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Fourier Series Approximations

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Part of Visual Mathematics Project

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Last updated Jan 30, 2026. Clone to remix or explore the blocks below.

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examples

Frequencies Everywhere

From MP3s to Instagram filters

Side-by-side comparison of time domain waveform and frequency domain spectrum showing peaks at different frequencies

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Time domain (left) vs frequency domain (right): same sound, two different views

Audio equalizer with multiple frequency sliders showing bass, mid, and treble controls

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Equalizer: each slider controls one frequency range independently (Fourier components)

Original high-quality image next to JPEG compressed version showing slight quality loss

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JPEG compression: remove high frequencies humans barely notice, keep low frequencies

Visual showing multiple sine waves of different frequencies adding up to approximate a square wave

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Square wave built from odd harmonics: 1st, 3rd, 5th, 7th... frequencies

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concept

🎵 How Your Phone Stores 1000 Songs

Your voice saying "hello" contains maybe 50 different frequencies mixed together.

MP3 compression: break the sound into frequencies, delete the ones you can't hear (above 20kHz, below 20Hz), keep the rest.

Result: 10% of original size, sounds identical.

The math behind this? Fourier decomposition — the visualization above.

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quiz

Apply Fourier Thinking

Think in frequencies for audio and images

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In the visualization, adding more circles (more terms in the Fourier series) makes the approximation better. Mathematically, the infinite series $\sum_{n=1}^{\infty}$ converges to the exact square wave. In practice, why stop at finite terms?