I received this Sunday gyan today:
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The knowledge- Sunday Reading
Studies suggest that walking in a park or near a water body reduces cortisol (stress hormone) by 30%.
Crazy fact: In just 1 hour, more solar energy hits Earth than humanity uses in a whole year. And in <8 hours, enough sunlight hits Earth to power us for a year.
i^i = 0.20787957635….. Imaginary number to the power of imaginary number is a real number be-cause of Euler's formula, which connects trigonometric functions with complex exponential func-tions.
...and more reading
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The third factoid was pretty interesting. I asked help of AI to find how i^i = 0.20787957... I added a bit of my own complexity just for fun. I asked AI thus: What would be i to the power of i? I saw somewhere that it would be 0.207.... This is amazing , is it not? A decimal fraction? What would be tetration of i , say 4 subscript i?
Now, tetration is an maths animal I had just heard about. Tetration of any number would be that number to the power of that number, to the power of that number, to the power of that number...and so on. In the case of 4 subscript i(or, ⁴i), it would be i to the power i, to the power of i, to the power of i, to the power of i, or in other words: i^(i^(i^i)), or,
Claude AI replied thus:
Yes, i^i is one of the loveliest surprises in complex analysis — a pure imaginary number raised to a pure imaginary power ends up being a real number.
Why i^i is real
Using Euler's formula, i = e^(iπ/2) (writing i in exponential form). So:
i^i =(e^(i*π/2))^i = e^(i*i*π/2) = e^(-π/2)
And e^(−π/2) ≈ 0.2078795...
It's a real number, but it's not a fraction in the sense of a ratio of integers — it's irrational (in fact, transcendental). 0.207... is just its decimal approximation, not a terminating or repeating fraction.
Now the tetration: ⁴i = i^i^i^i
Tetration stacks exponents from the top down: ⁴i = i^(i^(i^i)). Let's build it one level at a time, always using the principal branch:
Level 1: a₁ = i
Level 2: a₂ = i^i = e^(−π/2) ≈ 0.20788 (real, as above)
Level 3: a₃ = i^(a₂). Since a₂ is real, this is:
i^(0.20788) = e^(0.20788 * i*π/2) = cos(0.3266) + i*sin(0.3266) =approx 0.9472 + 0.3208i
Level 4: a₄ = i^(a₃), now with a complex exponent:
i^(a₃) = e^((a₃)*i*π/2))
Working through the complex arithmetic (multiplying a₃ by iπ/2, then exponentiating) gives:
⁴i= approx 0.0502 + 0.602i
Its magnitude is about 0.604, and it sits at an angle of roughly 85° from the positive real axis.
A fun fact: as you keep stacking more i's (⁵i, ⁶i, ...), the tower actually converges to a fixed complex number rather than blowing up or oscillating forever — infinite tetration of i converges to approximately 0.4383 + 0.3606i. That's a much rarer property; most bases don't have convergent infinite towers.
Explaining e^(−π/2): That line comes from calculating i^i.
Here's the reasoning behind it:
Step 1: Write i in exponential form.
Any complex number can be written as e^(iθ), where θ is the angle it makes on the complex plane. The number i sits at 90° (π/2 radians) on the unit circle, so:
i = e^(i*π/2)
Step 2: Raise it to the power i.
i^i = (e^(i*π/2))^i
When you raise a power to a power, exponents multiply:
= e^(i*π/2*i) = e^(-π/2)
since i² = −1. Notice the i's cancel out — that's the key step. What started as an imaginary exponent multiplying another factor of i becomes real, because i times i is just −1.
Step 3: Evaluate the number.
π/2 ≈ 1.5708, so we need e^(−1.5708). Since e ≈ 2.71828, this is 1 divided by e^1.5708, which works out to:
e^(-π/2), approx 0.2078795...
Why this feels surprising: you'd expect an imaginary number raised to an imaginary power to give you something imaginary or at least complex. Instead, the two imaginary-ness factors cancel each other through i² = −1, leaving a plain real decimal — and an irrational one at that, not a fraction, just a number that happens to start 0.207.../2) ≈ 0.2078795...further:
Me:
The fun fact, ' as you keep stacking more i's (⁵i, ⁶i, ...), the tower actually converges to a fixed complex number rather than blowing up or oscillating forever — infinite tetration of i converges to approximately 0.4383 + 0.3606i. That's a much rarer property; most bases don't have convergent infinite towers.' has got me intrigued. Have we found another e-like number as we tend tetration of i towards infinity, albeit in the complex plane?
Claude AI:
Not quite an analogue of e, but let's look at what makes e special versus what makes this fixed point special — the comparison is illuminating.
What actually makes e "e-like"
e is fundamental because it shows up as the unique solution to many independent, unrelated problems: the base where d/dx(eˣ) = eˣ, the limit of (1+1/n)ⁿ, the sum of 1/n!, and so on. It's a universal constant that keeps appearing across completely different branches of math — that ubiquity is what earns it the special status.
What the fixed point 0.4383 + 0.3606i actually is just the solution to one specific equation:
z = i^z
That's it. It's the fixed point of this particular map (raise i to the power of the input, repeat). It's a legitimate and interesting number — but it's not showing up independently all over mathematics the way e does. It's more like asking what does the sequence x, cos(x), cos(cos(x)), ... converge to? (answer: the Dottie number, ≈0.739085, the fixed point of cos(x) = x). That's a genuinely analogous situation — a real fixed point of a specific iterated function, interesting in its own right, but not a universal constant.
Where the real analogy lies
If you want the actual "complex-plane e," you don't need to look for a new number at all — e itself already lives on the complex plane and does the heavy lifting there via Euler's formula, e^(iθ) = cos θ + i sin θ. That's the bridge between e and complex numbers, and it's precisely the tool we used to compute i^i in the first place.
What you have found, though, is something worth appreciating on its own terms: a genuine example of a complex fixed point of an iterated exponential map. Your complex fixed point is the counterpart of that convergence question, just off the real axis. So: a nice number, a legitimate mathematical object, but its significance comes from being a fixed point of this one map — not from popping up everywhere the way e does.
Me:
Can this be geometrically created? Will this point be same point in space wherever we are on earth?
AI Claude:
Good questions — they touch two different ideas, so let's separate them.
Same point wherever we are on Earth?
Yes — but the reason is simpler than it might seem: this point isn't a location in physical space at all. It's a point in the complex plane, which is a purely mathematical construct — a 2D coordinate system where one axis represents real numbers and the other represents multiples of i. It has nothing to do with geography, gravity, or your position on the planet.
So 0.4383 + 0.3606i is the same abstract mathematical object whether you compute it in Bengaluru, Antarctica, or on the Moon — the same way "7" or "π" doesn't change depending on where you are. It's a number, not a place.
Can it be geometrically constructed?
Here there are two very different meanings of construct, and the answer differs for each:
1. Classical straightedge-and-compass construction — almost certainly no.
Classical constructibility (the kind used for bisecting angles or constructing regular polygons) only works for a specific class of algebraic numbers — ones reachable by a finite sequence of square roots. Our fixed point comes from solving z = i^z, a transcendental equation with no closed algebraic form. Numbers like this are expected to be transcendental (not the root of any polynomial with rational coefficients), which puts them firmly outside what compass and straightedge can ever reach — much like you can't classically square a circle because π is transcendental.
2. Iterative geometric construction — yes, and it's genuinely elegant.
You can construct it as a limiting picture through repeated geometric steps:
- Start at any point in the complex plane
- Apply the transformation z → i^z (a rotation by angle proportional to Re(z), combined with a scaling by a factor depending on Im(z))
- Plot the new point
- Repeat
If you connect the successive points, you get a spiral that winds tighter and tighter, converging visually toward 0.4383 + 0.3606i. This is the complex-plane version of a cobweb diagram, the same visualization technique used to show the real Dottie number (fixed point of cos x) spiraling into place. It never exactly arrives in finitely many steps — but it approaches arbitrarily closely, which is a legitimate and beautiful way to construct it geometrically, even though it's not the classical Euclidean sense of the word.
Want me to actually plot that spiral for you? It would visually be a satisfying way to see this whole conversation come together.
Me:
Yes please.
AI Claude:
Here it is:
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