the NE shore of the lake Kephisis (later Kopais) in Boeotia. The lake was drained around 1900, so the hill rises 25-40 m above the plain. Its shape is like a key or a pear and is about 850 m (West to East) amd 570 m (north to South).

The "Mycenaean" fortification around the hill is about 2650 m long. The polygonal walls are 4π MC (5.70 m) thick and have been built with large stones (now broken). In some places there were towers so the thickness there is greater.

The name of this important prehistoric city is unknown. The archaeologist Noack thought that it was the Homeric Arne. However, according to Stephanos Byzantios, Arne was the prehistoric name of Chaeronia, a city on the west side of this lake, west of Orchomenos.

Homer (Iliad 2, 500) mentions the most important Boeotian cities but none of them -except Hyle (Ύλη) fits in the NE end of the lake Kephisis. In Iliad (5, 708) he informs us that the hero Oresbios was living in the city Hyle next to the lake Kephisis. Although there is no proof, I believe that this acropolis is Hyle.

The first excavations on the hill were made by de Ridder (1894) and later by Threpsiades (1955-60) and Iacobides (1980-86). The large "Mycenaean" palace is built on the highest point and has two L-shaped wings with many rooms, corridors and sewers. The northern side is built over a precipice about 35 m above the plain.

The southern Gate

The walls of the Gate (ΑΔ and ΒΓ) are not exactly parallel, so the angle ΑΘΒ is not a right angle. AB is equal to the width of the lintel of the Treasury of Atreus in Mycenae.

There are two rooms for the guards on each side of the Gate. Both of them have the same width but the length of the one on the south side is twice its width.

General plan of the palace

The first part on the eastern side

The second part on the eastern side

The third part on the northern side

The north wall of this wing is built over a precipice about 35 m above the plain. There is a narrow path 1.5 m in width between the wall and the precipice, so it is easy to cross this side from H to K and Λ. The height of the wall above this path is 5-6 m. The NE corner at H is broken and most of the blocks have fallen down.

Although most thresholds are rectangular and their width is equal to the width of the walls, some of them have one of their sides rounded after the lines. The rounded part that is protruding is about 1 d lower than the rectangular part.

The fourth part

There are three thresholds between prodomos and domos without walls.

The perimeter of domos is 25π.

Tiryns

Measurements: May 19, 2002  - "Metron Ariston" (2003, p. 39).
(See also "The acropolis of Tiryns" posted on 11 May 2012).
Tiryns is a very old city situated about 7 km East of Argos. It was founded in the precataclysmic period by the Pelasgian hero and king Tiryns, a descendant "son" of Argos, "son" of Zeus and Niobe. Later Proetus (or Proitos) became a king of Tiryns and put the Cyclops to build the cyclopean (megalithic) fortification around the acropolis. The hill is about 300 x 100 m.

The Megaron on the acropolis is rectangular and consists of three rooms:  aethousa (hall), prodomos (meaning "before domos" and domos (the king's room). Outside the walls the length is 1599 d (25.93 m) and the width 773 d (12.535 m). Inside the length is 1504 d, but the width is 611 d in the north wall of the domos and 597 d in the beggining of aethousa. (* There is an important reason for all these inequalities in ancient architecture).


We observe that they have used many integral numbers in MC.  The length of the domos is 26 MC (11.804 m) and is divided by the columns into three sections of 7, 12 and 7 MC. The doorposts are 2 MC and the openings 128 d (see the Treasury of Atreus, south wall). Also, the two diagonal lines meet at M (middle) at the beggining of domos.

The Treasury of Atreus

First measurements: 24-26 April and 14 August 2002.

New measurements: 17 November 2007, 3 and 10 February 2008 and 2 March 2008.


1. Plan of the Treasury of Atreus.

There are 21 stones on the south wall of the dromos and 17 on the north (first row on the ground). Their length is given in d (ancient inches). The total length of the south wall is 2216 d and of the north wall 2206 d. So, the average in the middle (axis) is 2211 d, or 8π^2 MC (35.85 m).

The 15th stone on the north wall is huge and its length (387 d = 6.276 m) is about equal to the width of the dromos. This width is 390 d but in front of the entrance becomes 384 d (2 x 192 d). Now, 192 d is the height of the Gate of the Lions and its width inside (φ^4 MC). On the other side of  dromos (south wall) there are two other stones facing this one with a total length of 391 d.

2. The geometry of the facade projeted on the tholos.

OB is the radius of the tholos (450.5 d), equal to OF and OE.

The width of the entrance in the lintel is twice its thickness (height NM = 8/3 MC = 1.211 m)). The radius of the tholos (in this case OB) is equal to approximately 6 NM. The height of the tholos OH is 11 NM, or 88/3 MC (13.32 m). Some other important observations are given on the table on the left of this drawing.

3. Section of the Treasury


The north wall of the entrance consists of 23 large stones in nine rows. The length and the height of them is given in daktyloi (d). The total lenght of the north wall on the ground is 322.5 d (5.23 m) and the height (A' N) is 12 MC (5.45 m). The height of the tholos is 88/3 MC (13.32 m) and the upper part of the lintel (M) is in the middle (44/3 MC). The triangle above the lintel begins at E and the height
A" E is equal to the radius of the tholos on the ground (Ro = 16.09 MC = 7.306 m). The gigantic lintel occupies two rows of rings inside the tholos (10th and 11th). There are 33 rings and the lid on the top (at H) is the 34th (2 x 17). (The number 17 is very important).

The Megalithic Cubit

Copyright 2002 by Athanasios G. Angelopoulos
Published in 2003 in the book METRON ARISTON
ISBN 9608286069


In April 2002, I made the first precise measurements in the Treasury of Atreus and the Gate of the Lions in Mycenae, because I was interested in the unit of length that was used by the prehistoric architects. Some ancient Greek units of length that were given by archaeologists were not accurate (or even wrong) and did not agree with the measurements that existed at that time. Also, these measurements were very few and in most cases wrong, except those of Anastasios Orlandos for the Parthenon. For example, some books and encyclopedias wrote that the ancient Greek units of length were a foot of 0.3083 m (16 daktyloi = inches of 0.0193 m), a cubit of 0.4624 m (24 daktyloi) etc. But if an inch was 0.0193 m, then the foot was 0.3088 m and the cubit 0.4632 m.

After the measurements of the first three days (24-27 April), I found that the unit of length used by the Mycenaean architects was a cubit of 0.454 m. I called this unit "the Mycenaean Cubit". However, when in the following days and months I measured some other important monuments (including Parthenon), I found that this unit had been used by the Pelasgians long time before the Mycenaeans. Therefore, I changed the name and called it "the Megalithic Cubit" (MC). The MC was also used in historical times by all initiated architects.

Definition: The Megalithic Cubit is defined as 1: 14,000,000 of the Earth's polar radius (or 1: 28,000,000 of the Earth's polar diameter. Since the radius is 6,356,775 m, the MC is equal to 0.454055 m. It is also subdivided into 28 daktyloi, inches) of 0,01622 m). Now, this number (0.454) is a lot different from the one mentioned above (0.4624). Also, the reason they chose 14 or 28 is because these numbers are related to the period of the Moon. For example, there were 14 circles above the entrance of the Treasury of Atreus (not 16 as some artists draw) and 28 stones in the first row around the tholos.

Obviously, this means that many thousands of years ago, the Pelasgians and Mycenaeans knew precisely the size of the Earth and had advanced knowledge of geometry and mathematics. The Greek word γεωμετρία (geometria > geometry) means exactly "measurement of the earth".

A few examples:

1. The width of the entrance of the Treasury of Atreus is 6 MC (2.724 m). I made about ten measurements along the entrance and the width was 2.725 +/- 0.005 m. The height is 12 MC, or twice the width.

2. The first stone on the south wall of the same entrance is 5 MC (2.27 m).

3. The width of the Gate of the Lions on the threshold (and that of the similar Gate in Tiryns) is 2π MC (2.853 m). The width of the Gate of the Lions in the lintel is 6 MC - the same as the width of the entrance of the Treasury of Atreus.

4. The width of the doorposts in the front of this Gate is π/2 MC (0.713 ); the length of the lintel is π^2 MC (4.481 m).

5. The dimensions of the Pyramid of Akrisios and Proitos at Hellenikon, Argos, are 33 x 28 MC (14.984 x 12.714 m). The large square chamber inside the pyramid is 15 x 15 MC (6.81 x 6.81 m).

6. The dimensions of the Parthenon on the stylobates are 68 x 153 MC (30.876 x 69.47 m).

Ancient architects did not use only the number π, but also the numbers φ (golden number = 1.618034), the natural logarithm e (2.71828) and many square roots.