Dynasty 4
Two equilateral triangles in the Giza 3-pyramid site layout that mirror the 3-star 'Winter Triangle' asterism over the western and eastern horizons.
Two equilateral triangles with ~ 500 cubit sides, define the position of Khafre's pyramid (G2) relative to Khufu's pyramid (G1) and Menkaure's pyramid (G3) relative to G2.
Centre of south side of G1 to NE corner of G2, 500 cubits (261.855 m / 0.52371m = 500cubits), angle 59° 53' (7' off 60°)
Centre of G3 north side to SW corner of G2, 498.346 cubits (260.9484m / 0.522m = 500cubits); angle = 59° 39' (21' off 60°)
Centre of G3 west side to centre of G1 east side (red line), 2000 cubits, 45° N of E
The western equilateral triangle that defines the position of G2 relative to G3 mirrors the 3-star asterism over the western horizon. (Rigel 45° south of west; Sirius and Betelgeuse, similar height above the horizon):
The eastern equilateral triangle that defines the position of G2 relative to G1 mirrors the 'Winter Triangle' asterism in the sky over the eastern horizon (Betelgeuse and Sirius vertically aligned):
A pyramid with four equilateral triangles on a square base has an angle of incline of 54.74° and an arris angle of 45°.
The distinctive 'Winter Triangle' asterism of the bright stars Betelgeuse, Sirius and Procyon has the appearance of an equilateral triangle and may have inspired the initial design of Snefru's southern pyramid. According to Petrie’s survey, the angle of incline of the lower part of the southern pyramid is uncertain as the faces are slightly convex, and stone blocks have been removed from parts of the pyramid causing uneven subsidence. 54.76° is the average of Petrie’s measurements, which suggests a pyramid having four equilateral triangle faces, an angle of incline of 54.74°, and an arris (corner) angle of 45°.
The nearest whole number seked ratio to an equilateral triangle-based pyramid has an angle of incline of 54.46° (sqd 5), and an arris (corner) angle of 44.6°. A sqd 5 pyramid seems plausible for the final design, as the 44.6° arris angle closely corresponds to the angle of incline of both Snefru's southern pyramid's small subsidiary pyramid and his northern pyramid - the designer simply took the arris angle of the southern pyramid and used it for the design of the other two pyramids. In addition to an entrance in the north face, the southern pyramid has a 64.63m long passage with an entrance in the west face that has two angles of incline: 24.284° and 30.15° (Fakhry 1974 (1961): 93), added together = 54.43° which is almost identical to the angle of incline defined by sqd 5 (54.46°)
Orion's 3-star vertical alignment on the meridian due south:
The north face of the Snefru's northern pyramid aligned with Alnitak due south c.2590 BC
The north faces of Snefru's southern pyramid aligned with Betelgeuse, the uppermost star, and possibly with Alnitak, the middle star in the Orion vertical alignment c.2630 BC
Alnitak, the middle star in the vertical alignment was 44.5° above the horizon, and closely corresponds to the 44.6° arris angle of a sqd 5 southern pyramid, the angle of incline of the small subsidiary pyramid in the southern pyramid’s complex, and also to the northern pyramid.
The upper part of the southern pyramid has a slightly shallower angle of incline defined by sqd 7 1/2 (43.025°) than the small subsidiary pyramid and the northern pyramid, which is an anomaly. Why use the same angle of incline for the small subsidiary pyramid and the northern pyramid possibly defined by the arris angle of a sqd 5 southern pyramid, and not use the same angle of incline for the top part of the southern pyramid which in appearance is similar to the other two?
Other design considerations may have taken precedence - a trade-off between geometric schemes that harmonised the design of the southern and northern dual pyramids creating a coherent whole, and afterlife concerns that connections to the stars addressed.
The southern pyramid and the northern pyramid are clearly a pair - they both have ‘Snefru appears’ as part of the name:
‘Snefru appears, the southern pyramid’ (‘Snefru appears, the dual pyramid’ is also attested), and the northern pyramid was named ‘the pyramid, Snefru appears’. They both are the same height even though they have different angles of incline and different size bases.
The 'Winter Triangle' as it appeared over the south-western horizon - Sirius and Betelgeuse horizontally aligned
The western passage has an entrance/exit 13.7m south of the centre of the western face of the pyramid. It was facing an area of the sky where the distinctive Pleiades star cluster set due west and high above it, the twin stars Pollux and Castor:
Together, the Pleiades, Castor and Pollux resemble a celestial wAs-staff - “Its primary function in funerary contexts was to ensure the continued welfare of the deceased." ('The British Museum Dictionary of Ancient Egypt' 1995: 304)
The earliest astronomical ceiling that has survived in the architectural record is in a tomb of Senenmut and has a depiction of Orion (sAH) holding a wAs-staff:
Photo, Greg Reeder
In temple reliefs, wAs-staff were used to depict pillars standing on the earth and holding up the sky:
(R.H. Wilkinson1999 (1994): 139 Fig. 86)
Netjerikhet's 'Step' pyramid north/south axis is ~4.58° east of north and ~4.35° east of north of Snefru's southern pyramid, which means the centre north/south axis line extended ~9km southwards aligns with Snefru's southern pyramid.
Snefru's northern pyramid relative to his southern pyramid is ~2km to the north and offset ~300m west.
Snefru's southern pyramid relative to his northern pyramid:
A possible reason for this dual pyramid layout is found in the eastern sky as Sirius was rising and in vertical alignment with Betelgeuse. Castor and Pollux had a similar configuration to Snefru's dual pyramid layout.
Betelgeuse and Sirius in vertical alignment over the eastern horizon:
Detail of Castor and Pollux in the northeastern sky when Betelgeuse and Sirius in vertical alignment:
Another possibility has Sirius and Dubhe simultaneously on the north/south meridian with Sirius and Procyon aligned in a similar configuration to Snefru's dual pyramid layout.
Sirius on the meridian due south:
Dubhe simultaneously on the meridian due north:
Snefru's southern pyramid relative to his northern pyramid and to Netjerikhet's mortuary complex ~9km northeast is similar to the celestial appearance of the 'twin' stars Castor and Pollux relative to Capella:
Orion as viewed from Djedefre's pyramid along a 52.2° south of east sightline:
Giza site layout as viewed from Djedefre's pyramid:
Djedefre, Giza, Orion 52.2° alignments:
Betelgeuse, Sirius, Procyon triangle + Orion 3-star vertical alignment, a celestial spectacle on the north/south meridian possibly used by the surveyors to determine a north/south direction line to orientate 4th Dynasty pyramids to the cardinal directions:
