Editing 866: Compass and Straightedge
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{{w|Compass-and-straightedge construction|Compass and straightedge constructions}} are a class of problems in classical geometry. They take the form "Using only a compass and a straightedge, construct X", where X is a geometric figure such as a regular pentagon. The subject is typically covered in high school mathematics. Three such constructions ({{w|squaring the circle}}, {{w|Angle trisection|trisecting the angle}} and {{w|doubling the cube}}) remained unsolved for thousands of years before being shown impossible with the use of modern algebraic techniques. | {{w|Compass-and-straightedge construction|Compass and straightedge constructions}} are a class of problems in classical geometry. They take the form "Using only a compass and a straightedge, construct X", where X is a geometric figure such as a regular pentagon. The subject is typically covered in high school mathematics. Three such constructions ({{w|squaring the circle}}, {{w|Angle trisection|trisecting the angle}} and {{w|doubling the cube}}) remained unsolved for thousands of years before being shown impossible with the use of modern algebraic techniques. | ||
| − | The comic begins as if it were stating a problem in classical geometry but veers into an observation that no amount of technical knowledge can substitute for human companionship. An additional layer of humor is that [[Cueball]] is a stick figure so technically it is possible to create friends with a straightedge and a compass, a figure constructed like Cueball is | + | The comic begins as if it were stating a problem in classical geometry but veers into an observation that no amount of technical knowledge can substitute for human companionship. An additional layer of humor is that [[Cueball]] is a stick figure so technically it is possible to create friends with a straightedge and a compass, a figure constructed like Cueball is. |
{{w|Ferdinand von Lindemann}} was a German mathematician who showed in 1882 that pi is not a zero of any polynomial with rational coefficients, i.e. it is a transcendental number. Transcendental numbers cannot be constructed with straightedge and compass. This proves that {{w|squaring the circle}} (a problem where it is required to construct a square with the same area as a given circle) is impossible, being as the sides of the square would need to be √π times the radius of the circle, and pi is not constructible. | {{w|Ferdinand von Lindemann}} was a German mathematician who showed in 1882 that pi is not a zero of any polynomial with rational coefficients, i.e. it is a transcendental number. Transcendental numbers cannot be constructed with straightedge and compass. This proves that {{w|squaring the circle}} (a problem where it is required to construct a square with the same area as a given circle) is impossible, being as the sides of the square would need to be √π times the radius of the circle, and pi is not constructible. | ||
