Explique moi clairement et précisément : Our starting point was recent work on a key question in the geometric approach to conceptualization, to wit, what the naturalness of the concepts in actual use by humans consists in. The proposal we looked at answers this question in the spirit of Anderson’s and others’ rational analysis program, which seeks to explain cognitive phenomena in terms of optimal functioning. Douven and Gärdenfors (in press) proposed to define natural concepts as those represented by regions in an optimally designed similarity space. They identified and clarified a number of engineering principles to make concrete the notion of optimal design as it pertains to similarity spaces. Douven and Gärdenfors (in press) already pointed out experimental evidence showing that known conceptual systems obey certain of their principles, most notably, convexity, parsimony, and informativeness. Evidence for the principles concerning the placement of prototypes was however still lacking. According to these principles, prototypes are—- ideally speaking—placed in a similarity space so as to be both highly representative of the other items in their concept and easy to tell apart from other prototypes, all in order to minimize the chance of classification errors. To test this part of the design hypothesis, one needs detailed knowledge of a similarity space, including knowledge of the regions in the space representing the supposedly natural concepts and knowledge of where in the space the prototypes of those concepts are located. In this paper, we focused on a particular similarity space, viz., color space. The study we reported was meant to gain knowledge of the locations of the basic color prototypes in that space. While those locations are sufficient to calculate contrastiveness, we relied on the data from Jraissati and Douven (2018) concerning the regions representing basic color concepts to calculate how representative the various prototypes are. The outcomes were compared with millions of constellations of candidate color prototypes randomly created in computer simulations as well as with the constellations obtained by using a genetic algorithm to optimize representativeness and contrastiveness of color prototype constellations. The results arrived at strongly indicated that the actual color prototypes are placed where we could have found them had they been placed by clever engineers, attending to our cognitive limitations as well as to the pressures we face on a daily basis. In other words, we obtained evidence for the part of the design hypothesis that so far had remained entirely without empirical support. To be sure, our evidence is limited to color space. But note that the above can readily serve as a template for future studies. While, to our knowledge, at the moment there is no well-researched similarity space that is known in all its details to the extent that we now know color space—with knowledge of both the regions representing the concepts and the prototypes of those concepts—there are many similarity spaces that are partially known, such as the spaces mentioned in Note 1. To extend our knowledge of those spaces, in particular to determine which regions of those spaces represent which concepts and where in those regions we find the concepts’ prototypes, one could simply follow the playbook of the study reported here. With that knowledge available, it would be straightforward to subject the design hypothesis to further testing.