Difference between revisions of "Compass and straightedge"
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'''Compass and straightedge constructions''' played an important role in the [[history]] of [[mathematics]]. Some constructions accomplished by the ancients led to developments in abstract mathematics. Significantly, problems that the ancients ''couldn't'' solve by these methods led to even more revolutionary developments in abstract mathematics. | '''Compass and straightedge constructions''' played an important role in the [[history]] of [[mathematics]]. Some constructions accomplished by the ancients led to developments in abstract mathematics. Significantly, problems that the ancients ''couldn't'' solve by these methods led to even more revolutionary developments in abstract mathematics. | ||
| − | == | + | ==Background== |
| − | + | To appreciate the significance of these constructions, it is important to realize that the ancients thought of mathematics in terms of geometric figures rather than the numbers, equations, and abstract things that we use today. These geometric constructions may be considered amusements today, but they were once the lifeblood of mathematics. The transition from thinking in terms of geometry to thinking in terms of numbers, formulas, and sets, happened very slowly, and wasn't really complete until the 19th century. | |
| − | + | ==Rules of the game== | |
| + | You are given an abstract drawing surface (piece of parchment, clay tablet, patch of dirt, whatever), a writing implement (pencil, stylus, finger), a straightedge, and a compass. | ||
| + | *The straightedge is just a ruler with no markings. You can't measure; you can only draw straight lines. | ||
| + | *The compass can draw a circle through a given point, but, unlike modern compasses, it can't "remember" its radius from one circle to another. Once it is lifted off the paper, it "goes limp" and collapses. | ||
| − | The | + | The straightedge could be described as "infinitely long". A better description would be "as long as you need it to be", or "a construction will never fail because the straightedge wasn't long enough." The compass is similar—it can draw circles of any size. |
| − | + | The things you are allowed to do are | |
| + | *Given any two points (you can recognize a point as the place where two lines—straight or circular—intersect), you can place the straightedge against them and draw a line passing through both. | ||
| + | *Given any two points, you can place the point of the compass on one of them, adjust it so that the pencil touches the other, and draw a circle. That circle will be centered on one and pass through the other. | ||
| − | + | The ancients (the Greeks were most famous for this) were able to do a remarkable amount of mathematics through reasoning about these geometric figures. [[Euclid]]'s book ''The Elements'' is regarded as the best presentation of classical geometry. The methods of reasoning described by Euclid are widely used even today to teach logical reasoning and the principles of axioms, theorems, and proofs. | |
| − | The | ||
| + | The mathematical "operations" that the ancient Greeks could do with these constructions include what we would now, in our number-centric world, would call addition, subtraction, multiplication, division, and square roots. They could also solve problems of proportion, bisect arbitrary angles, find areas of a many shapes, and construct some, but not all, regular polygons. | ||
| + | |||
| + | In modern times, compass-and-straightedge constructions were rediscovered by [[Euler]], who resurrected them from the pages of Euclid's ''Elements'' and found that their simple geometric [[truth]]s were a pleasant diversion from the [[calculus wars]] then raging between the {{hs|Isaac|Newton}}ian British and the [[Leibniz]]ian German mathematical communities. In the twentieth century, compass-and-straightedge methods have become a popular form of [[recreational mathematics]], and the technique is taught as a one-week [[unit]] in many [[middle school]] [[math]] [[class]]es. | ||
| + | |||
| + | ==The classical unsolved problems== | ||
| + | There were three very famous geometrical problems that they couldn't solve. Listed by their popular names, they are: | ||
| + | *"[[Squaring the circle]]"—Given a circle, construct a straight line whose length is equal to the circle's circumference. Equivalently, construct a square with the same area. Equivalently, given a square (or rectangle), construct a circle with the same area or the same perimeter. This can be seen to be the problem of constructing the ratio <math>\pi</math>. | ||
| + | *"Duplicating the cube"—Given the length of the side of a cube, construct the length of the side of a cube with twice the volume. This can be seen to be the problem of constructing the cube root of 2. | ||
| + | *"Trisecting an angle"—Given an arbitrary angle, divide it into 3 equal angles. Some angles (e.g. 90 degrees) are easy to trisect, but the problem is to do it for any angle. The ancient Greeks could bisect arbitrary angles (hence divide them by 4, 8, 16, ...) but could not trisect them. [[Pythagoras]] himself was said to have remarked that trisecting the angle was the hardest thing he had ever attempted. This problem is equivalent to solving cubic equations. | ||
| + | |||
| + | <!-- baloney! They knew about Zeno's paradox. They understood infinite processes. | ||
| + | [[Archimedes]] attempted a "[[rationalism|rationalistic]]" approach to trisecting the angle, in which he repeatedly ''bisected'' the angle until the number of divisions became a multiple of 3; however, this does not work for most angles, and in any event it is only physically possible to bisect most angles seven or eight times, so Archimedes' approach does not [[scale]]. --> | ||
| + | |||
| + | {{math-m}} | ||
| + | |||
| + | It wasn't until nearly two thousand years later that mathematicians proved that the problems are in fact impossible. It can be shown that any quantity that can be constructed by compass and straightedge must be a number that is algebraic and has a minimal polynomial which is a power of 2. Pi is transcendental, so it is not the root of any polynomial at all. Duplicating the cube requires the solution to the equation x<sup>3</sup> - 2 = 0, which is cubic. Trisecting angles also requires cubic equations. | ||
| + | |||
| + | <!-- Not so. Algebraic numbers of order 8 can be solved by compass and straightedge, but octics can't be solved with square roots. | ||
These problems can't be solved with compass and straightedge; the analysis can be done with the help of [[Abstract Algebra]]. In short, starting with a fixed length ''L'', any two points that can be drawn by intersecting cicles and lines based on already existing points must lie at a distance that can be expressed as ''x L'', where ''x'' is a number whose expression uses only basic arithmetic operations and square roots - reciprocally, for every number ''x'' that can be thus expressed, it's possible to construct a line segment of length ''x L''. | These problems can't be solved with compass and straightedge; the analysis can be done with the help of [[Abstract Algebra]]. In short, starting with a fixed length ''L'', any two points that can be drawn by intersecting cicles and lines based on already existing points must lie at a distance that can be expressed as ''x L'', where ''x'' is a number whose expression uses only basic arithmetic operations and square roots - reciprocally, for every number ''x'' that can be thus expressed, it's possible to construct a line segment of length ''x L''. | ||
''Squaring the circle'' is equivalent to drawing a line segment of length <math>\pi L\,</math>, which can't be done, because <math>\pi\,</math> is not an [[algebraic number]]. | ''Squaring the circle'' is equivalent to drawing a line segment of length <math>\pi L\,</math>, which can't be done, because <math>\pi\,</math> is not an [[algebraic number]]. | ||
| − | ''Doubling the cube'' and ''trisecting an angle'' are equivalent to solving irreducible [[cubic equations]], whose solutions can't be expressed with only square roots - cubic roots (of the real number 2, in the doubling of the cube, or of complex numbers, in the case of the trisection) are required. | + | ''Doubling the cube'' and ''trisecting an angle'' are equivalent to solving irreducible [[cubic equations]], whose solutions can't be expressed with only square roots - cubic roots (of the real number 2, in the doubling of the cube, or of complex numbers, in the case of the trisection) are required. --> |
[[Category:Plane Geometry]] | [[Category:Plane Geometry]] | ||
Latest revision as of 01:51, July 12, 2019
| <math>x+3=7</math>
<math>x=?</math> |
This article/section deals with mathematical concepts appropriate for a student in early high school. |
Compass and straightedge constructions played an important role in the history of mathematics. Some constructions accomplished by the ancients led to developments in abstract mathematics. Significantly, problems that the ancients couldn't solve by these methods led to even more revolutionary developments in abstract mathematics.
Background
To appreciate the significance of these constructions, it is important to realize that the ancients thought of mathematics in terms of geometric figures rather than the numbers, equations, and abstract things that we use today. These geometric constructions may be considered amusements today, but they were once the lifeblood of mathematics. The transition from thinking in terms of geometry to thinking in terms of numbers, formulas, and sets, happened very slowly, and wasn't really complete until the 19th century.
Rules of the game
You are given an abstract drawing surface (piece of parchment, clay tablet, patch of dirt, whatever), a writing implement (pencil, stylus, finger), a straightedge, and a compass.
- The straightedge is just a ruler with no markings. You can't measure; you can only draw straight lines.
- The compass can draw a circle through a given point, but, unlike modern compasses, it can't "remember" its radius from one circle to another. Once it is lifted off the paper, it "goes limp" and collapses.
The straightedge could be described as "infinitely long". A better description would be "as long as you need it to be", or "a construction will never fail because the straightedge wasn't long enough." The compass is similar—it can draw circles of any size.
The things you are allowed to do are
- Given any two points (you can recognize a point as the place where two lines—straight or circular—intersect), you can place the straightedge against them and draw a line passing through both.
- Given any two points, you can place the point of the compass on one of them, adjust it so that the pencil touches the other, and draw a circle. That circle will be centered on one and pass through the other.
The ancients (the Greeks were most famous for this) were able to do a remarkable amount of mathematics through reasoning about these geometric figures. Euclid's book The Elements is regarded as the best presentation of classical geometry. The methods of reasoning described by Euclid are widely used even today to teach logical reasoning and the principles of axioms, theorems, and proofs.
The mathematical "operations" that the ancient Greeks could do with these constructions include what we would now, in our number-centric world, would call addition, subtraction, multiplication, division, and square roots. They could also solve problems of proportion, bisect arbitrary angles, find areas of a many shapes, and construct some, but not all, regular polygons.
In modern times, compass-and-straightedge constructions were rediscovered by Euler, who resurrected them from the pages of Euclid's Elements and found that their simple geometric truths were a pleasant diversion from the calculus wars then raging between the Newtonian British and the Leibnizian German mathematical communities. In the twentieth century, compass-and-straightedge methods have become a popular form of recreational mathematics, and the technique is taught as a one-week unit in many middle school math classes.
The classical unsolved problems
There were three very famous geometrical problems that they couldn't solve. Listed by their popular names, they are:
- "Squaring the circle"—Given a circle, construct a straight line whose length is equal to the circle's circumference. Equivalently, construct a square with the same area. Equivalently, given a square (or rectangle), construct a circle with the same area or the same perimeter. This can be seen to be the problem of constructing the ratio <math>\pi</math>.
- "Duplicating the cube"—Given the length of the side of a cube, construct the length of the side of a cube with twice the volume. This can be seen to be the problem of constructing the cube root of 2.
- "Trisecting an angle"—Given an arbitrary angle, divide it into 3 equal angles. Some angles (e.g. 90 degrees) are easy to trisect, but the problem is to do it for any angle. The ancient Greeks could bisect arbitrary angles (hence divide them by 4, 8, 16, ...) but could not trisect them. Pythagoras himself was said to have remarked that trisecting the angle was the hardest thing he had ever attempted. This problem is equivalent to solving cubic equations.
| <math>x^{2}-5x+6=0</math>
<math>x=?</math> |
This article/section deals with mathematical concepts appropriate for a student in mid to late high school. |
It wasn't until nearly two thousand years later that mathematicians proved that the problems are in fact impossible. It can be shown that any quantity that can be constructed by compass and straightedge must be a number that is algebraic and has a minimal polynomial which is a power of 2. Pi is transcendental, so it is not the root of any polynomial at all. Duplicating the cube requires the solution to the equation x3 - 2 = 0, which is cubic. Trisecting angles also requires cubic equations.