A Heronian triangle (from Hero of Alexandria) is a (planar) triangle with integer sides having also an integer area.
Note that all Pythagorean triangles are Heronian triangles, but they are not the only one. A superhero triangle is an Heronian triangle whose perimeter is the same as its area. Does that even exist? Or is there an infinity of them? How could an answer to such natural question be considered as conventional? The actual answer is that there are 5, and only 5 superhero triangles. Here is a cute video explaining why that is the case. It is rather simple, and you might lean a cute formula for the area of the triangle not involving its height. https://www.youtube.com/watch?v=UIjeCKPHbso Note how particular integers arise in this context. The interest in such triangle can be conventional, but the truth of their existence and of their properties cannot be. This is a different argument against computationalism than the usual argument I made already by mentioning the no-go theorem. If the mathematical reality was only a matter convention, I would decide that all triangles are superheroes! Bruno -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To view this discussion on the web visit https://groups.google.com/d/msgid/everything-list/D6308046-5483-4B48-A10B-5F8F18410987%40ulb.ac.be.
