I've been reading about the old geometry problems like squaring the circle, doubling the cube, etc., and I can't figure out what the limitations of the compass and straight-edge construction challenges were. What can't you do using the given instruments?
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Rules of compass and straight-edge constructions
- Thread starter cras
- Start date
I've been reading about the old geometry problems like squaring the circle, doubling the cube, etc., and I can't figure out what the limitations of the compass and straight-edge construction challenges were. What can't you do using the given instruments?
Of all the interesting problems from antiquity, duplicating the cube, squaring the circle, and trisecting the angle, attracted the most attention amongst Greek mathematicians to the point where they have attained a classical position in the history of geometry. Simply stated, the duplication of the cube requires the determination of the side of a cube whose volume is double that of a given cube; the squaring of a circle requires the determination of a square whose area is equal to that of a given circle; and the trisection of an angle requires the division of any angle into three equal parts. Each of the problems is to be solved by means of a geometrical construction involving the use of straight lines and circles only, that is by Euclidean geometry, meaning through the use of compasses and unmarked straight edges only. The compasses must be able to be opened as wide as desired and the straight edges must be of unlimited length.
With the Euclidean restriction all three problems are insoluble, the earliest rigorous proof of which is said to have been derived by P.L. Wantzel in 1837. To duplicate a cube the length of whose side is "a", you must find a line of length "x", such that x^3 = 2a^3. To trisect a given angle, you can find the sine of the angle, say "a", then, if "x" is the sine of an angle equal to one third of the given angle, you have 4x^3 = 3x - a. Clearly, from the analytical point of view, these problems require the solution of a cubic equation. However, since any constructions utilizing circles, whose equations are of the form x^2 + y^2 + ax + by + c = 0) and straight lines (whose equations are of the form of ax + by + c = 0) cannot be equivalent to the solution of a cubic equation, it is concluded that the problems are insoluble if we are restricted to the use of circles and straight lines in the constructions. The squaring of the circle requires the determination of the length "x" of a square whose area is equal to that of a given circle with radius "r." Since the area of the circle is Pi(r^2), you are left with the solution of x^2 = Pi(r^2). Every student of serious mathematics knows that one cannot obtain the square root of the irrational and transcendental number 3.14159....!
Probably the most pursued of the three problems is the trisection of the angle. Thousands of trisections have been proposed over the years, alas, none of which satisfy the given groundrules. I must admit to being one of the passionate pursuers of the "dream". I developed a method back in 1954, verified it numerically, but have yet to be able to prove it analytically. I take it out every now and then, recognizing full well that I am merely deluding myself into thinking that "someday, someone will do it."
There is an excellant discussion of all three of the problems in the fine book, "Mathematical Recreations and Essays by W.W. Rouse Ball and H.S.M. Coxeter, Dover Publications, Inc., 1987, pgs.338-359. The book has been a classic for over a century and has provided many stimulating hours of entertainment to the mathematically inclined. The problems posed often involve fundamental mathematical methods and notions, but their chief appeal is their capacity to tease and delight. You will find scores of "recreations" to amuse you and to challenge your problem-solving faculties--often to the limit.
If trisection interests you, I can heartily recommend the excellant book, "The Trisectors" by Underwood Dudley, Mathematical Association of America, 1994. It is a lighthearted treatment of the subject and contains a compilation of the many trisections offered over the years. It also contains detailed information about the personalities of trisectors and their constructions. It is a fun book.
Of all the interesting problems from antiquity, duplicating the cube, squaring the circle, and trisecting the angle, attracted the most attention amongst Greek mathematicians to the point where they have attained a classical position in the history of geometry. Simply stated, the duplication of the cube requires the determination of the side of a cube whose volume is double that of a given cube; the squaring of a circle requires the determination of a square whose area is equal to that of a given circle; and the trisection of an angle requires the division of any angle into three equal parts. Each of the problems is to be solved by means of a geometrical construction involving the use of straight lines and circles only, that is by Euclidean geometry, meaning through the use of compasses and unmarked straight edges only. The compasses must be able to be opened as wide as desired and the straight edges must be of unlimited length.
With the Euclidean restriction all three problems are insoluble, the earliest rigorous proof of which is said to have been derived by P.L. Wantzel in 1837. To duplicate a cube the length of whose side is "a", you must find a line of length "x", such that x^3 = 2a^3. To trisect a given angle, you can find the sine of the angle, say "a", then, if "x" is the sine of an angle equal to one third of the given angle, you have 4x^3 = 3x - a. Clearly, from the analytical point of view, these problems require the solution of a cubic equation. However, since any constructions utilizing circles, whose equations are of the form x^2 + y^2 + ax + by + c = 0) and straight lines (whose equations are of the form of ax + by + c = 0) cannot be equivalent to the solution of a cubic equation, it is concluded that the problems are insoluble if we are restricted to the use of circles and straight lines in the constructions. The squaring of the circle requires the determination of the length "x" of a square whose area is equal to that of a given circle with radius "r." Since the area of the circle is Pi(r^2), you are left with the solution of x^2 = Pi(r^2). Every student of serious mathematics knows that one cannot obtain the square root of the irrational and transcendental number 3.14159....!
Probably the most pursued of the three problems is the trisection of the angle. Thousands of trisections have been proposed over the years, alas, none of which satisfy the given groundrules. I must admit to being one of the passionate pursuers of the "dream". I developed a method back in 1954, verified it numerically, but have yet to be able to prove it analytically. I take it out every now and then, recognizing full well that I am merely deluding myself into thinking that "someday, someone will do it."
There is an excellant discussion of all three of the problems in the fine book, "Mathematical Recreations and Essays by W.W. Rouse Ball and H.S.M. Coxeter, Dover Publications, Inc., 1987, pgs.338-359. The book has been a classic for over a century and has provided many stimulating hours of entertainment to the mathematically inclined. The problems posed often involve fundamental mathematical methods and notions, but their chief appeal is their capacity to tease and delight. You will find scores of "recreations" to amuse you and to challenge your problem-solving faculties--often to the limit.
If trisection interests you, I can heartily recommend the excellant book, "The Trisectors" by Underwood Dudley, Mathematical Association of America, 1994. It is a lighthearted treatment of the subject and contains a compilation of the many trisections offered over the years. It also contains detailed information about the personalities of trisectors and their constructions. It is a fun book.