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Draw the diagonals in an irregular convex pentagon, forming a new, smaller, irregular pentagon. Repeat until the resulting area is vanishingly small. Does that point have any significant relationship to the original figure?

Roger
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2 Answers2

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Two kinds of experimental results (A and B) obtained through numerous experiments (using a Matlab program of my own) and a promising reference (C).

A) A positive result : the limit process (excepted in the exceptional case of a regular pentagon) has a characteristic feature : the different pentagons are becoming more and more flat giving rise not only to a point but as well to a privilegized direction. It is possible that this direction can be expressed in terms of the coordinates of the inital pentagon.

B) A negative result : I have tried to compare the limit point with two other points, with the aim of conjecturing some properties. These other points are "weighted" centers (see figure):

  • $V,$ the center of gravity of vertices (weights $1$ on each vertex), represented by a lozenge.

  • $E,$ the center of gravity of weighted edges (the midpoint of each edge reives a weight equal to the length of the corresponding edge), represented by a star.

The different attempts I have made don't give any plausible property of the limit point of the pentagons vs. the original figure or the other points $V$ and $E$.

C) I just discover (using adequate keywords) that very interesting studies have been done on this issue : see this article

enter image description here

Remark : of course, the question could have been settled in an equivalent way for a five-pointed star and its nested iterated internal stars.

Jean Marie
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  • Corollary 1 (iii) page 19 of the cited document states the (observed) total flattening of pentagons. – Jean Marie Feb 10 '20 at 14:32
  • see as well https://en.wikipedia.org/wiki/Pentagram_map and https://www.researchgate.net/publication/303457690_On_the_Diagonal_and_Inscribed_Pentagons_of_a_Pentagon – Jean Marie Feb 10 '20 at 14:59
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    Thanks, Jean Marie. Your references will be leading me in an interesting direction. – Roger Feb 11 '20 at 10:47
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This paper gives the limit you're looking for.

https://academic.oup.com/imrn/advance-article/doi/10.1093/imrn/rny110/5000002

As far as I'm aware it does not have any particular significance other than this property.

Owen
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