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Magic Mathworks Travelling Circus
Experiencing Mathematics through Sight, Sound, Touch and Movement

Masterclass: Archimedes’ best ideas

Archimedes masterclass session notes (pdf)

  • The ‘law of the lever’ (principle of moments), using mathematical balances.
  • The displacement principle and Archimedes’ principle: experiments on ‘the King’s Crown’ using the balances.
  • The stability of floating bodies. Simple experiment on centre of mass. Centre of buoyancy demonstrated. How principle of moments connects the two. Children asked to predict the orientation of a square wooden prism when released at 45 degrees, then make the experiment.
  • The Archimedean screw as an inclined plane coiled up.
  • That Archimedes used mirrors to ignite the Roman ships was probably just propaganda put out to alarm a navy already terrified by ‘the Claw’, but the children will shine a collimated beam at a strip of front-silvered polystyrene shaped to a parabola and observe the reflected ray passing through a single point.
  • The sphere’s surface area. The children will assemble two prefabricated models where the hemisphere is approximated by a stack of frusta topped by a cone. The smaller fits a true (acetate) hemisphere, which in turn fits inside the larger. If we made a better-and-better larger model and a better-and-better smaller model by using more- and-more layers, we would approach the true shape, and therefore the true area, from above and below. Down given meridia the children will insert glass-headed pins at equal intervals of latitude and join up the points. Counterintuitively, the regions so formed have equal area.
  • The sphere’s volume. The children will apply Pythagoras’ theorem to a vertical cross-section through a transparent model.With Polydron ‘Sphera’ pieces the children will shape wet sand into a cone and hemisphere of the same height and diameter. When they drop a matching cylinder over the top of the hemisphere and pack in sand from the cone, they will find it fits perfectly.
  • The area between a point on a parabola and a chord parallel to the tangent at that point is 4/3 x the area of the triangle with the chord as base and the point as apex. We shall use a well-known ‘proof without words’ to explain Archimedes’ result.

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