Uploaded December 2025 | Updated September 2026, 2 weeks ago
In this video, we show visualizations of ten classic mathematical constants:
1) The square root of 2 as the length of the diagonal of a 1 by 1 square.
2) Pi as the area of the circle with radius 1.
3) Tau as the circumference of the circle with radius 1.
4) The golden ratio as the number satisfying (x+1)/x=x/1.
5) The square root of 3 as the altitude or area of an equilateral triangle with side length 2.
6) Euler's number e as the ending x-coordinate yielding the area under the curve of 1/x starting at x=1.
7) The silver ratio as the horizontal length across the regular octagon with side length 1 or the area of the largest rectangle inside the same regular octagon.
8) The natural logarithm of 2, ln(2), as the sum of the convergent alternating harmonic series.
9) The plastic ratio as the limit of the ratio of side lengths in a spiral of equilateral triangles with side lengths dictated by the Padovan sequence.
10) The Euler-Mascheroni constant as the difference between the harmonic series and the area under the curve y=1/x from 1 to infinity.
If you like this video, consider subscribing to the channel or consider buying me a coffee: buymeacoffee.com/VisualProofs. Thanks!
Short form version:
youtube.com/shorts/UwVk-69nFkw?feature=share
#mathvideo #math #numbertheory #mtbos #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #iteachmath #constants #goldenratio #silverratio #plasticratio #squareroot #squareroot2 #squareroot3 #euler #eulerconstant #eulermascheroni #ln2 #logarithm #octagon #square #pi #tau
To learn more about animating with manim, check out:
https://manim.community
In this video, we show visualizations of ten classic mathematical constants:
1) The square root of 2 as the length of the diagonal of a 1 by 1 square.
2) Pi as the area of the circle with radius 1.
3) Tau as the circumference of the circle with radius 1.
4) The golden ratio as the number satisfying (x+1)/x=x/1.
5) The square root of 3 as the altitude or area of an equilateral triangle with side length 2.
6) Euler's number e as the ending x-coordinate yielding the area under the curve of 1/x starting at x=1.
7) The silver ratio as the horizontal length across the regular octagon with side length 1 or the area of the largest rectangle inside the same regular octagon.
8) The natural logarithm of 2, ln(2), as the sum of the convergent alternating harmonic series.
9) The plastic ratio as the limit of the ratio of side lengths in a spiral of equilateral triangles with side lengths dictated by the Padovan sequence.
10) The Euler-Mascheroni constant as the difference between the harmonic series and the area under the curve y=1/x from 1 to infinity.
If you like this video, consider subscribing to the channel or consider buying me a coffee: buymeacoffee.com/VisualProofs. Thanks!
Short form version:
youtube.com/shorts/UwVk-69nFkw?feature=share
#mathvideo #math #numbertheory #mtbos #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #iteachmath #constants #goldenratio #silverratio #plasticratio #squareroot #squareroot2 #squareroot3 #euler #eulerconstant #eulermascheroni #ln2 #logarithm #octagon #square #pi #tau
To learn more about animating with manim, check out:
https://manim.community
