Mechanical Linkages: Geometry Proofs


This unit is part of the special topic “Mechanical Linkages and Deductive Geometry”. Mechanical linkages – sets of hinged rods – form the basis of many everyday objects such as folding umbrellas and car jacks and are built using the geometry of triangles and quadrilaterals. They offer rich potential for investigating geometry, starting with real-life objects, then making working models and then using pre-prepared dynamic geometry software.

Designed for Year 10 (and 10A), this unit looks at more complex linkages, some invented by famous mathematicians to solve important problems of their time, such as turning circular motion into linear motion. Another lesson analyses a mechanical calculator in the shape of a monkey.. The lessons mostly use geometric facts from earlier years, and aim to assist students to develop their own proofs. Later lessons provide challenge for strong students. The lessons are independent of each other and can be taught in any combination.

Read the Teachers' Guide, also included in the unit download, for teaching advice and practical hints for the constructions.  


Lesson 1: Chebycheff’s Linkage

Students investigate Chebycheff’s linkage and determine whether it produces approximate or exact linear motion from circular motion. They make a physical model, observe the motion of points and try to explain them, and then observe a computer simulation. Visually, the motion appears as a very convincing straight line, so students are misled until measurements expose the small variation from linear motion. This establishes awareness of the need for proof in other geometry contexts. Students can also use Pythagoras’ theorem to establish some lengths.

Lesson 2: Scott Russell Linkage

Students investigate the design and operation of a car jack based on the Scott Russell linkage. As the horizontal screw is turned, the car attachment point moves vertically (perpendicular to the screw). Students make a physical model and operate a computer simulation. Students then use the geometry of the linkage to find if the motion is really vertical. This involves reasoning about angles in three connected triangles.

Lesson 3: Pascal’s Angle Machine

Students investigate the design and operation of an angle trisector invented by mathematician Blaise Pascal. The lesson has been deliberately named Pascal’s angle machine rather than Pascal’s angle trisector, so that students first explore the angle machine to find out what its purpose is, and then how it would be used. They use physical models and computer simulation to explore. Students then use the geometry of isosceles triangles and exterior angles to prove why the machine works.

Lesson 4: Consul

Students construct a physical model of Consul, a toy calculator, and use a computer simulation of Consul to explore how the geometric design enables the product of numbers between 2 and 12 to be displayed. Students may then develop a proof based on a sequence of deductive reasoning.

Lesson 5: Sylvester’s Pantograph

Students construct a physical model of Sylvester’s pantograph, a drawing instrument designed for copying drawings. They explore how the copied image compares with the original drawing and use a computer simulation of the pantograph to explore how the geometric design of the pantograph allows the pantograph to work. Students then develop a geometric proof based on a sequence of deductive reasoning in which they use their knowledge of rhombus properties and congruent triangles.

Lesson 6: Peaucellier’s Linkage

Peaucellier’s linkage converts circular motion to linear motion. Students explore how the linkage moves by constructing a physical model and using a computer simulation of the linkage. They discover how the geometric design of the linkage allows it to produce exact linear motion.


Last updated December 10 2018.