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Thread: geometry question on polygons

  1. #1
    Join Date
    Apr 2010
    Posts
    497

    geometry question on polygons

    OK here is the setup...

    lets say you draw a polygon of n sides, but extend those line segments into infinite lines.
    a polygon of 5 or 6 sides will generate what I will call a secondary triangle as the 'opposing' lines are not parallel (72degrees in a pentagon vs 90deg in a n=4 square).
    a polygon of 7 or 8 sides will generate secondary triangles, but also 'tertiary triangles' due to intersections
    a polygon on 9 or 10 sides will generate quaternary triangles
    a polygon of 11 or 12 sides will generate 5th order triangles etc

    my question is:
    why does increasing n in odd number of sides generate a higher order intersectional triangle?
    does this pattern occur indefinitely? .. or do other intersections become possible with more and more acute angles whereby other polygon sides start to intersect?

    many thanks-
    Screen Shot 2020-08-17 at 7.03.13 pm.png
    "It's only a model....?" :-)
    https://www.youtube.com/watch?v=m3dZl3yfGpc

  2. #2
    Join Date
    Jul 2005
    Posts
    19,670
    Imagine an axis of symmetry that divides your polygon through the centre of one face. For an even-numbered polygon, that axis will pass through the middle of the opposite face; for an odd-numbered polygon it will pass through the opposite vertex. But we can ignore the opposite side of the polygon, and just concentrate on the face divided by our axis of symmetry.
    For any order greater than five, flanking that face are two faces that are tilted towards each other, which will therefore generate a triangle if their lines are extended towards the axis of symmetry. For order greater than seven, there is another pair of converging faces, for another triangle ... and so on as the order increases. Each time you increase the odd number of faces by two, you add a new pair of symmetrically placed converging faces that can produce another triangle. But if you merely increase the odd number of faces by one, to produce an even-order polygon, you generate a pair of faces parallel to the axis of symmetry, which don't converge and don't produce a new triangle.
    So that's the progression. Starting with a pentagon, adding one face produces a figure with two non-converging faces parallel the axis of symmetry, so no new triangle. Adding two new faces adds a new pair of converging faces, so a new triangle. Repeat, ad infinitum.

    Grant Hutchison

  3. #3
    Join Date
    Apr 2010
    Posts
    497
    Thanks Grant

    actually, it seems that the intersections 1st produce triangles, but then produce polygons.... but you know what i mean...
    "It's only a model....?" :-)
    https://www.youtube.com/watch?v=m3dZl3yfGpc

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