Lissajous 3:2 Figure
Lesson notes
Overview
A 3:2 frequency ratio draws a closed knot-like curve. This topic is rated intermediate and is number 204 of 245 in Mathematics. Builds on the basics with relationships, methods and trade-offs.
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
With frequencies in ratio 3:2, x = sin 3t and y = sin 2t, the curve closes after one period and forms a looping knot-like figure. The ratio counts the tangent points: it touches the top and bottom of its bounding box 3 times along one side and 2 along the other. Simple ratios give clean closed figures, and when the ratio is not exactly right, the figure appears to rotate slowly, which is how engineers compare frequencies.
Reading the relationship
The expression "x = sin 3t, y = sin 2t" is a compact summary of the idea. It uses the symbols x, t, y; name what each one stands for before you start. Notice that it contains a trigonometric function, so the result repeats as the angle goes round. 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
Predict the outcome first, then test extreme slider values to find where the pattern changes or breaks.
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 “Projectile Path as Parametric Curve”, the topic before it in Mathematics. Once you are comfortable, continue with “Harmonograph”.
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 “Lissajous 3:2 Figure” 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 Lissajous 3:2 Figure?
A 3:2 frequency ratio draws a closed knot-like curve. It is an intermediate topic in Mathematics.
How do I use the Lissajous 3:2 Figure 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 x = sin 3t, y = sin 2t 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?
Builds on the basics with relationships, methods and trade-offs. If the quiz feels hard, reread the lesson notes and try the previous topic first.
What should I learn before this?
Start with “Projectile Path as Parametric Curve”, 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.