Opposite of Polygon: Open Shapes & Names

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SUMMARY

The discussion centers on identifying shapes that are the opposite of polygons, which are defined as closed figures formed by line segments. Participants suggest that the opposite includes curved shapes that may end in points, such as diamonds with inward-curving edges. The conversation emphasizes the need for a precise definition of curves, highlighting that a curve is characterized by a boundary with a changing derivative, indicating no straight lines are present. This distinction is crucial for understanding the geometric classification of shapes.

PREREQUISITES
  • Understanding of basic geometric concepts, including polygons and curves.
  • Familiarity with derivatives and their role in defining curves.
  • Basic knowledge of geometric figures and their classifications.
  • Awareness of conic sections and their properties.
NEXT STEPS
  • Research the mathematical definition of curves and their properties.
  • Explore the classification of geometric shapes beyond polygons, including conics and other curves.
  • Study the concept of derivatives in relation to geometric boundaries.
  • Investigate the properties of shapes with inward-curving edges and their mathematical implications.
USEFUL FOR

Students of geometry, mathematicians, and anyone interested in the classification of shapes and the properties of curves in geometry.

aspiring_one
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Sorry if this seems elementary but what shapes are the opposite of polygons (closed by line segments) and what do you call them? Sorry if my question is phrased weird.
 
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What do you mean by the opposite of a geometrical figure?
 
I guess I mean curved shapes that end in points. Not quite conics but a combination of them. One shape is a diamond with curved edges inward. I am no geometer nor have I taken any topology. Sorry if this is still vague
 
aspiring_one said:
Sorry if this seems elementary but what shapes are the opposite of polygons (closed by line segments) and what do you call them? Sorry if my question is phrased weird.

If I had to guess, I would have to see anything that has curves. If I had to define curve, I would define an object with a boundary that at all points has a changing derivative: in other words no straight lines on the objects boundary which means that the derivative is non-constant about any local region of the shapes boundary (I'm sorry I can't state a mathematical rule for this, but I'm sure there is one out there).
 

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