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Activities/Turtle Art/Tutorials/Euclid (view source)
Revision as of 01:07, 12 August 2011
, 01:07, 12 August 2011→Back to Euclid: Copying a line segment
Euclid did not proceed in the manner we just have, thinking about all of the points, lines, and circles that can be constructed from the three postulated operations without making any distinction among them. Euclid aimed to construct specific kinds of figure made from very specific points, lines, and circles. His goal in the Elements was the construction of all of the regular polyhedra discovered by the Pythagoreans (which are therefore ^_^ called the Platonic solids). However, at the very beginning, in Proposition 1 of Book I of the Elements, the one that Lewis Carroll's Tortoise mentioned above, Euclid did proceed much as we have done, because there was no alternative. Until we get to this point, there is nothing else we ''can'' construct. Proposition I,1 calls for constructing an equilateral triangle, as we did above, and then proving that that is what we did. Proof falls outside what we can do with Turtle Graphics, although at a much higher level it is possible and even useful to program theorem provers that work on propositions expressed in text, and not in images.
Euclid did not proceed in the manner we just have, thinking about all of the points, lines, and circles that can be constructed from the three postulated operations without making any distinction among them. Euclid aimed to construct specific kinds of figure made from very specific points, lines, and circles. His goal in the Elements was the construction of all of the regular polyhedra discovered by the Pythagoreans (which are therefore ^_^ called the Platonic solids). However, at the very beginning, in Proposition 1 of Book I of the Elements, the one that Lewis Carroll's Tortoise mentioned above, Euclid did proceed much as we have done, because there was no alternative. Until we get to this point, there is nothing else we ''can'' construct. Proposition I,1 calls for constructing an equilateral triangle, as we did above, and then proving that that is what we did. Proof falls outside what we can do with Turtle Graphics, although at a much higher level it is possible and even useful to program theorem provers that work on propositions expressed in text, and not in images.
Euclid, I,2. At a given point to construct a line equal to a given line.
Euclid, I,2. At a given point (A) to construct a line equal to a given line (BC).
Now, as you see, we have significantly greater scope, just from being able to draw an equilateral triangle. Similarly, the ability to copy lines will allow us to do vastly more again. This rapid expansion in geometric abilities is very similar to the expansion of abilities in programming as we define functions that we can use freely in further programming. It is unusual to start from such a small base in programming as Euclid does in geometry, but it can be done, as in Pure LISP (car, cdr, cons) or Unlambda (S, I, K). This may be more information than you require, but there will be some who are interested.
[[File:TAEuclidCopyLine.png]]
The construction is, beginning with the point A and the line segment BC as given,
* Join A and B (orange)
* Construct the equilateral triangle ABD (black)
* Draw the circle BC (cyan)
* Extend the line DB to E (blue)
* Draw the circle DE (blue)
* Extend the line DE to F (violet)
Then we have to prove that BC = AF, so that AF is the desired copy of BC, with one end at A.
==More questions==
==More questions==