construction of the regular 17-gon with straightedge and compass
E29366
classical geometry problem
constructible polygon problem
geometric construction
straightedge-and-compass construction
The construction of the regular 17-gon with straightedge and compass is a classical geometric achievement, first shown possible by Carl Friedrich Gauss, that exemplifies the link between constructible polygons and number theory.
Aliases (1)
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
classical geometry problem
→
constructible polygon problem → geometric construction → straightedge-and-compass construction → |
| hasAngleMeasure |
central angle 360°/17
→
interior angle 15·180°/17 → |
| hasConstructionType |
exact construction
→
|
| hasEducationalUse |
example in advanced Euclidean geometry courses
→
example in algebra and number theory courses → illustration of constructible numbers → |
| hasFieldExtensionDegree |
degree 8 over ℚ
→
|
| hasFirstProofYear |
1796
→
|
| hasHistoricalSignificance |
first new regular n-gon constructible in many centuries
→
influenced development of Galois theory → revived interest in Euclidean constructions → |
| hasKeyProperty |
cos(2π/17) is a constructible number
→
sin(2π/17) is a constructible number → |
| hasKeyQuantity |
cos(2π/17)
→
sin(2π/17) → |
| hasNumberOfSides |
17
→
|
| hasPolygonType |
regular 17-gon
→
|
| hasRelatedPolygon |
regular 257-gon
→
regular 3-gon → regular 5-gon → regular 65537-gon → |
| hasRelatedResult |
classification of constructible regular polygons
→
|
| hasStepType |
angle division via algebraic relations
→
|
| hasSymbolicMeaning |
bridge between classical geometry and modern algebra
→
symbol of Gauss's mathematical genius → |
| isConstructible |
true
→
|
| isDescribedIn |
Gauss's Disquisitiones Arithmeticae
→
|
| isExampleOf |
Gauss–Wantzel theorem
→
|
| isLinkedTo |
Galois theory
→
Gaussian periods → constructible numbers → cyclotomic fields → cyclotomic polynomial Φ₁₇(x) → number theory → roots of unity → |
| reliesOnProperty |
17 = 2^(2^2) + 1
→
17 is a Fermat prime → |
| satisfiesCriterion |
n is product of a power of 2 and distinct Fermat primes
→
|
| usesOperation |
nested square roots
→
successive quadratic extensions → |
| usesTool |
compass
→
straightedge → |
| wasProvedPossibleBy |
Carl Friedrich Gauss
→
|
Referenced by (2)
| Subject (surface form when different) | Predicate |
|---|---|
|
construction of the regular 17-gon with straightedge and compass
("Gauss–Wantzel theorem")
→
|
isExampleOf |
|
Carl Friedrich Gauss
→
|
notableWork |