Compass & Straightedge Construction and the Impossible Constructions
From Math Images
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|Image=HexagonConstructionAni.gif | |Image=HexagonConstructionAni.gif | ||
|ImageIntro=This image shows the step by step construction of a hexagon inscribed in the circle using a compass and a unmarked straightedge. | |ImageIntro=This image shows the step by step construction of a hexagon inscribed in the circle using a compass and a unmarked straightedge. | ||
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Let's assume we only have a compass and a unmarked straightedge. How could we construct certain geometric shapes and prove certain theorems? That was the problems that Euclid pondered not only because those were probably the only instrument that he had at his time but also he wanted to construct his theorems with as few assumptions, or axioms, as possible. In this picture, we want to divide the circle into six equal arcs and then connect consecutive points to form the hexagon. It seems to be a fairly simple construction but it should prompt you to ask a question: is every polygon constructible, that is able to be constructed using only compass and straightedge? To extend the question, what are constructible and what is not? The issue will be addressed later. | Let's assume we only have a compass and a unmarked straightedge. How could we construct certain geometric shapes and prove certain theorems? That was the problems that Euclid pondered not only because those were probably the only instrument that he had at his time but also he wanted to construct his theorems with as few assumptions, or axioms, as possible. In this picture, we want to divide the circle into six equal arcs and then connect consecutive points to form the hexagon. It seems to be a fairly simple construction but it should prompt you to ask a question: is every polygon constructible, that is able to be constructed using only compass and straightedge? To extend the question, what are constructible and what is not? The issue will be addressed later. | ||
|ImageDesc===The Compass and straightedge== | |ImageDesc===The Compass and straightedge== | ||
- | + | A Compass is a tool which can be used for drawing circles. It has two legs, one end of which is fixed on the plane of construction and the other end is of given distance away and maintains the distance throughout the construction. We all know what a normal form of compass is but there are many other variants, crude or precise. For example, a crude form will be pinning a thread of string on the plane and fixing a pencil/pen at a certain distance away from the pin. We usually see this kind of crude compass in military field operations. Look out for such a compass in HBO's TV series '''''Band of Brothers''''' second episode when the paratroopers were trying to figure out where they would be dropped on D-Day based on the distances of their preparatory flights. | |
- | A Compass is a tool which can be used for drawing circles. It has two legs, one end of which is fixed on the plane of construction and the other end is of given distance away and maintains the distance throughout the construction. | + | |
- | A straightedge is a tool which can be used for drawing straight lines, or segments thereof. It should be noted that the difference between a straightedge and a ruler is that the former has no graduations on it (and does not allow any markings as well) while the latter has divisions on them according to certain unit | + | A straightedge is a tool which can be used for drawing straight lines, or segments thereof. It should be noted that the difference between a straightedge and a ruler is that the former has no graduations on it (and does not allow any markings as well) while the latter has divisions on them according to certain unit system. |
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- | It should be carefully noted that Euclid started with two given points and produced a line segment, from where he could extend into a straight line if he pleases. Then from the original two points, he could use one point as center and ONLY spread the legs of compass the distance of the line segment to produce a circle. He could not say "I wanted to spread the legs of the compass <math>\pi</math> centimeters apart (or any specified denominations)" | + | It should be carefully noted that Euclid started with two given points and produced a line segment, from where he could extend into a straight line if he pleases. Then ONLY from the original two points, he could use one point as center and ONLY spread the legs of compass the distance of the line segment to produce a circle. He could not say "I wanted to spread the legs of the compass <math>\pi</math> centimeters apart (or any specified denominations)". Therefore, he could not (so couldn't we) however, choose any two points as he pleased. In all, Compass and Straightedge Constructions only allow us to start with points (and length) we have and create the ones we don't. |
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===Some Basic Constructions=== | ===Some Basic Constructions=== | ||
The constructions below are some basic ones from where many more constructions and operations are possible and they are by no means exhaustive. The resulting products are in red. The proofs for these constructions are relatively simple and only require the knowledge of congruent triangles. Congruency and theorems on congruent triangles are directly derived from Euclid's Postulates. Try proving the theorems yourself! | The constructions below are some basic ones from where many more constructions and operations are possible and they are by no means exhaustive. The resulting products are in red. The proofs for these constructions are relatively simple and only require the knowledge of congruent triangles. Congruency and theorems on congruent triangles are directly derived from Euclid's Postulates. Try proving the theorems yourself! | ||
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===='''Line Segment Bisection'''==== | ===='''Line Segment Bisection'''==== | ||
[[Image:CS1.png|border|550px|center]] | [[Image:CS1.png|border|550px|center]] | ||
- | + | {{HideShow|Given points <math>A</math> and <math>B</math> and the straight line passing through it. Construct a line that bisects line segment <math>AB</math>. | |
- | Given points <math>A</math> and <math>B</math> and the straight line passing through it. Construct a line that bisects line segment <math>AB</math>. | + | |
# Draw a circle centered at point <math>A</math> with radius equaling <math>AB</math>. | # Draw a circle centered at point <math>A</math> with radius equaling <math>AB</math>. | ||
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# Draw a line through points <math>C</math> and <math>D</math>. | # Draw a line through points <math>C</math> and <math>D</math>. | ||
- | <math>CD</math> intersects <math>AB</math> at the midpoint. It should be noted that <math>CD \perp AB</math> as well. Line <math>CD</math> is the perpendicular bisector of line segment <math>AB</math>. | + | <math>CD</math> intersects <math>AB</math> at the midpoint. It should be noted that <math>CD \perp AB</math> as well. Line <math>CD</math> is the perpendicular bisector of line segment <math>AB</math>.}} |
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===='''Angle Bisection'''==== | ===='''Angle Bisection'''==== | ||
[[Image:CS2.png|border|550px|center]] | [[Image:CS2.png|border|550px|center]] | ||
- | Given angle <math>\angle AOB</math>, construct a line that bisects the angle. | + | {{HideShow|Given angle <math>\angle AOB</math>, construct a line that bisects the angle. |
# Construct a circle centered at point <math>O</math> with radius <math>OA</math>. This circle should intersect at points <math>A</math> and <math>B</math>. | # Construct a circle centered at point <math>O</math> with radius <math>OA</math>. This circle should intersect at points <math>A</math> and <math>B</math>. | ||
# Keeping the same radius, draw a circle at points <math>A</math> and <math>B</math>. Where they intersect each other, call the point <math>C</math>. | # Keeping the same radius, draw a circle at points <math>A</math> and <math>B</math>. Where they intersect each other, call the point <math>C</math>. | ||
# Draw a line through points <math>C</math> and <math>O</math>. This line bisects <math>\angle AOB</math>. | # Draw a line through points <math>C</math> and <math>O</math>. This line bisects <math>\angle AOB</math>. | ||
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===='''Perpendicular through a point'''==== | ===='''Perpendicular through a point'''==== | ||
[[Image:Perp.png|border|450px|center]] | [[Image:Perp.png|border|450px|center]] | ||
- | Given a point <math>M</math> on a line <math>AB</math>, construct a line that is perpendicular to the given line through <math>M</math>. | + | {{HideShow|Given a point <math>M</math> on a line <math>AB</math>, construct a line that is perpendicular to the given line through <math>M</math>. |
# Draw a circle centered at <math>M</math>. | # Draw a circle centered at <math>M</math>. | ||
- | # Where the circle intersects the original line, construct a perpendicular bisector. | + | # Where the circle intersects the original line, construct a perpendicular bisector.}} |
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===='''Parallel'''==== | ===='''Parallel'''==== | ||
[[Image:CS4.png|border|550px|center]] | [[Image:CS4.png|border|550px|center]] | ||
- | Given two points, <math>A</math> and <math>B</math> and the straight line passing through them, construct a line that is parallel to the given line through another given point <math>C</math>. | + | {{HideShow|Given two points, <math>A</math> and <math>B</math> and the straight line passing through them, construct a line that is parallel to the given line through another given point <math>C</math>. |
# Draw a circle at <math>A</math>, crossing <math>C</math>. Where the circle <math>A</math> intersects <math>AB</math>, call the point <math>D</math>. | # Draw a circle at <math>A</math>, crossing <math>C</math>. Where the circle <math>A</math> intersects <math>AB</math>, call the point <math>D</math>. | ||
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# Connect points <math>C</math> and <math>E</math> with a line. | # Connect points <math>C</math> and <math>E</math> with a line. | ||
- | <math>CE</math> is parallel to <math>AB</math> | + | <math>CE</math> is parallel to <math>AB</math>}} |
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===='''Tangent Line to a Circle'''==== | ===='''Tangent Line to a Circle'''==== | ||
[[Image:CS5.png|border|550px|center]] | [[Image:CS5.png|border|550px|center]] | ||
- | Given a circle centered at <math>O</math> and another given point, <math>A</math>, construct a line that is tangent to the circle. | + | {{HideShow|Given a circle centered at <math>O</math> and another given point, <math>A</math>, construct a line that is tangent to the circle. |
# Connect the point <math>A</math>, with the center of the circle <math>O</math> | # Connect the point <math>A</math>, with the center of the circle <math>O</math> | ||
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- | To see that <math>AC</math> is tangent to the circle, connect <math>OC</math>. <math>\angle ACO</math> is an inscribed angle in the circle about <math>M</math>, so the inscribed <math>\angle ACO</math> is <math>90^\circ</math>. So <math>OC</math> is perpendicular to <math>AC</math>. Since the tangent is perpendicular to the radius to the point of tangency, by the uniqueness of the line through <math>C</math> perpendicular to <math>OC</math>, <math>AC</math> is tangent to the circle. | + | To see that <math>AC</math> is tangent to the circle, connect <math>OC</math>. <math>\angle ACO</math> is an inscribed angle in the circle about <math>M</math>, so the inscribed <math>\angle ACO</math> is <math>90^\circ</math>. So <math>OC</math> is perpendicular to <math>AC</math>. Since the tangent is perpendicular to the radius to the point of tangency, by the uniqueness of the line through <math>C</math> perpendicular to <math>OC</math>, <math>AC</math> is tangent to the circle.}} |
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===='''Euclid's Proof of Compass Equivalence Theorem'''==== | ===='''Euclid's Proof of Compass Equivalence Theorem'''==== | ||
- | This part | + | This part refers back to the previous section about the issue of compass being collapsible. Euclid's proof is presented in its originality. Additional comments are contained in the parenthesis. |
[[Image:CETpic.png|border|center|450px]] | [[Image:CETpic.png|border|center|450px]] | ||
- | + | {{HideShow|From a given point to draw a line segment equal to a given line segment. | |
- | From a given point to draw a line segment equal to a given line segment. | + | |
<u>Let <math>A</math> be the given point, and <math>BC</math> the given straight line : it is required to draw from the point <math>A</math> a straight line equal to <math>BC</math>.</u> | <u>Let <math>A</math> be the given point, and <math>BC</math> the given straight line : it is required to draw from the point <math>A</math> a straight line equal to <math>BC</math>.</u> | ||
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Wherefore from the given point <math>A</math> a straight line <math>AL</math> has been drawn equal to the given straight line <math>BC</math>.∎ | Wherefore from the given point <math>A</math> a straight line <math>AL</math> has been drawn equal to the given straight line <math>BC</math>.∎ | ||
- | It should be noted that from <math>AL</math>, we could "duplicate" <math>AL</math> in all directions by construct a circle centered at <math>A</math> with radius <math>AL</math>. | + | It should be noted that from <math>AL</math>, we could "duplicate" <math>AL</math> in all directions by construct a circle centered at <math>A</math> with radius <math>AL</math>.}} |
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==Algebraicization of Compass & Straightedge Constructions== | ==Algebraicization of Compass & Straightedge Constructions== | ||
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- | + | ''Algebraicization'' is the translation of any problem statements into algebraic problems. In the case of Compass & Straightedge construction, we algebraicize each step of a straightedge and compass construction, and consequently obtaining general results about the nature of constructibility. Hilda P. Hudson put it aptly in his lecture '''''Ruler & Compasses''''', | |
<blockquote>''"each step of a ruler and compass construction is equivalent to a certain analytical process; it is found that the power to use a ruler corresponds exactly to the power to solve linear equations, and the power to use compasses to the power to solve quadratics. For this reason, problems that can be solved with ruler only are called linear problems, and those that can be solved with ruler and compasses are called quadratic problems. Since each step of a ruler and compass construction is equivalent to the solution of an equation of the first or second degree, we consider that these algebraic processes can lead to , when combined in every possible way, and that enables us to answer the question before us and say that those problems and those problems alone can be solved by ruler only, which can be made to depend on a linear equation, whose root can be calculated by carrying out rational operations only; and that those problems and those problems alone can be solved by ruler and compasses, which can be made to depend on an algebraic equation, whose degree must be a power of 2, and whose roots can be calculated by carrying out rational operations together with the extraction of square roots only."''</blockquote> | <blockquote>''"each step of a ruler and compass construction is equivalent to a certain analytical process; it is found that the power to use a ruler corresponds exactly to the power to solve linear equations, and the power to use compasses to the power to solve quadratics. For this reason, problems that can be solved with ruler only are called linear problems, and those that can be solved with ruler and compasses are called quadratic problems. Since each step of a ruler and compass construction is equivalent to the solution of an equation of the first or second degree, we consider that these algebraic processes can lead to , when combined in every possible way, and that enables us to answer the question before us and say that those problems and those problems alone can be solved by ruler only, which can be made to depend on a linear equation, whose root can be calculated by carrying out rational operations only; and that those problems and those problems alone can be solved by ruler and compasses, which can be made to depend on an algebraic equation, whose degree must be a power of 2, and whose roots can be calculated by carrying out rational operations together with the extraction of square roots only."''</blockquote> | ||
- | Hudson lectured on this in the early 20th century and certain phrases of his could potentially cause confusion. The | + | Hudson lectured on this in the early 20th century and certain phrases of his could potentially cause confusion. The take-away from this paragraph is that in order to algebraicize straightedge and compass construction, we begin by choosing a specific length to be considered one unit. Then, every time we construct a segment of straight line or circle, we think of it instead as constructing ''numbers'', representing the coordinates of the points of intersection. From these points, we can then calculate the length between them. Compass & straightedge construction only allow us to construct points that are at intersections of lines and/or circles. This means that every time we do a construction, we can use equations of circles and lines from coordinate geometry to figure out what numbers are being constructed. In this way, a geometric process is translated into an algebraic process. |
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- | + | Firstly, we define "1" on a straight line by specifying the length between any two point. But some will say, hey, in Euclid's postulates, he did not say you can choose any points; a point is the intersection of two lines, two circles or one line and one circle. That is a good observation. To have a line, we have to have given points. Then we draw the straight line. Choose one of the two given points, and use that as the center for our compass. Now, spread the legs of the compass by a distance between the two given points. The intersection is a new point. With the new point, we now have one arbitrarily chosen points. Note, the "1" does not necessarily have to measure meter nor foot since we have chosen two random points. It could measure a new length of your choosing. Then, once you have chosen that length to be "1", you have to stick to this specification throughout your construction. | |
- | Firstly, we define "1" on a straight line by specifying the length between any two point. But some will say, hey, in Euclid's postulates, he did not say you can choose any points; a point is the intersection of two lines, two circles or one line and one circle. That is a good observation. To have a line, we have to have given points. Then we draw the straight line. Choose one of the two given points, and use that as the center for our compass. Now, spread the legs of the compass by a distance between the two given points. The intersection is a new point. With the new point, we now have one arbitrarily chosen points. | + | |
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- | Note, the "1" does not necessarily have to measure meter nor foot since we have chosen two random points. It could measure a new length of your choosing. Then, once you have chosen that length to be "1", you have to stick to this specification throughout your construction. | + | |
Next, it is very obvious that we could construct all the integers, that is <math>\cdots -3,-2,-1,0,1,2,3,\cdots</math> (or <math>x</math> = <math>\{x|- \infty < x < \infty,x \in \mathbb{Z}\}</math>). How so? Well, once we have the "1", all we have to do is to use the Compass Equivalence Theorem finite number of times to duplicate the length "1" that we previously defined. Now, that means that we could have any two random integers, <math>a</math> and <math>b</math>, and for the sake of this discussion and clarity, we are talking about positive integers here. Next, I will show that from <math>a</math> and <math>b</math>, we could construct <math>a \pm b</math>, <math>a \times b</math> and <math>\frac {a}{b}</math>. | Next, it is very obvious that we could construct all the integers, that is <math>\cdots -3,-2,-1,0,1,2,3,\cdots</math> (or <math>x</math> = <math>\{x|- \infty < x < \infty,x \in \mathbb{Z}\}</math>). How so? Well, once we have the "1", all we have to do is to use the Compass Equivalence Theorem finite number of times to duplicate the length "1" that we previously defined. Now, that means that we could have any two random integers, <math>a</math> and <math>b</math>, and for the sake of this discussion and clarity, we are talking about positive integers here. Next, I will show that from <math>a</math> and <math>b</math>, we could construct <math>a \pm b</math>, <math>a \times b</math> and <math>\frac {a}{b}</math>. | ||
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[[Image:A+-b.png|center|border|600px]] | [[Image:A+-b.png|center|border|600px]] | ||
+ | {{HideShow|To construct <math>a \pm b</math>, we will use <math>a</math> as the center and use <math>b</math> as radius. The two points of intersection with the line will be <math>a+b</math> and <math>a-b</math>.}} | ||
- | To construct <math>a \times b</math>, we have <math>0</math>, <math>1</math>, <math>a</math> and <math>b</math> on the straight line. | + | [[Image:Atimesb3.png|center|border|500px]] |
+ | {{HideShow|To construct <math>a \times b</math>, we have <math>0</math>, <math>1</math>, <math>a</math> and <math>b</math> on the straight line. | ||
# Draw any straight line through <math>0</math>, call it <math>l_1</math>. | # Draw any straight line through <math>0</math>, call it <math>l_1</math>. | ||
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# Construct circle centered at <math>0</math> with radius <math>0A</math>, intersecting <math>l_0</math> at a point. | # Construct circle centered at <math>0</math> with radius <math>0A</math>, intersecting <math>l_0</math> at a point. | ||
- | The distance between <math>0</math> and that point is <math>ab</math>. | + | The distance between <math>0</math> and that point is <math>ab</math>.}} |
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- | Similarly, we could construct <math>\frac {a}{b}</math>. | + | [[Image:Ab.png|center|border|620px]] |
+ | {{HideShow|Similarly, we could construct <math>\frac {a}{b}</math>. | ||
# Draw any straight line through <math>0</math>, call it <math>l_1</math>. | # Draw any straight line through <math>0</math>, call it <math>l_1</math>. | ||
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# Construct <math>l_3</math> that is parallel to <math>l_2</math>, intersecting <math>l_0</math> at a point. | # Construct <math>l_3</math> that is parallel to <math>l_2</math>, intersecting <math>l_0</math> at a point. | ||
- | The distance between <math>0</math> and that point is <math>\frac {a}{b}</math>. | + | The distance between <math>0</math> and that point is <math>\frac {a}{b}</math>.}} |
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I will leave the proofs to you since they are very simple using similar triangles. | I will leave the proofs to you since they are very simple using similar triangles. | ||
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[[Image:Sqrta.png|center|border|520px]] | [[Image:Sqrta.png|center|border|520px]] | ||
- | + | {{HideShow|# Construct <math>a+1</math>. | |
- | # Construct <math>a+1</math>. | + | |
# Construct <math>\frac {1}{2}(a+1)</math>. | # Construct <math>\frac {1}{2}(a+1)</math>. | ||
# Construct circle at <math>\frac {1}{2}(a+1)</math> with radius <math>\frac {1}{2}(a+1)</math>. | # Construct circle at <math>\frac {1}{2}(a+1)</math> with radius <math>\frac {1}{2}(a+1)</math>. | ||
# Draw perpendicular through <math>a</math>, intersecting the circle at point <math>A</math>. | # Draw perpendicular through <math>a</math>, intersecting the circle at point <math>A</math>. | ||
- | <math>aA=\sqrt {a}</math>.Again, I will leave the proof to you as well using similar triangles. | + | <math>aA=\sqrt {a}</math>.Again, I will leave the proof to you as well using similar triangles.}} |
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Revision as of 14:53, 7 July 2010
- This image shows the step by step construction of a hexagon inscribed in the circle using a compass and a unmarked straightedge.
Creating a regular hexagon with a ruler and compass |
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Contents |
Basic Description
Let's assume we only have a compass and a unmarked straightedge. How could we construct certain geometric shapes and prove certain theorems? That was the problems that Euclid pondered not only because those were probably the only instrument that he had at his time but also he wanted to construct his theorems with as few assumptions, or axioms, as possible. In this picture, we want to divide the circle into six equal arcs and then connect consecutive points to form the hexagon. It seems to be a fairly simple construction but it should prompt you to ask a question: is every polygon constructible, that is able to be constructed using only compass and straightedge? To extend the question, what are constructible and what is not? The issue will be addressed later.A More Mathematical Explanation
- Note: understanding of this explanation requires: *A little Geometry and Some Abstract Algebra
The Compass and straightedge
A Compass is a tool which can be used for drawing circles. It has tw [...]The Compass and straightedge
A Compass is a tool which can be used for drawing circles. It has two legs, one end of which is fixed on the plane of construction and the other end is of given distance away and maintains the distance throughout the construction. We all know what a normal form of compass is but there are many other variants, crude or precise. For example, a crude form will be pinning a thread of string on the plane and fixing a pencil/pen at a certain distance away from the pin. We usually see this kind of crude compass in military field operations. Look out for such a compass in HBO's TV series Band of Brothers second episode when the paratroopers were trying to figure out where they would be dropped on D-Day based on the distances of their preparatory flights.
A straightedge is a tool which can be used for drawing straight lines, or segments thereof. It should be noted that the difference between a straightedge and a ruler is that the former has no graduations on it (and does not allow any markings as well) while the latter has divisions on them according to certain unit system.
What is Compass & Straightedge Constructions
Introduction
Compass & Straightedge Construction is the construction of points, lengths, angles, and circles using only idealized straightedge and compass. The straightedge is infinite in length, has no markings on it and only one edge. The compass collapses when lifted from the page, so may not be directly used to transfer distances. However, it turns out that this restriction makes no difference due to the Compass Equivalence Theorem which was stated as Proposition II of Book I of Euclid's Elements. It stated that from a given point, it was possible to construct a straight line equal to a given straight line using collapsible compass. Euclid's proof for the Compass Equivalence Theorem will be presented after the section of Basic Construction.
We start by familiarizing ourselves with Euclid's three Postulates in his books Elements.
"Let it be grantedIt should be carefully noted that Euclid started with two given points and produced a line segment, from where he could extend into a straight line if he pleases. Then ONLY from the original two points, he could use one point as center and ONLY spread the legs of compass the distance of the line segment to produce a circle. He could not say "I wanted to spread the legs of the compass centimeters apart (or any specified denominations)". Therefore, he could not (so couldn't we) however, choose any two points as he pleased. In all, Compass and Straightedge Constructions only allow us to start with points (and length) we have and create the ones we don't.
- that a straight line may be drawn from any one point to any other point;
- that a line segment may be extended into a straight line;
- that given any straight line segment, a circle may be described having the segment as radius and one endpoint as center. "
Some Basic Constructions
The constructions below are some basic ones from where many more constructions and operations are possible and they are by no means exhaustive. The resulting products are in red. The proofs for these constructions are relatively simple and only require the knowledge of congruent triangles. Congruency and theorems on congruent triangles are directly derived from Euclid's Postulates. Try proving the theorems yourself!
Line Segment Bisection
Angle Bisection
Perpendicular through a point
Parallel
Tangent Line to a Circle
Euclid's Proof of Compass Equivalence Theorem
This part refers back to the previous section about the issue of compass being collapsible. Euclid's proof is presented in its originality. Additional comments are contained in the parenthesis.
Algebraicization of Compass & Straightedge Constructions
A simple derivation
From the few basic constructions, you would have probably realized that the different possibilities seems infinite. However, by intuition, we know that the possibility could not be infinite. Hence, mathematician are curious to find out what are constructible and what aren't and for this purpose, the language of pure geometry seems to have "limited vocabulary". Back in ancient times, mathematicians had limited algebraic knowledge and were more familiar with geometry. But in modern times, the reverse is true. Hence, today's mathematicians go back to their familiar realm of Algebra and try to find the link between geometry and algebra.
Algebraicization is the translation of any problem statements into algebraic problems. In the case of Compass & Straightedge construction, we algebraicize each step of a straightedge and compass construction, and consequently obtaining general results about the nature of constructibility. Hilda P. Hudson put it aptly in his lecture Ruler & Compasses,
"each step of a ruler and compass construction is equivalent to a certain analytical process; it is found that the power to use a ruler corresponds exactly to the power to solve linear equations, and the power to use compasses to the power to solve quadratics. For this reason, problems that can be solved with ruler only are called linear problems, and those that can be solved with ruler and compasses are called quadratic problems. Since each step of a ruler and compass construction is equivalent to the solution of an equation of the first or second degree, we consider that these algebraic processes can lead to , when combined in every possible way, and that enables us to answer the question before us and say that those problems and those problems alone can be solved by ruler only, which can be made to depend on a linear equation, whose root can be calculated by carrying out rational operations only; and that those problems and those problems alone can be solved by ruler and compasses, which can be made to depend on an algebraic equation, whose degree must be a power of 2, and whose roots can be calculated by carrying out rational operations together with the extraction of square roots only."
Hudson lectured on this in the early 20th century and certain phrases of his could potentially cause confusion. The take-away from this paragraph is that in order to algebraicize straightedge and compass construction, we begin by choosing a specific length to be considered one unit. Then, every time we construct a segment of straight line or circle, we think of it instead as constructing numbers, representing the coordinates of the points of intersection. From these points, we can then calculate the length between them. Compass & straightedge construction only allow us to construct points that are at intersections of lines and/or circles. This means that every time we do a construction, we can use equations of circles and lines from coordinate geometry to figure out what numbers are being constructed. In this way, a geometric process is translated into an algebraic process.
Firstly, we define "1" on a straight line by specifying the length between any two point. But some will say, hey, in Euclid's postulates, he did not say you can choose any points; a point is the intersection of two lines, two circles or one line and one circle. That is a good observation. To have a line, we have to have given points. Then we draw the straight line. Choose one of the two given points, and use that as the center for our compass. Now, spread the legs of the compass by a distance between the two given points. The intersection is a new point. With the new point, we now have one arbitrarily chosen points. Note, the "1" does not necessarily have to measure meter nor foot since we have chosen two random points. It could measure a new length of your choosing. Then, once you have chosen that length to be "1", you have to stick to this specification throughout your construction.
Next, it is very obvious that we could construct all the integers, that is (or = ). How so? Well, once we have the "1", all we have to do is to use the Compass Equivalence Theorem finite number of times to duplicate the length "1" that we previously defined. Now, that means that we could have any two random integers, and , and for the sake of this discussion and clarity, we are talking about positive integers here. Next, I will show that from and , we could construct , and .
I will leave the proofs to you since they are very simple using similar triangles.
Therefore, it has been proven that we could construction all the rational numbers since and are any arbitrary integers.
The natural question to ask right now is that what else is possible to construct? It is not hard to think of numbers that are not rational. For example, is constructible. Construct a unit square and the diagonal is of length . So is it possible to construct given any constructible number ? It turns out that we could. See below for method.
Next, we moved to the general solution of the problem.
Assume we have two points and with coordinates and . Take an arbitrary point on the line.
By similar triangle, .
Rearranging the above we have
.
Since , , and are constant we can express this as which is the general expression of a straight line.
Now, if we have two lines specified by four given points, ,...., with coordinates . The intersection of the two lines, will satisfy two equations
You may say that the there might not be a solution. True the two lines do not have to intersect. But if they do, we only need the operations of addition, subtraction, multiplication and division to find the point.
Now, we move onto circle. Say we have circle centered at some point with coordinates and radius . We know that the explicit expression for a circle is . Hence, if that circle intersects with one of the straight lines, then the points of intersection will satisfy
To solve for the points of intersection, we only need the operations of addition, subtraction, multiplication and division along with the extraction of square roots. Therefore, from this analysis, we have turned geometric problem into algebraic problem and come to the conclusion that a number is constructible if and only if it may be obtained from the integers by repeated use of addition, subtraction, multiplication, division and the extraction of square roots.
A Rigorous Proof
What I have presented above is a simplified version of the derivation towards the theorem. To see a rigorous proof of this theorem at a college level, refer to the text below which is mainly taken from I. N. Herstein's Topics in Algebra, Second Edition. You need some knowledge in Linear Algebra and/or Abstract Algebra. Also see Constructible Numbers. You should not be discouraged should you find it hard to understand. Instead, you should be marveled by the simplicity and elegance of the algebraic proof.
Why is it interesting?
What is Impossible to Construct (of course, using compass and straightedge alone)?
Below is the brief introduction of a few of the impossible constructions. Remember that a number is constructible if and only if it may be obtained from the integers by repeated use of addition, subtraction, multiplication, division and the extraction of square roots.
- is transcendental since it does not satisfy any rational polynomials. That means that is not a solution of any polynomials with rational coefficients. Too see complete proof that is transcendental, see Transcendental number and The 15 Most Famous Transcendental Numbers.
- From the above impossible construction, it follows that it is impossible to "square the circle (that is to construct a square that has the same area as a given circle)" because given a circle with radius 1 (the radius has to be constructible in the first place so that means it cannot, for example, be ), the area of the circle will be and we have to construct square with sides equal to which is not constructible.
- We could not double the volume of a given cube because we could not construct . It should be noted that we have to start with a cube whose sides are constructible in the first place. For example, we cannot even have a cube with sides equal to and thus volume 5, let alone doubling it. Prove that we cannot double the cube.
- We generally can not trisect any given angle because the process involves taking cube root. For example, it is impossible to trisect . See below for proof. For more, refer to Trisection of an Anglefor explanation in great detail. Prove that is impossible to trisect.
- There are certain polygons that are impossible to construct. See Constructible polygon for more detail.
Number 2, 3 and 4 are the so-called Geometric Problems of Antiquity. Though they have been proven impossible to construct with straightedge and compass, it does not deter amateur mathematicians to come up with false proofs even today.
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