I was reading the Wikipedia definition of a Geometric Simplex (https://en.wikipedia.org/wiki/Simplex):
We are told that a Simplex in more than 2 dimensions always has a triangular shape:
However, the equation of a simplex (i.e. thetha_o * u_o + ... + thetha_k * u_k ) seems to share the same general equation as that of a k-dimensional hyperplane (https://en.wikipedia.org/wiki/Hyperplane):
As we know, it is not necessary for a hyperplane to have a triangular shape - whereas a geometric simplex in more than 2 dimensions is always said to have a triangular shape.
Thus, can someone please explain the equation for a k-simplex is "always" has a triangular shape in more than 2 dimensions?
And how do we know that a "Simplicial Complex" (made by combinations of Simplexes) can produce any "shape"?
Thanks!
Note: I am told that these facts were important in the development of the "Simplex Algorithm" used in Linear Programming : A set of (linear) constraints will always form a Simplicial Complex, and we are interested in optimizing the objective function with respect to it's intersection with this Simplicial Complex : There are some theoretical properties (that I do not understand) which show that the Simplex Algorithm works by "navigating" (i.e. pivoting) down the vertices of this Simplicial Complex, until it finds the lowest vertex of this Simplicial Complex - and somehow, all this can be used to demonstrate the strong theoretical performance of the Simplex Algorithm in almost any Linear Programming problem.
You have asked several questions.
In the equation that specifies the points on a hyperplane the coefficients $\theta_i$ are arbitrary real numbers with a fixed sum. In the equation that specifies a simplex in that hyperplane the coefficients are restricted to nonnegative numbers that sum to $1$. When $\theta_i = 1$ and the other coefficients $\theta_j = 0$ you get the vector $u_i$ as the sum. Those $k+1$ vectors are the vertices of the simplex. For example, in the plane, the points where $x+y = 1$ is the line (hyperplane) joining the points $(1,0)$ and $(0,1)$. If you restrict to points where both $x$ and $y$ are nonnegative you get the line segment (a $1$-simplex) joining $(1,0)$ and $(0,1)$.
A $k$ dimensional simplex is the analogue in $k$ dimensional space of the triangle in the plane. It does not itself have a "triangular shape". It is true that every two dimensional face of a simplex is a triangle in the plane it lives in.
A simplicial complex is a finite collection of points and some subsets of those points. It's a combinatorial object. There is no geometry in its definition. But it can be used to study geometry. Any smooth surface (of any dimension) can be "trianglulated" - you can find a set of points on it and form a simplicial complex with those points such that the topology of the geometry of the surface is captured by the topology implicit in the simplicial complex. Think about studying the sphere by studying an icosahedron instead. From a topological perspective, the $12$ vertices, $30$ edges and $20$ triangular faces of the octahedron precisely mirror the topological "shape" of the sphere. (They don't capture the way the sphere is smooth.) See the wikipedia page on triangulation. The fact that modeling a high dimensional shape (surface) with simplexes is called "triangulation" is an instance of the analogy.
In a comment you ask why the simplex algorithm is so names. I do not know, but can hazard a plausible guess. The simplex algorithm finds the maximum of a function whose domain is a convex polytope. It works by walking along edges - the one dimensional faces of the polytope. In general (but not always), the $k-1$ dimensional faces of a $k$ dimensional polytope will be $k-1$ simplices: think about the triangular facets of the $3$ dimensional icosahedron.