I'm trying to find out who was the first person that discovered that the icosahedron and the dodecahedron have 43380 unique (nonisomorphic) nets. Also, how was this number first discovered? By brute force enumeration or some clever argument was used?
This article here lists two references, but I was unable to find them and, so, I'm unsure if they were the first to enumerate the unique nets and how the nets were counted.
Bouzette, S., Vandamme, F.: The regular Dodecahedron and Icosahedron unfold in 43380 ways (unpublished manuscript)
Hippenmeyer, C.: Die Anzahl der inkongruenten ebenen Netze eines regulären Ikosaeders. Elem. Math. 34, 61–63 (1979)
Thanks, Humberto.
The method was described by M. Jeger (Zürich) in Elemente der Mathematik in 1975. It consists of using Burnside's Lemma to count incongruent spanning trees along the edges. You find, over all isometries of the shape, the mean number of spanning trees which seem to remain in place. He showed explicitly how to count nets of the octahedron and cube. [PDF] (For an explanation in English, you can look at this paper by Goldstone and Valli from 2016.)
A note appeared in the same journal in 1979, from Ch. Hippenmeyer (Basel), who used the method to count the nets of the dodecahedron and icosahedron. [link], [PDF]
However, it is very difficult to check whether all these nets can be cut out of paper, as it is conceivable that some of the sides overlap when fully unfolded.
In 2011, T. Horiyama and W. Shoji programmed a computer to generate all of the nets. They confirmed for the first time that all of the nets are valid and have no overlapping parts. [PDF]
The manuscript by S. Bouzette and F. Vandamme dates from 1993; I found the date at the end of the paper "A Theory of Nets for Polyhedra and Polytopes" [PDF] in Designs, Codes and Cryptography, 10, 115–136 (1997).
For a very full treatment of different shapes, see: Buekenhout, F. and Parker, M. "The Number of Nets of the Regular Convex Polytopes in Dimension <=4." Disc. Math. 186, 69-94, 1998. [link]
Many other people have worked on the problem, but Hippenmeyer is probably the answer to your question. Unfortunately I cannot find his first name, but it is most likely to be Christian.