What Is a Vertex in Geometry?
Learn the definition of a vertex in geometry, from lines and polygons to polyhedrons. Explore its properties, related theorems, and real-world applications in architecture and design.
Alice Brooks
Dec 22, 2024 8 mins read
At least every day we encounter various geometric forms—buildings, for instance, whose architectural structures can be broken into points, lines, and planes; there may also be prominent points, such as pointed roofs, that meet specific conditions to form what are called "vertices," an essential concept in geometry defining points where two curves meet or meet at an intersection. Let's investigate further what vertices is and discover its properties!
Definition of a Vertex
Vertices are one of the fundamental concepts in geometry, whether in two- or three-dimensional spaces. Vertices play an essential part of these worlds - either as two- or three-dimensional planes.
For Lines and Rays
Definition:
Lines and rays are two essential building blocks in geometry: lines can extend in both directions while rays start at one point and continue indefinitely along their chosen course; vertices serve as points where two intersect or meet; such an intersection occurs where lines or rays meet or intersect at their termini.
Examples:
Here is an image depicting an angle formed by two rays coming together at one common point and meeting in their vertex.
For Polygons
Definition:
Polygons are flat, two-dimensional shapes with straight sides that form fully closed shapes. A vertex for polygons can be defined as the point where two of its sides intersect—these meeting points between line segments are known as vertices.
Examples of common polygons:
A square has four points where its sides intersect, but triangles only contain three, pentagons five, decagons ten, etc.; unlike traditional polygons, however, circles contain no vertex points whatsoever!
For Polyhedrons
Definition:
Polyhedrons (plural: polyhedrons) are three-dimensional solid shapes consisting of flat polygonal faces connected by straight edges to form their surfaces; in comparison, spheres (plural: spheres) are three-dimensional objects characterized by circular surfaces without edges or vertices, thus distinguishing them from polyhedrons in terms of construction methods and shapes.
Examples of common polyhedrons:
A cube contains eight vertices where its edges intersect; in comparison, pyramids feature five and triangular prisms six, respectively. Conversely, unlike polyhedrons (including spheres and cylinders), which consist of straight surfaces, only vertices don't exist for these shapes.
History of Vertices
Early Geometry
Vertices have always played an essential part in geometry history and contemporary theories alike, from ancient mathematical texts to contemporary theory; from understanding and defining geometric forms through ancient mathematical texts such as Euclid's Elements to current-day applications using them to identify key intersection points within geometric constructs like his convex polyhedrons or triadic solids.
Middle Ages
In the Middle Ages, famous mathematicians such as Al-Khwarizmi and Omar Khayyam used advanced geometric principles to expand on earlier geometric concepts and establish new geometric principles. They developed their knowledge on vertices' role in more intricate shapes, as well as developed algebraic methodologies for solving geometric issues involving them, enriching this field significantly. Al-Khwarizmi pioneered algebraic methods for solving equations, while Omar Khayyam advanced cubic equations and their geometric interpretations, bridging algebra and geometry for future mathematicians.
Modern Mathematics
Leonhard Euler made groundbreaking contributions to our understanding of vertices through his groundbreaking work in graph theory and topology. Euler's formulation illuminated geometric relationships within convex polyhedra that gave rise to topological concepts; his famous V-E+F=2 formula highlighted these interrelationships within convex polyhedra while creating new topological ideas; his work inspired Gustav Kirchhoff, who made significant contributions using graph theory when applied to electrical circuit analysis using graph theory, while Arthur Cayley expanded our understanding of trees further, cementing these concepts into modern geometry.
Properties of a Vertex
A figure or object that does not have any sides or edges does not have any vertices.
Vertices possess an essential characteristic: geometric figures without sides or edges—such as single points and circles—do not contain vertices inherently. Vertices form at the intersections between edges; therefore, figures without edges do not have any vertices to begin with.
Connectivity
Vertices serve an integral function within geometric figures. They connect edges in polygons and polyhedrons to form the framework for these figures while their connectivity ensures structural integrity and coherence of complex geometric forms.
Collinearity and Coplanarity
Vertices play an essential role in defining collinearity and coplanarity—two properties essential to understanding three-dimensional geometry spatial relationships. Collinear vertices lie along an identical straight line, while coplanar ones inhabit one plane. These properties help us better comprehend spatial relations.
Theorems related to vertices
Numerous essential theorems involve vertices and can offer insight into their properties and roles. Here we will primarily introduce two such theorems—the Two Ears Theorem and Euler's Formula—but feel free to explore further with Upstudy!
Two ears theorem
Ears and mouths in polygons:
"Ear" refers to any vertex with two adjacent edges forming an outward-facing convex angle; conversely, "mouth" refers to vertices where two adjacent edges form an inward-facing concave angle.
Explanation of theorem:
The Two Ears Theorem states that every simple polygon (one that does not intersect itself) with more than three vertices has at least two ears that can be clipped off one at a time in an iterative process called "ear clipping" in order to triangulate and break apart into non-overlapping triangles.
Example:
Consider a pentagon with five vertices labeled A, B, C, D, and E; removing vertex B from an octagon will produce an easier quadrilateral shape for further geometric analysis. By identifying and eliminating areas in more complex polygons, such as an octagon, we can simplify its geometry for further examination.
Euler’s formula
Explanation:
Euler's formula, V - E + F = 2, connects the number of vertices (V), edges (E), and faces (F) found within polyhedrons. This equation provides key insight into their intrinsic properties for topology and geometry purposes.
Example:
One common example is a cube, with its eight vertices, 12 edges, and six faces; using Euler's formula we get 8 - 12 + 6 = 2, verifying Euler's theorem. In similar fashion, consider a tetrahedron that has four vertices, six edges, and four faces; applying Euler's formula results in 4 - 6 + 4 = 2, which also proves its validity as a theorem.
Related Geometric Concepts
Vertex angle
Definition:
A vertex angle refers to any angle formed when two sides of a polygon meet at its vertex; in three-dimensional shapes, this term also encompasses any angles formed at polyhedron vertices.
Examples:
An equilateral triangle's vertex angle measures 60 degrees, reflecting the regular polygon's uniform angle distribution. Each vertex angle in a square measures 90 degrees due to perpendicular sides intersecting perpendicularly, while in a regular pentagon, each vertex angle stands at 108. By comparison, edge polygons don't display a uniform distribution of angles but instead use edge edges as points that measure 120 degrees each at their vertex angles and 90 degrees, respectively, at their vertex angles, reflecting regular distribution.
Edge
Definition:
An edge is defined as any linear segment connecting two vertices within a polygon or polyhedron. Edges are generally straight lines that define two-dimensional forms and three-dimensional solids, respectively, and play an essential part in shaping their structures and properties.
Examples:
While triangles comprise three edges connecting two of their three vertices, squares contain four edges with straight lines connecting adjacent vertices; similarly, a cube has twelve such connections among adjacent edges, while tetrahedrons have six such relationships between adjacent faces of four faces.
Face
Definition:
Faces are flat surfaces that form part of the boundary for solid objects. Faces usually consist of polygonal shapes bound by edges. Facies play an essential part in three-dimensional geometry as they define polyhedron shapes and structures.
Examples:
A cube has six square faces; an equivalent rectangular prism has six identical rectangular faces on either end; pyramids may contain various numbers of faces depending on their base shape; for instance, a square pyramid typically features five triangular-shaped surfaces as part of its composition.
Applications of Vertices in Real Life
Applications of Vertices in Architecture and Construction
Vertices play an essential part in architecture and construction, serving to define corners and intersections of structures. Architects rely on them when designing buildings to ensure the stability and integrity of design; for roof construction, they mark points where segments cross at different intervals to form peaks. Structural engineers utilize them when optimizing material use, both for safety purposes as well as cost; from Gothic cathedrals to contemporary structures, vertices help produce resilient yet visually appealing architecture designs.
Applications of Vertices in Art and Design
Vertices play an integral part in art and design. Artists use them as plot points that define shapes and forms ranging from basic polygons to intricate three-dimensional models, while digital designers utilize software manipulation of vertices for digital graphic creation as well as for animation purposes. Furthermore, 3D modeling utilizes them as the bones for objects, which allows designers to craft realistic three-dimensional models, or in video game landscape design, where understanding vertices allows precise yet engaging designs.