Any two polygons can be continuously "extended" or ....?

In summary, the polygon extension theorem states that any two polygons can be continuously extended or connected to form a larger polygon. A convex polygon has all interior angles less than 180 degrees, while a concave polygon has at least one angle greater than 180 degrees. A polygon cannot be both convex and concave. A regular polygon has all equal sides and angles, while an irregular polygon has different lengths and measures. A polygon cannot have an infinite number of sides, but the number of sides can approach infinity, such as in the case of a circle.
  • #1
I am looking for theorems/information related to the following statement: any polygon can be created by an infinite number of infinitely small "extensions" or "croppings" of any other polygon, such that the shape is always a polygon (after any amount of extensions of croppings). For example, I can have a square, and then "continuously" "push" a small piece out from the top such that the final polygon is like a small square on top of a larger square.
 
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  • #2

1. Can any two polygons be continuously extended or connected to form a larger polygon?

Yes, any two polygons can be continuously extended or connected to form a larger polygon. This is known as the polygon extension theorem and it states that if two polygons share a side, then the two polygons can be extended or connected to form a larger polygon.

2. What is the difference between a convex and concave polygon?

A convex polygon is a polygon in which all of its interior angles are less than 180 degrees, while a concave polygon has at least one interior angle that is greater than 180 degrees. In other words, a convex polygon has no "dents" or indentations, while a concave polygon has at least one "dent".

3. Can a polygon be both convex and concave?

No, a polygon cannot be both convex and concave. A polygon is either convex or concave, depending on the measure of its interior angles. If a polygon has at least one interior angle greater than 180 degrees, then it is concave. If all of its interior angles are less than 180 degrees, then it is convex.

4. What is the difference between a regular and irregular polygon?

A regular polygon is a polygon in which all of its sides and angles are equal, while an irregular polygon has sides and angles of different lengths and measures. In other words, a regular polygon is symmetrical, while an irregular polygon is not.

5. Can a polygon have an infinite number of sides?

No, a polygon cannot have an infinite number of sides. A polygon is a closed shape with finite sides and angles. However, the number of sides can be very large, approaching infinity, in the case of a circle which can be considered as a polygon with an infinite number of sides.

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