The popular old idea not only of a flat earth, but of a flat square earth, is evident in such ancient phrases as the 'four corners of the earth,' and Euclid's interest in the flat plane is best demonstrated in the classic text known as the Elements, in which we find 6 of 13 books devoted to flat plane figures, and a mere 3 books devoted to solids. As I have dealt with many by now who apparently rejoice in quibbling over minutiae rather than moving on to more practical points, let me assert, in order to prevent any further hysteria, that I am aware both that Euclid's writings were likely the result of many contributors and that starting with flat planes may indeed be a helpful start, but that solids are of greater utility in real matters of physics among all physicists I know, and Bucky's geometry of synergetics focuses more on solids than flat objects, and it also does not rely on circles, pi, and right angle themes as heavily as does Euclidean geometry (sometimes such themes are used not at all in synergetics, as in the case of pi); all of which I intend to show in a later paper as being critical to the virtues of synergetic geometry over Euclidean.
However, I also note that the term flat when applied to "space" means something different, and is considered in relation to something called the parallel postulate: if two parallel lines are extended to infinity and do not meet, the space is called flat. In this sense, synergetics is indeed flat and euclidean. The reason why we refer to synergetics as noneuclidean though, is because it does not rely on dimensionless points. Synergetics is fundamentally a finite rather than an infinite geometry.
For those of you who would tout the virtues of, say, noneuclidean spherical trigonometry, I would most vigorously assert that icosahedral trigonometry could accomplish the same tasks.