One Bit Too Many
Huffman coding is provably optimal and famously within one bit of entropy. On the right source that one bit is a twelve-fold overhead.
Page 6 of 9
Days 42 to 31
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Modes That Stop Being Harmonic
Twelve masses on springs. The first few modes are the harmonics you would expect. The last few are nowhere near.
sound - 40
The Shape Decides
The convex hull of n random points has far fewer than n vertices. How many fewer depends on the shape they came from, not on n.
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A Tone in Neither Ear
The same noise in both ears, with one narrow band phase-inverted on the right. Each ear alone is featureless. Together they produce a pitch.
sound - 38
Every Boundary Is All Three
Newton's method on z³ − 1 has three basins. Pick any point on any boundary between them and all three are arbitrarily close.
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The Wheel Goes Backwards
Two wheels on one shaft, one seen continuously and one at 24 frames a second. Half of all speeds make the second wheel spin the wrong way.
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A Filter That Lies
A Bloom filter's false-positive rate has a famous closed-form formula. The formula is right about filters in general and wrong about yours.
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Which Half Matters
A Fourier transform splits an image into sizes and offsets. Throw away the sizes and you can still read it. Throw away the offsets and you cannot.
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Cloth
No springs, no stiffness, no forces between the threads. Just moving the links back to the right length, over and over.
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The Best Fraction
22/7 is not just close to π. Nothing with a smaller denominator is closer, and there is a machine that finds such fractions.
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The Shadow A Sound Casts
A loud sound hides quiet ones for a fifth of a second afterwards, and for a few milliseconds beforehand.
sound - 31
The Angle A Sunflower Knows
Turn 137.507° between seeds and they pack perfectly. Turn 137.3° and it falls apart.