You can model a small triambic icosahedron in 3ds Max by combining an icosahedron and a trirectangular tetrahedron.
To construct the icosahedron go to Create, Extended Primitives, Hedra and select Dodec/Icos and set P = 1. Assuming that we want an edge length of 1.0 set the radius of the icosahedron to 0.951056 (see this link for a = 1).
A trirectangular tetrahedron is formed when the three edge angles at one of the vertices of a tetrahedron = 90°. Since we want the lengths of the base of the tetrahedron to equal 1.0 the radius is 0.61252 or ( 1 /(4 * 6^0.5)) Reference this link.
We need to adjust the height of the tetrahedron to form a trirectangular tetrahedron. To do this I created a plane on the world XY plane and then used Normal Align to place one face of the tetrahedron on the xy plane. This face will be referred to as the base plane.
Convert the tetrahedron to an editable mesh and move the vertex that isn't on the z = 0 plane to a value of z = 0.40805 (from 1/h^2 = 3/a^2 where a = 1). This will yield a tetrahedron with a base of sides = 1.0 and right angles for the edges not on the base plane. Reorient the pivot to the world so that its local z axis is perpendicular to the base plane.
Now use Normal align to position the base plane of the tetrahedron with one of the faces of the icosahedron.
Using object snaps position one vertex of the tetra with the icosahedron and then rotate about it local z axis to align the edges. You now have one of the twenty tetrahedrons properly positioned!
Using array and clone you can create and position the rest of the tetras (with some pain).
While working on this approach another thought came to me. An alternative approach is to create the icosahedron and then create a vertex in the middle of one of the faces (after converting to an editable mesh). Delete the face and use this vertex to create 3 faces as shown below.
You can now move the center vertex by 0.40805 to create in effect the trirectangular tetrahedron. I used a grid helper to define an axis perpendicular to the face.
I'll have to give this more thought to find an easier process.
Accepted solution
I did not like the solution I posted above. Here's a much better process.
- Create, Extended Primitives, Hedra, Dodec/Icos
P = 1.0
radius = 0.951056 - Position it at World 0,0,0
- Rotate the icosahedron by 31.7175° about the world x axis. This places vertices of at the top and bottom (rather than an edge).
- Find the midpoint for 2 faces, one on the top series and one in the middle series. This can be done creating two lines on the face from a vertex to an opposite midpoint.
- Trim one line in each face then add a point to note the center.
- Create a line from 0,0,0 to 1.16401,0,0 then adjust its pivot so that it is aligned to the world and located at the hedra center.
- Make a clone instance of the line.
- Add a Look At constraint to each of these two line with one point at one of the points and the other at the other point.
- Convert the hedra to an editable mesh then create 6 new faces referencing the existing face vertices and the end of the two radial lines. This yields 2 of the 20 tetrahedrons.
- Detach the 6 faces to form a new object the Array rotate the shape 5 times 360° around the z axis.
- Clone/Instance the collection and rotate them by 180° about world x.
- Using snap vertex move the group into position.
On a personal side note, back in the 1970's I was working for a small CAD company and we were looking to develop solid modeling capabilities for our 3D mechanical CAD system. We were considering buying one of the few solid modeling programs available. It was called Synthanvision and I visited the company MAGI (founded by 3 wise men). Their software was originally created to calculate radioactive penetration through various shaped 3D containment buildings. They eventually realize that they could substitute photons (light) for radioactive particles and therefore generate pictures of solid object. Synthavision was used as the solid modeler and renderer for some of the 1982 version of Tron.
.
Thanks for the video link! I watched it.
Incredible, I didn't realize you already have tablet and stylus in the 1970s. It must be quite an experience you had to work on 3D with these equipment. Amazing. And for Tron movie, I thought Tron CGI was made only by raw mathematics and keyboard!!!
But now that I think of it in the 1960s Ivan Sutherland created the Sketchpad which is the use of stylus directly on monitor. So indeed the technology was already there. And at the beginning of the 1980s some family microcomputer does included a stylus like the Thomson T07.
But even so, most of the 8bit microcomputer do not have stylus ,or mouse only keyboard hence my surprise by your experience in the 1970s. The first computer I got with a mouse was in 1988 with commodore Amiga 500...
Anyway, thank you again for the help 🙂 I completed the 4 Tron Bit state :
