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Abstract
Here are the main questions of Part I, Hermetic Geometry :
 
  		• What is a pyramid as a geometrical figure ?
     
  		• What is the new parameter A of a pyramid ?
     
    Some projections of the classical pyramid, which has an exact square as its base, are examined.
    New mathematical formulae for calculation of internal and external angles of a pyramid and their interrelation are shown in this section.
    Parameter A  is calculated, representing the maximum difference between the angle of the lateral side of the pyramid in relation to its base and the angle of the lateral edge of the pyramid in relation to its base.
    It has been found that parameter A equals 9.879 angular degrees for any given classical pyramid.


    Here are the main questions of Part II, The Sentinel of a Forgotten Legacy (Giza) :
     
  		• Why does the complex of Giza stand near the 30 degrees latitude of the northern hemisphere ?
     
  		• Why do the pyramids of Giza have so exact orientation in relation to the cardinal points ?
     
  		• Why did ancient builders construct the pyramids of Giza with an angle of the lateral slope close to 52 degrees ?
     
    It is known the Great Pyramid, the Second Pyramid and the Third Pyramid have the mean angle of the own lateral slope as 51°51' , 53°10' and 51°10' respectively.

    It demonstrated the important relationship between the meanings of the angles of the lateral slopes of the three pyramids in Giza and the altitude of the Sun at it's moment of crossing the direction of line due "east - west" by day on the summer solstice (21 June, modern Gregorian calendar).

    It was found that the Sun has an altitude near 52°40'  at the present time on the latitude of Giza  by day on the summer solstice when the Sun crosses a line due east (with an azimuth of 90 degrees) in the morning and also when the Sun crosses the line due west (with an azimuth of 270 degrees) in the evening respectively.

    When the Sun has the altitude near 52°40' on a line due east in the morning, the Sun’s rays strike downwards parallel to the western faces of the pyramids, and hence the pyramid's shadows vanish on the western faces at this moment.
    The pyramid's shadows appear on the eastern slopes of the pyramids after Sun's transit on a line due west in the evening when altitude of the Sun will less of 52°40' on horizon.

    The Giza pyramids had perfect polished limestone surfaces many years ago with amazingly exact orientation of their bases in relation to the cardinal points "north - south"  & "east - west".
    Hence the pyramids had a specific optical phenomenon as "vanish & appear" of shadows on it's western & eastern facets which possibly indicated the day of the summer solstice, an important day in the Egyptian year, when the Nile had a tendency to have it’s annual summer flood.  

    An assumption is made that the ancient builders had specially designed the pyramids with the angle of inclination near 52 degrees, specially gave so exact an orientation of the pyramids' bases, and also exactly positioned the Giza complex at a 30 degrees northern latitude for the observation of the summer solstice,  it constantly repeats every year and hence an exact determination of the solar year length.


    Here is the main question of Part III, The Gates of Time :
     
  		• When was the Giza Complex built ?
     
    The pyramids of Giza have various angles of inclination of their own slopes and hence an assumption is made about the various times of construction of these pyramids because the altitude of the Sun, when it crosses the line due east & west, depends from a meaning of the obliquity of the ecliptic.

    The obliquity of the ecliptic (eps), or inclination of the Earth's axis of rotation, is the angle between the ecliptic and the celestial equator.
    There are the mean and the true obliquity, the first term indicates that the correction for nutation is not taken into account.
    It is known empirical formula of cyclic variation of the mean obliquity of the ecliptic during long period of time which was first constructed by Newcomb (1906) and hence it is possible to find the possible time of construction of the monuments.
    The main period is around 41,000 years and maximum amplitude is around +/- 1.3 degrees.
    This method was first used by Lockyer for determination age of Stonehenge (1909), and then, Posnansky with Müller used it for investigation of age Tiahuanaco (1930).
    Current work uses modern numerical expressions for the mean obliquity of the ecliptic by two models of Edwardsson (2002) and Bretagnon (2003).

    There are three possible positions of the Sun on the eastern & western horizon (refer to angles of inclination of pyramids' slopes) when it crosses the line due "east - west" :
      1) altitude of the upper limb of the sun's disk equals angles of pyramids' inclination slopes ;
      2) altitude of the centre of the sun's disk equals angles of pyramids' inclination slopes ;
      3) altitude of the lower limb of the sun's disk equals angles of pyramids' inclination slopes ;
     
    The approximate time of construction of the Giza general monuments for second position (see above) of the Sun is found .

    The Great Pyramid was built :

    about 19,800 years BP (Edwardsson's model) ;
    about 19,400 years BP (Bretagnon's model) ;

    eps (I) = 23°.154

    The Second Pyramid was built :

    about 17,100 years BP (Edwardsson's model) ;
    about 17,000 years BP (Bretagnon's model) ;

    eps (II) = 23°.591

    The Third Pyramid was built :

    about 21,200 years BP (Edwardsson's model) ;
    about 20,500 years BP (Bretagnon's model) ;

    eps (III) = 22°.924
    Here are the main questions of Part IV, The Solar Cycle of the Egyptian Calendar. Initial Point of Return. :
     
  		• The solar cycle of the Egyptian calendar
     
  		• What is the meaning of the strange and enigmatic' phrase of Herodotus  "The Sun, however, had within this period of time, on four several occasions, moved from his wonted course, twice rising where he now sets, and twice setting where he now rises" ?
     
    The possible origin of the Egyptian calendar (three seasons of 120 days each one plus 5 additional days) is shown .

    There are three special points on pyramid's projection on eastern & western horizon according to next assumption :
    1) eastern horizon
      a) the projection of the left side (corner) of the pyramid's basement equals to summer solstice sunrise amplitude (with declination reference [+ eps] according to the centre of the sun's disk) ;
      b) the projection of the right side (corner) of the pyramid's basement equals to winter solstice sunrise amplitude (with declination reference [- eps] according to the centre of the sun's disk) ;
      c) the projection of the top of the pyramid equals position of the centre of the sun's disk with it's declination of [+ eps / 2] ;
    2) western horizon
      a) the projection of the left side (corner) of the pyramid's basement equals to winter solstice sunset amplitude (with declination reference [- eps] according to the centre of the sun's disk) ;
      b) the projection of the  right side (corner) of the pyramid's basement equals to summer solstice sunset amplitude (with declination reference [+ eps] according to the centre of the sun's disk) ;
      c) the projection of the top of the pyramid equals the position of the centre of the sun's disk with a declination of [+ eps / 2] ;
    Intervals of time by sun's annual movement between these positions are :
    1) eastern horizon
      a) from the projection of the left side (corner) of the pyramid's basement (with declination reference [ + eps ]) to the projection of the top of the pyramid (with declination reference [+ eps / 2]) is about 60 days ;
      b) from the projection of the top of the pyramid (with declination reference [+ eps / 2]) to the projection of the right side (corner) of the pyramid's basement (with declination reference [ - eps] ) is about 120 days ;
      c) from the projection of the right side (corner) of the pyramid's basement (with declination reference [ - eps] )  to the top of the pyramid (with declination reference [+ eps / 2]) is about 120 days ;
      d) from the projection of the top of the pyramid (with declination reference [+ eps / 2]) to the projection of the left side (corner) of the pyramid's basement (with declination reference [ + eps ])  is about 60 days ;
    There are plus 5 additional days for correction of the duration of the astronomical seasons and for completion of the annual solar cycle.
    2) western horizon
      a) from the projection of the right side (corner) of the pyramid's basement (with declination reference [ + eps ]) to the projection of the top of the pyramid (with declination reference [+ eps / 2]) is about 60 days ;
      b) from the projection of the top of the pyramid (with declination reference [+ eps / 2]) to the projection of the left side (corner) of the pyramid's basement (with declination reference [ - eps] ) is about 120 days ;
      c) from the projection of the left side (corner) of the pyramid's basement (with declination reference [ - eps] )  to the top of the pyramid (with declination reference [+ eps / 2]) is about 120 days ;
      d) from the projection of the top of the pyramid (with declination reference [+ eps / 2]) to the projection of the right side (corner) of the pyramid's basement (with declination reference [ + eps ])  is about 60 days ;
    There are plus 5 additional days for correction of the duration of the astronomical seasons and for the completion of the annual solar cycle.

    A new interpretation is given of the Herodotus' phrase :

    "The Sun, however, had within this period of time, on four several occasions, moved from his wonted course, twice rising and twice setting where he now rises and sets".

    It is known that the amplitude and altitude of the Sun on the eastern & western horizon depends upon the meaning of the obliquity of the ecliptic (eps).
    The amplitude and altitude of the Sun increase / decrease on eastern & western horizon when the obliquity of the ecliptic increase / decrease accordingly.
    For latitude 30°, summer / winter solstice sunrise / sunset amplitude (with declination reference [+/- eps]) has a variation of about 1° per 1° of variation of the obliquity of the ecliptic ; altitude of the Sun (with declination reference [+ eps/2] on line due "east & west" has a variation of about 1° per 1° of variation of the obliquity of the ecliptic too.
    Hence, the Sun can be "rising and setting" when it is situated  at three special points on pyramid's projection (see above) on eastern & western horizon.
     
     
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