lignograph
New member
- Joined
- Sep 26, 2014
- Messages
- 2
I am in need of some help. I am trying to construct some wooden models of a dodecahedron and an icosahedron. I can easily make a bunch of regular triangles and pentagons and just try to fudge it, but I would really like to know what the angled bevels or miters should be to assemble these together properly.
As an example, if I cut out 20 triangles as the faces of an icosahedron, and then I want to join them together properly, the 'inside' edges of each triangle would have to be mitered, or beveled (cut at an angle) so they fit together with no gap. I know that the angle of each cut is half of the total angle of the two triangles to one another. So, if the total angle was 90 degrees, I would have to cut each piece at 45 degrees; then when they are fastened together they form an angle of 90 degrees.
But I do not know how to calculate the angles. What is the formula to determine that?
If I had 30 triangles with 72 degree top angle and 54 degree angles at the base, and the sides of the triangles are 5", then the base of the triangles is 5.875". I figured that out well enough.
Likewise, I can make 12 pentagons for the dodecahedron. But again, I do not know how to figure out the angle from one pentagon to the next.
As both of these platonic solids are regular; every face is the same size, and they are identical no matter which face is up, etc; I assume the angle between any two adjacent faces will be identical as well. And I figure there has to be some formula to figure this out. But in searching online, all I can find are ways of 'unfolding' the various platonic solids, or calculating the total mass, etc. Nothing on figuring the angles of one face to another.
As an example, if I cut out 20 triangles as the faces of an icosahedron, and then I want to join them together properly, the 'inside' edges of each triangle would have to be mitered, or beveled (cut at an angle) so they fit together with no gap. I know that the angle of each cut is half of the total angle of the two triangles to one another. So, if the total angle was 90 degrees, I would have to cut each piece at 45 degrees; then when they are fastened together they form an angle of 90 degrees.
But I do not know how to calculate the angles. What is the formula to determine that?
If I had 30 triangles with 72 degree top angle and 54 degree angles at the base, and the sides of the triangles are 5", then the base of the triangles is 5.875". I figured that out well enough.
Likewise, I can make 12 pentagons for the dodecahedron. But again, I do not know how to figure out the angle from one pentagon to the next.
As both of these platonic solids are regular; every face is the same size, and they are identical no matter which face is up, etc; I assume the angle between any two adjacent faces will be identical as well. And I figure there has to be some formula to figure this out. But in searching online, all I can find are ways of 'unfolding' the various platonic solids, or calculating the total mass, etc. Nothing on figuring the angles of one face to another.