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Euclidean geometry, specifically Euclid's Elements, asks a sequence of questions about what can be constructed with only straight lines and circles, with as few assumptions as possible. We will examine that problem, from Euclid's point of view and some modern points of view, and we will also ask some other questions.  

 

 

 

Euclidean geometry, specifically Euclid's Elements, asks a sequence of questions with very definite answers about what can be constructed with only straight lines and circles, with as few assumptions as possible. We will examine that problem, from Euclid's point of view and some modern points of view, and we will also ask some other questions, for which there are no definite answers.

* For example, why did Euclid do that? Other Greek mathematicians addressed significantly different questions in the various books that have survived from their time, and in fragments of books that have not survived, quoted in books that did.

* For example, why did Euclid do that? Other Greek mathematicians addressed significantly different questions in the various books that have survived from their time, and in fragments of books that have not survived, quoted in books that did.

* Mathematics in other civilizations of the time was entirely different from Greek mathematics. Why is that?

* Mathematics in other civilizations of the time was entirely different from Greek mathematics. Why is that?

* Many important and fascinating questions were discovered later, well within the reach of Greek mathematics, that the Greeks apparently never asked. Why is that?

* Many important and fascinating questions were discovered later, well within the reach of Greek mathematics, that the Greeks apparently never asked, or in some cases asked and rejected as useless or meaningless. Why is that?

However, we cannot usefully ask those questions before we have some understanding of Euclid's work.

However, we cannot usefully ask ourselves those questions before we have some understanding of Euclid's work.

==The Essence of Euclidean Geometry==

==The Essence of Euclidean Geometry==

The Euclidean constructions are simply postulated, not explained in any way. You have to start somewhere.

The Euclidean constructions are simply postulated, not explained in any way. It turns out that you have to start somewhere, as Euclid was one of the first to make clear. There are three things that Euclid gives himself permission to do, without asking how they are done.

# Join two given points with a straight line segment.

# Join two given points with a straight line segment.

[[File:TAEuclideanConstructionPostulates.png]]

[[File:TAEuclideanConstructionPostulates.png]]

==Examples==

==Examples==

[[File:TAEuclid1point.png]]

[[File:TAEuclid1point.png]]

* Two points? Now we are getting somewhere. We can connect the two points with a straight line segment, and then extend the line segment.  

* How about two points? Now we are getting somewhere. We can connect the two points with a straight line segment, and then extend the line segment.  

[[File:TAEuclid2point.png]]

[[File:TAEuclid2point.png]]

[[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.

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, and that we will be able to mark another point where two lines cross, and extend several lines until they intersect both circles, and draw circles with every new line segment as radius, and so on.

[[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.

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.

Carrying out this construction to the limit (an idea not in the range of Greek thought, which rejected completed infinities) 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.

==Back to Euclid==

==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.

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.

 

 

==More questions==

==More questions==

==Euclid for Turtles==

==Euclid for Turtles==

Before we can get to copying lengths, however, we have to determine how to tell the turtle to do Euclid's three constructions. The starting point in Turtle Art is of course quite different from Euclid's starting point, with given points and lines. The Turtle has a position, some graphics properties, and a set of blocks for movement.

Before we can get to copying lengths, however, we have to determine how to tell the turtle to do Euclid's three constructions. The starting point in Turtle Art is of course quite different from Euclid's starting point, with given points and lines. The Turtle has a position, some graphics properties, and a set of blocks for movement.