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Activities/Turtle Art/Tutorials/Euclid
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~~Then we can draw more lines and ~~more ~~circles, and so ~~on ~~and on forever~~, ~~with segments as long as we want ~~and ~~as short as ~~we ~~want, by bisecting lines,~~+~~and ~~as ~~close as we want ~~to ~~any direction~~, ~~by bisecting angles~~. +~~[[File:TAEuclidBisectAngle.png]]~~+
−~~This is the smallest set of points that make up a non-trivial Euclidean geometry. Pythagoras thought that it included every point in the plane, but it was proved later on in Greek times that it omits infinitely many points, such as the cube root of 2. In the 19th century, Georg Cantor proved that point sets of this kind omit "almost all" points in the plane, since the number of points omitted is vastly greater than the number of points included. Infinitely greater, in fact. However, that is a topic for another time, after ~~we ~~lay a proper foundation for such ideas. That should not prevent you from asking yourself what it could mean for ~~one ~~infinite number to be infinitely larger than another~~, ~~a question ~~that ~~Greek mathematicians could not have dreamed of~~, ~~much less ask seriously~~.
−~~* How about three points, or a line segment and a new point, not one of the endpoints? Well, it may be that we can construct one of these points using just the other two, giving the same result as before. Or it may not be so, in which case we can construct many other new points, again continuing on forever.~~+~~The first ~~and ~~third ~~of the ~~Euclidean constructions ~~are ~~quite definite~~, ~~but "as far as needed" is rather vague. Usually it means "until ~~the ~~line intersects some line or circle ~~of ~~interest"~~. ~~This commonly means that ~~we ~~have ~~to ~~prove ~~that ~~they will intersect before ~~we can ~~invoke this construction~~, ~~or ~~that ~~the fact of intersection is something previously assumed~~.+~~Euclid made various assumptions. The most important, the Fifth Postulate, is about when straight lines intersect. This is known as the Parallel Postulate because it says when lines are not parallel, and is then used to determine when they are. (This sort of thing is considered perfectly ordinary in math.) However, he said nothing about other cases that could be assumed to be obvious, such as when circles intersect. ~~+

[[File:TAEuclid2point.png]]

[[File:TAEuclid2point.png]]

−We can't go anywhere in particular by doing so. But we can draw two circles, one with each of the given points as center, and the line segment joining them as radius~~. Those two circles intersect in more points that are not on the original line, and also intersect the extensions of the original line (which now does go somewhere of interest)~~.

+We can't go anywhere in particular by doing so. But we can draw two circles, one with each of the given points as center, and the line segment joining them as radius.

[[File:TAEuclid2circle.png]]

[[File:TAEuclid2circle.png]]

−Those two circles intersect in more points that are not on the original line. We can mark them as given for future constructions, and join them in pairs. Notice that we have two equilateral triangles.

[[File:TAEuclidBisectLine.png]]

[[File:TAEuclidBisectLine.png]]

+Some lines and angles in the diagram above are bisected in this construction, and it turns out that we can bisect any given line and any given angle, and then bisect the halves, and so on. So we can draw more lines and more circles; we can mark more points; and so on and on forever, with segments as long as we want and as short as we want. We can also approximate any direction we want as closely as we want with fractions of angles.

−Carrying out this construction to the limit (an idea not in the range of Greek thought) gives us the smallest set of points that make up a complete non-trivial Euclidean geometry. Pythagoras thought that it included every point in the plane, but it was proved later on in Greek times that it omits infinitely many points, such as the cube root of 2. In the 19th century, Georg Cantor proved that point sets of this kind omit "almost all" points in the plane, since the number of points omitted is vastly greater than the number of points included. Infinitely greater, in fact. However, that is a topic for another time, which requires a foundation not related to Euclid's constructions. That should not prevent you from asking yourself what it could mean for one infinite number to be infinitely larger than another, a question that Greek mathematicians could not have dreamed of, much less asked seriously.

−* If we started with just one given line segment, that would give us its two endpoints, with the same result as just above.

−* ~~If we started with just one given ~~line segment, that ~~would give us its ~~two ~~endpoints~~, ~~with ~~the same result as ~~just above~~.

+* How about three points, or a line segment and a new point, not one of the endpoints? Well, it may be that we can construct one of these points starting from just the other two, giving the same result as before. Or it may not be so, in which case we can construct many other new points, again continuing on forever.

−==Back to Euclid==

−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 figures 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, which Lewis Carroll's Tortoise mentioned to Achilles in the quotation given 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.

−==More questions==

More fundamentally, what does it mean for a point to be "given"? No comment. But it turns out that the three assumed constructions are all that we need to know about given points and line segments. We don't have to know how to give one or the other at the beginning of a problem, and we don't have to know how to do the three constructions. It is sufficient, in a proof, to just say, "Do it." What matters is that all of the constructions and proofs in Euclid's elements are built out of just those three constructions, and that we can take intersections of previously constructed lines and circles as given for doing further constructions.

More fundamentally, what does it mean for a point to be "given"? No comment. But it turns out that the three assumed constructions are all that we need to know about given points and line segments. We don't have to know how to give one or the other at the beginning of a problem, and we don't have to know how to do the three constructions. It is sufficient, in a proof, to just say, "Do it." What matters is that all of the constructions and proofs in Euclid's elements are built out of just those three constructions, and that we can take intersections of previously constructed lines and circles as given for doing further constructions.

This is often caller "ruler and compass" construction, but it is important to understand that the ruler is unmarked, and the compass maintains a radius only as long as the center is unchanged. So to begin with, we cannot copy lengths from one place to another. Euclid addresses that lack as quickly as possible, in Proposition 2 of Book 1.

This is often caller "ruler and compass" construction, but it is important to understand that the ruler is unmarked, and the compass maintains a radius only as long as the center is unchanged. So to begin with, we cannot copy lengths from one place to another. Euclid addresses that lack as quickly as possible, in Proposition 2 of Book 1.

+
+The first and third of the Euclidean constructions are quite definite, but "as far as needed" is rather vague. Usually it means "until the line intersects some line or circle of interest". This commonly means that we have to prove that they will intersect before we can invoke this construction, or that the fact of intersection is something previously assumed.

+
+Euclid made various assumptions. The most important, the Fifth Postulate, is about when straight lines intersect. This is known as the Parallel Postulate because it says when lines are not parallel, and is then used to determine when they are. (This sort of thing is considered perfectly ordinary in math.) However, he said nothing about other cases that could be assumed to be obvious, such as when circles intersect.

==Euclid for Turtles==

==Euclid for Turtles==

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