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Math Help - positioning points inside a triangle

  1. #1
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    positioning points inside a triangle

    hello people, i'm a 3d modeler and animator, and i have a question regarding triangles.

    this is a polyomavirus,

    it has a icosahedral geometry and 72 capsomeres (the flower things) arranged on its triangles, like this picture:

    if i pick a triangle, i can see that it has one capsomere in each vertex and three arranged inside the polygon. my question is how to properly arrange these 3 capsomeres in each triangle. if you take a closer look at the picture, you'll notice that each vertex' capsomere has 5 others capsomeres around it, and the other capsomeres (those arranged inside the triangles) have 6 others. they have to be distanced equally from all the capsomeres surrounding them.

    thank you in advance!
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  2. #2
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    Quote Originally Posted by fael097 View Post
    hello people, i'm a 3d modeler and animator, and i have a question regarding triangles.

    this is a polyomavirus,

    it has a icosahedral geometry and 72 capsomeres (the flower things) arranged on its triangles, like this picture:

    if i pick a triangle, i can see that it has one capsomere in each vertex and three arranged inside the polygon. my question is how to properly arrange these 3 capsomeres in each triangle. if you take a closer look at the picture, you'll notice that each vertex' capsomere has 5 others capsomeres around it, and the other capsomeres (those arranged inside the triangles) have 6 others. they have to be distanced equally from all the capsomeres surrounding them.
    This is a chiral structure. That is to say, it is not equal to its mirror image, but has a left-handed and a right-handed version. (You can see that the three capsomeres in each triangle are not symmetrically placed with respect to the edges of the triangle, but are closer to one end of the edge than the other.)

    There is a diagram of this geometric structure here (Fig.18 on p.198). It is a solid with 72 vertices and 140 triangular faces. Twelve of the vertices have five neighbours, and the remaining 60 vertices have six neighbours, just as you want. The author describes the solid as being obtained from a snub dodecahedron by a process of triangulation. He also mentions that it has been considered as a virus model (presumably with each vertex representing a capsomere).

    Having said that, I have no idea how you would construct this object mathematically. But I hope that you may find that reference helpful.
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    Quote Originally Posted by Opalg View Post
    This is a chiral structure. That is to say, it is not equal to its mirror image, but has a left-handed and a right-handed version. (You can see that the three capsomeres in each triangle are not symmetrically placed with respect to the edges of the triangle, but are closer to one end of the edge than the other.)

    There is a diagram of this geometric structure here (Fig.18 on p.198). It is a solid with 72 vertices and 140 triangular faces. Twelve of the vertices have five neighbours, and the remaining 60 vertices have six neighbours, just as you want. The author describes the solid as being obtained from a snub dodecahedron by a process of triangulation. He also mentions that it has been considered as a virus model (presumably with each vertex representing a capsomere).

    Having said that, I have no idea how you would construct this object mathematically. But I hope that you may find that reference helpful.
    you're correct when you say that they're positioned closer to one end of the edge than the other, but it's a question of rotation, not mirroring.
    and on the picture you sent, its almost that, but not the exact thing. the structure is a perfect icosahedron, and the three dimensional structure doesnt matter, it's all about the triangle. thanks for replying!
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  4. #4
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    Quote Originally Posted by fael097 View Post
    you're correct when you say that they're positioned closer to one end of the edge than the other, but it's a question of rotation, not mirroring.
    I think you're wrong there (in fact, I'm certain of it). In your smaller picture, each of the lines joining the yellow icosahedral capsomeres cuts through two of the grey capsomeres. As you go along that line, the first of the grey capsomeres is centred in the triangle to the left of the line, the second one is in the triangle to the right of the line. That holds on the line between any two yellow capsomeres, whichever direction you travel in, and no matter how you rotate the object. The one on the left always comes before the one on the right. That is a clear indication of a chiral structure.

    Your other picture, the large coloured picture of a virus, appears to show the enantiomorph or mirror image. In that picture, if you trace along the line between any two "pentagonal" capsomeres (those with five neighbours), you again pass across two of the "hexagonal" capsomeres. But this time, the one on the right of the direction of travel always comes first, before the one on the left. Again, this is not affected by any rotations, the one on the right will always come first. So this structure is intrinsically different from the one in the yellow/grey picture.
    Attached Thumbnails Attached Thumbnails positioning points inside a triangle-sv40_1w.gif  
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  5. #5
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    well yea, the red virus is wierd, not geometrically correct, but its just for reference

    there are a few more details here http://jon.visick.faculty.noctrl.edu...s/slides16.pdf

    but nothing that tells me where and how exactly to position the points in the triangle.
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  6. #6
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    As a follow-up to my previous comment, here's a possible construction for the three points in the icosahedral triangle where the capsomeres should be centred. The idea is to approach the problem backwards, starting with the three points and then determining where the triangle should go.

    Start with a tesselation of the plane by equilateral triangles, then draw the icosahedral triangle as in the attachment. The red triangle shows the left-handed version, the blue triangle shows the right-handed one. The key fact is that the sides of the small triangles in the tesselation should be 1/\sqrt7 times as long as the sides of the icosahedral triangle.
    Attached Thumbnails Attached Thumbnails positioning points inside a triangle-chiral.jpg  
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  7. #7
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    Like Opalg said.

    I would draw them as below with a rotation angle of +/- 19.11 deg.
    Attached Thumbnails Attached Thumbnails positioning points inside a triangle-virusstructure.png  
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  8. #8
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    well with these images it's getting clear, thank you. i'll try to build my structure according to this and see what happens, and i'll post the result later on.
    this is my virus with incorrect capsomere distribution, with and without the structure:



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  9. #9
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    Quote Originally Posted by Opalg View Post
    This is a chiral structure. That is to say, it is not equal to its mirror image, but has a left-handed and a right-handed version. (You can see that the three capsomeres in each triangle are not symmetrically placed with respect to the edges of the triangle, but are closer to one end of the edge than the other.)

    There is a diagram of this geometric structure here (Fig.18 on p.198). It is a solid with 72 vertices and 140 triangular faces. Twelve of the vertices have five neighbours, and the remaining 60 vertices have six neighbours, just as you want. The author describes the solid as being obtained from a snub dodecahedron by a process of triangulation. He also mentions that it has been considered as a virus model (presumably with each vertex representing a capsomere).

    Having said that, I have no idea how you would construct this object mathematically. But I hope that you may find that reference helpful.
    forgot to say, i see what you're saying opalg, and you're obviously right, i thought you were talking about different triangles on the same picture and not from both pictures, and yes, the first one is probably mirrored.

    and just to say, it works, that's exactly how it's done, thank you again!

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