Simplicial complex

In mathematics, a simplicial complex is a special way of organizing simple shapes like points, line segments, and triangles together. These simple shapes are called simplices.

Simplicial complexes are important because they help mathematicians study the shape and structure of more complicated objects. By breaking things down into simple pieces, they can understand how these pieces fit together to form larger shapes. This idea is used in many areas, from studying the surfaces of balls to modeling complicated shapes in computer graphics.

It’s also helpful to know that there is a related idea called a simplicial set, which is used in more advanced areas of math like homotopy theory.

Definitions

A simplicial complex is a special group of shapes called simplices. Simplices can be points, lines, triangles, or even shapes with more sides.

The rules are easy: if a smaller shape is part of a simplex, it must be in the group. When two shapes overlap, the part where they overlap must also be a shape in the group.

For example, if a triangle is in the group, its three corners and three sides must also be included. This helps make ordered shapes and patterns that mathematicians study. A pure simplicial complex is one where all the largest shapes are the same size, like a group of triangles with no bigger shapes.

Support

In a simplicial complex, every point in the space belongs to one simplex. This simplex can be a point, a line, or a triangle. These shapes cover the whole space without overlapping in a confusing way.

Closure, star, and link

In a simplicial complex, the closure of a set of simplices is the smallest part of the complex that includes all those simplices and their faces. It is like adding every piece that touches the chosen pieces to make a complete small group.

The star of a set of simplices includes all simplices that have the chosen pieces as part of their shape. For a single simplex, its star is all the simplices that include it as a face. The link of a set of simplices is a special part of the complex that connects to the chosen pieces but does not include the pieces themselves or their faces.

Algebraic topology

Main article: Simplicial homology

In algebraic topology, simplicial complexes help us study shapes. They let us break shapes into simple pieces like points, lines, and triangles. These pieces help us see how shapes are connected.

More advanced studies use other types of spaces, called CW complexes, for bigger or more complicated shapes. Simplicial complexes can also be seen as special kinds of shapes called polytopes that live inside Euclidean space.

Combinatorics

Combinatorialists study something called the f-vector of a simplicial complex. This is a list of numbers that shows how many points, lines, triangles, and other shapes are in the complex.

For example, the boundary of an octahedron has an f-vector of (1, 6, 12, 8). This means it has 1 point, 6 lines, 12 triangles, and 8 areas.

We can change this list into a special math expression called an f-polynomial. By adjusting this polynomial in a certain way, we get another list called the h-vector. This helps mathematicians learn more about the shape of the complex. For the octahedron, the h-vector is (1, 3, 3, 1).

Simplicial complexes can also help us understand how spheres fit together in space.

Triangulation

Main article: Triangulation (topology)

Triangulation is a way to break down shapes in math using simple parts like points, lines, and triangles. It helps us study tricky spaces by splitting them into smaller, easier pieces. Many important shapes, especially those that are not very complex, can be split this way.

Embedding

A special kind of math shape called a simplicial complex with d dimensions can always fit inside a space with 2d + 1 dimensions. This follows a famous math rule called the Whitney embedding theorem.

Computational problems

The simplicial complex recognition problem asks if a group of shapes, such as points and triangles, matches a certain geometric object. For some complicated shapes in higher dimensions, computers cannot solve this problem.