aphyer comments on Insights from Euclid’s ‘Elements’

aphyer 15 Feb 2021 1:57 UTC
2 points
A bit pedantic but still worth pointing out—unless I’m very mistaken or Euclid uses the term very differently from modern mathematicians, this bit:
Two triangles are similar when there exists such an affine transformation (i.e., you can scale as well).
is not the correct definition of ‘affine transformation’. If two triangles are similar, there will exist an affine transform between them, but the reverse is not necessarily true. The category of affine transformations is broader than you think—an affine transformation allows you to skew things as well (e.g. convert a square into a parallelogram), or to scale things differently along different axes (e.g. convert a square into a rectangle). The ratio between two distances in an affine transform must remain the same only if those distances are parallel. In point of fact, I believe affine transformations can convert any triangle into any other triangle.
(Checks online: Wolfram says yes).
- TurnTrout 15 Feb 2021 3:47 UTC
  4 points
  Parent
  Yes, this is correct; this phrasing was misleading. IMO, the most succinct formally correct characterization of similarities is:
  As a map $f : R^{n} \to R^{n}$ , a similarity of ratio $r$ takes the form $f (x) := r A x + t$ , where $A \in O_{n} (R)$ is an $n \times n$ orthogonal matrix and $t \in R^{n}$ is a translation vector.
  The only difference compared to congruence is that congruence requires $r = 1$ .