Lemniscate of Bernoulli
Lesson notes
Overview
The figure-eight curve that inspired the infinity symbol. This topic is rated beginner and is number 39 of 245 in Mathematics. Core ideas and first intuitions. No prior knowledge needed.
What you are seeing
The panel above shows a path traced by a parametric curve as its parameters change. The animation is drawn live from the model, so everything you see responds to the sliders. Press Space to pause, use Restart to replay from the start, and change the speed to slow down quick events.
In depth
The lemniscate is the figure-eight curve defined by (x² + y²)² = a²(x² − y²), where the product of the distances to two foci equals a constant. Jacob Bernoulli studied it in 1694, and the symbol ∞ was introduced soon after, though its link to the curve is a matter of debate. The curve crosses itself at the origin, and its arc length leads to elliptic integrals, an important chapter in the history of mathematics.
Reading the relationship
The expression "(x²+y²)² = a²(x²−y²)" is a compact summary of the idea. It uses the symbols x, y, a; name what each one stands for before you start. Notice that it contains a squared term, so doubling that quantity changes the result by a factor of four. Then use the sliders to see how changing one quantity changes the others.
A guided exploration
1. Run the animation once with the default settings and describe in words what happens. 2. Change one slider at a time and note which quantity responds most. 3. Push each slider to its extreme and decide whether the result still makes sense.
Try this
Start by only moving one slider at a time. Say aloud what you expect, then check.
Common pitfalls
A frequent slip is memorising the rule without the picture. If a result surprises you, sketch the graph or test a simple number by hand before trusting the algebra.
How it connects
It builds on “Cardioid”, the topic before it in Mathematics. Once you are comfortable, continue with “Spirograph Curves”.
Why it matters
Mathematical ideas like this one are the language that physics, engineering, computing and statistics are written in.
Where you might use it
- Revising “Lemniscate of Bernoulli” before a class, exam or certification
- Checking working on homework and exam questions by watching the graph respond
- Modelling growth, change and uncertainty in finance, data science and statistics
- Building the reasoning behind the algorithms and engineering tools people use every day
Frequently asked questions
What is Lemniscate of Bernoulli?
The figure-eight curve that inspired the infinity symbol. It is a beginner topic in Mathematics.
How do I use the Lemniscate of Bernoulli simulation?
The panel at the top shows a path traced by a parametric curve as its parameters change. Move the sliders one at a time, then press Space to pause the animation and look closer.
What does the formula mean?
The expression (x²+y²)² = a²(x²−y²) packs the idea into symbols. Name each symbol first, then change one quantity with the sliders and see which others respond.
Who is this topic for?
Core ideas and first intuitions. No prior knowledge needed. If the quiz feels hard, reread the lesson notes and try the previous topic first.
What should I learn before this?
Start with “Cardioid”, the previous topic in Mathematics. Each topic in a subject builds on the ones before it.
How do I track my progress?
Press “Mark as learned” on this page. Progress is stored only in this browser and is never sent anywhere. The Browse page can show topics you have learned or still need to learn.