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Thread: Dodecahedron

  1. #1

    Dodecahedron

    Greetings,

    I am an artist with a project that entails building a dodecahedron out of wood. I do not have much math but I visualize spatial relations pretty well. So the following is based on my "seeing" in my minds eye what I describe. I am hoping Sketchpad might have a way to verify this.

    I need to work out the angle that I would bevel the edges of the wood so that the hexagons will fit together accurately. Someone suggested the Sketchpad might be a tool that could help me solve this problem. I think that what I need to do is first specify the length of the sides of the hexagons, then work out the radius of a sphere that would encompass a dodecahedron made of hexagons with that side length and that would touch the vertices of the hexagons. Then if I construct an isosceles triangle using the radius of that sphere as two sides of the triangle and the length of the sides of the hexagons as the third side then the angle between one of the long sides and the short side I will call it angle A, would be what I need to cut the bevel if I halve the angle to account for the fact there will be two wood edges sharing that angle.

    So if I am not mistaken the above will get me to where I can cut the edges of my hexagons correctly but I still need to work out the angle where the vertices of the hexagons meet. If the above is correct would dividing angle A by 3 instead of 2 give me the angle I would need to miter the corners at so the points where the vertices meet would then fit accurately.

    If the above is an actual way to solve my problem and not simply my imagination then I need first to determine the circumference/radius of a sphere that would encompass a dodecahedron with X being the length of the sides of the hexagons that make up the dodecahedron. If I can work that out I can easily use Sketchpad to create the isosceles triangle that will provide the rest of the answer.

    Thank you for reading this any help will be greatly appreciated. Simply knowing if this is a useful line of enquiry and not a dead end will be really helpful.

    Will Tait

  2. #2

    Dodecahedron

    Hi Will,

    I've never built a physical dodecahdron. But the problem seems to me to be relatively simple, unless I'm missing something important -- or unless the dodecahedron you want to build is not regular. The regular dodecahedron has equally sized pentagonal faces, with three such faces meeting at each vertex. The dihedral angles (the angle formed by the plane of one face with the plane of an adjacent face) are about 116.5°, which means that each edge of a pentagonal face must be beveled at 31.75°:
    Click image for larger version

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    My intuition suggests that if the edges of the faces are properly beveled in this way, that the vertices, each with three faces coming together, will also fit perfectly.

    So I don't think that you need a Sketchpad dodecahdron, though I have to say that I enjoyed the challenge of building one after seeing your post. (I used Paul Kunkel's Perspective Tools and Solid Tools to do the construction.)
    Dodecahedron.gsp

    Have fun with your project!
    --Scott

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