प्य्रामिद ऎंड sphere

Pyramid and Spheres
Pyramids are polyhedra because they are made up of plane faces. Spheres are not polyhedra because they are curved.

Pyramids
If a line that is fixed at a point, called the vertex, is moved around the perimeter of a polygon, then a pyramid is formed.

The following solid is pyramid.



Pyramids are named after their base. So, the pyramid shown above is an example of a rectangular pyramid.

Some other pyramids are shown below.



A pyramid is said to be right when its vertex is directly above the centre of the base. The cross-sections parallel to the base are the same shape but have different sizes.




The net of the above pyramid consists of:

a rectangular base; and
four triangular faces


Note:
The cross-sections parallel to the base have the same shape as the base but different sizes.
The net of a pyramid consists of a base polygon and a number of triangular faces.
A regular pyramid has a base that is a regular polygon and has faces which are isosceles triangles.

Spheres
If a semicircle is revolved about its diameter, then a sphere is formed.

The following solid is a sphere.



The cross-sections of a sphere are circles. The cross-sections that pass through the centre of the sphere are called great circles whilst other cross-sections are called small circles.



A hemisphere is a half sphere.











www.piyushdadriwalamaths.co.in

pyramid

The curious synchronicities of the pyramids



The use of the cross and plumb line reveal several mathematical designs that co ordinate with time speed and distance of the motions of the earth, sun and moon against the zodiac thereby implying that the designer of the complex understood astronomical mathematics not considered to have appeared historically until several thousand years later.

Click on thumbnails

The Giza Complex near Cairo in Egypt offers many mysteries to solve and is laid out in a complex of 3 larger pyramids and six small ones amounting to a total of 9



Essentially, the complex is located at 30° north Latitude

Did the architect astronomer design the pyramids to reflect their own latitude?



In essence the reducing circumference of a small circle as you progress from the south equator to the north pole south to north is 240 nautical miles per degree

A degree is equal to 60 nautical miles at the equator and 1 nautical mile = 1 arc minute of time/speed/ distance

Speed of the sun over a great circle is 900 nautical miles per hour and is directly related to the speed of earth rotation

360° x 60 minutes = 21600 Nautical miles earth circumference at the equator this number is directly related to 2160 years in an astrological age because of the symbiotic geometry

The pyramid is at 30 degrees North

30 x 240 = 7200 - 21600 = 14400 nautical miles earth circumference at 30 degrees latitude

Each face of the pyramid is 180 degrees

180 x 4 = 720 take the seen and the unseen as in the squaring of the circle exercise

720 x 2 = 1440 x 10 equals 30 latitude at Giza of a small circle circumference of 14,440 Nautical miles.


Did the architect astronomer show in the pyramids a knowledge of Precession?

Motion of the Precession of the Equinoxes

Multiply by three main pyramids 720 X 3 = 2160 an astrological age

9 pyramids in the complex X 720 x 4 = 25920 years or 12 Zodiac signs

Did the Astronomer Architect measure the motion of the moon?

25920 - 21600 = 4320 x 4 = 17280 years

The moon is retrograde and only travels at 720 miles per hour and is related to the base line of one side of the great pyramid as 720 feet

in 24 hours it travels 17280 miles on a great circle

1 hour for the sun in distance of rotation 900 Sea Miles

1 hour for the moon in distance 720 sea miles

900 - 720 = 180° = or 1 face of the pyramid in degrees

52° x 52° x 76 °= 180° squares the circle and incorporates the Golden Section and perfect spherical geometry relating to

As Above so below.

Did the Astronomer Architect need an instrument capable of measuring degrees as a sidereal observation?

Measuring the path of a celestial object with a cross and plumb line



Are the Dixon Relics the missing Architects Instrument?





Dixon relics assembled

Was the Pyramid a clock?

On the matter of time
The earth spins at 900 nautical miles every hour (Unit of Horus) or 60 minutes of arc

60 ÷ 900 = 0.0666 minutes of time
1 minute of arc at any latitude = 0.0666 minutes of time

24 hours x 60 minutes = 1440 minutes

24 hours x 60 minutes x 60 seconds = 86,400 seconds ÷ 4 = 21600

1440 ÷ 21600 = 0.0666

Earth measurement is the secret of time and that is governed by cycles of life, death and rebirth
These cycles were measured by observing the constellation Draconis in the North as it spins known as Shiva, The Serpent, The Lord of the Dance, the worm, the dragon, the millstone and Reret.
There is much more of course, but it was the 1872 Dixon Relics that allowed Crichton E M Miller to re construct the astronomer, navigator architects instrument to rediscover this curiosity of number and synchronicity in the Giza Complex

piyushdadriwala
www.piyushdadriwalamaths.co.in

pyramids

Geometry > Solid Geometry > Polyhedra > Pyramids
Recreational Mathematics > Interactive Entries > LiveGraphics3D Applets


Pyramid








A pyramid is a polyhedron with one face (known as the "base") a polygon and all the other faces triangles meeting at a common polygon vertex (known as the "apex"). A right pyramid is a pyramid for which the line joining the centroid of the base and the apex is perpendicular to the base. A regular pyramid is a right pyramid whose base is a regular polygon. An -gonal regular pyramid (denoted ) having equilateral triangles as sides is possible only for , 4, 5. These correspond to the tetrahedron, square pyramid, and pentagonal pyramid, respectively.

A pyramid is self-dual, corresponding to the fact that a pyramid's skeleton (a wheel graph) is a self-dual graph.

An arbitrary pyramid has a single cross-sectional shape whose lengths scale linearly with height. Therefore, the area of a cross section scales quadratically with height, decreasing from at the base () to 0 at the apex (assumed to lie at a height ). The area at a height above the base is therefore given by

(1)

As a result, the volume of a pyramid, regardless of base shape or position of the apex relative to the base, is given by

(2)
(3)
(4)

Note that this formula also holds for the cone, elliptic cone, etc.

The volume of a pyramid whose base is a regular -sided polygon with side is therefore

(5)

Expressing in terms of the circumradius of the base gives

(6)

(Lo Bello 1988, Gearhart and Schulz 1990).

The geometric centroid is the same as for the cone, given by

(7)

The lateral surface area of a pyramid is

(8)

where is the slant height and is the base perimeter.

Joining two pyramids together at their bases gives a dipyramid, also called a bipyramid.


piyushdadriwala
www.piyushdadriwalamaths.co.in