Overview
String art creates curves from straight lines — a mathematical phenomenon that produces some of the most visually striking geometric art you can make by hand. By connecting numbered points around a circle or angle with straight thread, a smooth curve called an envelope emerges from the overlapping lines. It's the same mathematics that describes the edge of a shadow cast by a curved surface.
In this project you'll prepare a wood board, hammer nails in precise geometric arrangements, and wrap embroidery thread in mathematical patterns to create finished artwork. Two core designs: the parabolic curve (connecting points on two straight lines) and the circle pattern (connecting points around a single circle to create an inner curve).
What You'll Learn
- How a mathematical envelope emerges from a family of straight lines
- How the multiplier rule (n → kn mod N) generates different curves
- Why evenly spaced points and consistent counting are essential for the curve to appear
- The connection between string art patterns and differential geometry
- How to plan a two-color or layered string art design
Background
The curves that appear in string art are mathematical envelopes — curves that are tangent to every line in a family of straight lines. When you connect point 1 on one line to point 10 on another, then point 2 to point 9, point 3 to point 8, and so on, the resulting web of lines outlines a parabola. Each individual string is a tangent to the curve — the curve itself is never drawn directly.
This is the same concept used in differential geometry to define curves and surfaces. The cardioid (heart-shaped curve) can be generated by connecting points around a circle using a doubling rule: from point n, string goes to point 2n. The epicycloid patterns in complex string art are all generated by variations of this multiplier relationship.
The mathematical precision required — evenly spaced nails, exact pattern counting — means that any counting error propagates visibly through the finished work. This makes string art an unusual intersection of mathematical rigor and visual art.
Safety note: Hammering nails requires care — a hammer hitting a finger is painful. Supervise younger builders. Nails are sharp after hammering — be aware of the points sticking up from the board. Use eye protection when hammering.
Materials
Everything on this list is either free or available at a dollar store. No lab equipment required.
Step-by-Step Procedure
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1
Paint and Mark the Board
Paint your board with two coats of dark paint (black or deep navy). Allow to dry completely. Dark backgrounds make thread colors pop dramatically.
Choose your design and mark the nail positions lightly in pencil. For the circle design: use a compass to draw a circle in the center of the board. Mark 36 equally-spaced points around the circumference (every 10 degrees — use a protractor or print a template). Number them 0–35 with a pencil.
For the parabolic arch: draw two straight lines meeting at a right angle. Mark 20 equally-spaced points on each line, numbering from the corner outward.💡Exact spacing is critical — unequal spacing creates an uneven curve. Print a template and tape it to the board rather than measuring freehand if you want perfect results.
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2
Hammer the Nails
Wear your safety glasses. Hammer a nail at each marked point, leaving about 10mm of nail protruding above the board surface. Keep nails vertical — nails that lean will cause thread to slip off during wrapping. Tap firmly but don't countersink.
For the circle: 36 nails. For the arch: 40 nails (20 per line).💡Start with lighter taps to seat the nail, then drive with firm blows. Nails hammered at an angle are very difficult to straighten after the fact.
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3
Learn the Wrapping Pattern
For the cardioid (circle design): tie your thread to nail 0. Take the thread to nail 0×2=0? No — use the formula: from nail n, go to nail (n×2) mod 36. So from nail 1, go to nail 2. From nail 2, go to nail 4. From nail 3, go to nail 6. From nail 18, go to nail 36 mod 36 = 0. Continue until you return to nail 0. Wrap the thread around each nail once before moving to the next.
For the parabola: number each line 1–20 from the corner. Connect line-A nail 1 to line-B nail 20. Line-A 2 to line-B 19. Line-A 3 to line-B 18. Continue: the curve appears when you connect n on one line to (21−n) on the other.💡Print the number sequence on paper before you start wrapping. Looking up at the board and down at your count sheet is easier than trying to remember the formula mid-wrap.
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4
Wrap the Full Design
Complete the full circuit of your pattern. Keep tension consistent — thread should be taut but not pulling nails sideways. When you complete one full pass, tie off securely and cut. For a denser, more opaque result, repeat the same pattern 2–3 times with the same or a contrasting color.
For a two-color design: wrap the inner portion (nails 0–17) in one color and the outer portion (nails 18–35) in a contrasting color, overlapping slightly in the middle. -
5
Finish and Frame
Once all wrapping is complete, check for any nails where the thread slipped off — rewrap those. Trim all thread ends close to the knots. The finished board can be hung as-is, or you can add a simple frame of thin wood molding around the edges.
Stand back from your finished piece: the straight individual threads are still clearly visible up close, but the smooth mathematical curve dominates from any distance. This is the envelope in full effect.
Expected Results
A 40-nail circle with the doubling multiplier (n → 2n mod 40) should produce a clear cardioid within 30 minutes of wrapping. The parabolic arch design using two 15-nail lines produces a crisp parabola. Colors look best when single-color designs use 2–3 thread layers for opacity, or when contrasting colors are layered on the same pattern.
If the curve looks uneven or jagged, recount your nail assignments — a single miscount creates a visible kink in the envelope.
Your results may differ — and that's valuable data, not a mistake. Document what you actually observed, not what you expected.
Going Further
Try Different Multipliers
Repeat the circle design with multiplier 3 instead of 2 (n → 3n mod 36). The resulting curve is a nephroid (two-lobed). Try multiplier 4 (three-lobed) and multiplier 5 (four-lobed). How does the number of lobes relate to the multiplier minus 1?
Layer Multiple Colors
Complete a cardioid in one color. Without removing any thread, wrap a second pattern at a different multiplier in a contrasting color. Where the two patterns intersect creates a third visual element. How many distinct regions can you create with three colors?
Simulate in Code
Use Processing (free at processing.org) or p5.js to simulate your string art on screen. Draw a circle of points, connect n to (n×k) mod N with lines. Run the program with different values of k and N. Can you find multipliers that produce patterns you've never seen in physical string art?
