Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

3D shapes with perfect polygons

  1. Sep 30, 2006 #1
    Hello, (edit should be regular polygons in title)

    I have been thinking a lot recently about 3D shapes formed by 2D regular polygons. I was asking myself if there would be any way to calculate the minimium number of regular polygons to form a complete 3D shape. It is fairly easy with an equilateral triangle, which requires 4 sides, and for a square, with 6, but when I got to pentagons many problems arised.

    For one I was trying to calculate the angle the corners of 3 pentagons would have to be placed up agaisnt each other relative to a plane (see drawing). I ended up just assuming that the angle was 12, though I really have no idea, and continued with my estimates. So assuming that x (in the diagram) is 12 degrees, then I can calculate that the angle of each face to the other is about 156 degrees. With 156 I found the number of sides of a ploygon with those angles to be 15. From there I guesstimated that a pentagon forming a 3D shape by itself must have a minimum of around 80 faces.

    Now I have several questions. Is my assumption that the angle x is 12, correct, if not how do I preform a correct calculation? Also what is the true number of sides a regular pentagon must have to form a 3D shape? Is there an equation which can identify the number of faces required for certain polygons to form 3d shapes? Finally, is there any way a single regular ploygon can approminate a sphere? Beyond a hexagon it is impossible to form 3D shapes since the measure of internal angles goes beyond 120, which means 3 corners cannot intersect.


    Hopefully you can read this.

    Attached Files:

    Last edited: Sep 30, 2006
  2. jcsd
  3. Sep 30, 2006 #2
  4. Oct 1, 2006 #3
    Yeah, I knew there would be only a few. But I still have one question. What is the angle that 3 corners of a pentagon have to be bent relative to a plane, (see drawing) to fit together?

  5. Oct 1, 2006 #4
    I can't see the drawing because it hasn't been approved yet :tongue2:. If you're talking about the angle I assume you're talking about then there is an expression for it on that page (search for "dihedral angle").
  6. Oct 2, 2006 #5
    Thanks for the links and info Data.

  7. Oct 3, 2006 #6

    If you allow more than one type of regular polygon, you can get a few more. Look up: Archimedean solids.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?