Geometric Figures & Constructions¶

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Abstractions about geometric objects, figures, and foundational constructions, covering polygons and polyhedra (hexagons, kites, cyclic quadrilaterals, dual and ideal polyhedra), coordinate and metric constructions (orthants, quasi-isometries, Vincenty's formulae), and number-theoretic foundations like surreal numbers.

32 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.

Abstract polytope — In mathematics, an abstract polytope is an algebraic partially ordered set which captures certain combinatorial properties of a traditional polytope without specifying purely geometric properties such as the position of vertices.
Affine hull — In mathematics, the affine hull or affine span of a set S in Euclidean space \mathbb{R}^n is the smallest affine set containing S , or equivalently, the intersection of all affine sets containing S .
Cake number — In mathematics, the cake number, denoted by C n , is the maximum of the number of regions into which a 3-dimensional cube can be partitioned by exactly n planes.
Characterization (mathematics) — In mathematics, a characterization of an object is a set of conditions that, while possibly different from the definition of the object, is logically equivalent to it.
Construction of the Real Numbers — A consequence of the axioms is that this structure is unique up to an isomorphism, and thus, the real numbers can be used and manipulated, without referring to the method of construction.
Cyclic quadrilateral — In geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral (four-sided polygon) whose vertices all lie on a single circle, making the sides chords of the circle.
Divisor summatory function — In number theory, the divisor summatory function is a function that is a sum over the divisor function.
Dual polyhedron — In geometry, every polyhedron is associated with a second dual structure, wherein the vertices of one correspond to the faces of the other and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other.
Ellipse — A closed planar curve whose distances to two fixed foci sum to a constant greater than their separation.
Formal theorem — In mathematics and formal logic, a theorem is a statement that has been proven, or can be proven.
Heronian triangle — In geometry, a Heronian triangle (or Heron triangle) is a triangle whose side lengths , , and and area are all positive integers.
Hexagon — In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon.
Ideal polyhedron — In three-dimensional hyperbolic geometry, an ideal polyhedron is a convex polyhedron all of whose vertices are ideal points, points "at infinity" rather than interior to three-dimensional hyperbolic space.
Incidence (geometry) — In geometry, an incidence relation is a heterogeneous relation that captures the idea being expressed when phrases such as "a point lies on a line" or "a line is contained in a plane" are used.
Kite (geometry) — In Euclidean geometry, a kite is a quadrilateral with reflection symmetry across a diagonal.
Magic square — In mathematics, especially historical and recreational mathematics, a magic square is a square array of numbers, usually positive integers, where the sums of the numbers in each row, each column, and both main diagonals are the same.
Newton–Gauss line — In geometry, the Newton–Gauss line (or Gauss–Newton line) is the line joining the midpoints of the three diagonals of a complete quadrilateral.
Non-Archimedean geometry — In mathematics, non-Archimedean geometry is any of a number of forms of geometry in which the axiom of Archimedes is negated.
Orthant — In geometry, an orthant or hyperoctant is the analogue in n-dimensional Euclidean space of a quadrant in the plane or an octant in three dimensions.
Polyconic Projection Class — Polyconic as a class refers to those projections whose parallels are all non-concentric circular arcs, except for a straight equator, and the centers of these circles lie along a central axis.
Prototile — In mathematics, a prototile is one of the shapes of a tile in a tessellation.
Quasi-Isometry — In mathematics, a quasi-isometry is a function between two metric spaces that respects large-scale geometry of these spaces and ignores their small-scale details.
Real point — In geometry, a real point is a point in the complex projective plane with homogeneous coordinates for which there exists a nonzero complex number such that , , and are all real numbers.
Regular Polygon — A Euclidean polygon whose equally spaced vertices follow one constant edge-connection step, giving a connected rotationally symmetric boundary.
Ribbon Theory — In differential geometry, a ribbon (or strip) is the combination of a smooth space curve and its corresponding normal vector.
Simplicial sphere — In geometry and combinatorics, a simplicial (or combinatorial) d-sphere is a simplicial complex homeomorphic to the d-dimensional sphere.
Spherical trigonometry — Spherical trigonometry is the branch of spherical geometry and trigonometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions.
Supplementary Angles — When summing two angles (either adjacent or separated in space), three special cases are named complementary, supplementary, and explementary angles.
Surreal number — In mathematics, the surreal number system is a totally ordered proper class containing not only the real numbers but also infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number.
Vertex (curve) — In the geometry of plane curves, a vertex is a point of where the first derivative of curvature is zero.
Vincenty's formulae — Vincenty's formulae are two related iterative methods used in geodesy to calculate the distance between two points on the surface of a spheroid, developed by Thaddeus Vincenty (1975a).
Weakly o-minimal structure — In model theory, a weakly o-minimal structure is a model-theoretic structure whose definable sets in the domain are just finite unions of convex sets.