Cosmology That ClicksEpisode 5 · the math-friendly edition

Episode 4 let you feel "stepping in points" → This time: can those points be told apart from a smooth wave?

Can You Tell a
Record from a CD? The metaphor running through the series — "record = continuous, CD = discrete" — collected in math this time.
The figures below are real: coarsen the sampling and watch a fake wave being born.

Tools you'll need: sine waves, frequency, division Sampling theorem: \(f_s > 2 f_{\max}\)

Throughout this series we've repeated that the continuous universe (\(c=\text{constant}\)) and the discrete one (the computer's view of \(c\cdot t=\text{constant}\)) are "like owning the same music on a record and a CD." Episode 4 let you experience chopping a smooth curve into points. Now, the heart of it — how far can you tell apart a CD chopped into points from a smooth record? The answer rests on exactly the same math that makes music-on-CD work: the sampling theorem. And that theorem draws a clear line between where discrete and continuous "can be told apart" and where they "can never be told apart."

STEP 01First, record a smooth wave as "points"

Sound is a vibration of air — a wave. A CD records this wave by measuring "what's the height now?" tens of thousands of times per second, storing a sequence of numbers. This "how many times per second you measure" is the sampling frequency \(f_s\). A CD uses \(f_s = 44100\) (44,100 times per second).

A naive question arises. If you only record discrete points, why can the original smooth sound be reconstructed? You didn't record anything between the points. In the figure below, first watch the "original wave (blue)" being recorded as "points (orange)."

Fig. 1: Recording a smooth wave (blue) as evenly spaced points (orange). Point spacing = coarseness of sampling
original smooth wave recorded points

If there are enough points, just connecting them smoothly brings the original wave back into view. The question is "how many is enough." Here, a beautiful theorem gives the answer.

STEP 02The sampling theorem — "measure faster than twice, and you can reconstruct it perfectly"

The sampling theorem (Nyquist–Shannon)

Let the highest frequency in the wave be \(f_{\max}\). If the sampling frequency satisfies

$$f_s > 2\, f_{\max}$$

then from the discrete points you can reconstruct the original smooth wave perfectly, with zero error.

This is a startling claim. Even though "you recorded nothing between the points," as long as the condition holds, it's 100% restored, including the in-between information. A CD uses \(f_s=44100\) because the upper limit of human hearing is about 20,000 hertz (20 kHz), and this value is set with a little margin above twice that (40,000). In other words —

Try it — why is a CD 44,100?

The upper limit of human hearing

\(f_{\max}\approx 20{,}000\ \mathrm{Hz}\) (20 kHz). By the theorem, the required sampling is

$$f_s > 2\times 20{,}000 = 40{,}000\ \mathrm{Hz}.$$

The CD's actual value

\(f_s = 44{,}100\ \mathrm{Hz}\) — just above 40,000. So a CD can, in principle, perfectly record all sound within human hearing. That you can't tell a record (continuous) from a CD (discrete) by ear is neither imagination nor placebo — it's a fact guaranteed by this theorem.

The reveal (the heart of the series) This is the mathematical backing for what the series has said all along. For a signal with an upper bound on its bandwidth (the frequencies it contains), the discrete (points) and the continuous (smooth) are perfectly equivalent as information. There is, in principle, no way from the inside to decide which is "the real one." This is the basis for what Episode 1 said: "a \(c=\text{constant}\) universe and a \(c\cdot t=\text{constant}\) universe can't be told apart."

STEP 03Run it — break the theorem and a "fake wave" is born

So what happens if you break the theorem? What if \(f_s\) is smaller than \(2f_{\max}\) — that is, you measure too slowly? This is the most interesting figure of the episode. Below, keep the original wave's frequency fixed and reduce only the sampling (the number of points).

Fig. 2: Measure too slowly…? Sample the original (blue, fast) wave coarsely and the points look like "a different, slow wave (red)"
The condition holds (f_s > 2·f_max)
original wave (real, fast) recorded points fake wave seen from the points (alias)

As you lower \(f_s\), the moment you cross a threshold (\(f_s = 2f_{\max}\)), the row of points starts to look like not the original fast wave (blue) but an entirely different slow wave (red). This is aliasing — a fast wave that the too-slow recording couldn't capture "disguises" itself as a fake slow wave. When this happens on a CD, you'd hear a low tone that isn't really there. That's why, when making a CD, everything above 20 kHz is cut before recording, to prevent this disguise.

The fake wave's frequency can be computed

The alias frequency

For an original wave of frequency \(f\) sampled at \(f_s\), the fake wave you end up seeing has frequency

$$f_{\text{alias}} = |\,f - f_s\,|\quad(\text{nearest case})$$

For example, \(f=30\), \(f_s=40\) gives a fake wave of \(|30-40|=10\). The fast wave (30) disguises itself as a slow wave (10) — match the numbers in Fig. 2 and you'll see this relation.

◇ ◇ ◇

STEP 04So — discrete and continuous part ways only "at the edge"

The line the sampling theorem draws can be summarized like this.

Discrete vs. continuous — where can they be told apart?

Inside the band (sound at or below \(f_{\max}\)) → discrete and continuous are perfectly equivalent, indistinguishable.
At the edge of the band and beyond (components too fast) → the difference appears, in the form of aliasing.

A record and a CD can't be told apart when listened to in your room (inside the band) — guaranteed by the theorem. The difference only matters at the edge of the highest pitches almost no one can hear (the edge of the band). What audiophiles argue about with "hi-res" is precisely this edge region.

And this is the message of the whole series. Episode 1's "continuous universe with \(c=\text{constant}\)" and "the discrete view of \(c\cdot t=\text{constant}\)" have exactly the same structure — within the range we can touch (inside the band), they're perfectly equivalent and indistinguishable. The difference can matter in principle only at the very finest edge of the universe (the unimaginably microscopic world called the Planck scale). That, no one has yet been able to check by experiment.

The reveal (a bridge to the cosmos) What corresponds to sound's "highest frequency \(f_{\max}\)" is, for the universe, thought to be a "shortest length" = the Planck length (about \(10^{-35}\) meters). If the universe really has this "bandwidth limit," then the universe is a "band-limited signal," and writing it as continuous or as discrete is exactly the same — the record-and-CD relationship would apply to the universe itself.
The honest line (kept light, just one)

"Indistinguishable" is strictly a statement inside the band. Whether the universe truly has a bandwidth limit (a shortest length), and how discrete and continuous part ways at the edge, is still an unsolved frontier of physics. So while "it can be written as either continuous or discrete" is correct, "which the universe really is" can't be decided by this theorem alone — that's a door for your generation to open by experiment.

Practice problems (solvable with just this episode's formulas)
  1. A sound has a highest frequency \(f_{\max}=15{,}000\ \mathrm{Hz}\). What minimum sampling frequency is needed to record it perfectly?
    Show answer
    \(f_s > 2\times 15{,}000 = 30{,}000\ \mathrm{Hz}\). Measure more than 30,000 times per second and you can reconstruct it with zero error.
  2. For \(f=35\), \(f_s=40\), what is the frequency of the fake wave (alias) you end up seeing?
    Show answer
    \(f_{\text{alias}}=|35-40|=5\). The fast wave (35) appears disguised as a slow wave (5). Check it in Fig. 2.
  3. Restate "a record and a CD can't be told apart" in terms of this series' cosmology, in one sentence.
    Show answer
    e.g. "Inside the band (the fineness we can see), the continuous universe and the discrete view carry exactly the same information and can't be told apart; they part ways only at the edge called the Planck scale." A restatement of Episode 1's conclusion in the language of the sampling theorem.

Wrap-upThe same music on two discs — the theorem that guarantees it

The sampling theorem says: for a signal with a bandwidth limit, just measuring at \(f_s>2f_{\max}\) lets you perfectly reconstruct the original smooth wave from discrete points. So a record (continuous) and a CD (discrete) can't be told apart within the reach of the ear. It's the same music, held on two media.

This theorem is the mathematical identity of the metaphor running through the series. The \(c=\text{constant}\) universe and the discrete view of \(c\cdot t=\text{constant}\) are perfectly equivalent inside the band we can touch. Whichever disc you listen to, the music playing is one — that's what we've wanted to say since Episode 1.

This is Episode 5 of "Cosmology That Clicks," a reading piece for curious high-schoolers. Every figure is real data computed on the spot in your browser, visualizing the sampling theorem (Nyquist–Shannon) and aliasing (\(f_{\text{alias}}=|f-f_s|\), nearest case). A CD's 44.1 kHz sampling frequency and the ~20 kHz range of hearing are real values. Whether the universe has a shortest length (bandwidth limit) is an unsolved problem in physics, and the equivalence of discrete and continuous holds only inside the band. — To print, use your browser's Print → Save as PDF (in the printed version, sliders and answers are static/hidden).

Print / save as PDF: ⌘+P (Ctrl+P on Windows). On screen, sliders recompute the figures and you can watch aliasing appear. "Show answer" opens the solutions.