Pentagonal Hexecontahedron Variation

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Jonathan Lawrence
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Joined: Fri Aug 07, 2020 10:29 am

Pentagonal Hexecontahedron Variation

Post by Jonathan Lawrence » Fri Aug 07, 2020 10:56 am

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In the three-step process to convert a Snub Dodecahedron into the Irregular Pentagonal Hexecontahedron [in the picture above], how do I build pyramids on sets of identical faces [simultaneously?] so that the new pyramid-faces are co-planar with the faces at their bases? Note, this is also one way to build the dual of the Snub Dodecahedron with Symmetrical Pentagons.

I've just acquire Great Stella and I'm a sculptor used to thinking geometrically, but working with my hands. Thanks!

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robertw
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Re: Pentagonal Hexecontahedron Variation

Post by robertw » Sat Aug 08, 2020 4:49 am

As you may have found, if you use augmentation, there isn't a way to make the new faces coplanar with existing ones.

However there's an easy way to create this model, using stellation.

Make sure you have a stellation view open:
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Also select the option to show the stellation plane on the selected face, rather than just highlighting it:
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Then when you select one of the triangular faces you'll see this:
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Hold down Shift and look at the available mouse actions in the bottom right corner:
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What you want to do is Shift+Right-click on the central triangle, to select the stellation cell above that triangle, which will add the pyramid you want.

Stellation is all about intersecting facial planes. Here, the neighbouring facial planes will extend over the triangle to enclose a region above it.

Do the same for the pentagon and you've got the model you wanted:
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Finally, if you want, hit the left-and-down button in the top right corner of the stellation view to convert the stellation into the new base model:
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Jonathan Lawrence
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Joined: Fri Aug 07, 2020 10:29 am

Re: Pentagonal Hexecontahedron Variation

Post by Jonathan Lawrence » Sat Aug 08, 2020 5:28 am

Thanks very much! I thought you might be interested in some virus geometry, first on a geodesic, then as variations of the [irregular] pentagonal hexecontahedron:

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I dunno if the SARS Cov-2 virus has this form... that would be interesting.

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