Borbin the 🐱

¹

More than two thousand years ago, Archimedes² tried to calculate the circumference of a circle. It was already known that larger circles have both a larger diameter and a larger circumference, but it was not clear whether the ratio between them would always remain the same.

By enclosing a circle between polygons with an increasing number of sides, he was able to show that this ratio is the same for every circle, and that its value can be bounded within steadily tightening intervals. Using a polygon of 96 sides, he proved that the value is contained in the interval

$$\frac{223}{71} < \pi < \frac{22}{7}$$

Archimedes narrowing the bounds

After Euler adopted the notation in 1736, the world denoted this value as $\pi$ ³

As an irrational number, its decimal expansion never repeats and never ends. And the search for more digits has never stopped.

This raises a simple question.

How many of those digits do we actually need to describe something real?

The circumference of a circle provides exactly that test.

The circumference of a circle is

$$C = \pi D$$

So any inaccuracy in $\pi$ directly affects the result.

Example: The Earth

The diameter of the Earth is approximately

$$D_{Earth} \approx 12742\ km$$

We want the circumference to be correct within

$$1\ mm = 0.001\ m$$

This means the allowed error in the circumference is

$$\Delta C \leq 0.001\ m$$

Since

$$C = \pi D$$

any error in $\pi$ causes

$$\Delta C = D \cdot \Delta \pi$$

So the required accuracy of $\pi$ must satisfy

$$D \cdot \Delta \pi \leq 0.001$$

Solving for $\Delta \pi$ leads to

$$\Delta \pi \leq \frac{0.001}{12742\cdot1000}$$

$$\Delta \pi \leq 7.85 \times 10^{-11}$$

That corresponds to roughly 10 decimal places of $\pi$. Using 10 places results in an error of about $10^{-11}$, which is within the required bound.

So if you use

$$\pi_{10} = 3.1415926536$$

you can calculate the circumference of the entire Earth with an error smaller than one millimeter.

Example: A Pizza

The diameter of a pizza is approximately

$$D_{Pizza} \approx 0.30\ m$$

This time we set the target accuracy to the diameter of an atomic nucleus

$$1\ \text{fm} = 10^{-15}\ m$$

Applying the same formula leads to

$$\Delta \pi \leq \frac{10^{-15}}{0.30}$$

$$\Delta \pi \leq 3.3 \times 10^{-15}$$

That corresponds to roughly 15 decimal places of $\pi$.

So if you use

$$\pi_{15} = 3.141592653589793$$

you can calculate the circumference of a pizza with an error smaller than the diameter of an atomic nucleus.

Example: The Observable Universe

Now let us scale things up.

The diameter of the observable universe is estimated at roughly

$$D_{Universe} \approx 8.8 \times 10^{26}\ m$$

Using the same formula with a target accuracy of 1 mm, as in the Earth example, leads to

$$\Delta \pi \leq \frac{0.001}{8.8 \times 10^{26}}$$

$$\Delta \pi \leq 1.14 \times 10^{-30}$$

That corresponds to roughly 30 decimal places of $\pi$.

So if you use

$$\pi_{30} = 3.141592653589793238462643383279$$

you can calculate the circumference of the entire observable universe with an error smaller than one millimeter.

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Conclusion

Ten digits of $\pi$ are enough for planetary scale.

Fifteen digits are enough to measure a pizza to the width of an atomic nucleus.

Thirty digits are enough for cosmic scale.

Beyond that, the digits describe no physical quantity we can measure, only the internal consistency of mathematics itself.

Fifteen digits are also exactly what NASA uses to navigate spacecraft across the solar system, the same precision that measures a pizza to the width of an atomic nucleus.
They have not yet missed a planet.

Interplanetary navigation with finite precision

See also Engineering Pi Day

Zwischen Rückblick und Ausblick liegt dieser Augenblick.

1/30s f/5,6 ISO 800/30° 16-50mm f/2,8 VR f=50mm/75mm

1/125s f/2,8 ISO 2800 16-50mm f/2,8 VR f=33mm/49mm

1/30s f/5,6 ISO 3200/36° 16-50mm f/2,8 VR f=33mm/49mm

Combine pictures with PTGui, Focus stacking

Alfred Hitchcock's North by Northwest (1959, deutscher Titel: Der unsichtbare Dritte) is a masterclass in suspense, style, and cinematic precision. But even the most iconic thrillers have their unscripted moments. Sometimes, spontaneity finds its way into the frame.

While watching the film, an unintended moment came to light: just after Roger Thornhill (Cary Grant) exits the train, he steps out of the washroom in his suit, having removed the porter's uniform. At that exact moment, a mother and her two children pass by, and the kids cheerfully wave into the camera. The girl walking ahead of them also looks directly into the camera. It is brief and easy to miss. A moment that does not belong, but remains visible to those who look closely.

For comparison, I have included another well-known moment from the Mount Rushmore visitor center scene, where a young boy preemptively plugs his ears before Eve Kendall (Eva Marie Saint) fires a gun. That one's been noted before. The train station wave? Not yet.

Station Scene: Unscripted Awareness at 01:01:25

As Roger Thornhill exits the washroom and steps into the station, a mother walks past with her two children. The kids wave directly toward the camera crew, and the mother, visibly amused, allows herself a brief smile. She knows they are in a film and chooses not to interfere.

Check the Scene – Unexpected Interaction

And for comparison, there is a moment that is already familiar:

Visitor Center: Predictable Surprise at 1:40:55

In the Mount Rushmore visitor center, a young boy plugs his ears just before the gun is fired. The moment is well-known among film enthusiasts and serves as a quiet reminder that not everyone on screen follows the director's timeline. Whether anticipating a loud bang or waving at the director, they rarely wait for the cue.

Check the Scene – Visitor Center Timing

Director's cut, The One Master Frame

A visual analysis of North by Northwest through all 1,566 I-frames⁴, arranged in a 54×29 mosaic.
Runtime: 2h 10m 30s⁵. Rated: frame-heavy.
7,830 seconds of cinema, with one I-frame captured every 5 seconds. A mosaic of suspense, one frame at a time, for continuity analysis and visual structure.

Zoom in for more continuity errors⁶

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More slips from the same classic

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Details everywhere, even in the last frame.

THCUO RPQVR IHIEZ EQBJS UELFW BGNCK CXBEW BGQEY SDH

In der alten Bibliothek fand man drei Manuskripte: I, II und III. Jeder Abschnitt war mit AAA markiert – ein merkwürdiger Zufall, denn auch die letzten Seiten trugen erneut das Kürzel AAA. Manche glauben, es handle sich um ein Werk, das Gedanken in Zeichen verwandelte – und Zeichen in Schweigen.
Amgine erscheint dort, wo der Weg zurück zum Ausgang führt.

I often pass the Montlake Bridge on my way to UW, but today was different: the bridge opened for water traffic just as I got there. The light turned red, and the bars descended, signaling the bridge's closure. I watched as the bridge slowly lifted to allow a sailboat to pass through, but I only saw the sailboat's mast passing by. Remarkable to see the massive structure rise and then descend back into place.

14:00:56, Bridge starting to open

14:01:36

14:01:52

14:02:06

14:02:36

14:03:06, Bridge fully opened

14:04:52, Bridge closing

14:05:08

14:05:50, Bridge completed its cycle and traffic resumes in a few seconds

A few days later, I was on the other side of the canal while traffic stood still, watching the bridge rise and settle once more.

A few months later, I was first in line on that same route with a perfect view as the bridge began its cycle. The timing could not have been better. The red lights flashed, the gates descended, and the massive structure started its graceful ascent. Positioned at the very front, I could see every detail: the roadway lifting, the two halves opening, and the quiet pause at full height before the bridge eased back into place.

A while later, I checked the traffic and noticed the red lines of waiting cars on the map. Sure enough, the bridge was operating. Here are some screenshots from a different perspective:

Gates open

Traffic resumes

See also Lake Washington Bridges.