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.

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All labels observed (2)

Label	Occurrences
Gauss–Wantzel theorem	1
construction of the regular 17-gon with straightedge and compass canonical	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	Disquisitiones Arithmeticae ⓘ surface form: Gauss's Disquisitiones Arithmeticae
isExampleOf	construction of the regular 17-gon with straightedge and compass self-linksurface differs ⓘ surface form: 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)

Full triples — surface form annotated when it differs from this entity's canonical label.

Carl Friedrich Gauss → notableWork → construction of the regular 17-gon with straightedge and compass ⓘ

construction of the regular 17-gon with straightedge and compass → isExampleOf → construction of the regular 17-gon with straightedge and compass self-linksurface differs ⓘ

this entity surface form: Gauss–Wantzel theorem