The Absolute Dimensions Are Special
The original question identifies the ratio 7/11 as a √φ/2 convergent. But there's more: the absolute dimensions 280 × 440 cubits are also number-theoretically distinguished.
The 7th convergent of √(7/11)
The convergents of √(7/11) are: 0, 1, 3/4, 4/5, 67/84, 71/89, 280/351, 631/791, ...
The 7th convergent is exactly 280/351, where 280 is the pyramid height.
The pair {c₇, (7/11)/c₇} = {280/351, 351/440} brackets √(7/11), with 440 being the pyramid base.
Why k=40 is unique
For a pyramid with a ratio 7/11, the dimensions are 7k × 11k for some scaling factor k. But only k=40 makes the dimensions appear directly in the √(7/11) convergent bounds:
| k | Dimensions | 7k is convergent numerator? |
|---|---|---|
| 10 | 70 × 110 | ✗ (70 not in sequence) |
| 20 | 140 × 220 | ✗ (140 not in sequence) |
| 30 | 210 × 330 | ✗ (210 not in sequence) |
| 40 | 280 × 440 | ✓ (280 = 7th convergent numerator) |
| 50 | 350 × 550 | ✗ (350 not in sequence) |
Convergent numerators of √(7/11): 0, 1, 3, 4, 67, 71, 280, 631, ...
For k ≠ 40, the pyramid still has a ratio 7/11, but the dimensions don't appear in the continued fraction structure of √(7/11).
What about other convergents?
For dimensions 7k × 11k to appear in convergent bounds, the numerator must be divisible by 7. This happens only for every 6th convergent:
| Convergent # | k | Pyramid size |
|---|---|---|
| 7 | 40 | 280 × 440 cubits ≈ 147 × 230 m |
| 13 | 28 080 | ≈ 103 × 162 km |
| 19 | 19 712 120 | larger than Earth |
k=40 is the unique physically realizable solution.
Connection to x² − 77y² = 1
The scaling k=40 equals the fundamental y-solution of x² − 77y² = 1, where 77 = 7 × 11.
Fundamental solution (Brahmagupta, Brāhmasphuṭasiddhānta, 628 CE): (x, y) = (351, 40)
This gives: Height = 7 × 40 = 280, Base = 11 × 40 = 440.
Comparison with other Giza pyramids
| Pyramid | Ratio | Actual k | Brahmagupta y | Match? |
|---|---|---|---|---|
| Khufu | 7/11 | 40 | 40 | ✓ |
| Khafre | 2/3 | 137 | 2 | ✗ |
| Menkaure | 5/8 | 25 | 3 | ✗ |
Only Khufu's dimensions satisfy this number-theoretic constraint.
Summary
The original observation: pyramid ratios are √φ/2 convergents.
New observation: Khufu's absolute dimensions are the unique physically realizable values where:
- The dimensions appear directly in √(7/11) convergent bounds
- The scaling factor equals the Brahmagupta-Bhaskara fundamental solution
Whether this reflects lost Egyptian number theory or a coincidence remains open.