Geometry

Geometry studies shapes, positions and spatial relationships through the structures and transformations under consideration. Euclidean geometry measures lengths and angles; projective geometry emphasizes incidence; differential geometry studies smooth spaces using tangent spaces and curvature. A circle can be stretched into an ellipse by an affine map, but generally not by a Euclidean isometry. Their curves are also homeomorphic, illustrating how topology preserves a different level of structure.

A convex polytope is the convex hull of finitely many points in a finite-dimensional real vector space. Equivalently, it is a bounded intersection of finitely many closed half-spaces. Its affine dimension may be smaller than that of the ambient space. Broader uses of polytope, including abstract or nonconvex versions, must specify their convention.

The simplex is the convex hull of affinely independent points; a square is a two-dimensional polytope with four vertices. The cube $[0,1{]}^n$ can be described either by its $2^n$ vertices or by the inequalities $0\le x_i\le 1$. These are its vertex and half-space descriptions.

A linear objective on a nonempty polytope attains its maximum at some vertex: every point is a convex combination of vertices, and its objective value is the same combination of vertex values. This links the geometry to linear programming. The maximizing vertex need not be unique, and an entire face may attain the optimum.

Doubling the cube is the classical construction task of producing a cube with twice the volume of a given cube. If the original edge has length $a>0$, the new edge must have length $a\sqrt[3]{2}$, since volume scales as the cube of edge length.

Starting from a unit segment, exact straightedge and compass construction cannot produce this length. The polynomial $x^3-2$ is irreducible over $\mathbb{Q}$ by the rational root theorem, so $[\mathbb{Q}(\sqrt[3]{2}):\mathbb{Q}]=3$. A real constructible number lies in a tower of quadratic extensions, and its degree over $\mathbb{Q}$ must therefore be a power of two. Three fails that necessary condition.

This is an impossibility for the stated construction tools, not for approximation or every enlarged tool set. A decimal approximation can be made as accurate as desired; allowing additional construction operations changes which lengths are exactly obtainable.

Angle trisection is the construction of an angle one-third as large as a given angle. Some angles can be trisected by straightedge and compass construction, but there is no such construction for every angle.

For a concrete obstruction, trisecting $60^{\circ }$ would construct $20^{\circ }$. Set $x=2\cos 20^{\circ }$. The triple-angle identity givesThe only possible rational roots of this monic cubic are $\pm 1$, and neither is a root. The cubic is therefore irreducible over $\mathbb{Q}$, giving $x$ degree three. A constructible number has degree a power of two, so this angle cannot be constructed from a unit length and the given constructible $60^{\circ }$ angle using those tools.

In contrast, $90^{\circ }$ can be trisected by constructing $30^{\circ }$ from an equilateral triangle. The failure concerns a general construction procedure; it does not prohibit all individual trisections, numerical approximations, or constructions with other tools.

A measure of separation between locations or objects. In a metric space, distance is specified by a metric satisfying positivity, symmetry and the triangle inequality. The familiar straight-line separation in Euclidean space is Euclidean distance; other metrics can assign different distances to the same pair of points.

The surface area of an object divided by its volume.

Two smooth curves have tangency at a shared point when their tangent lines agree. A tangent to a circle is perpendicular to its radius at the contact point: at $(r,0)$ on $x^2+y^2=r^2$ the tangent is $x=r$. For smooth surfaces the analogous condition uses tangent planes. Tangency is local and does not preclude crossing: $y=0$ and $y=x^3$ have the same tangent line at zero but pass through one another there.

A bounding box is a rectangle, or its higher-dimensional counterpart, containing a specified set. For finitely many points, the smallest axis-aligned box is ${\prod }_j[{\min }_i{x}_{ij},{\max }_i{x}_{ij}]$. For $(1,4),(3,-1),(2,2)$ this is $[1,3]\times [-1,4]$. Every coordinate extreme forces the corresponding box endpoint, proving minimality for the chosen axes. Allowing a rotation changes the optimization problem. Disjoint bounding boxes certify disjoint enclosed objects and accelerate geometric searches.

A corner of a polygon or polyhedron. In a polygon, it is the shared endpoint of consecutive polygon edges.