Euclidean & Geometric Structures

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Abstractions about geometric objects and their defining properties — triangle and polygon classifications, symmetry groups, conics, and singularities like the tacnode — together with related integral transforms and combinatorial-geometric problems such as kissing numbers, foliations, and quadrisecants.

24 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Acute and obtuse triangles — A Euclidean triangle classification based on whether all angles are below a right angle or exactly one angle exceeds it.
• Datum reference — A declared geometrically significant point, axis, plane or feature used as the origin or constraint for defining and measuring other features of an object.
• Equichordal point problem — The plane-geometry question whether a convex body can have two distinct interior points through each of which every chord has the same point-specific length, answered negatively.
• Fixed points of isometry groups in Euclidean space — The common-fixed-point set of a Euclidean isometry group, which is either empty or an affine subspace.
• Foliation — A decomposition of a manifold into connected immersed submanifolds of equal dimension that locally look like parallel coordinate slices.
• Kissing number — The maximum number of nonoverlapping congruent spheres that can simultaneously touch one congruent central sphere in a specified space or dimension.
• Law of sines — Relate each side of a triangle to the sine of its opposite angle through one common ratio equal to the circumdiameter in Euclidean geometry, enabling triangle solution while preserving the side-side-angle ambiguous case and curvature-specific variants.
• Octahedral symmetry — The finite symmetry group of a regular octahedron, equivalently a cube, comprising 24 rotations and 48 full isometries when reflections are included.
• One-seventh area triangle — The inner triangle formed by three one-third cevians of a triangle, whose area is exactly one seventh of the original.
• Orthocentric system — A planar set of four points in which each point is the orthocenter of the triangle formed by the other three.
• Orthoptic (geometry) — The locus of points from which two tangents to a given curve meet at a right angle.
• Parabola — A conic curve whose points are equidistant from a fixed focus and directrix, equivalently a nondegenerate quadratic curve of eccentricity one.
• Pi — The dimensionless mathematical constant equal to a Euclidean circle's circumference divided by its diameter, equivalently characterized through analysis, with an irrational transcendental value near 3.14159.
• Quadrisecant — A line meeting a spatial curve or related geometric set in four distinct points, a maximal generic multi-secant whose existence and ordering carry knot- and algebraic-geometric information.
• Radon transform — An integral transform mapping a function to its integrals over hyperplanes, with inversion reconstructing the original under suitable conditions.
• Reuleaux polygon — A convex constant-width curve assembled from an odd number of equal-radius circular arcs, with each arc centered at an opposite vertex of its generating 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.
• Simplex — The convex hull of n+1 affinely independent points in n-dimensional space, generalizing a point, segment, triangle and tetrahedron as the simplest full-dimensional polytope.
• Star polyhedron — A nonconvex polyhedron with systematic star-like self-intersection or alternating salient and reentrant geometry.
• Tacnode — A plane-curve double singularity where two smooth local branches share the same tangent, canonically modeled by y²=x⁴ and carrying higher contact than an ordinary node.
• Tangential quadrilateral — A convex quadrilateral whose four sides are tangent to one inscribed circle.
• Translation surface — A surface assembled from Euclidean polygons by identifying parallel boundary edges through translations, yielding a flat metric away from finitely many conical singularities.
• Triangle group — A reflection group generated by the sides of a spherical, Euclidean or hyperbolic triangle with angles π/l, π/m and π/n, acting on the corresponding tessellation.
• Uniform polyhedron — A polyhedron with regular polygonal faces and a symmetry group transitive on vertices, including convex, star and self-intersecting examples.