#topology #simplex In topological terms, the term "simplex" refers to a geometric object that is the generalization of a line segment (1-dimensional simplex), a triangle (2-dimensional simplex), and a tetrahedron (3-dimensional simplex) to higher dimensions. A simplex in topology is defined as the convex hull of a set of points in Euclidean space. It can be thought of as the simplest possible polytope in any given dimension. For example, a 0-dimensional simplex is simply a point, and a 1-dimensional simplex is a line segment connecting two points. In general, an n-dimensional simplex consists of n+1 vertices and all the line segments connecting these vertices. These line segments form the edges of the simplex. Furthermore, each vertex is incident to exactly n edges, each edge is incident to exactly two vertices, and each face (higher-dimensional analogs of triangles) is incident to exactly n edges. Simplexes have several important properties in topology. For instance, they are used as building blocks for constructing more [[Complex|complex]] topological spaces through techniques like triangulation. Additionally, simplicial complexes are often used to approximate more general topological spaces in computational geometry and computer graphics. ## Simplex in Cubical Type Theory In the context of [[Robert Harper]]'s [[Cubical Type Theory]], simplexes from topology offer a valuable analogy for understanding the geometric representation of types and logical relationships in programming. The relationship between Cubical Type Theory and the concept of simplexes can be explored through their shared emphasis on dimensional structures and connections. Cubical Type Theory fundamentally uses the notion of cubes instead of simplexes. However, the idea of a simplex as a geometric object—the generalization of a point, line segment, triangle, and tetrahedron to higher dimensions—parallels how Cubical Type Theory deals with higher-dimensional cubes. In Cubical Type Theory, these cubes serve as the basic building blocks to represent types and logical relations in a higher-dimensional, computational framework. A simplex, defined as the convex hull of a set of points in Euclidean space, embodies the simplest form of a polytope in any given dimension, which conceptually supports the basic elements in Cubical Type Theory. Although Cubical Type Theory primarily operates with cubes (n-dimensional hypercubes to be specific), the foundational idea from simplexes about connecting vertices and forming higher-dimensional shapes via simpler components is directly relevant. In topology, simplexes are used for constructing complex topological spaces through techniques like triangulation, and in computational tasks, they approximate shapes and spaces. Similarly, in Cubical Type Theory, cubes (analogous to simplexes but with orthogonal edges) are used to structure data and encode proofs within a logical framework, permitting a systematic exploration of computational properties and relations in a topologically coherent manner. Moreover, the connection between vertices, edges, and faces in a simplex mirrors the way dimensions, directions, and connections are considered in [[Cubical Type Theory]]. This analogy extends to how computational relationships and type dependencies are mapped out in programming languages, utilizing a geometrically intuitive model that aligns with Harper’s views on the subject. Each dimension in a cube or simplex can represent a different variable or type, allowing complex logical constructs to be visualized and manipulated geometrically, which is central to the philosophy of [[Cubical Type Theory]]. # Connection with Platonic Solids There's a fascinating connection between the topological notion of simplexes and the [[Platonic solids]]: Also see [[Platonic solids and Topological Simplexes]]. **Simplexes as Building Blocks:** - **Simplexes:** In topology, simplexes are the simplest possible shapes in each dimension: a point (0-simplex), a line segment (1-simplex), a triangle (2-simplex), a tetrahedron (3-simplex), and so on. They are the fundamental building blocks for constructing more complex shapes. [1. Simplex - Wikipedia](https://en.wikipedia.org/wiki/Simplex) [](https://en.wikipedia.org/wiki/Simplex) - **Platonic Solids:** Platonic solids are three-dimensional shapes where all faces are identical regular polygons and the same number of faces meet at each vertex. There are only five: tetrahedron, cube, octahedron, dodecahedron, and icosahedron. [1. 7.5: Platonic Solids - Mathematics LibreTexts](https://math.libretexts.org/Bookshelves/Applied_Mathematics/Mathematics_for_Elementary_Teachers_(Manes)/07%3A_Geometry/7.05%3A_Platonic_Solids#:~:text=All%20vertices%20are%20also%20identical,solids%20(named%20for%20Plato).) [](https://math.libretexts.org/Bookshelves/Applied_Mathematics/Mathematics_for_Elementary_Teachers_(Manes)/07%3A_Geometry/7.05%3A_Platonic_Solids#:~:text=All%20vertices%20are%20also%20identical,solids%20(named%20for%20Plato).) [2. Platonic solid | Regular polyhedron, 5 elements & symmetry - Britannica](https://www.britannica.com/science/Platonic-solid#:~:text=Also%20known%20as%20the%20five,580%E2%80%93c.) [](https://www.britannica.com/science/Platonic-solid#:~:text=Also%20known%20as%20the%20five,580%E2%80%93c.) **The Connection:** - **Faces of Platonic Solids:** The faces of Platonic solids are themselves simplexes. For example: - The faces of a tetrahedron are triangles (2-simplexes). [1. Number of faces in a tetrahedron are - BYJU'S](https://byjus.com/question-answer/number-of-faces-in-a-tetrahedron-are/#:~:text=A%20Tetrahedron%20has%204%20triangular%20faces.&___-,Q.,and%20vertices%20of%20a%20tetrahedron.&Q.,-A%20tetrahedron%20has) [](https://byjus.com/question-answer/number-of-faces-in-a-tetrahedron-are/#:~:text=A%20Tetrahedron%20has%204%20triangular%20faces.&___-,Q.,and%20vertices%20of%20a%20tetrahedron.&Q.,-A%20tetrahedron%20has) - The faces of a cube are squares, which can be divided into two triangles (2-simplexes). - The faces of an octahedron are triangles (2-simplexes). [1. Octahedron - Shape, Meaning, Formula, Examples - Cuemath](https://www.cuemath.com/geometry/octahedron/#:~:text=Face%3A%20An%20octahedron%20consists%20of,octahedron%20consists%20of%2012%20edges) [](https://www.cuemath.com/geometry/octahedron/#:~:text=Face%3A%20An%20octahedron%20consists%20of,octahedron%20consists%20of%2012%20edges) - The faces of a dodecahedron are pentagons, which can be divided into three triangles (2-simplexes). - The faces of an icosahedron are triangles (2-simplexes). [1. Icosahedron](https://en.wikipedia.org/wiki/Icosahedron#:~:text=The%20best%20known%20is%20the,faces%20are%2020%20equilateral%20triangles.) [](https://en.wikipedia.org/wiki/Icosahedron#:~:text=The%20best%20known%20is%20the,faces%20are%2020%20equilateral%20triangles.) - **Triangulation:** Any polyhedron (including Platonic solids) can be broken down or "triangulated" into a collection of simplexes. This is a key concept in computational geometry and computer graphics, where complex shapes are often represented as collections of triangles for rendering and analysis. **In Summary:** Platonic solids, while geometrically distinct, can be understood as being built from simpler topological elements - the simplexes. This connection highlights the fundamental role simplexes play in the construction and representation of shapes in mathematics and beyond. # Conclusion Overall, simplexes play a fundamental role in topological discussions due to their simplicity and ability to represent higher-dimensional objects using lower-dimensional building blocks. # References ```dataview Table title as Title, authors as Authors where contains(subject, "simplex") or contains(subject, "simplicial surface") or contains(subject, "Simplicial Complexes") or contains(subject, "Simplex") sort title, authors, modified ```