Fine Points on Polygons and Polyhedra

Last time, looking at degenerate polygons, I mentioned some other issues pertaining to the definition of a polygon. Let’s take the opportunity to look at them. This post supplements what was said previously in What is a Polyhedron … Really?

Degenerate Polygons

We’ve been looking at degenerate figures, starting with the most interesting case, degenerate conic sections. But other things can also be degenerate, so we should take a look at some of these, which perhaps arise even more often. We’ll examine triangles that aren’t triangles, rectangles that aren’t rectangles, and bigger polygons – or smaller polygons! …

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Degenerate Conics I: Mystery of the Missing Case

Degenerate cases are instances of a concept that are just on the edge of fitting its definition. They occur when we stretch a definition to its limits, at which point some of the original properties remain, but others break. We’ll start here with common instances of the phenomenon, in conic sections, pursuing the elusive case …

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Two Tricky Questions on Tangent Lines

(A new question of the week) Sometimes we have lots of quick questions and a number of long discussions, neither of which seems suitable for a post. This time I’ve chosen to combine two distantly related questions, one recent and one from several months ago, both involving tangent lines to functions.

Two Worlds of Relations

Terminology and definitions in mathematics sometimes vary according to context. Here we’ll look at the ideas of relations, functions, and their domains, and discover that they look different from different perspectives.

Cartesian Product of Sets

(A new question of the week) I had a long discussion recently about the Cartesian product of sets, answering questions like, “How is it Cartesian?” and “How is it a product?” I like discussions about the relationships between different concepts, and people who ask these little-but-big questions. We’ll be looking at about a quarter of …

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What is a Set … Really?

Sometimes the more basic an idea is, the harder it is to define it. It is also very hard to understand a definition in English when you are not a native speaker! We have had some interesting discussions of such issues recently with a student who asks very basic and yet very challenging questions of …

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Sines Without Right Triangles

(A new question of the week) A recent question dealt with an apparent conflict between the right-triangle definition of sines and cosines, and the unit-circle definition, pertaining to multiples of 90° (angles on the axes). This provides an opportunity to look closely at the relationship between those two definitions. Two definitions Recall that the right-triangle …

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