Dr. Loveless Curiosity Lab

Weaving Curves

Closed Loops, Pairing Rules, String Art, and Curves in 2D and 3D

How can straight strings suggest smooth curves? This page is an exploration space for string art, weaving rules, and visual patterns. We are using the graphs to make examples, notice patterns, and collect questions worth proving later.

The story starts with Concept 3, a draggable open-string model with four movable points. Concept 4 came next after a short literature review, when we tried to recreate some of the shapes we were seeing. Concepts 1 and 2 are the focused 2D and 3D exploration galleries that grew from those first experiments.

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Explore the Weaving Rule

Start with the original draggable experiment, then compare it with closed-loop versions and 2D/3D weaving explorations. The goal is to notice: What shapes appear? What changes when the spacing changes? Which visuals seem to have a clean mathematical explanation?

Concept 1: 2D Weaving
Planar loops: ellipse, Lissajous curve, rose curve, and epicycle curve.

Pick a curve and connection rule. Look for patterns, symmetries, dense regions, and possible hidden curves. Unused parameters are greyed out for the selected curve.

Connection rule parameters

Curve parameters

Feel free to click on and move the graph around.

Loading 2D weaving graph...
Concept 2: 3D Weaving
Spherical loops: tilted great circle, spherical spiral, spherical Lissajous, and spherical epicycloid.

The same visual idea now moves through 3D space. Instead of trying to name every surface, explore how straight connectors create twisting, ribbon-like, or sculpture-like forms.

Connection rule parameters

Curve parameters

Feel free to click on and move the graph around.

Loading 3D weaving graph...
Concept 3: Draggable Envelope Builder
Drag the four boundary points, adjust spacing, and watch how the suggested envelope changes.

This graph is Marcus's open string-art model. Drag points A, B, C, and D directly in the graph. The red-orange curve is a candidate envelope traced by the family of strings.

Spacing rule: start with evenly spaced values \(U=0,1/n,2/n,\ldots,1\). The left pegs use \(U^p\) and the right pegs use \(U^q\). When \(p=q=1\), the spacing is even.

Click and drag A, B, C, D in the graph. Use the toggle buttons to compare the strings with or without the candidate envelope.

Loading draggable envelope graph...
Concept 4: Closed-Loop Envelope Experiments
After a short literature review, we tried to recreate some of the shapes we were seeing.
Visual goal: compare three circle rules, roses, and Lissajous loops. Which presets look like known caustics? Which look like new questions?

Use the preset buttons first, then animate \(R_0\) or \(N\). See below the graph for the connection rules.

Loading closed-loop envelope graph...

Connection rules and curve equations

Circle 1 \(R(t)=(\cos t,\sin t)\) connect \(R(t)\) to \(R(t+R_0)\)
Circle 2 and Circle 3 Circle 2: connect \(R(t)\) to \(R(2t)\) Circle 3: connect \(R(t)\) to \(R(3t)\)
Rose \(R(t)=(\cos(A_0t)\cos(B_0t),\cos(A_0t)\sin(B_0t))\) Here \(A_0\) changes the radial rhythm and \(B_0\) changes the spin.
Lissajous \(R(t)=(\sin(A_0t),\sin(B_0t))\) Here \(A_0\) and \(B_0\) are the two frequencies.

Where It Started

This began with the classic string-art question: why do straight strings seem to draw a curve?

Marcus's first visual was the draggable open-string model in Concept 3. The closed-loop graph in Concept 4 came next after a short literature review, when we tried to recreate some of the shapes we were seeing. Concepts 1 and 2 became the focused exploration gallery built from those experiments.

Going Further

The next step is to gather many visual examples and turn them into good questions: Which patterns are stable? Which depend strongly on the pairing rule? Which have a clean hidden curve?

Later versions can add curved connectors, caustic-style reflected rays, reverse construction problems, physical peg-board templates, and 3D-printable sculpture ideas.

Further Reading