Opposite of Polygon: Open Shapes & Names

  • Context: High School 
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Discussion Overview

The discussion revolves around identifying shapes that could be considered the opposite of polygons, specifically focusing on open shapes and their nomenclature. Participants explore the characteristics of these shapes, including their boundaries and curvature, while expressing uncertainty about definitions and classifications.

Discussion Character

  • Exploratory, Conceptual clarification, Debate/contested

Main Points Raised

  • One participant questions what is meant by the "opposite" of a geometrical figure, indicating a need for clarification on the term.
  • Another participant suggests that the opposite may refer to curved shapes that end in points, distinguishing them from polygons.
  • A specific example is provided of a diamond shape with inward-curving edges, though the participant admits to a lack of formal training in geometry or topology.
  • A further elaboration on the concept of curves is presented, defining a curve as an object with a boundary that has a changing derivative, implying that such shapes do not contain straight lines.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the definitions or classifications of shapes that could be considered the opposite of polygons. Multiple interpretations and uncertainties remain regarding the characteristics and terminology of these shapes.

Contextual Notes

There are limitations in the discussion regarding the definitions of terms like "curved shapes" and "opposite," as well as the mathematical rules that might apply to the characterization of these shapes.

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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