How to think like Archimedes

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A brief set of slides that attempt to speculatively identify key principles and techniques present in the thought process of Archimedes. The slides also include a brief biography and succinct outline of Archimedes major works and inventions.

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How to think like Archimedes

  1. 1. Archimedes How to Think like Archimedes
  2. 2. Archimedes (287 – 212 BC)      The greatest mathematician of the ancient world Authored several treatise on mathematics and pioneer in the field of calculus Founded new scientific disciplines including hydrostatics Inventor Defender of Syracuse against Roman attack Figure 1.
  3. 3. Biography      Born in Syracuse in 287 BC Son of Pheidias, the astronomer Reputedly related to King Hieron of Syracuse No known biography survived Plutarch comments on Archimedes in his work, “Parallel Lives” Figure 2.
  4. 4. Biography    Likely Archimedes studied in Alexandria in his youth with the followers of Euclid Commenced lifelong correspondence with the mathematicians “Erasthosthenes” and “Dositheus” For example, a greeting from Archimedes to Dositheus proceeds the propositions in the treatise “On the Sphere and the Cylinder” Figure 3.
  5. 5. Mathematical Treatise  Author of several mathematical treatise including: − On the equilibrium of planes − Quadrature of the Parabola − On the Sphere and the Cylinder − On Spirals − On Conoids and Spheroids − On Floating Bodies − Measurement of a Circle − The Sand Reckoner Figure 4.
  6. 6. Archimedes Inventions   The Archimedes Screw – developed for pumping water Reputedly invented by Archimedes while in Alexandria Figure 5.  The Archimedes Claw – overturned Roman ships attacking Syracuse Figure 6.
  7. 7. Sacking of Syracuse   Archimedes perished in the sacking of Syracuse in 212 BC. Killed by a Roman soldier after supposedly saying “Stand away, fellow, from my diagram”. Figure 7.  According to Plutarch, The Roman General Marcellus “was most of all afflicted” by Archimedes death suggesting that Marcellus wished to spare Archimedes perhaps in order to profit from Archimedes' military ingenuity.
  8. 8. Archimedes Palimpsest     Discovered in 1906 by Johan Ludvig Heiberg in Constantinople. The parchments of a manuscript containing Archimedes writings had been reused as a prayer book by Byzantine monks in 1229 Figure 8. The Archimedes Palimpsest project established in 1998 to conserve and study the palimpsest Unique and previously unknown translations of “The Method”, “The Stomachion” and “On Floating Bodies”
  9. 9. How to Think like Archimedes From known parts to the unknown whole     Archimedes used known geometry to calculate the properties of unknown geometrical figures He often built up new theorems using geometrical laws discovered in earlier theorems An example is how Archimedes arrived at his calculation of Pi Using known inscribed and circumscribed polygons to estimate the area of the unknown circle Figure 9.
  10. 10. How to Think like Archimedes Continuous Contemplation & Reflection    Archimedes continually contemplated his works even during mundane daily tasks According to Plutarch, Archimedes would draw “geometrical figures in the ashes of the fire, or, when anointing himself, in the oil on his body” Another example of this is the Eureka story of Archimedes running naked through Syracuse after discovering the principle of displacement in the bath Figure 11.
  11. 11. How to Think like Archimedes Visualisation & Conceptual Diagrams    Annotated diagrams were an integral part of Archimedes theorems. The diagrams while assisting in the explanation of the theorems may also have aided the development of the theorem and the conceptualising of complex geometrical propositions The previous slide referring to Archimedes continual sketching of geometry suggests it played a strong role in his thinking process Figure 11.
  12. 12. How to Think like Archimedes Observation of Phenomena    Archimedes discovered the principle of displacement while observing the rise in the level of the water as he emersed himself to bathe While contemplating a problem one should therefore observe the related phenomena in the world around us at small and large scales Such observations may act as catalysts for new ideas as they occur unexpectedly at random and therefore force the researcher to view a problem in ways they otherwise would not have considered
  13. 13. How to Think like Archimedes Solve by Approximation     Archimedes employs this technique in his treatise to approximate the calculation of an unknown area or a value. The inscribed and circumscribed polygons cited earlier to approximate the area of a circle is an example of this technique. Another example is the division of a sphere into an infinite number of discs to calculate its volume A general principle can be extracted for problem solving as: define the limits which contain a problem and then repeatedly subdivide those limits until the exact properties of the problem are defined
  14. 14. How to Think like Archimedes Deduction     In many of Archimedes propositions he proves that some property A is equal to another property B by first proving, that A is not greater than B, and A is not less than B This is a process of iteratively proving what is false to arrive at what is true The researcher should therefore, as far as possible, list all possible conclusions, and then seek to disprove each in turn in order to arrive at a solution An example of this is Proposition No.1 of “Measurement of a Circle” where Archimedes proves that the area of a circle is equal to that of a triangle, K, by first proving that the area of the circle is not greater or less than K.
  15. 15. Image References  Figure 1 - http://www.math.nyu.edu/~%20crorres/Archimedes/Pictures/ArchimedesPictures.html, (Accessed December 23rd 2013).  Figure 2 - http://www.historyandcivilization.com/Maps---Tables---Ancient-Greece---the-Aegean.html, (Accessed December 23rd 2013).  Figure 3. - http://faculty.etsu.edu/gardnerr/Geometry-History/abstract.htm, (Accessed December 23rd 2013).  Figure 4 - Heath T.L (2002), The Works of Archimedes  Figure 5 - http://www.math.nyu.edu/~crorres/Archimedes/Screw/ScrewEngraving.html, (Accessed December 23rd 2013).  Figure 6 - http://www.math.nyu.edu/~crorres/Archimedes/Claw/illustrations.html, (Accessed December 23rd 2013).  Figure 7 - http://www.math.nyu.edu/~crorres/Archimedes/Death/DeathIllus.html, (Accessed December 23rd 2013).  Figure 8 - http://archimedespalimpsest.org/about/, (Accessed December 23rd 2013).  Figure 9 http://www.mathworks.com/matlabcentral/fileexchange/29504-the-computation-of-pi-by-archimedes/content/html/ComputationOfPiByArchimedes.html, (Accessed December 23rd 2013).  Figure 10 - http://ed.ted.com/lessons/mark-salata-how-taking-a-bath-led-to-archimedes-principle#watch, (Accessed December 23rd 2013).  Figure 11 - Heath T.L (2002), The Works of Archimedes

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