Hexagon¶

In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon.

Version: v1 · 2026-09-28 · History
Domain-specific #: 9855
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomains: Geometry, Polygons → Mathematics

Core Idea¶

Hexagon is treated here as the recurring mathematics_logic_statistics identity summarized by this source-grounded definition: In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon.

In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°. A regular hexagon is defined as a hexagon that is both equilateral and equiangular.

In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. The Lemoine hexagon is a cyclic hexagon (one inscribed in a circle) with vertices given by the six intersections of the edges of a triangle and the three lines that are parallel to the edges that pass through its symmedian point. Pascal's theorem (also known as the "Hexagrammum Mysticum Theorem") states that if an arbitrary hexagon is inscribed in any conic section, and pairs of opposite sides are extended until they meet, the three intersection points will lie on a straight line, the "Pascal line" of that configuration.

For Hexagon, the abstraction is narrower than the article's general subject matter: a positive case must preserve In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. Retaining only the name, a familiar example, or a downstream effect is insufficient. The specialist roles and tests remain anchored in mathematics_logic_statistics, which is why this identity is domain-specific rather than prime.

Structural Signature¶

Sig role-phrases:

Defining carrier — The Lemoine hexagon is a cyclic hexagon (one inscribed in a circle) with vertices given by the six intersections of the edges of a triangle and the three lines that are parallel to the edges that pass through its symmedian point.
Constitutive relation — Let ABCDEF be a hexagon formed by six tangent lines of a conic section.
Operating condition — For the regular hexagon these are given by a = r, and p {} = 6R = 4r\sqrt{3} , so.
Recognition evidence — Each subgroup symmetry allows one or more degrees of freedom for irregular forms.
Admissible variation — Hexagons of symmetry g2, i4, and r12, as parallelogons can tessellate the Euclidean plane by translation.
Characteristic consequence — The 6 roots of the simple Lie group A2, represented by a Dynkin diagram , are in a regular hexagonal pattern.
Failure boundary — The 12 roots of the Exceptional Lie group G2, represented by a Dynkin diagram are also in a hexagonal pattern.

What It Is Not¶

Not the whole field of mathematics_logic_statistics. The node requires the specific identity stated by In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon.
Not an over-broad reading. However, the regular hexagon can also be considered as cutting off the vertices of an equilateral triangle, which can also be denoted as \mathrm{t}{3} .
Not an over-broad reading. The 12 roots of the Exceptional Lie group G2, represented by a Dynkin diagram are also in a hexagonal pattern.
Not an over-broad reading. A skew hexagon is a skew polygon with six vertices and edges but not existing on the same plane.
Not automatically Right triangle. Retrieval proximity does not establish equivalence; the two identities must be compared by carrier, operation, and failure boundary.

Scope of Application¶

Hexagon applies literally inside mathematics_logic_statistics wherever the source-defined carrier and relation can be established. Its documented habitats include:

ConstructionSymmetry. Each subgroup symmetry allows one or more degrees of freedom for irregular forms.
Regular hexagon. A regular hexagon is defined as a hexagon that is both equilateral and equiangular.
Regular hexagon. However, the regular hexagon can also be considered as cutting off the vertices of an equilateral triangle, which can also be denoted as \mathrm{t}{3} .
Regular hexagon. A regular hexagon is bicentric, meaning that it is both cyclic (has a circumscribed circle) and tangential (has an inscribed circle).
Regular hexagon. The common length of the sides equals the radius of the circumscribed circle or circumcircle, which equals \tfrac{2}{\sqrt{3}} times the apothem (radius of the inscribed circle).
Measurement. The longest diagonals of a regular hexagon, connecting diametrically opposite vertices, are twice the length of one side.

Outside mathematics_logic_statistics, the name should be retained only when these same operational conditions survive; otherwise the comparison belongs to the broader parent Pattern or should be marked as analogy.

Clarity¶

A clear use of Hexagon names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The strongest recognition evidence in the frozen account is: Each subgroup symmetry allows one or more degrees of freedom for irregular forms. A report should distinguish that evidence from a proxy, consequence, or common implementation. It should also state the qualification However, the regular hexagon can also be considered as cutting off the vertices of an equilateral triangle, which can also be denoted as \mathrm{t}{3} . so that a reader can reproduce the classification rather than infer it from topical resemblance.

Manages Complexity¶

Hexagon compresses multiple mathematics_logic_statistics details into a stable diagnostic relation. The source shows both the central mechanism—let ABCDEF be a hexagon formed by six tangent lines of a conic section.—and the practical consequence—the 6 roots of the simple Lie group A2, represented by a Dynkin diagram , are in a regular hexagonal pattern. This compression makes cases comparable while leaving parameters, conventions, exceptions, and evidential quality explicit. It is lossy by design: local history and implementation details may be omitted only when they do not alter the defining relation.

Abstract Reasoning¶

Type the carrier. Identify the mathematics_logic_statistics entities to which the claim applies.
State the relation. Use the source-grounded identity: In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon.
Check operation and conditions. For the regular hexagon these are given by a = r, and p {} = 6R = 4r\sqrt{3} , so.
Demand recognition evidence. Each subgroup symmetry allows one or more degrees of freedom for irregular forms.
Test variation. Change an implementation or setting while preserving hexagons of symmetry g2, i4, and r12, as parallelogons can tessellate the Euclidean plane by translation.
Run the collapse test. Remove the defining operation; if the label still seems equally apt, only a topic or correlate was retained.
Reduce cautiously. When the specialist conditions cannot be carried, route the residual comparison to Pattern.

Knowledge Transfer¶

Within the home domain. Knowledge about Hexagon transfers literally when a new case preserves the same carrier type, relation, and recognition test. Each subgroup symmetry allows one or more degrees of freedom for irregular forms. A regular hexagon is defined as a hexagon that is both equilateral and equiangular.

Beyond the home domain. No canonical parent is asserted for Hexagon. An outside case receives the specialist name only when the same typed roles and rejection conditions can be filled literally; otherwise the comparison remains an analogy pending later graph densification.

Examples¶

Canonical¶

In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. This case is canonical because it supplies a concrete carrier and lets the defining relation be checked rather than merely named.

Mapped back: carrier → the entities in the documented case; operation → In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon; recognition evidence → Each subgroup symmetry allows one or more degrees of freedom for irregular forms

Applied / In Practice¶

If a hexagon has vertices on the circumcircle of an acute triangle at the six points (including three triangle vertices) where the extended altitudes of the triangle meet the circumcircle, then the area of the hexagon is twice the area of the triangle. The applied case shows how the identity is used under a second setting or qualification while keeping the same operative relation.

Mapped back: changed setting → Cyclic hexagon; invariant → In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon; boundary → the case exits the class when however, the regular hexagon can also be considered as cutting off the vertices of an equilateral triangle, which can also be denoted as \mathrm{t}{3}

Structural Tensions¶

T1 — Stable identity versus admissible variation. However, the regular hexagon can also be considered as cutting off the vertices of an equilateral triangle, which can also be denoted as \mathrm{t}{3} . The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Which changes preserve the defining relation, and which replace it?

T2 — Recognition versus proxy. The 12 roots of the Exceptional Lie group G2, represented by a Dynkin diagram are also in a hexagonal pattern. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the cited evidence establish the identity or only a correlated sign?

T3 — Definition versus implementation. A skew hexagon is a skew polygon with six vertices and edges but not existing on the same plane. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Is the observed implementation constitutive, optional, or merely common?

T4 — Scope versus overextension. There is no Platonic solid made of only regular hexagons, because the hexagons tessellate, not allowing the result to "fold up". The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Can every claimed application fill the same typed roles without metaphor?

T5 — Transfer versus domain accent. The Lemoine hexagon is a cyclic hexagon (one inscribed in a circle) with vertices given by the six intersections of the edges of a triangle and the three lines that are parallel to the edges that pass through its symmedian point. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the receiving case instantiate Hexagon literally, co-instantiate Pattern, or only resemble it?

T6 — Autonomy versus reduction. Let ABCDEF be a hexagon formed by six tangent lines of a conic section. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: What does Hexagon distinguish that the broader parent Pattern leaves together?

Structural–Framed Character¶

Hexagon is structural-leaning. Its structural side is the repeatable organization summarized by In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. Its framed side is the mathematics_logic_statistics vocabulary that fixes the carrier, evidence, exceptions, and admissible transformations.

Evaluative weight: the identity can be stated descriptively even when applications carry practical stakes. Human-practice dependence: the source-grounded carrier determines whether the relation exists independently or is constituted by a practice. Institutional origin: disciplinary conventions stabilize the name and test. Vocabulary portability: For the regular hexagon these are given by a = r, and p {} = 6R = 4r\sqrt{3} , so. Import versus recognition: literal transfer requires the same mechanism; shape alone is analogy.

Its portable skeleton is Pattern. Its character: a recurring specialist identity whose thin organization can be abstracted, while its operational meaning remains domain-bound.

Structural Core vs. Domain Accent¶

What is skeletal. In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The stable skeleton is the typed relation expressed in that definition and the entry's recognition and collapse tests. The source identifies these operative conditions: The Lemoine hexagon is a cyclic hexagon (one inscribed in a circle) with vertices given by the six intersections of the edges of a triangle and the three lines that are parallel to the edges that pass through its symmedian point. Let ABCDEF be a hexagon formed by six tangent lines of a conic section. It further constrains recognition and variation through: For the regular hexagon these are given by a = r, and p {} = 6R = 4r\sqrt{3} , so. Each subgroup symmetry allows one or more degrees of freedom for irregular forms.

What is domain-bound. mathematics logic statistics supplies the operative entities, technical vocabulary, warrants, and exceptions that make Hexagon literal. Its documented scope includes the condition that Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Another bounded application condition is that A regular hexagon is defined as a hexagon that is both equilateral and equiangular. These are not decorative examples; they determine which carrier and evidence can fill the abstraction's roles.

Why no parent is asserted. Removing those specialist details does not currently yield one live catalog node that is a necessary genus for every instance. The entry is therefore approved as unparented rather than attached by topical resemblance. Its collapse evidence remains specific—Hexagons of symmetry g2, i4, and r12, as parallelogons can tessellate the Euclidean plane by translation.—and future graph densification may discover a defensible relation only if it preserves that boundary.

This entry is a kind of Polygon.

Approved unparented node. No current live node supplies a defensible necessary genus or structural prerequisite for Hexagon. The reviewed identity is: In geometry, a hexagon (from Greek,, meaning "six", and,, meaning "corner, angle") is a six-sided polygon. The accelerated suggestion was declined because topical or lexical similarity does not establish hierarchy; the node is admitted without a parent pending later graph densification.
Related reasoning operations. Evidence, representation, comparison, classification, transformation, or evaluation may participate in particular cases, but participation does not make any one of them a necessary parent of every instance.

Relationships to Other Abstractions¶

Current abstraction Hexagon Domain-specific

Parents (1) — more general patterns this builds on

Hexagon is a kind of Polygon Domain-specific

It is a six-sided polygon.

Hierarchy path (1) — routes to 1 parentless root

Hexagon → Polygon

Neighborhood in Abstraction Space¶

Hexagon sits in a sparse region of the domain-specific corpus (75^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Geometric Figures & Constructions (32 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With¶

Pattern. The parent omits the specialist differentia. Tell: Can the case establish In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon?
Right triangle. Specify a triangle with exactly one right angle, making the opposite side the hypotenuse and coupling its sides, acute angles, altitude, circumcircle, and similarity relations through Euclidean constraints. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Acute and obtuse triangles. A Euclidean triangle classification based on whether all angles are below a right angle or exactly one angle exceeds it. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Corner-point grid. A three-dimensional hexahedral grid whose cells are defined by ordered pillars and independently positioned corner points along those pillars. Tell: Which entry's carrier, operation, and failure condition are satisfied?
A measurement, proxy, or consequence. Those may provide evidence without being the identity. Tell: Would Hexagon remain present if the detector or downstream effect changed?
A metaphorical analogue. A similar shape outside mathematics_logic_statistics lacks the specialist mechanism. Tell: Do the native roles transfer literally, or only the parent Pattern?

References¶

Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Hexagon (revision 1370160831).
Preserved source candidate: https://deimel.org/images/plain_cube.gif
Preserved source candidate: https://books.google.com/books?id=N8lX2T-4njIC&pg=PA9
Preserved source candidate: https://web.archive.org/web/20160102075753/https://books.google.com/books?id=N8lX2T-4njIC&pg=PA9
Preserved source candidate: https://www.rgnpublications.com/journals/index.php/cma/article/view/1420/1065
Preserved source candidate: https://books.google.com/books?id=koUfrlgsmUcC&pg=PA92
Preserved source candidate: https://books.google.com/books?id=zyRXEAAAQBAJ&pg=PA26
Preserved source candidate: http://forumgeom.fau.edu/FG2014volume14/FG201424index.html
Preserved source candidate: https://web.archive.org/web/20141205210609/http://forumgeom.fau.edu/FG2014volume14/FG201424index.html

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.