Amphitheatre Pompeii The Amphitheatre of Pompeii is the oldest surviving Roman amphitheatre. It is situated in the ancient Roman city of Pompeii and was buried by the eruption of Vesuvius in 79 ad. It is located to the east of the city and the edge of the eruption. It measures 135m long and 104m wide, the arena's pit is measured at 6m below ground level and was built in 70 bc. Coordinates 40.751827,14.494962 Measurements 135m x 104m (341ft x 443ft) 76.2m x 38.1m ( 250ft x 125ft) Angled at 60° towards northwest Area π×341×443= 474578 π×250×125= 98175 474578× 98175= 46591695150 46591695150÷223= 208931368 π×341×450= 482077 482077× 98175= 47327909475 47327909475÷223= 212232777 Roman Units The Roman cubit (cubitum) was 1½ Roman feet (17.48 inches) Five Roman feet made the pace (passus), 1.48 metres (58.23 inches). If considering a standardised unit, the Roman foot was probably used here, making the above whole units in 5's as if a passus. If applying again, the amph
Pyramid - Construction Ratios
With the construction of the Giza Pyramids, was there anything more to it than just a build ratio of 5.5×7 giving different slopes of between 51°/53°?
It has been shown that the connection for pi and golden ratio in a circle is volumetric with a ratio of √φ. The connection to π and φ in a standing wave (this is included as it connected to pi÷5=36° inside a circle).
2×cos(π÷5)=φ
φ=√((5+√5)÷(5-√5))
Standard Measurements
2 cubits as a length are equal to 1.046m, a standard unit of measure (φ÷√2) has a ratio of 16.5:1 with the φ.
It has been shown that the connection for pi and golden ratio in a circle is volumetric with a ratio of √φ. The connection to π and φ in a standing wave (this is included as it connected to pi÷5=36° inside a circle).
2×cos(π÷5)=φ
φ=√((5+√5)÷(5-√5))
Standard Measurements
2 cubits as a length are equal to 1.046m, a standard unit of measure (φ÷√2) has a ratio of 16.5:1 with the φ.
1.046−1.144=0.098
1.618÷0.098=16.5
The φ ratio, π and cubit connections for a φ circle of radius 1.618.
Area= (φ÷(2 cubits-(φ÷√2)))÷2=8.22
Area of two circles plus separation distance
2((φ÷(2 cubits-(φ÷√2)))÷2)+(3÷60)=16.5
1.618÷0.098=16.5
The φ ratio, π and cubit connections for a φ circle of radius 1.618.
Area= (φ÷(2 cubits-(φ÷√2)))÷2=8.22
Area of two circles plus separation distance
2((φ÷(2 cubits-(φ÷√2)))÷2)+(3÷60)=16.5
A ⅓ of a 108° shape has a value of one.
(πφ÷5)−(1÷60)=1
(πφ÷5)−(1÷60)=1
Circumference of a circle is 2πr but including the φ is,
Circumference= 1÷(π-φ²)×5.32
Pyramids
In a triangle (a pyramid) with sides 1.618 and slope 51.34° (Menkaure's pyramid), the area is approximately √φ and perimeter 10 cubits. If this was 1.618m, then the error off would be 1.91 cms, and the base would be 2m.
Increasing the slope in factors of ×10 would increase the area by 10².
At slope=φ the area=√φ and the base=2
In a triangle with a 51.867° slope (Khufu's pyramid), the ratio gives a difference of 3182 or 100 × 31.82 number.
The slope of 1.618 is much nearer to an area of √1.618 and perimeter of 10 cubits.
The √1.618×½base is equal to the height.
√(1.618)×115.15=146.47m
Also, the best angle to keep a φ is 51.827°, 0.04° less than it has been measured at, at the slope length it is a difference of 13 cms.
A cubit to a pyramid ratio, this is Khufu's pyramid,
(16÷5−0.523)+(16÷5+0.523)=6.4
Circumference= 1÷(π-φ²)×5.32
Pyramids
In a triangle (a pyramid) with sides 1.618 and slope 51.34° (Menkaure's pyramid), the area is approximately √φ and perimeter 10 cubits. If this was 1.618m, then the error off would be 1.91 cms, and the base would be 2m.
Increasing the slope in factors of ×10 would increase the area by 10².
At slope=φ the area=√φ and the base=2
In a triangle with a 51.867° slope (Khufu's pyramid), the ratio gives a difference of 3182 or 100 × 31.82 number.
The slope of 1.618 is much nearer to an area of √1.618 and perimeter of 10 cubits.
The √1.618×½base is equal to the height.
√(1.618)×115.15=146.47m
Also, the best angle to keep a φ is 51.827°, 0.04° less than it has been measured at, at the slope length it is a difference of 13 cms.
A cubit to a pyramid ratio, this is Khufu's pyramid,
(16÷5−0.523)+(16÷5+0.523)=6.4
With
16÷5=3.2
3.2+3.2=6.4
From Nabta Playa, the numbers as sums produce two numbers, 9.128 and 2.68. These applied in the same ratio equals 1.43. (9.128+√(2.68))÷(9.128−√(2.68))=1.43
These then connect in the ratio.
The value of (31.82÷100)×π=1 this equates to the % size of the pyramid shafts to the constellation of Orion. (√((5+√5)÷(5−√5)))÷(π×0.36)=1.43
(√((5+√5)÷(5−√5)))×(31.82)=0.05
In construction, these numbers would have been applied in the following ways. Applying these factors to pyramids, the lower blocks on Khufu's pyramid are 1×1×2.5 in metres, the CoG=1.43634m with a Perimeter of= 3.28284.
For a 1×1 block in metres, if the 2.3 metric ton blocks (1000kg) would actually be 99 cms and 5.6 cms off from 2 cubits, then the measurements used for density and equation would be,
2.323 sandstone
2.692 granite
Volume=mass÷density
The construction process was not limited to stones being of the same dimensions.
16÷5=3.2
3.2+3.2=6.4
From Nabta Playa, the numbers as sums produce two numbers, 9.128 and 2.68. These applied in the same ratio equals 1.43. (9.128+√(2.68))÷(9.128−√(2.68))=1.43
These then connect in the ratio.
The value of (31.82÷100)×π=1 this equates to the % size of the pyramid shafts to the constellation of Orion. (√((5+√5)÷(5−√5)))÷(π×0.36)=1.43
(√((5+√5)÷(5−√5)))×(31.82)=0.05
In construction, these numbers would have been applied in the following ways. Applying these factors to pyramids, the lower blocks on Khufu's pyramid are 1×1×2.5 in metres, the CoG=1.43634m with a Perimeter of= 3.28284.
For a 1×1 block in metres, if the 2.3 metric ton blocks (1000kg) would actually be 99 cms and 5.6 cms off from 2 cubits, then the measurements used for density and equation would be,
2.323 sandstone
2.692 granite
Volume=mass÷density
The construction process was not limited to stones being of the same dimensions.
In building two different ways were used, (1) a foreman would use a predetermined measure length for choosing stones that arrived from quarries, (2) the height of stones were predetermined at quarries and measured in ratios. The casing stones and sandstones for the outside would adhere to smaller tolerances. The slopes would later be carved to an angle of 51-53° with similar coloured mortar being used in binding.
There are various measurements for the casing stones, but using the numbers (link below) 685mm x depth 860mm x height 546mm x length of sloping side 612mm.
For these stones with a slope of 612mm, can be considered as a block and triangle.
There are various measurements for the casing stones, but using the numbers (link below) 685mm x depth 860mm x height 546mm x length of sloping side 612mm.
For these stones with a slope of 612mm, can be considered as a block and triangle.
The block is 0.476m×0.476m and a triangle of 0.476m×0.385m×0.612m.
This gives a side area of 1.14m against a slope of 51°, although with a height of 0.546m, the slope increases to 54°.
Width/height 685×546
Depth 860
Slope 612
Area (0.685×0.546)×0.532×10=1.99
Depth 0.861×0.532=1.618
Slope 0.612×0.532×10=3.2
Khufu's pyramid casing stones are said to be in ratios of 1×1.33×2.66 instead of 1×1×2.
This gives a side area of 1.14m against a slope of 51°, although with a height of 0.546m, the slope increases to 54°.
Width/height 685×546
Depth 860
Slope 612
Area (0.685×0.546)×0.532×10=1.99
Depth 0.861×0.532=1.618
Slope 0.612×0.532×10=3.2
Khufu's pyramid casing stones are said to be in ratios of 1×1.33×2.66 instead of 1×1×2.
The volume multipled by density gives weight, but divided gives one plus a cubit.
3.5378×2.323=8.22 tons
3.5378÷2.323=1.523
Using exact measurements for Khafre's pyramid for slope and construction with measurements in cubits, then using an overall ratio of 7 by 5.5, gives an angle of 51.843°.
3.5378÷2.323=1.523
Using exact measurements for Khafre's pyramid for slope and construction with measurements in cubits, then using an overall ratio of 7 by 5.5, gives an angle of 51.843°.
This shows that although proportions internally were 7 by 5.5 with the slope of the steps, the casing stones could vary slightly, and the difference of the final angle is very near to √φ.
Slope= 342.97322
Height= 274.3786
Base= 205.7839
Inclination=53.13°
Considering that these are the current day measurements with only a few casing stones left, the different proportions of 10.46/7 (5.23/7) could have been a possibility in construction.
Slope= 342.97322
Height= 274.3786
Base= 205.7839
Inclination=53.13°
Considering that these are the current day measurements with only a few casing stones left, the different proportions of 10.46/7 (5.23/7) could have been a possibility in construction.
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