Arrange 4-Gon Figures: Find Descendants w/o Descendants

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Discussion Overview

The discussion revolves around the classification of 4-gon figures, specifically focusing on the concept of "descendants" within geometric shapes, such as rectangles and squares. Participants explore the relationships and properties of these shapes, including their hierarchical structure and symmetry.

Discussion Character

  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant suggests that a rectangle is a descendant of a parallelogram, and a square is a descendant of a rectangle, prompting a question about which 4-gon figures have no descendants.
  • Another participant asks for the rules that determine these "descendants," indicating a need for clarity on the classification criteria.
  • A participant describes the relationship between squares and rectangles, noting that while every square is a rectangle, not every rectangle is a square, and introduces the idea of "father brothers" to describe rectangles that are not squares.
  • One participant proposes that a square is the figure being sought, as it can only be classified as a square and does not have further descendants.
  • A later reply clarifies that the term "descendants" can be understood as subsets, explaining that squares are a subset of rectangles, which are in turn a subset of parallelograms, and highlights the unique symmetry of squares compared to other quadrilaterals.

Areas of Agreement / Disagreement

Participants express differing views on the classification of figures and the concept of descendants. While some agree on the hierarchical relationships, there is no consensus on the specific figures that qualify as having no descendants.

Contextual Notes

Participants discuss the properties and relationships of geometric figures without fully resolving the definitions or criteria for "descendants." The discussion reflects varying interpretations of these concepts.

roni1
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If I arrange the 4-gon polygon by order as:
a rectangle is a descendant of parallelogram and a square is a descendant of rectangle.
What are the descendant that no have any descendants (in 4-gon figures)?
 
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What are the rules for determining descendants?
 
the property that the figure has like the father but you can say that the son have but so "fathers" (let call them father brothers) haven't.

square is the son of rectangle.

Not every rectangle can be a square,
But every square is a rectangle.
O.K.
The square is the son and the rectangle is the father. The brothers of the rectangle are rectangle that are not square.
 
Then I'd say a square is what your looking for; a square can only be a square.
 
roni said:
If I arrange the 4-gon polygon by order as:
a rectangle is a descendant of parallelogram and a square is a descendant of rectangle.
What are the descendant that no have any descendants (in 4-gon figures)?

Hi roni.

The mathematical term for what you call “descendants” is subsets. Thus, the set of all rectangles is a subset of the set of all parallelograms, and the set of all squares is a subset of the set of all rectangles.

The subset you’re looking for is the set of all squares because the square is the “most symmetrical” of all quadrilaterals – in the sense that its symmetry group has order 8 and no other quadrilateral has a symmetry group of the same or higher order.
 
Last edited:

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