See previous posts on my blog if you’re confused:

Part 1: Normal numbers in base tenPart 2: Numbers in base φ and πPart 3: One is normal in base π

I'm only posting Part 3 (the punchline) here on LW; Parts 1 and 2 explain things in much more detail.

One is a normal number.

“No it’s not.”

In base π, I mean.

“No, it’s still just 1.“

Don’t be so greedy. Start with 0. and go from there.

“So it’s 0.222… or 0.333… or something?”

Nope, it’s 0.3011021110…

“That’s… bad. Does it repeat?”

I think it’s normal, but that seems hard to prove.

“So each digit occurs equally often?”

No, normal numbers in base π are about a 37%-30%-29%-4% split of 0, 1, 2, 3.

“Where did that distribution come from?”

Integrals of this weird function:

(Specifically, integrals over [0, 1/π], [1/π, 2/π], [2/π, 3/π], and [3/π, 1].)

“And where did that come from?”

It’s the distribution of remainders when computing a random number in base π.

“What are the x coordinates of those discontinuities?”

Sequence of remainders when computing 0.3011021110…

“um”

which are dense in [0, 1], by the way.

“Can you prove that?”

No, and neither can you.

“…And the y coordinates?”

The discontinuities get smaller by a factor of π each time.

“And almost all numbers have this distribution of remainders?”

Yup.

“Including 1, if you use the 0.3011021110… representation.”

“So the histogram of x-coordinates of the N largest discontinuities in this function approaches… this function itself, as N goes to infinity.”

“Which has derivative zero almost everywhere, but has a dense set of discontinuities.”

Yup!

“Any other neat facts about it?”

It’s a fractal.

“How?”

Stretch out the colored regions by a factor of π horizontally, shrink by π vertically, and add them to each other. You get the original function back:

(Some imagination required.)

"And this is all because π is normal, or transcendental, or something?"

Nope, I think analogous statements are all true for base 3.5.

See previous posts on my blog if you’re confused:

Part 1:Normal numbers in base tenPart 2:Numbers in base φ and πPart 3: One is normal in base πI'm only posting Part 3 (the punchline) here on LW; Parts 1 and 2 explain things in much more detail.

One is a normal number.

“No it’s not.”

In base π, I mean.

“No, it’s still just

1.“Don’t be so greedy. Start with

0.and go from there.“So it’s

0.222…or0.333…or something?”Nope, it’s

0.3011021110…“That’s… bad. Does it repeat?”

I think it’s normal, but that seems hard to prove.

“So each digit occurs equally often?”

No, normal numbers in base π are about a 37%-30%-29%-4% split of 0, 1, 2, 3.

“Where did

thatdistribution come from?”Integrals of this weird function:

(Specifically, integrals over [0, 1/π], [1/π, 2/π], [2/π, 3/π], and [3/π, 1].)

“And where did

thatcome from?”It’s the distribution of remainders when computing a random number in base π.

“What are the x coordinates of those discontinuities?”

Sequence of remainders when computing

0.3011021110…“um”

which are dense in [0, 1], by the way.

“Can you prove that?”

No, and neither can you.

“…And the y coordinates?”

The discontinuities get smaller by a factor of π each time.

“And almost all numbers have this distribution of remainders?”

Yup.

“Including 1, if you use the

0.3011021110…representation.”Yup.

“So the histogram of x-coordinates of the N largest discontinuities in this function approaches… this function itself, as N goes to infinity.”

Yup.

“Which has derivative zero almost everywhere, but has a dense set of discontinuities.”

Yup!

“Any other neat facts about it?”

It’s a fractal.

“How?”

Stretch out the colored regions by a factor of π horizontally, shrink by π vertically, and add them to each other. You get the original function back:

(Some imagination required.)

"And this is all because π is normal, or transcendental, or something?"

Nope, I think analogous statements are all true for base 3.5.