In this short, we show animations of four of the most famous proofs without words: the formula for the sum of the first n integers; the pythagorean theorem using negative space/sliding rectangles/Chou pei suan ching; the formula for the sum of the first n odd integers; and the infinite geometric series of positive powers of 1/2 (using a rectangle/square dissection of a unit area square). We include brief justifications for these visual proofs.
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For a slightly different version of this short without words, see:
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For longer videos with related animations, see
Summing Integers: [ Ссылка ]
Pythagorean Theorem: [ Ссылка ] (and old version: [ Ссылка ])
Summing Odds: [ Ссылка ]
Geometric Series: [ Ссылка ]
The first proof was known to the ancient greeks (cited by Martin Gardner), the second proof is adapted from the Chou pei suan ching (around 200 BCE according to Roger Nelsen), the third is attributed to Nicomachus of Gerasa by Roger Nelsen, and the final one is attributed to Warren Page (from the September 1981 issue of Mathematics Magazine, page 201 - [ Ссылка ] ). The first three can all be found in Roger Nelsen's first compendium, "Proofs Without Words: Exercises in Visual Thinking: [ Ссылка ] .
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To learn more about animating with manim, check out:
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