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I have the following decahedron:

But as you can see below, its faces are not "flat" but rather curve towards the inside:

How can I make it so that all the faces become flat kites, without adding edges or vertices?

EDIT: To be clear, I want to end up with a shape similar to this:

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    $\begingroup$ Hello, your 4 vertices don't seem to be on the same plan so it hardly could become flat I guess? $\endgroup$
    – moonboots
    Jan 19, 2021 at 9:10
  • $\begingroup$ Perhaps I haven't been clear. I'm not trying to project the whole solid to a flat surface (what point would there be in circling that specific part in my second screenshot?). You can see on each of the kite-shaped faces of the solid that they seem to have an "invisible bending edge" vertically along the middle line, so that each kite is split into two halves. I would like that not to happen, so that there is no depression within each of the faces. $\endgroup$
    – theberzi
    Jan 19, 2021 at 9:20
  • $\begingroup$ oh ok sorry I understand now ;) $\endgroup$
    – moonboots
    Jan 19, 2021 at 9:34
  • $\begingroup$ You can flatten the edges onto one plane for one face, but you'll not be able to do it for all the faces at the same time because the delta position on one vertice may not be helpfu in other transformation. It's much better to change modeling strategy. $\endgroup$
    – Carlo
    Jan 19, 2021 at 19:15

3 Answers 3

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You're creating a pentagonal trapezohedron. Its height is constrained by the height of the antiprism which is the base of its composition, so flattening faces won't necessarily work, and measurement might not be the best way to precision... here's a construction from scratch:

First, create the antiprism:

enter image description here

Left, above, with 'Pivot' set to the 3D cursor at World 0 ..

  • Create a fan-filled five-sided circle at the world origin
  • In Edit mode, GZ move it a little way up.. and ..
  • ShiftDS-1 Create the bottom face by duplicating, and scaling through the origin

Right, above..

  • CtrlE Edge Menu > Bridge Edges of the perimeters, followed by
  • CtrlT Triangulate Faces

enter image description here.

  • Using the little '+' in the Transform Orientation dropdown, in Face mode, create a transform orientation from one of the upward-sloping triangles

  • EX Extrude the top edge, with the new orientation active. take it far enough to cover the center

  • Repeat the last 2 steps for one of the downward-sloping triangles

  • With Snap set to 'Face', GZ in Global orientation, drag the central vertices up and down until they snap to the extruded faces

  • Delete the guide faces, and any unwanted edges.

enter image description here

The result is good for the whole family of polyhedra.. if you scale all in Z, the kites remain planar.

For some reason, even if checked, I can't get Z -scaling the central verts to the guide faces to snap? That would have cut out a couple of steps.

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  • $\begingroup$ Thank you so much, this was the easiest solution I found to get the result I wanted. However, I don't quite understand what the guide faces were for—following the instructions, the central vertices did not snap to them anyway. $\endgroup$
    – theberzi
    Jan 20, 2021 at 21:15
  • $\begingroup$ Another note: I found myself having to repeat the first three steps to recreate the above and bottom faces, because the following two steps somehow deleted them. $\endgroup$
    – theberzi
    Jan 20, 2021 at 21:16
  • $\begingroup$ I got a snap OK, couldn't really believe it, so repeated a few times to check.. snapping with 'Active' vertex. Should have mentioned that. The snap is to ensure the central vertex is at the right height for its triangle to be coplanar with the triangle it has sprung from. How did you make yours coplanar? Steps 1-3, a bit of a sneaky move, scaling by -1 through the 3D cursor.. again, works for me? $\endgroup$
    – Robin Betts ♦
    Jan 20, 2021 at 21:53
  • $\begingroup$ I'm... not too sure how I did it, I just followed the instructions and when the next step wasn't clear I went by trial and error with what seemed sensible to try. In the end I just deleted the horizontal edges left on the middle part and resized the elongated parts. I must admit my final result doesn't look quite as good as yours, if I try to make it too short the middle part "breaks", but it's good enough for my purposes. $\endgroup$
    – theberzi
    Jan 21, 2021 at 14:02
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    $\begingroup$ @FedericoS Ahh.. OK, I've added a link to mine in the answer. Bit messy re. naming, etc. but the object is there. $\endgroup$
    – Robin Betts ♦
    Jan 21, 2021 at 14:31
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A very unscientific way to do it would be to first create this shape:

enter image description here

Use the Array modifier in Object Offset mode, rotate the offset object 72° on the Z axis:

enter image description here

Rotate your mesh in Edit mode so that if fits perfectly with its instances:

enter image description here

Apply the modifier, duplicate, rotate the copy 180° on the Y axis, rotate 36° on the Z axis, merge the 2 meshes:

enter image description here

Bevel if you want more geometry:

enter image description here

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  • $\begingroup$ The key to solving the issue seems to be in that vertical edge slicing the kites in two halves: perhaps there's a way to obtain that from my existing mesh? I wouldn't mind manually modifying the faces as long as I can do it exactly on each. This is probably basic but I'm a beginner, sorry. $\endgroup$
    – theberzi
    Jan 19, 2021 at 11:11
  • $\begingroup$ A way to fix your current mesh would be to move the top vertex until all the faces look flat, which seems a bit hazardous to me. Maybe someone will find a better idea. My way would be to start from scratch. $\endgroup$
    – moonboots
    Jan 19, 2021 at 11:25
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To create a true decahedron in that shape you need to connect these points.

enter image description here

Select 2 at a time and hit the J key. It will make new edges, but this is the only way to flatten out the faces.

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  • $\begingroup$ My solid already has 10 faces. If I do that wouldn't it result in it having more faces, since each of the kites would be split into two triangles? For reference, check this image of a ten-sided die: cdn11.bigcommerce.com/s-70184/images/stencil/1280x1280/products/… This is roughly what I want to obtain, and the kites of that solid are flat, not concave, and are not two triangles pointing in different directions as I believe your solution would result in. $\endgroup$
    – theberzi
    Jan 19, 2021 at 9:31
  • $\begingroup$ so maybe put this reference image in your first post $\endgroup$
    – moonboots
    Jan 19, 2021 at 9:37
  • $\begingroup$ This is a true decahedron with the triangular look. subjectcoach.com/tutorials/math/topic/math-definitions-letter-d/… If you want that dice look that's a different question. $\endgroup$
    – Psyonic
    Jan 19, 2021 at 9:39
  • $\begingroup$ A decahedron is a solid with 10 faces. There are many different kinds of decahedron (32300 according to Wikipedia), including yours and mine both. I could have used the term "pentagonal trapezohedron", but given that there are screenshots and that I mention the faces are kite-shaped, I don't think that level of precision is necessary when the question could have been about many other possible solids presenting the same issue. $\endgroup$
    – theberzi
    Jan 19, 2021 at 10:49
  • $\begingroup$ @FedericoS Your faces aren't actual faces since they're not flat. They have a bend in them. If they were flat Blender would display them as flat. $\endgroup$
    – user253751
    Jan 19, 2021 at 20:52

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