# The Irrational Mathematicians of Giza, Part 2

In Part 1 we dealt with numbers and ratios that were not unknown in the ancient world, even if we still think it was the Greeks that invented π and φ and the whole Pythagoras theorem etc.

Now we introduce ℯ (2.71828…), the base of the natural logarithm, which is defined as the limit of (1 + 1/n)n  as n approaches infinity. There is no evidence that the Egyptians knew of this number, but here it is:

ⅇ in Giza plan

The angle is 132.340, we want 132.437, out by 0.096 degrees, accuracy is 99.927%

ⅇ and φ are nicely balanced:

ⅇ and φ in Giza plan

And now for something much more controversial … logarithms in base ⅇ. We have no indication that the Egyptians knew of them, but we can demonstrate both ln(2) and ln(3) here, once more based off of the great pyramid and linking all three:

ln(2) and ln(3) in Giza plan

NameRatioPoint 1Point 2CalculatedDesiredAbsDiffPercentageError %
360/ⅇP3 TLP1 C132.340132.4360.096299.9270.073
ln2360(ln2)P1 BRP2 TR248.957249.5330.57699.7690.231
ln3360/ln3P1 TLP3 TL327.518327.6860.16899.9490.051

Here’s ln(π):

ln(π) in Giza plan

and π/ⅇ:

π/ⅇ in Giza plan

And πⅇ:

πⅇ in Giza plan

The accuracy is quite worrying.

NameRatioPoint 1Point 2CalculatedDesiredAbsDiffPercentageError %
ln(π)360/ln(π)P2 BRP1 BR314.550314.4850.066100.0210.021
π/ⅇ360/(π/ⅇ)P2 BRP3 BR311.672311.4920.180100.0580.058
πⅇ360/πⅇP3 TLP2 TR42.21342.1560.057100.1350.135

Which leads us to this beautifully symmetrical arrangement of ln(π) and π/ⅇ

ln(π) and π/ⅇ in Giza plan

ⅇ and ln(π) are also nicely balanced:

ⅇ and ln(π) in Giza plan

Or this way around:

ⅇ and ln(π) in Giza plan

Or π and  ln(π):

π and ln(π) in Giza plan

And ⅇ and π/ⅇ :

ⅇ and π/ⅇ in GIza plan

And not forgetting πⅇ and π/ⅇ :

πⅇ and π/ⅇ in Giza plan

Then we have π^ⅇ and ⅇ^π… (not really happy with π^ⅇ but will have to do for now):

π^ⅇ ⅇ^π in Giza plan

For π^ⅇ, the angle is 21.940, we want 22.459, out by 0.519 degrees, accuracy is 97.690%

For ⅇ^π, the angle is 23.076, we want 23.141,  out by 0.065 degrees, accuracy is 99.718%

The Omega constant Ω also belongs in this section. The Omega constant is the number for which Ωⅇ^Ω = 1, which is Ω = 0.567143290…. :

Ω in Giza plan

The angle is 204.060, we want 204.172, out by 0.111 degrees, accuracy is 99.945%

And lastly, we have √τⅇ = √2πⅇ  (4.1327313541…), which like the Omega constant above, I’m not entirely sure where the practical use is, but it’s on Wikipedia’s list of mathematical constants so I guess it does have a use somewhere.

√τⅇ in Giza plan

The angle is 87.003, we want 87.109, out by 0.107 degrees, accuracy is 99.878%

NameRatioPoint 1Point 2CalculatedDesiredAbsDiff °
360/ⅇP3 TLP1 C132.340132.4370.096
ln2360(ln2)P2 TRP1 BR248.957249.5330.576
ln3360/ln3P3 TLP1 TL327.518327.6860.168
ln pi360/(ln pi)P2 BRP1 BR314.550314.4850.066
π/ⅇ360/(π/ⅇ)P3 BRP2 BR311.672311.4920.180
πⅇ360/πⅇP3 TLP2 TR42.21342.1560.057
pi^epi^eP2 TLP1 BL21.94022.4590.519
e^pie^piP2 CP3 BR23.07623.1410.065
Omega360ΩP3 BRP1 TL204.060204.1720.111
√τⅇ360/√τⅇP3 C 45°P1 BR87.00387.1090.107
Average0.194

So if we can quote Robert Bauval again, how many pigeons do we need before we decide it’s not co-incidence?

The best explanation I can think of for these curious alignments (apart from the builders simply showing off) is to send a message down through the ages… Yes, we are intelligent, and yes, we can do mathematics….

A message down through the ages …

Continues in Part 3