Settlers, Hipsters, and Similar Triangles

A couple days ago I wrote about some observations from reading Little House on the Prairie for the first time by myself. I read the books aloud to my daughters, but not alone. One thing that stood out was the settlers’ attitude toward technology: they gladly used the technology they could get their hands on, but were also willing to do without it as necessary.

The settlers, at least as portrayed in this one example, didn’t try to avoid technology per se, unlike hipsters or Luddites. Nor did they seem to be concerned with the spiritual effects of technology, unlike the Amish. They didn’t seem to think about technology much at all.

(The relation of the Amish to technology is commonly misunderstood. They don’t want to be physically connected to cities via power lines for fear that they would be culturally connected. They don’t necessarily oppose electricity or modern technology in general. See Kevin Kelly’s post on Amish hackers.)

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If two triangles have the same perimeter and the same area, are the triangles congruent? Or similar?

My latest post gives a couple reasons why the conjecture is plausible, then shows that it is false. But as always when a conjecture is disproved, there are still related conjectures to explore.