[Paper Review] A compilation of LEGO Technic parts to support learning experiments on linkages

Zoltán Kovács, Benedek Kovács|arXiv (Cornell University)|Dec 1, 2017

Teaching and Learning Programming2 references3 citations

TL;DR

This paper presents a curated LEGO Technic parts compilation to enable hands-on learning of planar linkages in STEM education. Using physical models and dynamic geometry software (GeoGebra), it demonstrates how linkages like Chebyshev’s, Watt’s, and Hart’s mechanisms produce approximate curves—such as near-straight lines or lemniscates—enabling students to explore geometric loci, algebraic curves, and mechanical accuracy through tactile and digital validation.

ABSTRACT

We present a compilation of LEGO Technic parts to provide easy-to-build constructions of basic planar linkages. Some technical issues and their possible solutions are discussed. To solve questions on fine details---like deciding whether the motion is an exactly straight line or not---we refer to the dynamic mathematics software tool GeoGebra.

Motivation & Objective

• To develop an accessible, low-cost method for teaching geometric and mechanical concepts using LEGO Technic parts.
• To address the challenge of visualizing and verifying the accuracy of linkage motions, especially near-straight lines or closed curves.
• To bridge physical construction with digital mathematical validation using GeoGebra for deeper conceptual understanding.
• To provide a didactic framework that supports learning across multiple mathematical levels, from circles to polynomial factorization and automated theorem proving.
• To offer practical, reproducible instructions for building and troubleshooting key linkages, enhancing classroom implementation.

Proposed method

• Designing and compiling a minimal set of LEGO Technic parts (beams, connectors, pen refills) to construct five core planar linkages: compass, Chebyshev’s, Chebyshev’s λ, Watt’s, and Hart’s inversor.
• Using a pen refill inserted into a connection peg to trace motion on paper, enabling direct observation of geometric loci.
• Employing dynamic mathematics software GeoGebra to analyze and verify the mathematical accuracy of traced curves, especially distinguishing between exact and approximate straight lines.
• Creating digital models with LEGO Digital Designer (LDD) to document step-by-step assembly, ensuring precise alignment of holes and connectors.
• Applying mechanical solutions such as Pythagorean triangles (3-4-5) to achieve accurate right angles and using half-beams to replace unavailable full beams.
• Introducing a method to extend Hart’s inversor’s motion by temporarily reconfiguring the antiparallelogram into a parallelogram to access additional curve branches.

Experimental results

Research questions

• RQ1 How can LEGO Technic parts be systematically compiled to support the construction of fundamental planar linkages for educational purposes?
• RQ2 To what extent can physical LEGO models accurately represent mathematical curves such as near-straight lines or lemniscates, and how can discrepancies be detected?
• RQ3 How can dynamic geometry software like GeoGebra be integrated with physical models to validate the algebraic and geometric properties of linkage motions?
• RQ4 What technical challenges arise during LEGO linkage construction (e.g., missing beam lengths, restricted movement), and how can they be resolved using available parts or alternative configurations?
• RQ5 In what ways can the combination of physical construction and digital analysis enhance STEM learning in geometry, algebra, and mechanics?

Key findings

• Chebyshev’s linkage produces a curve that closely resembles a straight line but is not mathematically exact, with accuracy dependent on geometric parameters.
• The Chebyshev λ mechanism successfully draws the full closed sextic curve in one continuous motion, providing visual evidence that neither the circle-like nor line-like segments are exact.
• Watt’s linkage traces a lemniscate-like curve, which is not a perfect straight line, but remains widely used in engineering due to its practical accuracy.
• Hart’s inversor can trace a sextic curve, but its motion is limited by mechanical constraints; the path can be extended by reconfiguring the antiparallelogram into a parallelogram.
• Using a 3-4-5 Pythagorean triangle setup enables precise right angles in linkages, improving mechanical stability and accuracy.
• Digital modeling with LEGO Digital Designer (LDD) enables accurate, step-by-step documentation of complex linkages, especially critical for aligning connector pegs with holes in multi-part assemblies.