Inspired by reading Andreas Speiser's group theory textbook, which includes applications to crystallography and decorative ornaments, Burckhardt continued his studies in 1924 at the University of Zurich.
There he heard lectures by Andreas Speiser, Rudolf Fueter, Erwin Schrödinger, and the astronomer Alfred Wolfer (1854–1931).
At ETH Zurich Burckhardt listened to Hermann Weyl, George Pólya (whose seminar he attended), and the mineralogist Paul Niggli.
In 1927 Burckhardt passed the qualifying examination for teaching in higher education and received his doctorate in mathematics with advisor Andreas Speiser and thesis Die Algebren der Diedergruppen (The algebras of dihedral groups).
At Göttingen he also met Bartel Leendert van der Waerden and Otto Neugebauer, both of whom later became well-known mathematicians.
After Burckhardt habilitated in 1933 at the University of Zurich with the work Zur Theorie der Bewegungsgruppen[2] (Theory of space groups), he was a Vertreter (visiting teacher) at the Zurich University of Applied Sciences/ZHAW and at the Höhere Töchterschule der Stadt Zürich (later renamed the Kantonsschule Hohe Promenade).
Burckhardt investigated (partly with van der Waerden[11]) medieval Islamic astronomers' writings (such as the planet tables of Al-Khwarizmi.[12][13]).
He was the editor, with Karl Matter and Edmund Hoppe, of volume III/2, Rechenkunst (Geneva 1942), of Euler's collected works.
In this context, he edited some of Euler's physical treatises and was involved in the compilation of the list of correspondence (Series IV A, Volume 1, 1975).
He, in collaboration with Emil Schubarth, translated Leonard Dickson's 1923 Algebras and their arithmetics into German as Algebren und ihre Zahlentheorie (Orell Füssli, Zürich 1927) — as an assignment from Andreas Speiser.