The author reviews the main properties of blossoms along with their important repercussions in all aspects of geometric design. More precisely, he points out the implicit presence of blossoms in the work of Laguerre. This is proved in three different ways, involving various aspects of polynomials blossoms: the first proof is based on their algebraic nature, while the second only uses their geometric properties. Finally, combining both aspects proves to the particularly efficient in the third proof.
Next, the author deals with possible ways of extending blossoms beyond the polynomial framework. More precisely, an algebraic approach, modelled on the polynomial case, is developed. De Boor-type algorithms and B-spline-like functions emerge from this approach, along with spline spaces defined in a purely algebraic way.
Afterwards, the geometrical extension directly inspired by the geometric interpretation of polynomial blossoms, is developed. The author recalls how, through subblossoms, this enables to use all the results developed in the algebraical approach presented.
Finally, he compares the two approaches and gives a survey on how to practically ensure their validity; this leads to the extended Chebyshev or quasi-Chebyshev spaces. Such spaces can be characterized either by the existence of blossoms or existence of B-spline bases.