Discover and read the best of Twitter Threads about #50famouscurves
Most recents (12)
The composition of two harmonic motions with frequencies ω and ω' produces the beautiful Lissajous curves: x=cos(ωt), y=sin(ω't). [Wiki bit.ly/2HJB4v8] Inspired by @juliomulero. #50FamousCurves
How to draw Lissajous curves? Build a pendulum whose length is different in perpendicular directions. The pendulum will trace a Lissajous curve. [Source: ]
Lissajous curves on an oscilloscope #50FamousCurves
This morning I've met a group of highschoolers and we had a great time playing with some of the #50FamousCurves #LFoS19 
It was a 50-minute maths session organized as part of the Leeds Festival of Science leeds.ac.uk/festivalofscie…. #LFoS19 @STEMatLeeds
I showed them the Fibonacci sprial and told them why insect are attracted to lights. (Spoiler: they're not, but they don't have a choice.)
Here is how you draw the Witch of Agnesi. It's a nice bell-shaped curve that is used as the probability density for the Cauchy distribution. Surprisingly, the region between the curve and its asymptote has no balance point (centroid) [Wiki bit.ly/2TjqYGA] #50FamousCurves
Here is the reason it doesn't have a balance point: The x-coordinate of the centroid of a region between the x axis and the graph of a function y=y(x), A<x<B is x̅ = ∫ x y(x)dx / ∫ y(x)dx, where A and B are the bounds of the definite integrals.
1/4
1/4
In the case of the Witch of Agnesi, we have y(x)=a³/(x²+a²) with some fixed a>0 and A=-∞, B=∞. The indefinite integrals are easy to calculate:
∫ y(x)dx = ∫ a³/(x²+a²)dx = a² arctan(x/a)+C
and
∫ x y(x)dx = ∫ x a³/(x²+a²)dx = a³ ln(x²+a²)/2+C'.
2/4
∫ y(x)dx = ∫ a³/(x²+a²)dx = a² arctan(x/a)+C
and
∫ x y(x)dx = ∫ x a³/(x²+a²)dx = a³ ln(x²+a²)/2+C'.
2/4
If you trace a point on the circle rolling inside another circle with four times its radius you get a star-shaped* curve called the astroid. ✨ [Wiki bit.ly/2U0A0oW] #50FamousCurves
*I know, I know, stars aren't actually spikey (bit.ly/2EkZrw3).
*I know, I know, stars aren't actually spikey (bit.ly/2EkZrw3).
Drawing an astroid with a do-nothing machine. #50FamousCurves
By the way, do-nothing machines can be used to draw ellipses as well.
Wishing you all a Happy Valentine's Day! ❤️ Share this heart curve (cardioid) and your love for mathematics with friends, loved ones and someone special your life revolves around! 😊 #ValentinesDay #50FamousCurves
🎶 Don't play with my heart 🎶 play with @moenig's heart instead in this cool @SnapCloud project.
snap.berkeley.edu/snapsource/sna…
snap.berkeley.edu/snapsource/sna…
Follow a point on the rim of a wheel as it rolls (without slipping) on a flat road and you'll get a cycloid. It's a beautiful curve with all sorts of interesting applications. [Wiki bit.ly/2St3vmA] #50FamousCurves
Fun fact: The area under one arch of a cycloid is exactly three times the area of its generating circle. #50FamousCurves
Fun fact: The period of a cycloidal pendulum is independent of its amplitude. (This isn't the case for the simple pendulum.) This was discovered by Huygens in his search for more accurate pendulum clock designs ≈350 years ago. #50FamousCurves
[Gif source: Wikipedia, Rem088roy]
[Gif source: Wikipedia, Rem088roy]
Take two gears with 3:1 ratio and a handle of length 3 on the smaller gear. The end of the handle traces a circle and the corresponding point on the larger gear traces a trifolium. What do you think happens with other gear ratios? [Wiki bit.ly/2Sadb5o] #50FamousCurves
The trifolium belongs to the family of rose curves defined by the polar equation r=a·sin(kφ) associated with gear ratio k:1. If k is odd, there are k petals, but if k is even, you get 2k petals. Here is the case when k=2. #50FamousCurves
Table of rose curves r=sin(kφ) with k=n/d (gear ratio n:d) and n,d=1,2,...,7. #50FamousCurves
What's the shape of a hanging chain? Galileo knew that it's something like a parabola, but not quite. This curve is called the catenary and its the graph of a hyperbolic cosine function cosh(x)=(eˣ+e⁻ˣ)/2. [Wiki bit.ly/2RmyP15] #50FamousCurves
Here's a derivation via Noether's theorem. tamasgorbe.wordpress.com/2015/05/27/wha…
Fun fact: Smooth travel on square wheels is possible as long as the road consists of upside down catenary arcs (the length of which equals the side length of the wheel). [Source: ] #50FamousCurves
How to draw ∞? Make a three-bar linkage with length ratios 1:√2:1 and the ends fixed at distance √2 apart. As white rods go around, the midpoint of the longer bar traces a curve called Bernoulli's Lemniscate. It's infinitely nice! [Wiki bit.ly/2sFDUaS] #50FamousCurves
Fun fact: The word "lemniscate" comes from the Ancient Greek λημνίσκος (lēmnískos) meaning "ribbon". #50FamousCurves
How to draw ∞?
1) Take the unit sphere S: x²+y²+z²=1.
2) Draw the hyperbola H: x²-y²=a² in the (x,y) plane.
3) Project H to S by shining light from the North Pole (0,0,1).
4) Turn the curve on S upside down.
The rotated curve projects to Bernoulli's lemniscate.
#50FamousCurves
1) Take the unit sphere S: x²+y²+z²=1.
2) Draw the hyperbola H: x²-y²=a² in the (x,y) plane.
3) Project H to S by shining light from the North Pole (0,0,1).
4) Turn the curve on S upside down.
The rotated curve projects to Bernoulli's lemniscate.
#50FamousCurves
Dogs on leashes dragging their owners. Surely, this has nothing to do with maths! Well, if they are running along a line perpendicular to the initial direction of the leash, the owners will be dragged along a curve called tractrix.🐶 [Wiki bit.ly/2stpcUl] #50FamousCurves
Look at them doggos go. Aren't they cute? Unfortunately, I couldn't find the source of these cute drawings, hence the lack of image credit. If anyone finds the OP let me know!
Fun fact about the tractrix: The envelope of lines perpendicular to the tractrix is a hyperbolic cosine, which is also called a catenary, because it's the curve describing the shape of a hanging chain. #50FamousCurves
Ever seen insects spiralling to a lamp? They actually want to fly in a straight line by looking at the light source at a constant angle. This would work with the Sun or Moon, but lamps fool them into flying along logarithmic spirals. [Wiki bit.ly/2Vzt2sf] #50FamousCurves
That "φ" should be a "t" in the parametric equations.
Fun fact about logarithmic spirals: They appear in the Mandelbrot set. Namely, the Seahorse Valley (the region between the "head" and the "body" of the set) is full of logarithmic spirals. #50FamousCurves
If you trace a point on the circle rolling around another circle of equal radius you get a heart-shaped curve called the cardioid. ❤️ It has all sorts of cool properties. This week on #50FamousCurves we'll be circling around this beauty! 😀 [Wiki bit.ly/2GEL2Oy]
Fun fact about the cardioid: Its name comes from the Greek καρδία "heart". #50FamousCurves
How to draw a cardioid? You only need two cups, a pencil and a couple of rubber bands. #50FamousCurves
