Why Use 'Singular' to Describe Singular Simplex?

[navigator]

Why do we use the term "singular" to describe singular simplex? Are there any relations between singular matrix (or singular point)and singular simplex?

 

[quasar987]

I ran a google search for singular and found http://www.thefreedictionary.com/singular. Definitions 3 and 4 read

3. Being beyond what is ordinary or usual; remarkable.
4. Deviating from the usual or expected; odd. See Synonyms at strange.

and I believe it is the meaning appropriate for "singular simplex". Indeed, whereas in simplicial homology, the objects of interest are actual simplices (the n-dimensional generalisation of a triangle), in singular homology, the objects of interest are merely continuous images of simplices. In particular, these images may be very degenerate (a point for instance) and not resemble a simplex at all. That is, they may deviate from what is expected of something called a simplex.

 

[zhentil]

To answer the other question, singular matrices and singular simplices have nothing to do with each other, unless you really stretch :)

(The big stretch: if you have a degenerate simplex in the top dimension represented by a differentiable map, its Jacobian would be singular)

 

[navigator]

Thank for your replies.

 

[blue_raver22]

The term "singular" is used to describe singular simplex because it refers to a geometric object that has a unique and simple structure. In mathematics, a simplex is a geometric figure with a minimal number of vertices, which can be thought of as the simplest form of a polytope. The term "singular" is used to emphasize the simplicity and uniqueness of the simplex.

There is a relation between singular matrix (or singular point) and singular simplex, as both refer to a state of being unique and simple. A singular matrix is a square matrix that does not have an inverse, meaning it cannot be inverted to obtain a unique solution. Similarly, a singular point is a point in a geometric figure that has no neighbors, making it a unique and simple point. In both cases, the term "singular" is used to describe the unique and simple nature of the object.