THE ANALYTIC STRUCTURE OF CONSTANTS

How singularities and symmetry determine the speed of numerical approximation

Some mathematical constants are easy to approximate. Others converge painfully slowly. A few remain stubborn even after centuries of work. This variation is not random. It reflects the analytic structure of the functions that define the constants.

The central idea of this article is simple:

The ability of a function to continue analytically beyond the real line determines how fast any basic approximation method can converge. The location of singularities and the presence of global symmetries influence the decay of coefficients in Taylor, Fourier, or related expansions, and that decay controls the speed of computation.

This gives us a clear way to understand why certain constants are intrinsically slow and why others allow rapid algorithms once the right structure is identified.

1. Local and Global Analytic Structure

Constants inherit their computational difficulty from the analytic behaviour of the functions behind them.

Local structure

Some functions have singularities very close to the real axis. For example:

• arctan has singularities at ±i

• 1/x has a pole at 0

• algebraic functions have branch points near their roots

Such functions have a limited radius of convergence for their power series. Their coefficients decay only at a polynomial rate, and this restricts how fast any elementary approximation can converge. By “elementary,” we mean methods that use:

• Taylor expansions

• Euler–Maclaurin corrections

• Riemann sums and trapezoidal rules

• simple algebraic transformations

• Machin-type arctan decompositions

These methods rely solely on real-line information and do not use any global structures such as periodicity or modular symmetry.

A brief historical aside

The contrast between “local” and “global” structure is not just a theoretical classification. When modular-form formulas for π were discovered and refined, the speed was so extraordinary that the Chudnovsky brothers built a home-made supercomputer in their New York apartment in the 1990s specifically to exploit them. The machine, assembled from spare parts and cooled with improvised plumbing, set world records for digits of π. It remains one of the clearest demonstrations of how global analytic structure can translate directly into raw computational power.

Global structure

Other functions behave nicely over large regions of the complex plane. Examples include:

• sin(πx), which is entire and periodic

• modular forms, which are analytic on the upper half-plane and satisfy transformation laws

• elliptic functions, which are doubly periodic

Their Fourier or spectral coefficients decay exponentially or faster, and this creates the possibility of very rapid convergence. Algorithms that use these structures are not elementary in the sense defined above. They rely on analytic continuation and global symmetry.

2. Why Analytic Structure Determines Convergence

The mechanism behind the phenomenon is classical. If a function is analytic inside a disk of radius R, then its Taylor coefficients are bounded by M divided by R to the power n. This means:

• a nearby singularity (small R) leads to slow coefficient decay

• entire behaviour (large R) gives exponential decay

• modular or elliptic symmetries can create even faster decay

Since all basic approximation schemes ultimately depend on expansions of this sort, the rate of coefficient decay sets a hard limit on the speed of convergence.

This is a precise mathematical fact, not a heuristic.

3. Constants Limited by Local Singularities

These constants can only be reached slowly with elementary methods.

π through arctan

The singularities of arctan at ±i are at distance 1 from the real axis. Its Taylor coefficients behave like 1/n, which gives convergence of order 1/n for the usual Gregory series. This proves that real-line Taylor methods for π must be slow.

Machin-type formulas help only because arctan(1/q) moves the singularities farther away, but the convergence is still polynomial.

e and the logarithm

The standard definitions through integrals or ODEs involve local behaviour. Any Riemann-sum or Euler–Maclaurin approach remains slow for the same analytic reason.

γ (Euler–Mascheroni)

The constant γ is the limit of Hₙ minus ln n. The defining function 1/x has a singularity at 0, so any elementary method that uses derivative information of 1/x, including Euler–Maclaurin, can only achieve polynomial convergence. There is no known elementary method that gives exponential decay of coefficients.

4. Constants that Become Fast Once Their Global Structure Is Recognized

ζ(2)

The naive series 1 + 1/2² + 1/3² + … converges slowly. This is exactly what the coefficient-decay principle predicts.

The situation changes completely once ζ(2) is linked to the sine function. The infinite product for sin(πx) is entire and periodic, so its associated coefficients decay exponentially. Fourier expansions and spectral methods then provide rapid convergence and lead directly to the closed form π²/6.

This is the clearest example of how identifying the right global structure can transform a slow constant into a fast one.

The Analytic Speed Limit

Bars show digits gained per iteration. Local singularities (red) cap progress; global symmetries (green) accelerate it.

Current Iteration

0

Step Size

100

Local (polynomial)

Global (exponential)

Click Run 100 repeatedly to see divergence.

5. Constants With No Known Usable Global Structure

ζ(3)

The constant ζ(3) is analytically well-defined, and many series exist for it, but none of the known representations produce exponentially decaying coefficients using elementary constructions. At present there is no known periodic expansion, no simple entire product, and no modular-form identity that generates a rapidly convergent expression. Some series converge reasonably well, but never in a truly exponential way without heavy analytic work.

Catalan and elliptic constants

These constants are connected to functions with branch cuts and deep symmetries that are difficult to exploit. No simple representation with rapid coefficient decay is known.

6. The Mechanistic Pattern

The behaviour of constants now follows a very simple pattern:

Local singularities produce polynomial convergence. Examples include π via arctan, e, the logarithm, γ, and the naive series for ζ(2) and ζ(3).

Global periodicity or entire behaviour produces exponential convergence once the structure is used. Examples include ζ(2) through the sine product, and fast π algorithms based on modular forms.

Deep analytic structure without accessible symmetry produces no known fast elementary convergence. Examples include ζ(3), Catalan’s constant, and elliptic integrals.

The pattern is not historical. It is a direct consequence of standard complex analysis.

7. Why Modular Forms Create Fast Algorithms for π

Modular forms satisfy transformation laws that relate values at different points in the upper half-plane. By moving to regions where q = exp(2πiτ) is extremely small, one obtains series whose coefficients fall away at a superexponential rate. This behaviour is the reason the Chudnovsky and Ramanujan series converge so quickly. They harness global symmetry that elementary methods cannot access.

This explains why polygon-based approximations are slow and why modular methods are exceptionally fast. The analytic behaviour is fundamentally different.

Chudnovsky π Calculator

Digits:

Ready.

8. Counterexamples and Edge Cases

BBP formulas for π

Although the BBP series looks elementary, its derivation relies on analytic continuation of polylogarithms and special algebraic identities. It does not fall under the elementary methods described here.

Euler–Maclaurin for γ

The method improves constants but not the overall rate. It remains polynomial.

Continued fractions

Some continued fractions converge quickly for algebraic constants, but analytic limitations prevent them from giving exponential speed for transcendental constants like π or γ without global structure.

Nothing here contradicts the mechanism.

9. Why These Ideas Matter

The analytic structure of a constant provides a practical guide to its computational difficulty. It tells us:

• no simple fast algorithm for γ exists unless new global structure is found • ζ(3) will not yield rapid convergence without discovering symmetry now unknown • every fast algorithm for π must rely on entire or modular behaviour

These are clear predictions grounded in complex analysis.

The principle is concise. The decay of coefficients controls convergence. The analytic continuation of a function controls the decay of its coefficients.

Local structure gives slow convergence. Global structure gives fast convergence. Deep structure remains inaccessible without heavy machinery.

This is why some constants are easy and others are not, and why the discovery of global analytic structure has such dramatic computational consequences.

https://thinkinginstructure.substack.com/p/the-analytic-structure-of-constants

December 12, 2025

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February 14, 2012

How to remember the Schrödinger Equation without really trying

This essay is part of a three-part series on mathematical structures that survive the collapse of their original worldviews. Part I — Schrödinger (this essay) Part II — Hamilton Part III — Maxwell

There’s a familiar pop-science ritual for deriving the Schrödinger equation:
start with a wave, add a dash of Planck, differentiate once, and watch the structure fall neatly into place.

And mathematically, it really does.

But whenever something in physics looks too natural, it usually means we’ve chosen a language that makes it natural.
The elegance is real — but it’s purchased.

What follows isn’t a derivation.
It’s an examination of why the derivation looks so clean, and why that neatness shouldn’t be mistaken for inevitability.

1. Interference demands complex numbers — and we quietly accept that

A wave must oscillate, carry a phase, and combine linearly with other waves.

Complex exponentials do this flawlessly:

e^{i\omega t}

Add two of them and interference simply happens.

This feels like clever bookkeeping, but it isn’t trivial.
It’s a commitment to:

linear superposition
phase as physically meaningful
smooth, generator-based time evolution

We rarely stop to notice that these commitments shape everything downstream.

Wave Interference & Complex Exponentials

Section 1: “Interference demands complex numbers — and we quietly accept that”

Wave 1 Frequency (ω₁): 2.0 Wave 1 Phase (φ₁): 0.0

Wave 2 Frequency (ω₂): 2.5 Wave 2 Phase (φ₂): 1.6

Lines: Wave 1, Wave 2, and their sum (interference).

2. Introduce quantisation — and notice how smoothly it fits

Planck gave us the relation:

E = \hbar \omega

Insert that relation into the exponential:

e^{-iEt/\hbar}

Now the wave’s phase evolves at a rate set by its energy.

It fits so naturally that we barely register how much structure is being inherited.
We’ve carried over from classical mechanics the principle that energy generates time evolution — and that inheritance shapes everything that follows.

Still, the machinery hums along perfectly.

3. Differentiate once and admire the elegant fit

Differentiate:

\frac{d}{dt}\!\left[e^{-iEt/\hbar}\right] = -\frac{iE}{\hbar}\, e^{-iEt/\hbar}

Multiply both sides by iħ:

i\hbar\, \frac{dU}{dt} = E\,U

It’s compact, well-behaved, and looks like it’s been waiting to be written down.

Generalise from one exponential to a superposition.
Replace the number E with the operator H (the Hamiltonian).
And out drops the familiar equation:

i\hbar\, \frac{d|\psi\rangle}{dt} = H|\psi\rangle

At this point most treatments declare victory:

“Look, the Schrödinger equation emerges naturally.”

But the historical Schrödinger equation did not emerge from this reasoning — and that matters.

4. Schrödinger wrote down the right equation for the wrong theory

When Schrödinger introduced his equation, he believed he had discovered a classical wave description of matter.

His papers describe ψ as a literal physical field spreading smoothly through space.
Wave packets, he hoped, would behave like particles.

They didn’t.

Packets spread — relentlessly, mathematically, inevitably.
A “particle-like” lump at one moment dissolves into a diffuse cloud the next.

The equation worked spectacularly.
But it did not describe what Schrödinger thought it described.

The modern view — ψ as a probability amplitude, with |ψ|² giving outcomes and phase controlling interference — came later.
The interpretation was not contained in the math; physicists imposed it because nothing else matched the data.

The author of the equation didn’t understand what the equation meant.

That tells us something important:
the apparent inevitability is retrospective.

Re(ψ) — oscillating wave with a widening envelope

What Schrödinger hoped was “the thing itself”

Speed

t = 0.0

This panel shows only the real part (a visual proxy). The oscillations ride inside an envelope that spreads; the “particle-like bump” does not stay put.

5. The derivation is clean because we selected the framework that makes it clean

Consider each “natural” step:

Complex numbers → preserve linear superposition
Linearity → required for interference
Hermitian generators → guarantee real energy values
Momentum as -iħ∇ → enforces chosen commutation relations
Multiply by iħ → ensures unitary time evolution

None of these is forced by nature.
They are forced by the conceptual architecture we want the theory to inhabit.

The Schrödinger equation looks inevitable because we have internalised the worldview that makes it inevitable.

Schrödinger himself had not yet internalised that worldview — which is why he misunderstood his own equation.

Both truths coexist:

Within the quantum framework, the equation really is the only one that behaves properly.
But the framework wasn’t dictated. It was chosen, refined by experiment, constrained by symmetry, and later retold as if it led directly to the equation.

The derivation works because the scaffolding had already been built.

Conclusion: the equation is simple. Choosing the equation was not.

The Schrödinger equation is elegant, compact, and structurally satisfying.

But that elegance is the product of hindsight.
We adopted the mathematical tools that make the equation look natural, then retold the story as if the mathematics compelled the physics.

The reality is subtler:

The math feels inevitable only because the worldview behind it isn’t.

And in that gap — between the equation’s tidy form and the theory’s conceptual strangeness — lives the entire modern mystery of quantum mechanics.

https://thinkinginstructure.substack.com/p/why-schrodingers-equation-

February 3, 2012

Fourier Series Are Just Coordinates

Why an infinite wiggle can be treated like a point in a room you can’t draw

Fourier series often look like formulas you’re supposed to memorise.

They aren’t.

The idea is almost embarrassingly simple:

A function is a point in an infinite-dimensional space.
Sines and cosines are orthogonal axes.
Fourier coefficients are just the coordinates of that point.

Once you see this, the formulas stop looking mystical.
They become inevitable.

1. Coordinates: machinery your brain already owns

On a sheet of paper, a point has two coordinates.
In 3D, it has three.
In 10D, it has ten.

Nothing new happens — you just add axes.

Given a vector $v$ and an axis $e$ , the coordinate of $v$ along $e$ is always

$text{coordinate} = frac{v cdot e}{e cdot e}$

That’s it.

Dot product over length squared.
Projection, nothing more.

Every coordinate system you’ve ever used works this way.

2. Swap points for functions. The geometry survives.

Functions behave like vectors:

you can add them
you can scale them

So they live in a vector space.

What replaces the dot product?
An integral: $langle f, g rangle = int_{-pi}^{pi} f(x),g(x),dx$

Two functions are orthogonal if this integral is zero.

This isn’t exotic.
It works for essentially every function encountered in physics, engineering, and applied maths. (If you want the precise functional-analysis caveats, they exist — but they don’t change the picture.)

Once you have:

a vector space
an inner product
orthogonal axes

coordinates are automatic.

So the only real question is:

What are the axes?

3. Why sines and cosines are orthogonal (the symmetry doing all the work)

The orthogonality of sine and cosine functions is not a miracle.
It’s symmetry.

Two facts do everything:

Sine is odd, cosine is even
The integral of an odd function over $[-pi, pi]$ is zero

So:

sine × cosine integrates to zero
different frequencies cancel perfectly
only matching frequencies survive

This gives an infinite family of mutually perpendicular axes:

$1, cos x, cos 2x, cos 3x, dots$ $sin x, sin 2x, sin 3x, dots$

That’s not decoration.

That is the coordinate frame.

4. Fourier coefficients are literally coordinates

Once the axes are fixed, the coordinates of a function $f(x)$ are forced:

$a_n = frac{langle f, cos(nx) rangle}{langle cos(nx), cos(nx) rangle} quad b_n = frac{langle f, sin(nx) rangle}{langle sin(nx), sin(nx) rangle}$

For $n > 0$ : $langle cos(nx), cos(nx) rangle = pi,quad langle sin(nx), sin(nx) rangle = pi$

For the constant axis: $langle 1, 1 rangle = 2pi$

That’s where the famous formulas come from:

$a_0 = frac{1}{2pi} int f(x),dx$ $a_n = frac{1}{pi} int f(x)cos(nx),dx$ $b_n = frac{1}{pi} int f(x)sin(nx),dx$

The factors of $pi$ are not magic.
They are just the squared lengths of the axes.

Remember the idea, not the formula:

Coordinates = projections.

Everything else is arithmetic.

5. Reconstruction is not a metaphor

Once you have coordinates, you rebuild the vector:

$f(x) = a_0 + sum_{n=1}^{infty} left( a_n cos(nx) + b_n sin(nx) right)$

This is not an analogy.
It is literal reconstruction from coordinates in an orthogonal basis.

A Fourier series is not “approximating” a function in some vague sense.
It is expressing the same point in a different coordinate system.

6. Smoothness, corners, and why convergence looks the way it does

This geometric picture explains convergence instantly.

A smooth function lies mostly along low-frequency directions
→ large $a_1, a_2$ tiny high- $n$ coefficients
A function with a corner points diagonally across the space
→ energy spread across many axes
→ slow coefficient decay

Sharp edges require high-frequency directions.

That is the real reason behind:

Gibbs phenomenon
ringing
slow convergence near discontinuities

No mystery.
Just geometry.

7. Example: $f(x) = x$ on $[-pi, pi]$

The function is odd, so all cosine coefficients vanish.

The sine coefficients are: $b_n = frac{1}{pi} int x sin(nx),dx = frac{2(-1)^{n+1}}{n}$

So the series builds as:

$S_1(x) = 2sin x$ $S_2(x) = 2sin x – sin 2x$ $S_3(x) = 2sin x – sin 2x + frac{2}{3}sin 3x$

Fourier Series: Functions as Coordinates

A function is a point in infinite-dimensional space. Fourier coefficients are its coordinates.

The Core Idea: Just like a point in 3D space has (x, y, z) coordinates, a function has coordinates along sin(x), sin(2x), sin(3x)… The Fourier series is reconstructing the function from its coordinates in an orthogonal basis.

Target Function: f(x) = x

Fourier Reconstruction (Adding Coordinates)

Number of Terms (Axes)

5

b₁ (sin x coordinate) 2.00

b₂ (sin 2x coordinate) -1.00

b₃ (sin 3x coordinate) 0.67

Mean squared error (distance in function space)

0.000

f(x) ≈ 2sin(x) – sin(2x) + 0.67sin(3x) + …

Each term adds another coordinate.

Geometrically, this is no different from reconstructing a 3D point by first adding its $x$ x coordinate, then $y$ , then $z$ .
The picture sharpens as you include more axes.

8. The room you’ll never draw — but always use

Every $sin(nx)$ and $cos(nx)$ is a perpendicular direction.

Your function is a point in that infinite-dimensional room.

Fourier coefficients are its coordinates.

The Fourier series is the reconstruction.

Once you internalise “Fourier = coordinates”:

the formulas stop looking arbitrary
convergence becomes geometric
smoothness becomes visible
memorisation disappears

You don’t remember Fourier series.

You remember one rule:

A coordinate is a projection onto an axis.

Everything else follows.
Everything else is arithmetic.

https://thinkinginstructure.substack.com/p/fourier-series-are-just-coordinates

November 6, 2011

How to remember Bayes’ Theorem without really trying

Bayes’ Theorem crops up a lot. There’s even a picture of it in neon tubes on the Wikipedia page.

Its beauty is that it relates the probability of one event occurring after another to its inverse. That is, it relates
$P(A \mid B)$ — the probability of $A$ given $B$
to
$P(B \mid A)$ — the probability of $B$ given $A$ .

The standard textbook example goes like this.

Let $A$ be the event that an individual in a population has a disease (say cancer). Let $B$ be the event that a medical test for that disease comes back positive.

We are given:

$P(A)$ : the prevalence of the disease (say 1%).
$P(B \mid A)$ : the probability the test is positive given the disease is present (the true positive rate, say 98%).
$P(B \mid \neg A)$ : the probability the test is positive given the disease is absent (the false positive rate, say 3%).

Now imagine you — yes, you — take the test, and it comes back positive.

What is the probability you actually have cancer?

It’s not 99%.

Using Bayes’ Theorem, the answer in this case is only 24.8% — far lower than most people would guess.

Unfortunately, if Bayes’ Theorem is learned as a magic formula into which numbers are blindly plugged, it’s very easy to get confused by all the $P(A)$ ’s, $P(B)$ ’s, and $P(A \mid B)$ ’s. Which is a shame, because the underlying ideas are extremely simple.

In fact, if you understand what’s going on geometrically, you don’t need to memorise any formulae at all.

Just the pictures.

The Picture

Think of a sample space, represented by a rectangle. This rectangle contains every possible outcome.

Inside the rectangle are two ovals:

Oval A (blue): event $A$
Oval B (yellow): event $B$

Where they overlap is green — this region represents $A \cap B$ .

The area of each shape represents probability. If an oval is large, the event is likely. If it’s small, the event is rare.

Imagine closing your eyes and randomly sticking a pin into the rectangle. The probability of landing in any region is proportional to its area.

To understand Bayes’ Theorem, we only need three scenarios.

Scenario 1: No Overlap

In this case, events $A$ and $B$ do not intersect at all.

$P(A \cap B) = 0$

Imagine you stick the pin into oval $B$ (you test positive). The moment the pin lands, you shrink to the size of an ant and a ten-foot barbed-wire fence shoots up around oval $B$ . This fence is now the boundary of your world.

You’re terrified to look down.

If you see green, you have cancer.
If you see yellow, you’re safe.

But in this scenario, there is no green inside BBB. The events don’t overlap.

So: $P(A \mid B) = 0$

A positive test guarantees safety (an unrealistic test, but a useful limiting case).

Scenario 2: $B$ Entirely Inside $A$

Now imagine oval $B$ B lies completely inside oval $A$ .

Every point in $B$ is green.

Again, you stick the pin into $B$ B. The fence rises. You look down.

This time you are definitely standing on green.

$P(A \cap B) = P(B)$

and therefore $P(A \mid B) = 1$

A positive test guarantees disease.

Scenario 3: The Mixed Case (The Real World)

This is the interesting case.

Oval $B$ partially overlaps oval $A$ .

You stick the pin into $B$ , shrink to ant size, and the fence rises. Inside the fence, some floor is green, some is yellow.

What is the probability you’re standing on green?

It is simply: $P(A \mid B) = \frac{P(A \cap B)}{P(B)} \tag{1}$

This single formula covers all three scenarios.

Bayesian “Fence” Visualizer

The total box is the population. The colored rectangles are people who tested positive.

Disease Prevalence (P(A)): 10%

Test Sensitivity (P(B|A)): 98%

False Positive Rate (P(B|¬A)): 5%

Probability you have the disease given the positive test:

0.0%

Ratio of Green to (Green + Yellow)

Swapping the Question

So far we’ve been asking:

Given a positive test, what is the probability of disease?

Now let’s reverse it.

Suppose we know we have the disease. What is the probability the test is positive?

That is $P(B \mid A)$ .

By symmetry, swapping $A$ and $B$ in (1) gives: $P(B \mid A) = \frac{P(B \cap A)}{P(A)} \tag{2}$

But $P(B \cap A) = P(A \cap B)$ , so combining (1) and (2) gives:

$P(A \mid B) = \frac{P(B \mid A)\, P(A)}{P(B)}$

This is the compact form of Bayes’ Theorem — the one in neon lights.

The Extended Form

We’re not quite finished, because $P(B)$ is often unknown.

In the test example, $P(B)$ is the probability that a random person tests positive — either because they have the disease or because of a false positive.

From the diagram:

$P(B) = P(A \cap B) + P(\neg A \cap B)$

Using conditional probabilities:

$P(B) = P(B \mid A) P(A) + P(B \mid \neg A) P(\neg A)$

Substituting this into Bayes’ formula gives the extended version:

$P(A \mid B) = \frac{P(B \mid A)\, P(A)} {P(B \mid A)\, P(A) + P(B \mid \neg A)\, P(\neg A)}$

The Numbers

In our example:

$P(A) = 0.01$
$P(B \mid A) = 0.98$
$P(B \mid \neg A) = 0.03$

Plugging these in:

$P(A \mid B) = \frac{0.98 \times 0.01} {(0.98 \times 0.01) + (0.03 \times 0.99)} = 24.8\%$

There is hope — a positive test is not the near-certainty most people imagine.

Final Note

Many people find it helpful to lay these probabilities out in a table to avoid mistakes. There are excellent examples of that approach online.

But ultimately, Bayes’ Theorem isn’t about algebra.

It’s about areas, fences, and what information actually does to your world.