When Shapes Are the Same

Topology · Axiom Academy

INTRO When Are Two Shapes "The Same"? Discover the surprising world of topological equivalence In geometry, we learn that a circle is different from a square. They have different angles, different perimeters, different everything... right? But what if I told you that in topology, a circle and a square are actually the same shape ? Two shapes are topologically equivalent (or homeomorphic ) if one can be continuously deformed into the other by stretching, bending, or twisting—but without tearing, cutting, or gluing . Think of shapes as being made of infinitely stretchy rubber. If you can mold one shape into another without breaking or connecting new points, they're topologically equivalent. Let's look at some examples. All of these shapes are topologically equivalent to each other: They're all simple closed curves —one continuous loop with no holes or breaks. You can morph any of them into any other. But here's the critical insight: a circle is NOT topologically equivalent to a figure-8! In the next lessons, you'll learn how to prove whether two shapes are homeomorphic, and discover the deep properties that topology preserves.

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