https://www.quantamagazine.org/the-useless-perspective-that-transformed-math... "Representation theory is a way of taking complicated objects and “representing” them with simpler objects." In no way does the article support this thesis; in fact, it supports the opposite and more reasonable: "Representation theory is a way of taking [simple] objects and “representing” them with [more complex] objects." The article does give a graphical definition of the triangular dihedral group--almost as simple as algebraic letters with some prescribed abstract meaning. It does not list out six matrices such as: MatrixForm /@ Join[RotationMatrix[2 Pi/3 #] & /@ Range[3], Dot[IdentityMatrix[2] {1, -1}, RotationMatrix[2 Pi/3 #] ] & /@ Range[3]] Out[]={{{-(1/2), -(Sqrt[3]/2)}, {Sqrt[3]/2, -(1/2)}}, {{-(1/2), Sqrt[3]/ 2}, {-(Sqrt[3]/2), -(1/2)}}, {{1, 0}, {0, 1}}, {{-(1/2), -(Sqrt[3]/2)}, {-(Sqrt[3]/2), 1/2}}, {{-(1/2), Sqrt[ 3]/2}, {Sqrt[3]/2, 1/2}}, {{1, 0}, {0, -1}}} Nor does it explain that matrix representations are coordinate dependent, nor that they have inverses and close under matrix multiplication, nor does it explain how the idempotent projectors and their characters would be derived from a matrix representation. Even worse, the article does not explain that Representation theory came into acceptance through its applications in physics. Why even mention Andrew Wiles proof? If you are trying to learn representation theory, the Quanta article looks to be useless, especially because it only references other Quanta articles (probably with similar problems). For a better course on why Representation Theory isn't a useless perspective, try reading William Harter's book: https://modphys.hosted.uark.edu/markup/PSDS_Info.html Which explains in detail how to calculate character tables, and how to use them for something productive. --Brad PS. Don't forget the beetle's have their own spectrum and a very special technique for controlling the Stokes vector: https://www.inaturalist.org/observations/49263113 https://modphys.hosted.uark.edu/pdfs/PSDS_Pdfs/PSDS_Ch.7_(4.21.10).pdf ( See also page 85 / 636 )