At Mind Matters News: Nobelist Roger Penrose talks about his impossible triangle

CD, Wenger, then quite active, and the lockback was a specific model. I also have a champ and a couple of others. KF kairosfocus

You know, you could have just said Swiss Army knife so we didn’t have to look up Victorinox lockback …. :-) chuckdarwin

F/N: Apparently, it is both Penroses, Father [a psychiatrist] and Son [a Mathematician]. kairosfocus

PS, the Penrose triangle [is it the same Penrose?] shows how our optical system expresses computational processing and how its context-sensitive response can lead to optical illusions. Each part seems to be part of a 2-d structure but they are incongruous and no such structure can be built. kairosfocus

Folks, of course, mathematically perfect triangles are abstract entities that will be manifested in any possible world as part of its necessary, embedded logic of structure and quantity. The two 7" speed squares [one, Al, the other fibreglass plastic] next to me as I type, illustrate how pervasive triangle properties are . . . especially the protractor side. At the same time, they show that approximation and controlled tolerance are key engineering phenomena, their points are rounded, not razor sharp, their markings are builder-coarse not set square fine, but not even my TD set squares and tee square are abstractly exact, never mind that their corners are close enough to needle sharp to require care. Extending, optical precision of the zoom lens on a camera also next to me, are also less than mathematically exact but are good enough for purpose. Then, magic step, the logic and structure of tolerance, subject matter of one of my uni textbooks, "errors of observation and their treatment," is itself a highly ordered mathematical discipline. And of course, as my Victorinox lockback shows, even a razor sharp edge is not a mathematical point. But, engineering approximation is itself pervaded by a deeper level of Mathematics. KF kairosfocus

Triangles In Nature – Why? Triangular shapes are everywhere in Nature. They show up in geology, biology, chemistry and physics; from the sub-atomic scale to the cosmic. But is it significant? Connect any three points and it makes a triangle – it’s hard to avoid. Triangles are bound to appear in Nature, because it’s … well, natural. […] Good lord, there’s triangles everywhere. Not kinda triangular, but sharp-edged and consistently angled, that repeat, over an over. It’s amazing, really. And to think this could happen from millions of independent, random forces acting over millions of years. It’s a wonder.

Seversky

We’ve been here before—you keep forgetting the Bermuda Triangle. Tell all those lost seamen triangles don’t exist. And then, also, what about the thousands of “YIELD” signs virtually everywhere……? chuckdarwin

There are no triangles in the real world only in the mind. However, all these abstract non real world concepts are incredibly useful in the world where they do not exist. What would we do without them? That is the amazing thing. jerry