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Cardboard Spirograph Drawing Machine

Roll a toothed cardboard gear inside a ring and trace repeating curves from different pen-hole positions.

One circle rolls inside another while the pen rides off-center. Tooth counts and pen radius decide whether the curve closes quickly or draws a dense repeating pattern.

Difficulty
Intermediate
Build time
75-110 min
Estimated cost
$0-$6
Age range
11-17
Workspace
A clear table about 90 cm wide

The finish line

What you will build

The moving gear rolls through a complete closed pattern without tooth skipping, and three pen positions produce visibly different curves.

Learning goals

  • Identify how hand-guided rolling gear motion produces repeating pen curve.
  • Construct and explain a rotary rolling-to-planar drawing path system.
  • Measure how the pen-hole radius changes performance.
  • Diagnose losses caused by tooth friction and paper drag.

Before you build

Materials, tools, and safety

Reuse-material cost: $0-$3 with reused materials. Supervision: Adult help recommended for sharp or heated tools.

Tools

  • Ruler
  • Pencil
  • Scissors
  • Low-temperature glue gun or tape
  • Compass
  • Adult-operated craft knife

Low-cost swaps

  • Use clean shipping-box cardboard instead of buying sheets.
  • Replace hot glue with strong tape and folded tabs.
  • Use printed circles with evenly spaced peg holes and a string-guided rolling disk.

Project-specific safety

  • An adult should handle craft knives and make difficult starter cuts.
  • Let hot glue cool before pressing a joint or testing moving parts.
  • Use capped washable markers and have an adult cut internal teeth or pen holes.

Orient the build

Place the build so hand-guided rolling gear motion is on your left and repeating pen curve is on your right. Call the side facing you the front, the far side the back, the tabletop the bottom, and the opposite face the top.

Build it

Step-by-step instructions

  1. Step 1

    Draw the fixed ring

    Mark two concentric circles and divide the inner edge into equal tooth spaces.

    Keep the ring at least 3 cm wide.

  2. Step 2

    Laminate and cut the ring

    Glue layers with crossed corrugation and cut smooth internal teeth.

    Tape the outer rim to a drawing board.

  3. Step 3

    Build the moving gear

    Choose a smaller tooth count that matches the same pitch and laminate it.

    Test rolling around the full ring without a pen.

    Builder checkpoint: After build the moving gear, the first subassembly should stay aligned when handled gently.

  4. Step 4

    Add pen positions

    Mark holes at 25, 50, and 75 percent of gear radius.

    Reinforce each with a small tape patch.

    Watch for: If this stage binds or drifts, inspect hand wobble before adding more parts.

  5. Step 5

    Set the drawing surface

    Place paper under the ring and secure all corners.

    Mark the gear and ring start teeth.

  6. Step 6

    Trace the first curve

    Insert the marker loosely and roll the gear without lifting or sliding.

    Continue until start marks meet again.

    Builder checkpoint: After trace the first curve, operate the build slowly and confirm that repeating pen curve begins without binding.

  7. Step 7

    Change pen radius

    Repeat on clean paper using a farther hole.

    Keep gear and ring tooth counts unchanged.

  8. Step 8

    Change gear ratio

    Use the second moving gear and predict closure before tracing.

    Label each drawing with counts and pen radius.

    Builder checkpoint: At the final checkpoint, The moving gear rolls through a complete closed pattern without tooth skipping, and three pen positions produce visibly different curves.

See the engineering

Why it works

Input
hand-guided rolling gear motion
Output
repeating pen curve
Motion
rotary rolling-to-planar drawing path
Energy losses
tooth friction, paper drag, gear flex, hand wobble
Cardboard Spirograph Drawing Machine concept diagram with labeled input, output, and motion arrows.
The rotary rolling-to-planar drawing path motion path, with the main efficiency losses called out.

Why this works

Periodic rolling geometry

A small gear rolling inside a fixed ring combines rotation about its own center with orbit around the ring center. An off-center pen traces a hypotrochoid whose repeat depends on tooth-count common factors.

Look for: Mark one moving-gear tooth and count its rotations before the pen returns to the start point.

Where the energy goes

Efficiency and losses

The ideal model leaves out tooth friction, paper drag, gear flex, hand wobble. These effects turn some input energy into heat, sound, vibration, or unwanted motion, so measured performance will be lower than an ideal calculation.

Look for: Run the build slowly and locate the first place where tooth friction becomes visible or audible.

Math bite

Predict rotations to close

Formula: ring cycles to close = moving teeth / gcd(ring teeth, moving teeth)

  • Ring = 60 teeth
  • Moving gear = 24 teeth
  • gcd = 12

Substitute: cycles = 24/12 = 2

Result: The pattern closes after two trips around the ring.

Different common factors change repeat length.

Tooth skipping or hand slip can prevent exact closure.
Brick-building meme reading: Chuck Norris does not build LEGO; he roundhouses the bricks into sculptures.
The curve almost closed, which is how one skipped tooth signs its artwork.Image supplied by the site owner.

Make it behave

Test, troubleshoot, and tune

Controlled test

Start here: Roll the moving gear around the ring once without a marker.

Success looks like: The gear stays engaged and the traced pattern closes within one tooth of the start mark.

Measure: Closure error and number of ring cycles.

Change: the pen-hole radius

Keep constant: ring, moving gear, tooth counts, paper, marker, and start marks

  1. inner pen hole
  2. middle pen hole
  3. outer pen hole
Troubleshooting guide
SymptomLikely causeConfirm itFix
The gear skips teethTooth pitch differs or ring flexesRoll slowly at the failing sectionCorrect pitch and brace the ring
The pattern will not closeA tooth skipped or start was misidentifiedUse marked teeth and count cyclesRetrace slowly from aligned marks
The marker tears paperHole grips tightly or hand force is highMove with the marker lifted slightlyWiden hole and reduce pressure
The moving gear liftsHand pushes inward or ring edge is unevenPractice one dry orbitGuide from the gear center and smooth edges

Choose your tradeoff

Accurate tooth pitch matters more than decorative shape. Keep the marker loose and vertical; larger pen radius creates wider curves but magnifies wobble and tooth error.

Keep experimenting

Try another version

Easier

Smooth rolling circles

Trace with a string-linked pen before adding teeth.

Performance

Closure catalog

Compare gear pairs with different greatest common factors.

Creative

Layered patterns

Overlay two colors from different pen radii.

Build together

Classroom and access options

Classroom version

Teams can compare the pen-hole radius while keeping ring, moving gear, tooth counts, paper, marker, and start marks. Assign builder, tester, recorder, and explainer roles; have each team predict the result before collecting three trials.

Access adaptations

  • Pre-cut repeated pieces and mark fold lines with high-contrast ink.
  • Use large tabs, binder clips, and tape for easier one-handed assembly.
  • Use a wide marker grip, high-contrast paper, and large gears with tactile start marks.

Reflect on the design

  1. How did the pen-hole radius change the measured result?
  2. Where did tooth friction affect the build most strongly?
  3. What evidence shows that periodic rolling geometry explains the motion?
  4. Which change would improve repeating pen curve without creating a new problem?
Glossary
Periodic rolling geometry
A small gear rolling inside a fixed ring combines rotation about its own center with orbit around the ring center.
Input
The action or energy supplied to a system; here it is hand-guided rolling gear motion.
Output
The useful response produced by a system; here it is repeating pen curve.
Efficiency
The fraction of input energy that becomes useful output instead of friction, sound, heat, or unwanted motion.

Build your dreams

One build can start the next.

Share what you learned, change one variable, and help another builder understand what worked.

Explore more guides

Sources and build notes

An original BrickLabClips cardboard machine with dimensionally specified construction.

  • Cardboard design verification: Dimensions, fold allowances, repeated-motion joints, and likely load paths received an editorial geometry review.

Written and edited by BrickLabClips. Published 2026-07-22; updated 2026-07-22.

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