The following information is due to Sergey Fomin.

Pentagramma Mirificum

Here are historical roots of the Pentagramma Mirificum. It was discovered by Napier around 1600, and fully explained by Gauss more than two centuries later, see below.

Section 1.1 of

provides a lot of references.

As Fomin points out, it would be a stretch to say that Gauss discovered cluster algebras: the pentagon equation, in its algebraic form, was found by W. Spence in 1809. In the context of Pentagramma Mirificum, the realization that the 5-cycle represents a nontrivial algebraic identity seems to appear first in print by Arthur Cayley; see [A. Cayley, On Gauss's Pentagramma Mirificum, Philosophical Magazine, vol. XLIL (1871), 311-312; The collected mathematical papers of Arthur Cayley. Vol. 7].

In the 20th century, the topic was popularized by H. S. M. Coxeter; see [Kaleidoscopes, pages 85-103] and [Non-Euclidean geometry, Section 12.7].

From: Bruce Director: From Plato.s Theaetetus to Gauss.s Pentagramma Mirificum: A Fight for Truth.

Some further references and sources:

The pages from the Nachlass of Gauss (Band 3 der Werke):